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Cooperative binding
0
160
1460
2012-08-29T21:26:17Z
Mstefan
25
Created page with "this is going to be the cooperative binding page"
5l7bi896fmq69x34hchkrheqxkpb0ak
this is going to be the cooperative binding page
1542
1460
2012-08-30T21:23:03Z
Mstefan
25
9ddi28y4cio3l2f9vfxc94afvt2gy1y
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions <ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904)
Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref>. When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen <ref name=Bohr1904/>. This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== References ==
<references />
1544
1542
2012-08-30T21:24:13Z
Mstefan
25
szwxpgem0p8dfpqpwrh027csvpforuc
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions <ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904)
Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref>. When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen <ref name=Bohr1904/>. This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
== References ==
<references />
1545
1544
2012-08-30T21:26:46Z
Mstefan
25
/* The Hill equation */
ijw7w3wuz4ddj863dvrk75xij8qstzw
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions <ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904)
Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref>. When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen <ref name=Bohr1904/>. This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th<sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>)
\citep[reviewed in][]{Wyman1990}. The first description of cooperative binding to a multi-site protein was developed by \citet{Hill1910}. Drawing on observations of oxygen binding to h{\ae}moglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the ``Hill coefficent'':
== References ==
<references />
1546
1545
2012-08-30T21:29:25Z
Mstefan
25
/* The Hill equation */
o0yg9k0nb9y0qnkgx0jl0d105yeq6zx
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions <ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904)
Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref>. When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen <ref name=Bohr1904/>. This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by Hill <ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to h{\ae}moglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
== References ==
<references />
1547
1546
2012-08-30T21:32:59Z
Mstefan
25
/* The Hill equation */
068svkqsed7n27zbeiyftnso7525bz4
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions <ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904)
Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref>. When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen <ref name=Bohr1904/>. This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to h{\ae}moglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
== References ==
<references />
1548
1547
2012-08-30T21:33:51Z
Mstefan
25
/* History and definitions */
f8wgmxvb1bsc10g0jgh2wih1h04i7gp
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen. <ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to h{\ae}moglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
== References ==
<references />
1549
1548
2012-08-30T21:34:27Z
Mstefan
25
/* History and definitions */
ha71ihsik33b3q428tsevs9epb77imo
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to h{\ae}moglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
== References ==
<references />
1550
1549
2012-08-30T21:35:39Z
Mstefan
25
/* The Hill equation */
j9o7ovk71bapu96nn8c2igqj42fdzin
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
== References ==
<references />
1551
1550
2012-08-30T21:36:44Z
Mstefan
25
/* The Hill equation */
ihv3rs5rcra8a22blxa3ycpo5ds3nd2
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
== References ==
<references />
1552
1551
2012-08-30T21:39:03Z
Mstefan
25
/* The Hill equation */
4cpxsyt2hqy9grybntk3180lgc7m1xx
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, \ie it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== References ==
<references />
1553
1552
2012-08-30T21:39:27Z
Mstefan
25
/* The Hill equation */
ciysrkjyaoh7dwv88rtcmg9p0arkih4
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== References ==
<references />
1554
1553
2012-08-30T21:40:15Z
Mstefan
25
/* The Hill equation */
korbrtj63ptg6azsga6fwquul8lthcr
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== References ==
<references />
1555
1554
2012-08-30T21:41:01Z
Mstefan
25
8xon43lxqhuykj90w53j0gznsix3dz1
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
== References ==
<references />
1556
1555
2012-08-30T21:49:02Z
Mstefan
25
/* The Adair equation */
6o7g7h1319ipu0efw7p0hpwkpnyb2er
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
== References ==
<references />
1557
1556
2012-08-31T22:08:52Z
Mstefan
25
/* The Adair equation */
g795ztmnnee6fi86p8sz2qm7zk72obu
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X -> PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
== References ==
<references />
1558
1557
2012-08-31T22:09:25Z
Mstefan
25
/* The Adair equation */
pdip8s5srelerbltcyxfrtag8jyewvz
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
== References ==
<references />
1559
1558
2012-08-31T22:10:46Z
Mstefan
25
/* The Adair equation */
dphiefv7200qiezabteew14ntxfvoj8
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{equation}
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
\end{equation}
</math>
== References ==
<references />
1560
1559
2012-08-31T22:10:58Z
Mstefan
25
/* The Adair equation */
1g6bnb7p3xx1fejhpiwozp2pernf40e
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
== References ==
<references />
1561
1560
2012-08-31T22:11:16Z
Mstefan
25
/* The Adair equation */
gcz3u5oum9q2pdomf5vw2qf169ne99f
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.2 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
== References ==
<references />
1562
1561
2012-08-31T22:11:29Z
Mstefan
25
/* The Adair equation */
6ltodkuzqde0i0mqu37gr8uallmia8u
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
== References ==
<references />
1563
1562
2012-08-31T22:12:03Z
Mstefan
25
/* The Adair equation */
lp9kf495hnnrkmwc6htr3hf2kwueyal
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
== References ==
<references />
1564
1563
2012-08-31T22:12:44Z
Mstefan
25
/* The Adair equation */
8683yykibdipvdqumjsyimi8pvsyi98
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
== References ==
<references />
1565
1564
2012-08-31T22:13:49Z
Mstefan
25
/* The Adair equation */
q3kcwi2b6jlu66yclpe6vkqb7lhx6u3
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where n denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of i ligand molecules.
== References ==
<references />
1566
1565
2012-08-31T22:14:19Z
Mstefan
25
/* The Adair equation */
osypzano0agb2202h3rb927smxm214i
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
== References ==
<references />
1567
1566
2012-08-31T22:15:13Z
Mstefan
25
eapb76cu1j65xv710r35qrf6qbnsp1j
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
== References ==
<references />
1568
1567
2012-08-31T22:16:38Z
Mstefan
25
/* The Klotz equation */
celrmzoc5veawwyt7wl0r7qnt2c5v7m
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004/>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, \(K_1\) is the association constant governing binding of the first ligand molecule, \(K_2\) the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For \(\bar{Y}\), this gives
== References ==
<references />
1569
1568
2012-08-31T22:19:51Z
Mstefan
25
/* The Klotz equation */
g3vfe8vo2tswaz1lzwrl3v4d22llabe
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
== References ==
<references />
1570
1569
2012-08-31T22:20:31Z
Mstefan
25
/* The Klotz equation */
rpbpv6e0gqvx6qroe9af4v2fijs4yhc
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
== References ==
<references />
1571
1570
2012-08-31T22:22:45Z
Mstefan
25
/* The Klotz equation */
b2czdps6ts4eyunb0ld4ach0nud7puh
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K</math>, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
== References ==
<references />
1572
1571
2012-08-31T22:23:00Z
Mstefan
25
/* The Klotz equation */
fw24ovlme6boxbaqaag9rkr8vkv2sg5
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K</math>, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
== References ==
<references />
1573
1572
2012-08-31T22:23:25Z
Mstefan
25
/* The Klotz equation */
046yl85qmr541xgbz40hhzw9fh791jg
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
== References ==
<references />
1574
1573
2012-08-31T22:24:15Z
Mstefan
25
/* The Klotz equation */
msr10tydc7ehv2kv0rzaoi2oupmbibf
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== References ==
<references />
1575
1574
2012-08-31T22:24:52Z
Mstefan
25
/* The Klotz equation */
9x8jyp1y1w31ftj0ryrmzsj8k99hwmo
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
== References ==
<references />
1576
1575
2012-08-31T22:31:50Z
Mstefan
25
/* The KNF model */
3zh9t07y26u5b7jr714sa4u54i4cte8
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup>} century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as ``induced fit''. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== References ==
<references />
1577
1576
2012-08-31T22:32:06Z
Mstefan
25
/* The KNF model */
30y44dj0rk32cv045qip3e7j59mdwh8
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as ``induced fit''. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== References ==
<references />
1578
1577
2012-08-31T22:32:33Z
Mstefan
25
/* The KNF model */
2bunh2hkhhcsmg3hyv5g5642pa1itot
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== References ==
<references />
1579
1578
2012-08-31T22:34:52Z
Mstefan
25
/* The KNF model */
gkjzc9l1nkpn8sow1lprcynwzipxc2l
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
== References ==
<references />
1580
1579
2012-08-31T22:35:57Z
Mstefan
25
/* The MWC model */
dmpk20y0ficx3ppx5z788p4sa6mjhkm
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
== References ==
<references />
1581
1580
2012-08-31T22:39:56Z
Mstefan
25
/* The MWC model */
h766oqabg32cj5sxizu1viw37453tcu
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
== References ==
<references />
1583
1581
2012-08-31T22:41:06Z
Mstefan
25
/* The MWC model */
nygsq1e4q6elbh8kzv09rs0csuytfec
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
== References ==
<references />
1584
1583
2012-08-31T22:43:04Z
Mstefan
25
/* The MWC model */
1qync5fsz0afuow78wjqb5sedu1pzan
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
== References ==
<references />
1585
1584
2012-08-31T22:43:52Z
Mstefan
25
/* The MWC model */
899lej50q9a49hwtspm35qx15fh2m3v
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
== References ==
<references />
1586
1585
2012-08-31T22:45:50Z
Mstefan
25
/* The MWC model */
q14i24szpxax78cckozadvq4eiwgzj2
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
== References ==
<references />
1587
1586
2012-08-31T22:54:15Z
Mstefan
25
/* The MWC model */
mgo2g4qfy3d17rgm6bghwdoum4x237w
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
== References ==
<references />
1588
1587
2012-08-31T22:56:53Z
Mstefan
25
/* The MWC model */
krde7e10onqo7gx3bhrwac113tpoign
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
== References ==
<references />
1589
1588
2012-08-31T22:57:36Z
Mstefan
25
/* The MWC model */
ckw8cxpk0fngbo64vdgh2zwhh2hs5dw
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
== References ==
<references />
1590
1589
2012-08-31T23:03:08Z
Mstefan
25
/* The MWC model */
ckbmt2hzv73shlgd5l3fmzmam7y4yng
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== References ==
<references />
1591
1590
2012-08-31T23:04:30Z
Mstefan
25
/* The MWC model */
qob0tstd44l1jgujuu1w9qdngo0b4qe
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
== References ==
<references />
1673
1591
2012-09-18T12:51:06Z
Mstefan
25
/* Conformational Spread */
2jlhpkdpqt2c9r52hh3vr1opxlifede
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. \citet{Wyman1969} proposed such a model with ``mixed conformations'' (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1675
1673
2012-09-18T12:52:23Z
Mstefan
25
1sd6jn1ore4xy751d66l4p79lnt2axk
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>{{cite pmid|5357210}}</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. Wyman <ref name=Wyman1969/>
\citet{Wyman1969} proposed such a model with ``mixed conformations'' (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1676
1675
2012-09-18T12:56:14Z
Mstefan
25
/* The Hill equation */
a8fvg55x3wv6l3vo0zt5fndy31trvpe
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. Wyman <ref name=Wyman1969/>
\citet{Wyman1969} proposed such a model with ``mixed conformations'' (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1678
1676
2012-09-18T12:57:08Z
Mstefan
25
/* Conformational Spread */
j031h7grorsi51ok17nwbl1snk1j09y
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. Wyman <ref name=Wyman1969> </ref>
\citet{Wyman1969} proposed such a model with ``mixed conformations'' (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1680
1678
2012-09-18T12:58:12Z
Mstefan
25
/* Conformational Spread */
o3wx41rk7ifrihc8gk4c153rftajahl
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref>
\citet{Wyman1969} proposed such a model with ``mixed conformations'' (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1681
1680
2012-09-18T12:59:02Z
Mstefan
25
/* Conformational Spread */
nzzstkw537e4to7wrmldw2ctjme73h3
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
== References ==
<references />
1682
1681
2012-09-18T13:00:17Z
Mstefan
25
/* Conformational Spread */
0yb52ur0ykyqxpmy53t53wyl63ftl46
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001/>
\citet{Duke2001} subsumes both the KNF and the MWC model as special cases. [NLN: Yes, but this subsumption was already presented by Eigen and Perutz long before.] In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can ``spread'' around the entire complex.
== References ==
<references />
1683
1682
2012-09-18T13:01:31Z
Mstefan
25
/* Conformational Spread */
p0kn7rctcynwt5uacjbhrtswvg55iwf
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref>
subsumes both the KNF and the MWC model as special cases. [NLN: Yes, but this subsumption was already presented by Eigen and Perutz long before.] In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can ``spread'' around the entire complex.
== References ==
<references />
1684
1683
2012-09-18T13:02:27Z
Mstefan
25
/* Conformational Spread */
j4y0sqgldrnilznooey6pren93o879t
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
1685
1684
2012-09-18T13:03:22Z
Mstefan
25
/* Conformational Spread */
s1grauv6u7ogdgxpeuo8czfvxuty3iq
Cooperative binding occurs if the binding of a ligand increases non-linearly with the concentration of ligand. This can be due for instance to an affinity for the ligand depending on the amount of ligand bound. Cooperativity can be positive or negative. A protein that does not display cooperativity is called non-cooperative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2013
1685
2012-11-23T15:57:23Z
Nlenovere
36
ktfh27agcd8ig4jizn5ss4hs29th9vi
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A protein is said to exhibit cooperative binding if ligand binding scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the protein's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the protein is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2014
2013
2012-11-23T16:02:26Z
Nlenovere
36
/* History and definitions */
2uybxodu25yh38277m1242qnz02ed6v
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, we have homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a protein with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2017
2014
2012-11-23T16:05:00Z
Nlenovere
36
fc7m4nnovcn975asnz10baoa61t7u7q
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the "Bohr effect".
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2018
2017
2012-11-27T16:04:25Z
Nlenovere
36
/* History and definitions */
27jzxgu89qdtq02qxy8sqxkwv4pq9po
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [http://en.wikipedia.org/wiki/Bohr_effect Bohr effect].
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2020
2018
2012-11-27T16:35:10Z
Nlenovere
36
/* History and definitions */
mlb1ouyy1hxf2l6r0yxm220dc6mbne0
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect Bohr effect]].
[[File:Bohr_effect.png |thumb|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2021
2020
2012-11-27T16:36:34Z
Nlenovere
36
/* History and definitions */
kbuob69lnor1uwqwqcoms2du5w06yaq
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2023
2021
2012-11-27T16:48:02Z
Nlenovere
36
/* History and definitions */
5nwrd0ufvji7nay8imhglphe00buwj9
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2024
2023
2012-11-27T16:54:36Z
Nlenovere
36
/* History and definitions */
5r4oirmqg3dz539c40h9ofyzfmo5f23
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied hemoglobin binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the partial pressure of oxygen, he obtained a sigmoidal (or "S-shaped") curve {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>).
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the extent of activation does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence.
[FIGURE: Rbar != Ybar]
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2025
2024
2012-11-27T17:06:16Z
Nlenovere
36
/* History and definitions */
jvqjerh6rpaq2vc8vsz4za7knrq1nda
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional occupancy" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{[X]^{n_H}}{K_d+ [X]^{n_H}}
</math>
where K<sub>f</sub> denotes an apparent dissociation constant, [X] denotes ligand concentration and n<sub>H</sub> is the Hill coefficient.
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2026
2025
2012-11-27T17:10:54Z
Nlenovere
36
/* The Hill equation */
9w9brn524sy36b59f0b8axebjj5lisv
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional occupancy" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a phenomenological equation that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead).
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2027
2026
2012-11-27T17:17:00Z
Nlenovere
36
/* The Hill equation */
85x7hdh0d615pzdfetp1v4y93kp3dcl
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional occupancy" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by A.V. Hill.<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead).
If n<sub>H</sub><1, the system exhibits negative negative cooperativity, and if n<sub>_H</sub>>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n<sub>H</sub>. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n<sub>H</sub> and intercept log(K). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2028
2027
2012-11-27T17:31:15Z
Nlenovere
36
/* The Hill equation */
rdh7rigcuifglh68bsoxpnypgb98h8h
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional occupancy" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
2042
2028
2012-11-27T22:38:54Z
Spencer Bliven
1
Adding [[Category:PLoS Computational Biology drafts]]
7oucyzqanvs1pnc1xq9n05jkl75kyrl
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional occupancy" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2101
2042
2012-12-11T10:26:14Z
Nlenovere
36
/* History and definitions */
i1j91r2tci4h2jdsr2z3fwv9kmgd8qt
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleauges<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2102
2101
2012-12-11T10:27:22Z
Nlenovere
36
/* The KNF model */
ecf2q8zx3yoqrtzvz4b6ihenanqhv6e
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and definitions ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2103
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/* History and definitions */
omli2of1mkht1i5d4s87duta1voxqfk
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the number of ligand-bound binding sites divided by the total number of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>) does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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36
/* Christian Bohr and the origin of cooperative binding */
f6apdl8jhcznlpws0knrycchcq2unhq
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands (reviewed in <ref name=Endler2012> Endler, L. and Stefan, M.I. and Edelstein, S. and Le Novère, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Novère (ed)</ref>). [NLN: this is not the proper reference. We should quote a real reference on ligand interaction and cooperativity.]
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
== The Hill equation ==
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>). The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref> Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2105
2104
2012-12-11T10:43:10Z
Nlenovere
36
7jqpgzjwrxzd10r9a1was9kyqjdkkpj
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]] that linked fractional saturation to ligand concentration, an association constant, and an exponent now called the "Hill coefficent":
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}}
</math>
where n<sub>H</sub> is the Hill coefficient, [X] denotes ligand concentration and K<sub>a</sub> denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n<sub>H</sub><1, the system exhibits negative cooperativity, while if n<sub>_H</sub>>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n<sub>H</sub>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2106
2105
2012-12-11T11:25:14Z
Nlenovere
36
/* The Hill equation */
kmsfhxhek4fnfbryml4borksxtz7t6l
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear for the Hill equation, with slope n<sub>H</sub> and intercept log(1/K_a). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2107
2106
2012-12-11T11:29:28Z
Nlenovere
36
/* The Hill equation */
f46o5am01fzbg44ld6j1ytusz56l3j6
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2108
2107
2012-12-11T11:43:47Z
Nlenovere
36
l3ge1ste38c0qmy3jsxaqcbp7kyngua
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.
Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]looking at this plot alone.
Many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to haemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|12}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2109
2108
2012-12-11T11:48:03Z
Nlenovere
36
rmb3hixkpeowzddavg0q42sb8xhsuzl
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
[FIGURE: Hill plot]
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2110
2109
2012-12-11T11:48:33Z
Nlenovere
36
/* The Hill equation */
lzgixum47u3wtqml4yvfkt0924tezg1
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
The work of Adair<ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> changed this view of cooperativity, and recognised that cooperativity was not a fixed term, but dependent on ligand saturation.
G.S. Adair<ref name=Adair1925/> also considered hemoglobin binding to oxygen, but took into account more information about the nature of hemoglobin, and in particular, the existence of four oxygen binding sites. He also worked from the assumption that fully saturated hemoglobin is formed in stages and that intermediate forms with one, two, or three bound oxygen molecules exist. The formation of each intermediate stage from unbound hemoglobin can be described using a macroscopic association constant K<sub>i</sub>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} & = \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2111
2110
2012-12-11T11:58:20Z
Nlenovere
36
/* The Adair equation */
m6z2djdjt2dqjj9b8ukz51lxldfk16o
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, G.S. Adair <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K<sub>A</sub>, we have <math>[PX]=[P][X]K_A</math> at equilibrium. This allows us to express <math>\bar{Y}</math> in terms of ligand concentration and the K<sub>i</sub>:
<math>
\begin{array}{llcl}
\bar{Y} & = & \frac{PX+2PX_2+3PX_3+4PX_4}{4(P+PX+PX_2+PX_3+PX_4)} \\[0.5 cm]
& = & \frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4k_{IV}X^4}{4(1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4)}
\end{array}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2112
2111
2012-12-11T12:01:00Z
Nlenovere
36
/* The Adair equation */
mqazaiahpqy4jcjgvxzqb8z9ngtbro7
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, G.S. Adair <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> and later Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2113
2112
2012-12-11T12:49:48Z
Nlenovere
36
/* The Adair equation */
dclkkguec18xrl325rc4nscqp6upnby
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, G.S. Adair <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Klotz<ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2114
2113
2012-12-11T12:50:53Z
Nlenovere
36
/* The Klotz equation */
msuinnkiglj9mh766s8l23n086x5a84
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, Christian Bohr studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, G.S. Adair <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each K<sub>i</sub> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages. In his framework, K<sub>1</sub> is the association constant governing binding of the first ligand molecule, K<sub>2</sub> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants K<sub>1</sub>, K<sub>2</sub>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant K, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if K<sub>i</sub> lie above these expected values for i>1.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2115
2114
2012-12-11T13:04:22Z
Nlenovere
36
o0njtqaipgk9n4787istl7pg558qyt6
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. Koshland<ref name=Koshland1958>{{cite pmid|16590179}}</ref> and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit". Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2116
2115
2012-12-11T13:08:38Z
Nlenovere
36
/* The KNF model */
refs0jy30l3tf8d3tdvfvnvzrg3zcmr
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [wp:Daniel_E._Koshland,_Jr. | Daniel Koshland] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The Monod-Wyman-Changeux (MWC) model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. While the Adair-Klotz framework traditionally operates with association constants, the MWC framework has traditionally been described using dissociation constants. The ratio of dissociation constants for the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same ligand affinity and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2117
2116
2012-12-11T14:31:20Z
Nlenovere
36
oaw8kqr24y1ig1goomxkxlturwirem2
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC) model]] for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
[FIGURE: energy diagram]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. According to the MWC model,<ref name=Monod1965/> fractional occupancy is described as follows:
<math>
\bar{Y} = \frac{\frac{[X]}{K_d^R}\left(1+\frac{[X]}{K_d^R}\right)^{n-1}+Lc\frac{[X]}{K_d^R} \left(1+c\frac{[X]}{K_d^R}\right)^{n-1}}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
with <math>K_d^R</math>, <math>L</math> and <math>c</math> as described above. Setting <math>\alpha = \frac{[X]}{K_d^R}</math>, this can also be written in a more readable form as
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the R state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>
according to the MWC model<ref name=Monod1965/> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity).
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised (reviewed in <ref name=Endler2012/>). Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2118
2117
2012-12-11T14:47:48Z
Nlenovere
36
/* The MWC model */
bwotilqj6tewe1x1okeujo1o065stn6
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC) model]] for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha\left(1+\alpha\right)^{n-1}+Lc\alpha\left(1+c\alpha\right)^{n-1}}{\left(1+\alpha\right)^n+L\left(1+c\alpha\right)^n}
</math>
[FIGURE: energy diagram]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{\left(1+\frac{[X]}{K_d^R}\right)^n}{\left(1+\frac{[X]}{K_d^R}\right)^n+L\left(1+c\frac{[X]}{K_d^R}\right)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2119
2118
2012-12-11T14:52:50Z
Nlenovere
36
/* The MWC model */
nh34pi5sfijct1n1k9e2s8zgiht4km8
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC) model]] for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
[FIGURE: energy diagram]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2120
2119
2012-12-11T15:16:49Z
Nlenovere
36
jd2ug0wwm51aai44gy2486r6xxd30js
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC) model]] for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} ]]
[FIGURE: energy diagram]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2121
2120
2012-12-11T15:47:16Z
Nlenovere
36
/* The MWC model */
9hku08gbnaf4elqfedxrht6f5ko7egu
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
[FIGURE: MWC model]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} ]]
[FIGURE: energy diagram]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2122
2121
2012-12-11T16:12:39Z
Nlenovere
36
/* The MWC model */
nlshq6se71mqk1m2up7cpyfqc2x1yk3
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>.
[[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2123
2122
2012-12-11T16:13:36Z
Nlenovere
36
/* The Hill equation */
f41salkts1c3jfzee2bh97bcyvxgsra
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the Conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
[FIGURE illustrating conformational spread?]
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2124
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/* Conformational Spread */
sc6dxqu1u9ksxucv4toc8c5542uelrp
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]]studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
What Bohr observed in hemoglobin were two different kinds of cooperative binding. A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.
In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
When studying cooperativity, it is useful to consider the "fractional occupancy" <math>\bar{Y} </math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone. Note that many proteins are said to be activated or in other ways regulated by cooperative ligand binding. It is important to note though, that the "fractional activity" (sometimes called <math>\bar{R} </math>), that is the quantity of active proteins divided by the total quantity of proteins, does not necessarily coincide with <math>\bar{Y}</math>, nor is there a simple dependence between both.
Drawing on observations of oxygen binding to hemoglobin, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
Also considered hemoglobin binding to oxygen, [[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot was not a straight line, and hypthesized that cooperativity was not a fixed term, but dependent on ligand saturation. He took into account the existence of four oxygen binding sites and also worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Irving Klotz<ref name=Klotz1946a> Klotz, I.M. (1946) The Application of the Law of Mass Action to Binding by Proteins. Interactions with Calcium. Arch Biochem 9:109-117</ref> ref name=Klotz1946a/> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2125
2124
2012-12-11T16:42:18Z
Nlenovere
36
7jjai3pagqn865o1mdeh5t4jqqh2c0r
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Adair<ref name=Adair1925/> and Klotz.<ref name=Klotz1946a/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2132
2125
2012-12-12T11:54:01Z
Nlenovere
36
2u56exkc4d00j02m7moobn2qf3h25x6
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
Pauling<ref name=Pauling1936>{{cite pmid|16587956}}</ref> derived from Adair. Express K1-4 as combination of a unique binding constant K
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2133
2132
2012-12-12T11:59:30Z
Nlenovere
36
/* Pauling equation */
qjci7johjxqhly031dtgelft6tqn9ni
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
[[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> derived from Adair. Express K1-4 as combination of a unique binding constant K
== The KNF model==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2134
2133
2012-12-12T14:02:25Z
Nlenovere
36
j90y360pdsycbswolc46mec8xzx0lqa
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2X^2+3\alpha{}^3K^3X^3+\alpha{}^6KK^4X^4}{1+K[X]+6\alpha{}K^2X^2+4\alpha{}^3K^3X^3+\alpha{}^6KK^4X^4}
</math>
== The KNF model==
[[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2135
2134
2012-12-12T14:03:08Z
Nlenovere
36
/* Pauling equation */
enj0s0rs0nbfn089pr6hj2saecc3hrw
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2X^2+3\alpha{}^3K^3X^3+\alpha{}^6K^4X^4}{1+K[X]+6\alpha{}K^2X^2+4\alpha{}^3K^3X^3+\alpha{}^6K^4X^4}
</math>
== The KNF model==
[[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2136
2135
2012-12-12T14:04:08Z
Nlenovere
36
/* Pauling equation */
jqi8z5nqinjayxub0dnd9sbm6wch3vs
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
[[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> offered a tentative biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2137
2136
2012-12-12T14:09:41Z
Nlenovere
36
/* The KNF model */
t7h66j6xpw7zz6h8u52k5aey0cia693
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
</math>
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2138
2137
2012-12-12T14:24:51Z
Nlenovere
36
/* The KNF model */
50jxqoshyow5lf87i20q7bx9y280zmw
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_SK_t[X])+12K_{AB}^4K_{BB}(K_SK_t[X])^2+12K_{AB}^3K_{BB}^3(K_SK_t[X])^3+4K_{BB}^6(K_SK_t[X])^4}{1}
</math>
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the lignad from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The MWC model */
lq93u2xt8s8de6zrf5mkt4qyxzbjw15
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K_a\cdot{}[X]^{n_H}}{1+ K_a\cdot{}[X]^{n_H}} = \frac{[X]^{n_H}}{K_d + [X]^{n_H}}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n_H<1</math>, the system exhibits negative cooperativity, while if <math>n_H>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_SK_t[X])+12K_{AB}^4K_{BB}(K_SK_t[X])^2+12K_{AB}^3K_{BB}^3(K_SK_t[X])^3+4K_{BB}^6(K_SK_t[X])^4}{1}
</math>
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2140
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2012-12-12T14:33:03Z
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0kf2ap2r9v5oxnjxdsfvv1ql6zhsul9
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2141
2140
2012-12-12T14:39:06Z
Nlenovere
36
/* The KNF model */
r9yyssso7s6mfw7jhwqwvmz0mtggzjy
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> the relative stability of a pair of neighbouring subunits relative to a pair where both subunits are in the A state.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2142
2141
2012-12-12T14:40:36Z
Nlenovere
36
/* The KNF model */
6q4njjtw05f09qvabt4jt2cgeaxkm3h
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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qubu910nsomxystc7wvnr3c51pbiu53
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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t596v5vuq1nmejds5i3ritsor30kh85
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== Christian Bohr and the origin of cooperative binding ==
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
== The Hill equation ==
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
== The Adair equation ==
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
==The Klotz equation ==
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
== Pauling equation ==
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
== The KNF model==
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
== The MWC model==
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Conformational Spread and binding cooperativity ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2145
2144
2012-12-12T15:04:35Z
Nlenovere
36
k3iidhzniz0k40td5sjncb7sbcnyf1s
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Example of cooperative binding ==
Famous examples
=== Multimeric enzymes ===
Aspartate trans-carbamylase
Threonine-deaminase
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecule ===
Calmodulin
=== Transcription factor binding ===
Lambda phage
== Conformational Spread and binding cooperativity ==
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2146
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2012-12-12T15:09:55Z
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36
ev0d6sk3m3vignug32seqvz3vm6j7yr
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Example of cooperative binding ==
Famous examples
=== Multimeric enzymes ===
Aspartate trans-carbamylase
Threonine-deaminase
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecule ===
Calmodulin
=== Transcription factor binding ===
Lambda phage
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2147
2146
2012-12-12T16:05:31Z
Nlenovere
36
/* Example of cooperative binding */
8yxdwbj99v94pctoxqia64n7iykxbuk
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]]
Threonine-deaminase
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2148
2147
2012-12-12T17:29:03Z
Nlenovere
36
/* Examples of cooperative binding */
3okl660tocoxl0h7ycd1lq178m84088
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]]
Cooperativity shown <ref name=Ackers1982> {{cite pmid|13897943}}</ref>
Structure shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2149
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2012-12-12T17:50:30Z
Nlenovere
36
/* Examples of cooperative binding */
hoczmlh3x0dgmdnbotpbc80cfbxd32e
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt. Skandinavisches Archiv Für Physiologie, 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested to behave similarly to hemoglobine<ref name=Changeux963></ref> was [[wp:Threonine_ammonia-lyase | Threonine deaminase]].
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2150
2149
2012-12-12T17:51:50Z
Nlenovere
36
5v890y4vnvgsdjb3fizf6s6frkvnlst
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) Binding and linkage. Functional chemistry of biological molecules. University Science Books, Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J Physiol 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin. J Biol Chem 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested to behave similarly to hemoglobine<ref name=Changeux963></ref> was [[wp:Threonine_ammonia-lyase | Threonine deaminase]].
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | Threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channel
IP3 receptor
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
5tbv2v78t53pndb98b4roqhzb7x7m58
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | Threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2155
2152
2012-12-13T09:32:14Z
Nlenovere
36
/* Multimeric enzymes */
ayo5wr0le4d1syuk8et86kn16ydjrgn
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2156
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2012-12-13T10:03:44Z
Nlenovere
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/* Examples of cooperative binding */
8ocquy81qgqfyc9c7qr461qby9lacq7
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2157
2156
2012-12-13T13:41:38Z
Nlenovere
36
/* Ionic channels */
sj6ukka1k054dqdbfdkc2kzy0tmthtq
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2012-12-13T13:51:29Z
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/* Transcription factor binding */
4i22d8ccvr8mmdkhcrk8t3f66mzo5cv
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
In proteins, conformational change is often associated with activity, or activity towards specific targets. It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
The MWC framework has since been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The MWC model */
f01i9ky0d14xmeaabfjn50ilt1q4qyq
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
It is important to note that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2161
2160
2012-12-14T14:15:03Z
Nlenovere
36
/* The MWC model */
j4ake1q7ci7zrs09rpdleyre4exiql7
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2162
2161
2012-12-14T14:22:02Z
Nlenovere
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/* The MWC model */
rrdkhy4zcu4zgw2b11td4mr4xjlpnle
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
367dk2kps8ne9cjg774oinu4laf9rh9
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
First MWC model <ref name=Karlin1967> {{cite pmid|6048545}}</ref>.
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2164
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2012-12-14T17:34:03Z
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36
/* Examples of cooperative binding */
dgd2syprizpcr8yvke2sp2vulgnnpnx
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
First MWC model <ref name=Karlin1967> {{cite pmid|6048545}}</ref>.
First pentameric structure of an homologous protein<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
Cooperative
Structure
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2012-12-14T17:48:49Z
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/* Multi-site molecules */
c5dwcghhkfprolq0sdg7u67sws3r3mj
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
First MWC model <ref name=Karlin1967> {{cite pmid|6048545}}</ref>.
First pentameric structure of an homologous protein<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
Cooperative
Teo1973 4353626
Structure
Babu1980 3990807
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2166
2165
2012-12-14T17:49:54Z
Nlenovere
36
/* Examples of cooperative binding */
7jd9v6fo6wby0wrc48ssi3486eiqa4c
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assembly exhibiting cooperative binding is very large and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
The most prominent example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. However, many other molecular assemblies exhibiting cooperative binding have been studied in details.
=== Multimeric enzymes ===
[[wp:Aspartate_transcarbamoylase | Aspartate trans-carbamylase]] was one of the first enzymes shown to bind ligands cooperatively<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
The other enzyme suggested early on to behave similarly to hemoglobine<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine
Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504
</ref> was [[wp:Threonine_ammonia-lyase | threonine deaminase]]. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
=== Ionic channels ===
Cys-loop ion channels
First MWC model <ref name=Karlin1967> {{cite pmid|6048545}}</ref>.
First pentameric structure of an homologous protein<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors
Cooperativity<ref name=Meyer1988> {{cite pmid|2452482}}</ref>.
Four IP3 binding sites<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Calmodulin
Cooperative
<ref name=Teo1973> {{cite pmid|4353626}}</ref>.
Structure
<ref name=Babu1980> {{cite pmid|3990807}}</ref>.
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2203
2166
2012-12-19T10:57:48Z
Nlenovere
36
/* Examples of cooperative binding */
tf2e68bmlo99yaiz8khfbhaq3zcovuq
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<a name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2204
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2012-12-19T11:05:28Z
Nlenovere
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/* Multimeric enzymes */
1nra0oy6ad9hy9keh05us4ewf8yl0lv
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
[[wp:Lambda_phage | Lambda phage]]
Discovery repressor cooperativity<ref name=Ptashne1980> {{cite pmid|6444544}}</ref>
Quantitative model <ref name=Ackers1982> {{cite pmid|6461856}}</ref>
Other example of positive cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/17498746
n=1.6
Examples of negative cooperativity
http://www.ncbi.nlm.nih.gov/pubmed/22093184
n=0.56
http://www.ncbi.nlm.nih.gov/pubmed/15967460
n=0.7
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2205
2204
2012-12-19T11:54:29Z
Nlenovere
36
/* Transcription factor binding */
7r163q6f6wxsyk54j5bed0tdk8ypgjq
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K_a</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2206
2205
2012-12-20T13:06:47Z
Nlenovere
36
/* The Hill equation */
tdgqzlug7fk814ul9eevadwmrkerrrz
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that cooperativity is solely a property of ligand molecules, rather than a property of binding sites, which always exhibit the same affinity.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2207
2206
2012-12-20T13:09:00Z
Nlenovere
36
/* The Hill equation */
6i8lox9l58pndlpj3ux49rxtboq7zqp
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Pauling equation */
cl32y82aeg2pwtg7kn91ivl70u1vq0z
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on resutls showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+6K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The KNF model */
lle7kf5qh1l65sbmj2e2e01r97l5sjd
{{author
|first1=Melanie I.
|last1=Stefan^
|department1=
|institution1=
|address1=
|username1=User:
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2210
2209
2013-01-03T21:14:06Z
Mstefan
25
added my affiliation
7spymlkv34dac0zamjvgf8i3ywzkthy
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the origin of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another binding of ligand molecule) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a reflect of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2211
2210
2013-01-03T21:18:43Z
Mstefan
25
changed "origin" to "concept" (because the origin was a few Million years before Bohr)
dvtliqi1kz0d2gsoatt6sd4kmhcaxio
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot", obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math> is in the case of the Hill equation a line with slope <math>n_H</math> and intercept <math>-log(K_d)</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2212
2211
2013-01-03T21:21:43Z
Mstefan
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/* The Hill equation */
rma8ho5oooyw9rypvpfps761poz71xe
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to expressed the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2213
2212
2013-01-03T21:22:58Z
Mstefan
25
/* The Klotz equation */
f60d4srm8n5m5kulqt6hucxl0psw94q
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2214
2213
2013-01-03T21:23:57Z
Mstefan
25
/* Pauling equation */
cbgyfwygtyyxtwsho6u8ldorjmo9goo
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptots and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extends of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2215
2214
2013-01-03T21:26:50Z
Mstefan
25
/* The MWC model */
ivu4qv27ti5la9yvqbh6uk2x9xp6a0l
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron caring four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great details.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2216
2215
2013-01-03T21:30:19Z
Mstefan
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/* Examples of cooperative binding */
lvh3ojj4o8drv3lc5a3ndbbzngn1gq5
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave as hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was latter shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2217
2216
2013-01-03T21:32:02Z
Mstefan
25
/* Multimeric enzymes */
ig2tlv1p9e6n285vkw0ff8f70fq1h14
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in the biological membranes. It is not surprising that several classes of such channels which opening is regulated by ligands exhibit cooperative binding of them.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2218
2217
2013-01-03T21:33:35Z
Mstefan
25
/* Ionic channels */
defx3b5cmaf35ucb8r1db9ml1butk1z
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds cooperatively four calcium ions<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2219
2218
2013-01-03T21:34:42Z
Mstefan
25
/* Multi-site molecules */
s1rwxmd9q8ergigc2ngcuuolz2k9n98
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, that occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibits positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2220
2219
2013-01-03T21:35:41Z
Mstefan
25
/* Transcription factor binding */
5yq0cew7j4pooml7h2f68r34ss0nmhw
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math>. This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2221
2220
2013-01-03T21:40:29Z
Mstefan
25
/* The Hill equation */
1r8ts6613mj23qsw1m3w6r731n3pfex
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition".
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2222
2221
2013-01-03T21:42:22Z
Mstefan
25
/* The MWC model */
5ocubii3vhtahg6c8ja4fxshdycfhxm
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2223
2222
2013-01-03T21:43:29Z
Mstefan
25
/* The MWC model */
py9r8x46vrdynew2ygzktj5n0m06946
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|4}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen. Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2224
2223
2013-01-03T21:45:17Z
Mstefan
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/* Examples of cooperative binding */
emnvpj0al6nx3d2pz2tzbb1pna6neap
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} ]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|4}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2225
2224
2013-01-03T22:50:22Z
Nlenovere
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/* Christian Bohr and the concept of cooperative binding */
c6w35lsaokd386sj02cqxj1apoliij8
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>
Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|4}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The Hill equation */
4c1gzf3kguw1wlyk5q70gmr9nyjh1xl
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin, and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]:
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. [[File:Hill_Plot.png |thumb|right|{{Figure|2}} ]]
The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|4}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2227
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Nlenovere
36
/* The Hill equation */
mof3bij653gy1lkm27rrf6ykx2abdcv
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} ]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilise the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|4}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|4}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-01-03T23:36:27Z
Nlenovere
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/* The MWC model */
8tjptpf7wpd7g9x0d85g6d10xtpd9u9
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
[[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} ]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The MWC model */
ef92i0rofg7681555o1dtuxxrfksxfe
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|5}} ]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|5}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
Other examples of cooperative lattices: Chemotaxis?IP3R?
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-01-03T23:49:11Z
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/* Examples of cooperative binding */
2ewvp8ujm7qddyg9d0x3e86wdai9qrp
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
Many [[wp:enzyme | enzymes]] are multimeric and carry several binding sites for regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>. It was later shown to be a tetrameric protein<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>.
The other enzyme suggested early to bind ligands cooperatively was [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]]<ref name=Gerhart1962> {{cite pmid|13897943}}</ref>. Although initial models were consistent with four binding sites<ref name="Changeux1968">{{cite pmid|4868541}}</ref>, its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>.
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Multimeric enzymes */
ircdvrnfcetveqwhsd3udc7inbmylbu
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling<ref name=Pauling1936/>. The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit"<ref name=Koshland1958>{{cite pmid|16590179}}</ref>. Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The KNF model */
hjdkq7vtf1gu7jr930klkvypk9uupeb
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}. The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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Nlenovere
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/* The MWC model */
ifpqfhis5sgtdl0sqyyix4iazebzyhf
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction<ref name=Perutz1960> {{cite pmid|18990801}}</ref>, exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain<ref name=Brejc2002> {{cite pmid|11357122}}</ref>.
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding<ref name=Meyer1988> {{cite pmid|2452482}}</ref>. The structure of those receptors shows four IP3 binding sites symmetrically arranged<ref name=Seo2012> {{cite pmid|22286060}}</ref>.
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmoduline | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively<ref name=Teo1973> {{cite pmid|4353626}}</ref>. Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref>. Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid| 22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalised MWC mechanism, in which the transition between R and T state is not necessarily synchronised across the entire protein<ref name=Changeux1967> {{cite pmid|16591474}}</ref>. In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
6k2z0apzt8xny1xt1r1qix533m86547
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound acetylcholine in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
IP3 receptors form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-01-04T16:57:24Z
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/* Ionic channels */
m8bybl7r7sx321riy2x6wjat64zjm85
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Multi-site molecules */
328ghhz044ntxutz9n4omr3dxy19njb
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin,]]
As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains.
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
ei1w2y5jw52gxp3r6m90sokahlosrsn
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The MWC model */
t4algzma63ci4k1jjm4vxwexlyaau9l
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor binding ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
blwnmcq9hsag3ej97sfscaay406xlfk
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
Cooperative binding occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the binding of a ligand increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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ftxx5fu6bj9t8rqzgrnr1yrevmjnvw6
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] increases non-linearly with its concentration. This can be due for instance to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive or negative. Cooperative binding is most often observed in proteins. However, nucleic acids can also exhibit cooperative binding, for instance of transcription factors.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] increases non-linearly with its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be, due for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca2+ (PDB id: 1EXR). This figure was made by PyMol.]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2247
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2013-01-09T22:34:30Z
Daniel Mietchen
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/* Multi-site molecules */
qrebapk5nyj4rhghd5ydt6nktg06t5e
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] increases non-linearly with its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be, due for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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Daniel Mietchen
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] increases non-linearly with its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] increases non-linearly with its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factor ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Transcription factor */
ctuf9rdqw7qoacznu52x0knjng4riv4
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>(-log(K_d))</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The Hill equation */
4ve28fcvegnn361chsfniy42t2t52vf
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The Klotz equation */
7kkq01a0h8ach3bmadqmkjpsef27pyv
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{4K_{AB}^3(K_XK_t[X])+12K_{AB}^4K_{BB}(K_XK_t[X])^2+12K_{AB}^3K_{BB}^3(K_XK_t[X])^3+4K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The KNF model */
eteqw54edvx4zcwci0c5lyjoljnaonv
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state.
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2274
2273
2013-02-01T11:39:53Z
Nlenovere
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/* The KNF model */
mchs6g0v9c1wuckococw6ecc2fkg75w
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ionic channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Ionic channels */
3ll8lwv1w9afy5f9cmszp2avg11hd7v
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | Aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | Lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
clkq1in2ol4wrvv3glbv4xg7n12l0he
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Christian Bohr and the concept of cooperative binding */ - changed the sentence about the Bohr effect
1emccvukt0hsgrclqj2fo7afk0dfbli
{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
'''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of its concentration<!-- phrasing still not ideal. suggestions? link to [[wp:molecular binding]]?-->. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound<!--add ref-->. Cooperativity can be positive or negative<!--add brief explanation and ref-->. Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]<!--add ref-->.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan^
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère^
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]. However, [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr | Christian Bohr ]] studied [[wp:hemoglobin | hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure | partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill | A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry) | phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair | G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling | Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr. | Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model | Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme | enzymes]] is regulated by [[wp:Allosteric regulation | allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase | Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase | aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb | William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors | nicotinic acetylcholine receptors]] bound [[wp:acetylcholine | acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor | Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin | calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand | EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage | lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904> Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412 </ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925> Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref> (reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960> {{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998> {{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962> {{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982> {{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967> {{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002> {{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988> {{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012> {{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973> {{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980> {{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980> {{cite pmid|6444544}}</ref><ref name=Ackers1982> {{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007> {{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011> {{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967> {{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969> {{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref>When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub>pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub>make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/>This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math>of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math>at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub>reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup>century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref>Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math>is the "Hill coefficient", <math>[X]</math>denotes ligand concentration, <math>K</math>denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math>is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math>versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math>and intercept <math>log(K_d)</math>(see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref>found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math>is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math>is the association constant governing binding of the first ligand molecule, <math>K_2</math>the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math>, ... do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math>(that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math>lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup>century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref>reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math>in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math>below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref>refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref>Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math>is the constant of association for X, <math>K_t</math>is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math>and <math>K_{BB}</math>are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref>went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math>state. As the energy diagram illustrates, <math>\bar{R}</math>increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math>and <math>\bar{R}</math>do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref>with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref>proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref>or several types of allosteric modulators <ref name=Najdi2006/>and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref>exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref>and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref>It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref>Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref>its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref>(when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref>and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref>The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref>Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup>([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref>Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref>In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref>proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref>subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2543
2542
2013-03-25T10:46:33Z
Daniel Mietchen
5
formatting
t911h7w5zuslpyojfmcqc39rr0ak5t3
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{Figure|1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T10:53:16Z
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/* Christian Bohr and the concept of cooperative binding */
ide1q8f0q22wnf8hxul8xuyal1v0kzc
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to oxygen under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing CO<sub>2</sub> pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). {{See Figure|1|Figure 1}} is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T10:56:18Z
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/* Christian Bohr and the concept of cooperative binding */
q48ueljsok3wjkcymfipxs8jujzim4i
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin in function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be homotropic, if a ligand influences the binding of ligands of the same kind, or heterotropic, if it influences binding of other kinds of ligands.In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe binding of ligand to a protein with more than one binding site and cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T11:02:11Z
Daniel Mietchen
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/* Christian Bohr and the concept of cooperative binding */
6b878xpf1cgo93r0fwde2zqt8uxz5n3
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed in <ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2547
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2013-03-25T11:03:24Z
Daniel Mietchen
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/* Christian Bohr and the concept of cooperative binding */
3f6m9mcxd7wm7oqn1pqy15r6ue9u45t
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a [[wp:Hill_equation_(biochemistry)|phenomenological equation]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2548
2547
2013-03-25T11:05:23Z
Daniel Mietchen
5
/* The Hill equation */
d5gmy5ecd2l70bkrhdzl2bmzvh9elis
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, while if <math>n>1</math>, cooperativity is positive. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2549
2548
2013-03-25T11:07:09Z
Daniel Mietchen
5
/* The Hill equation */
su3lvht8nlpsejmcbqye11qp9dg4vkt
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T11:10:15Z
Daniel Mietchen
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/* The Hill equation */
kknl69dcl1gye91l13fhrzlrd009yal
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n_H</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n_H</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2552
2550
2013-03-25T12:17:09Z
Daniel Mietchen
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either always n_H or always n. choosing the latter for the moment.
njq7u5jp4cg99wk732a2gwiiga8m3ii
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T12:27:10Z
Daniel Mietchen
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/* The Hill equation */
rnfkc5bk31roexc5efwaxal6jzxw67j
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2554
2553
2013-03-25T12:31:46Z
Daniel Mietchen
5
/* The Adair equation */
t1faz1iwubmk3bg7ibfyhoezg84xiq5
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation. <ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2555
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2013-03-25T12:33:21Z
Daniel Mietchen
5
/* The Adair equation */
sdk5slk2f6nli9vef05grmj7a2ee6uu
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz<ref name=Klotz1946a>{{cite pmid|21009581}}</ref>(reviewed in <ref name=Klotz2004>{{cite pmid|14604979}}</ref>) deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law. In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2556
2555
2013-03-25T12:35:30Z
Daniel Mietchen
5
/* The Klotz equation */
9jysnven7on6bva5tk65frwfvc26u9c
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T12:37:35Z
Daniel Mietchen
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/* The Klotz equation */
mbeomiov89nqbttnjkgxm1vy7nkr80w
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]]<ref name=Pauling1936>{{cite pmid|16587956}}</ref> reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below). Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Pauling equation */
3fmon7gvi9f9pw69p6fr4hegljcd1ci
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/>The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2559
2558
2013-03-25T12:40:45Z
Daniel Mietchen
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/* The KNF model */
4lkh0amy01x26nvt1rsuxzkdcx6ma82
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents Ns, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T12:41:53Z
Daniel Mietchen
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/* The KNF model */
b2425gcaetd3562gfkk5gbrw00zewdh
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2561
2560
2013-03-25T12:44:28Z
Daniel Mietchen
5
/* The MWC model */
cd0lz166bko0aikykz7mfuz5t5bh5kd
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allows to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2562
2561
2013-03-25T12:46:43Z
Daniel Mietchen
5
/* The MWC model */
li7hnxhadvjb2ya5us5x3w558zl1ea8
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math>is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2563
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2013-03-25T12:47:50Z
Daniel Mietchen
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/* The MWC model */
oblvym1dgomm3yspw5f0p9sskvajovh
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. If we set <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2564
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2013-03-25T12:52:16Z
Daniel Mietchen
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/* The MWC model */
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant and/or what is experimentaly measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2565
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2013-03-25T12:53:42Z
Daniel Mietchen
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/* The MWC model */
lkdpi0sqa304d60eaq7oxvkvvxmq8gk
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref>i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different (An extreme case is provide by the bacteria flagella motor<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> with a Hill coefficient of 1.7 for the binding and 10.3 for the activation). The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called allosteric modulators.
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2566
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2013-03-25T12:57:34Z
Daniel Mietchen
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/* The MWC model */
px0jclyx3dhf6r0nr25wgis0edi4vb0
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, and a comprehensive list would not be of great use. However, some examples are noticeable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T12:59:42Z
Daniel Mietchen
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/* Examples of cooperative binding */
1k8bgkb0s81dwjo5fbmg2gzrnis4ye4
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2568
2567
2013-03-25T13:00:34Z
Daniel Mietchen
5
/* Examples of cooperative binding */
bcnliyu6ft52mj57dlivqpe46hi4t3w
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is regulated by [[wp:Allosteric regulation|allosteric effectors]]. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues. <ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T13:02:51Z
Daniel Mietchen
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/* Multimeric enzymes */
omwsxj421xkqfh3b26snuw9aqe7z4u5
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. It is not surprising that several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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Daniel Mietchen
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/* Ion Channels */
ino1h7d1p9t50866nh3gs6aauxodcas
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]]<ref name=Babu1980>{{cite pmid|3990807}}</ref>, each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2571
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2013-03-25T13:06:05Z
Daniel Mietchen
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/* Multi-site molecules */
kxwpcg52u0rhmgq74j9odgte0so3cgy
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref>(n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref>(n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the Pseudomonas putida cytochrome P450cam hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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2013-03-25T13:11:19Z
Daniel Mietchen
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/* Transcription factors */
14djptfpxb84k6az64nvbxv56zxdwuz
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues <ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunit. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2574
2573
2013-03-25T13:13:00Z
Daniel Mietchen
5
/* Conformational Spread and binding cooperativity */
kgvjf35pfw6qnticxenua5vxiex0irc
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational Spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2575
2574
2013-03-25T13:13:47Z
Daniel Mietchen
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/* Conformational Spread and binding cooperativity */
njq3nbqc8lkkydkgq7mrlq1z4jp6459
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.{{citation needed}}
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The Klotz equation */
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]]{{citation needed}} but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]].{{citation needed}} Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.{{citation needed}}
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, wherea, cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* The Hill equation */ - fixed a typo
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}X^2+3K_{III}X^3+4K_{IV}X^4}{1+K_1[X]+K_{II}X^2+K_{III}X^3+K_{IV}X^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2604
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Mstefan
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/* The Adair equation */ - added missing brackets around X in the first version of the equation
mud8yfa0o5km098tbc11jtg4rrof5em
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_1[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2605
2604
2013-04-12T19:27:58Z
Mstefan
25
/* The Adair equation */ - also K_I rather than K_1
nsvdyui42ik77i60nqagu30x35icnm1
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
2606
2605
2013-04-12T22:03:17Z
Mstefan
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/* The MWC model */
cl8x67f7i5hgtg9g8r1974ck1cxtbay
{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}.
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:Hemoglobin_t-r_state_ani.gif |thumb|right|{{Figure|6}} Animation showing the transition between T and R states of hemoglobin.]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:Calmodulin-Ca.png |thumb|right|{{Figure|7}} Calmodulin structure with Ca<sup>2+</sup> ([[wp:Protein Data Bank|PDB]] id: 1EXR). This figure was generated using [[wp:PyMol|PyMol]].]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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/* Examples of cooperative binding */
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{{author
|first1=Melanie I.
|last1=Stefan
|department1= Division of Biology
|institution1=California Institute of Technology
|address1= Pasadena, CA 91125, USA
|username1=User:mstefan
|first2=Nicolas
|last2=Le Novère
|institution2= and [[wp:Babraham_Institute|Babraham Institute]]
|address2=Babraham, Cambridge CB22 3AT, UK
|username2=User:Nlenovere
}}
----
[[wp:molecular binding|Molecular binding]] is an interaction between molecules that results in a stable association between those molecules. '''Cooperative binding''' occurs if the number of [[wp:binding sites|binding sites]] of a macromolecule that are occupied by a specific type of [[wp:ligand|ligand]] is a non-linear function of this ligand's concentration. This can be due, for instance, to an affinity for the ligand that depends on the amount of ligand bound. Cooperativity can be positive (supra-linear) or negative (infra-linear). Cooperative binding is most often observed in [[wp:protein|proteins]] but [[wp:nucleic acid|nucleic acids]] can also exhibit cooperative binding, for instance of [[wp:transcription factor|transcription factors]]. Cooperative binding has been shown to be the mechanism underlying a large range of biochemical and physiological processes.
== History and mathematical formalisms ==
=== Christian Bohr and the concept of cooperative binding ===
In 1904, [[wp:Christian_Bohr|Christian Bohr ]] studied [[wp:hemoglobin|hemoglobin]] binding to [[wp:oxygen|oxygen]] under different conditions.<ref name=Bohr1904>Bohr, C., Hasselbalch, K., and Krogh, A. (1904) '''Ueber einen in biologischer Beziehung wichtigen Einfluss, den die Kohlensäurespannung des Blutes auf dessen Sauerstoffbindung übt.''' ''Skandinavisches Archiv Für Physiologie'', 16(2): 402-412</ref> When plotting hemoglobin saturation with oxygen as a function of the [[wp:Partial_pressure|partial pressure]] of oxygen, he obtained a sigmoidal (or "S-shaped") curve, see {{See Figure|1}}. This indicates that the more oxygen is bound to hemoglobin, the easier it is for more oxygen to bind - until all binding sites are saturated. In addition, Bohr noticed that increasing [[wp:Carbon dioxide|CO<sub>2</sub>]] pressure shifted this curve to the right - i.e. higher concentrations of CO<sub>2</sub> make it more difficult for hemoglobin to bind oxygen.<ref name=Bohr1904/> This latter phenomenon, together with the observation that hemoglobin's affinity for oxygen increases with increasing pH, is known as the [[wp:Bohr_effect |Bohr effect]].
[[File:Bohr_effect.png |thumb|right|{{Figure|1}} Original figure from Christian Bohr, showing the sigmoidal increase of oxyhemoglobin as a function of the partial pressure of oxygen.]]
A receptor molecule is said to exhibit cooperative binding if its binding to ligand scales non-linearly with ligand concentration. Cooperativity can be positive (if binding of a ligand molecule increases the receptor's apparent affinity, and hence increases the chance of another ligand molecule binding) or negative (if binding of a ligand molecule decreases affinity and hence makes binding of other ligand molecules less likely). Figure 1 is a chart of the "fractional occupancy" <math>\bar{Y}</math> of a receptor with a given ligand, which is defined as the quantity of ligand-bound binding sites divided by the total quantity of ligand binding sites:
<math>
\bar{Y}=\frac{[\text{bound sites}]}{[\text{bound sites}]+[\text{unbound sites}]} = \frac{[\text{bound sites}]}{[\text{total sites}]}
</math>
If <math>\bar{Y}=0</math>, then the protein is completely unbound, and if <math>\bar{Y}=1</math>, it is completely saturated. If the plot of <math>\bar{Y}</math> at equilibrium as a function of ligand concentration is sigmoidal in shape, as observed by Bohr for hemoglobin, this indicates positive cooperativity. If it is not, no statement can be made about cooperativity from looking at this plot alone.
The concept of cooperative binding only applies to molecules or complexes with more than one ligand binding sites. If several ligand binding sites exist, but ligand binding to any one site does not affect the others, the receptor is said to be non-cooperative. Cooperativity can be [[wp:Allosteric_regulation#Homotropic|homotropic]], if a ligand influences the binding of ligands of the same kind, or [[wp:Allosteric_regulation#Heterotropic|heterotropic]], if it influences binding of other kinds of ligands. In the case of hemoglobin, Bohr observed homotropic positive cooperativity (binding of oxygen facilitates binding of more oxygen) and heterotropic negative cooperativity (binding of CO<sub>2</sub> reduces hemoglobin's facility to bind oxygen.)
Throughout the 20<sup>th</sup> century, various frameworks have been developed to describe the binding of a ligand to a protein with more than one binding site and the cooperative effects observed in this context (reviewed by Wyman, J. and Gill, 1990<ref name=Wyman1990>Wyman, J. and Gill, S. J. (1990) '''Binding and linkage. Functional chemistry of biological molecules'''. ''University Science Books'', Mill Valley.</ref>).
=== The Hill equation ===
The first description of cooperative binding to a multi-site protein was developed by [[wp:A_V_Hill|A.V. Hill]].<ref name=Hill1910>Hill, A. V. (1910) ''''The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves'''. ''J Physiol'' 40: iv-vii.</ref> Drawing on observations of oxygen binding to hemoglobin and the idea that cooperativity arose from the aggregation of hemoglobin molecules, each one binding one oxygen molecule, Hill suggested a phenomenological equation that has since been [[wp:Hill_equation_(biochemistry)|named after him]]: [[File:Hill_Plot.png |thumb|right|{{Figure|2}} Hill plot of the Hill equation in red, showing the slope of the curve being the Hill coefficient and the intercept with the x-axis providing the apparent dissociation constant. The green line shows the non-cooperative curve.]]
<math>
\bar{Y} = \frac{K\cdot{}[X]^n}{1+ K\cdot{}[X]^n} = \frac{[X]^n}{K_d + [X]^n}
</math>
where <math>n</math> is the "Hill coefficient", <math>[X]</math> denotes ligand concentration, <math>K</math> denotes an apparent association constant (used in the original form of the equation) and <math>K_d</math> is an apparent dissociation constant (used in moderm forms of the equation). If <math>n<1</math>, the system exhibits negative cooperativity, whereas cooperativity is positive if <math>n>1</math>. The total number of ligand binding sites is an upper bound for <math>n</math>. The Hill equation can be linearized as:
<math>
\log \frac{\bar{Y}}{1-\bar{Y}} = n\cdot{}\log [X] - \log K_d
</math>
The "Hill plot" is obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus <math>\log [X]</math>. In the case of the Hill equation, it is a line with slope <math>n_H</math> and intercept <math>log(K_d)</math> (see {{See Figure|2}}). This means that cooperativity is assumed to be fixed, i.e. it does not change with saturation. It also means that binding sites always exhibit the same affinity, and cooperativity does not arise from an affinity increasing with ligand concentration.
=== The Adair equation ===
[[wp:Gilbert_Smithson_Adair|G.S. Adair]] found that the Hill plot for hemoglobin was not a straight line, and hypothesized that cooperativity was not a fixed term, but dependent on ligand saturation.<ref name=Adair1925>Adair, G.S. (1925) '''The hemoglobin system. IV. The oxygen dissociation curve of hemoglobin'''. ''J Biol Chem'' 63:529-545</ref> Having demonstrated that hemoglobin contained four hemes (and therefore binding sites for oxygen), he worked from the assumption that fully saturated hemoglobin is formed in stages, with intermediate forms with one, two, or three bound oxygen molecules. The formation of each intermediate stage from unbound hemoglobin can be described using an apparent macroscopic association constant <math>K_i</math>. The resulting fractional occupancy can be expressed as:
<math>
\bar{Y} = \frac{1}{4}\cdot{}\frac{K_I[X]+2K_{II}[X]^2+3K_{III}[X]^3+4K_{IV}[X]^4}{1+K_I[X]+K_{II}[X]^2+K_{III}[X]^3+K_{IV}[X]^4}
</math>
Or, for any protein with ''n'' ligand binding sites:
<math>
\bar{Y}=\frac{1}{n}\frac{K_I[X] + 2K_{II}[X]^2 + \ldots + nK_{n} [X]^n}{1+K_I[X]+K_{II}[X]^2+ \ldots +K_n[X]^n}
</math>
where ''n'' denotes the number of binding sites and each <math>K_i</math> is a combined association constant, describing the binding of ''i'' ligand molecules.
=== The Klotz equation ===
Working on calcium binding proteins, Irving Klotz deconvoluted Adair's association constants by considering stepwise formation of the intermediate stages, and tried to express the cooperative binding in terms of elementary processes governed by mass action law.<ref name=Klotz1946a>{{cite pmid|21009581}}</ref><ref name=Klotz2004>{{cite pmid|14604979}}</ref> In his framework, <math>K_1</math> is the association constant governing binding of the first ligand molecule, <math>K_2</math> the association constant governing binding of the second ligand molecule (once the first is already bound) etc. For <math>\bar{Y}</math>, this gives:
<math>
\bar{Y}=\frac{1}{n}\frac{K_1[X] + 2K_1K_2[X]^2 + \ldots + n\left(K_1K_2 \ldots K_n\right)[X]^n}{1+K_1[X]+K_1K_2[X]^2+ \ldots +\left(K_1K_2 \ldots K_n\right)[X]^n}
</math>
It is worth noting that the constants <math>K_1</math>, <math>K_2</math> and so forth do not relate to individual binding sites. They describe ''how many'' binding sites are occupied, rather than ''which ones''. This form has the advantage that cooperativity is easily recognised when considering the association constants. If all ligand binding sites are identical with a microscopic association constant <math>K</math>, one would expect <math>K_1=nK, K_2=\frac{n-1}{2}K, \ldots K_n=\frac{1}{n}K</math> (that is <math>K_i=\frac{n-i+1}{i}</math>) in the absence of cooperativity. We have positive cooperativity if <math>K_i</math> lies above these expected values for <math>i>1</math>.
The Klotz equation (which is sometimes also called the Adair-Klotz equation) is still often used in the experimental literature to describe measurements of ligand binding in terms of sequential apparent binding constants.<ref name=Klotz1946a>{{cite pmid|21115073}}</ref>
=== Pauling equation ===
By the middle of the 20<sup>th</sup> century, there was an increased interest in models that would not only describe binding curves phenomenologically, but offer an underlying biochemical mechanism. [[wp:Linus_pauling|Linus Pauling]] reinterpreted the equation provided by Adair, assuming that his constants were the combination of the binding constant for the ligand (<math>K</math> in the equation below) and energy coming from the interaction between subunits of the cooperative protein (<math>\alpha</math> below).<ref name=Pauling1936>{{cite pmid|16587956}}</ref> Pauling actually derived several equations, depending on the degree of interaction between subunits. Based on wrong assumptions about the localisation of hemes, he opted for the wrong one to describe oxygen binding by hemoglobin, assuming the subunit were arranged in a square. The equation below provides the equation for a tetrahedral structure, which would be more accurate in the case of hemoglobin:
<math>
\bar{Y} = \frac{K[X]+3\alpha{}K^2[X]^2+3\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}{1+4K[X]+6\alpha{}K^2[X]^2+4\alpha{}^3K^3[X]^3+\alpha{}^6K^4[X]^4}
</math>
=== The KNF model===
Based on results showing that the structure of cooperative proteins changed upon binding to their ligand, [[wp:Daniel_E._Koshland,_Jr.|Daniel Koshland]] and colleagues<ref name=Koshland1966>{{cite pmid|5938952}}</ref> refined the biochemical explanation of the mechanism described by Pauling.<ref name=Pauling1936/> The Koshland-Némethy-Filmer (KNF) model assumes that each subunit can exist in one of two conformations: active or inactive. Ligand binding to one subunit would induce an immediate conformational change of that subunit from the inactive to the active conformation, a mechanism described as "induced fit".<ref name=Koshland1958>{{cite pmid|16590179}}</ref> Cooperativity, according to the KNF model, would arise from interactions between the subunits, the strength of which varies depending on the relative conformations of the subunits involved. For a tetrahedric structure (they also considered linear and square structures), they proposed the following formula:
<math>
\bar{Y} = \frac{K_{AB}^3(K_XK_t[X])+3K_{AB}^4K_{BB}(K_XK_t[X])^2+3K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}{1+4K_{AB}^3(K_XK_t[X])+6K_{AB}^4K_{BB}(K_XK_t[X])^2+4K_{AB}^3K_{BB}^3(K_XK_t[X])^3+K_{BB}^6(K_XK_t[X])^4}
</math>
Where <math>K_X</math> is the constant of association for X, <math>K_t</math> is the ratio of B and A states in the absence of ligand ("transition"), <math>K_{AB}</math> and <math>K_{BB}</math> are the relative stabilities of pairs of neighbouring subunits relative to a pair where both subunits are in the A state (Note that the KNF paper actually presents <math>N_s</math>, the number of occupied sites, which is here 4 times <math>\bar{Y}</math>).
=== The MWC model ===
[[File:MWC_structure.png |thumb|right|{{Figure|3}} Reaction scheme of a Monod-Wyman-Changeux model of a protein made up of two protomers. The protomer can exist under two states, each with a different affinity for the ligand. L is the ratio of states in the absence of ligand, c is the ratio of affinities.]][[File:MWC_energy.png |thumb|right|{{Figure|4}} Energy diagram of a Monod-Wyman-Changeux model of a protein made up of two protomers. The larger affinity of the ligand for the R state means that the latter is preferentially stabilized by the binding.]]
The [[wp:MWC_model|Monod-Wyman-Changeux (MWC)]] model for concerted allosteric transitions<ref name=Monod1965>{{cite pmid|14343300}}</ref> went a step further by exploring cooperativity based on thermodynamics and three-dimensional conformations. It was originally formulated for oligomeric proteins with symmetrically arranged, identical subunits, each of which has one ligand binding site. According to this framework, two (or more) interconvertible conformational states of an allosteric protein coexist in a thermal equilibrium. The states - often termed tense (T) and relaxed (R) - differ in affinity for the ligand molecule. The ratio between the two states is regulated by the binding of ligand molecules that stabilizes the higher-affinity state. Importantly, all subunits of a molecule change states at the same time, a phenomenon known as "concerted transition". The MWC model is illustrated in {{See Figure|3}}.
The allosteric isomerisation constant ''L'' describes the equilibrium between both states when no ligand molecule is bound: <math>L=\frac{\left[T_0\right]}{\left[R_0\right]}</math>. If ''L'' is very large, most of the protein exists in the T state in the absence of ligand. If ''L'' is small (close to one), the R state is nearly as populated as the T state. The ratio of dissociation constants for for the ligand from the T and R states is described by the constant ''c'': <math>c = \frac{K_d^R}{K_d^T}</math>. If <math>c=1</math>, both R and T states have the same affinity for the ligand and the ligand does not affect isomerisation. The value of ''c'' also indicates how much the equilibrium between T and R states changes upon ligand binding: the smaller ''c'', the more the equilibrium shifts towards the R state after one binding. With <math>\alpha = \frac{[X]}{K_d^R}</math>, fractional occupancy is described as:
<math>
\bar{Y} = \frac{\alpha(1+\alpha)^{n-1}+Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
The sigmoid Hill plot of allosteric proteins (shown in {{See Figure|5}}) can then be analysed as a progressive transition from the T state (low affinity) to the R state (high affinity) as the saturation increases (see {{See Figure|4}}). The Hill coefficient also depends on saturation, with a maximum value at the inflexion point. The intercepts between the two asymptotes and and the y-axis allow to determine the affinities of both states for the ligand.
[[File:Hill_Plot_MWC_model.png |thumb|right|{{Figure|5}} Hill plot of the MWC binding function in red, of the pure T and R state in green. As the conformation shifts from T to R, so does the binding function. The intercepts with the x-axis provide the apparent dissociation constant as well sas the microscopic dissociation constants of R and T states.]]
In proteins, conformational change is often associated with activity, or activity towards specific targets. Such activity is often what is physiologically relevant or what is experimentally measured. The degree of conformational change is described by the state function <math>\bar{R}</math>, which denotes the fraction of protein present in the <math>R</math> state. As the energy diagram illustrates, <math>\bar{R}</math> increases as more ligand molecules bind. The expression for <math>\bar{R}</math> is:
<math>
\bar{R}=\frac{(1+\alpha)^n}{(1+\alpha)^n+L(1+c\alpha)^n}
</math>
A crucial aspect of the MWC model is that the curves for <math>\bar{Y}</math> and <math>\bar{R}</math> do not coincide,<ref name=Rubin1966>{{cite pmid|5972463}}</ref> i.e. fractional saturation is not a direct indicator of conformational state (and hence, of activity). Moreoever, the extents of the cooperativity of binding and the cooperativity of activation can be very different: an extreme case is provide by the bacteria flagella motor with a Hill coefficient of 1.7 for the binding and 10.3 for the activation.<ref name=Cluzel2000>{{cite pmid|10698740}}</ref><ref name=Sourjick2002>{{cite pmid|12232047}}</ref> The supra-linearity of the response is sometimes called [[wp:ultrasensitivity|ultrasensitivity]].
If an allosteric protein binds to a target that also has a higher affinity for the R state, then target binding further stabilises the R state, hence increasing ligand affinity. If, on the other hand, a target preferentially binds to the T state, then target binding will have a negative effect on ligand affinity. Such targets are called [[wp:allosteric modulator|allosteric modulators]].
Since its inception, the MWC framework has been extended and generalised. Variations have been proposed, for example to cater for proteins with more than two states,<ref name=Edelstein1996>{{cite pmid|8983160}}</ref> proteins that bind to several types of ligands <ref name=Mello2005>{{cite pmid|16293695}}</ref><ref name=Najdi2006>{{cite pmid|16819787}}</ref> or several types of allosteric modulators <ref name=Najdi2006/> and proteins with non-identical subunits or ligand-binding sites.<ref name=Stefan2009>{{cite pmid|19602261}}</ref>
== Examples of cooperative binding ==
The list of molecular assemblies that exhibit cooperative binding of ligands is very large, but some examples are particularly notable for their historical interest, their unusual properties, or their physiological importance.
[[File:HemoglobinConformations.png |thumb|right|{{Figure|6}} Cartoon representation of the protein hemoglobin in its two conformations: "tensed (T)" on the left corresponding to the deoxy form (derived from [[wp:Protein Data Bank|PDB]] id:11LFL) and "relaxed (R)" on the right corresponding to the oxy form (derived from [[wp:Protein Data Bank|PDB]] id:1LFT).]] As described in the historical section, the most famous example of cooperative binding is [[wp:hemoglobin|hemoglobin]]. Its quaternary structure, solved by [[wp:Max_Perutz |Max Perutz]] using X-ray diffraction,<ref name=Perutz1960>{{cite pmid|18990801}}</ref> exhibits a pseudo-symmetrical tetrahedron carrying four binding sites (hemes) for oxygen (see {{See Figure|6}}). Many other molecular assemblies exhibiting cooperative binding have been studied in great detail.
=== Multimeric enzymes ===
The activity of many [[wp:enzyme|enzymes]] is [[wp:Allosteric regulation|regulated]] by allosteric effectors. Some of these enzymes are multimeric and carry several binding sites for the regulators.
[[wp:Threonine_ammonia-lyase|Threonine deaminase]] was one of the first enzymes suggested to behave like hemoglobin<ref name=Changeux1961>{{cite pmid|13878122}}</ref> and shown to bind ligands cooperatively.<ref name=Changeux963>Changeux, J.-P. (1963) '''Allosteric Interactions on Biosynthetic L-threonine Deaminase from E. coli K12'''. ''Cold Spring Harb Symp Quant Biol'' 28: 497-504</ref> It was later shown to be a tetrameric protein.<ref name=Gallagher1998>{{cite pmid|9562556}}</ref>
Another enzyme that has been suggested early to bind ligands cooperatively is [[wp:aspartate_transcarbamoylase|aspartate trans-carbamylase]].<ref name=Gerhart1962>{{cite pmid|13897943}}</ref> Although initial models were consistent with four binding sites,<ref name="Changeux1968">{{cite pmid|4868541}}</ref> its structure was later shown to be hexameric by [[wp:William_Lipscomb|William Lipscomb]] and colleagues.<ref name=Honzatko1982>{{cite pmid|6757446}}</ref>
=== Ion Channels ===
Most [[wp:ion_channel|ion channels]] are formed of several identical or pseudo-identical monomers or domains, arranged symmetrically in biological membranes. Several classes of such channels whose opening is regulated by ligands exhibit cooperative binding of these ligands.
It was suggested as early as 1967<ref name=Karlin1967>{{cite pmid|6048545}}</ref> (when the exact nature of those channels was still unknown) that the [[wp:nicotinic_receptors|nicotinic acetylcholine receptors]] bound [[wp:acetylcholine|acetylcholine]] in a cooperative manner due to the existence of several binding sites. The purification of the receptor<ref name=changeux1970 >{{cite pmid|5274453}}</ref> and its characterization demonstrated a pentameric structure with binding sites located at the interfaces between subunits, confirmed by the structure of the receptor binding domain.<ref name=Brejc2002>{{cite pmid|11357122}}</ref>
[[wp:IP3_receptor|Inositol triphosphate (IP3) receptors]] form another class of ligand-gated ion channels exhibiting cooperative binding.<ref name=Meyer1988>{{cite pmid|2452482}}</ref> The structure of those receptors shows four IP3 binding sites symmetrically arranged.<ref name=Seo2012>{{cite pmid|22286060}}</ref>
=== Multi-site molecules ===
Although most proteins showing cooperative binding are multimeric complexes of homologous subunits, some proteins carry several binding sites for the same ligand on the same polypeptide. One such example is [[wp:calmodulin|calmodulin]]. One molecule of calmodulin binds four calcium ions cooperatively.<ref name=Teo1973>{{cite pmid|4353626}}</ref> Its structure presents four [[wp:EF_hand|EF-hand domains]],<ref name=Babu1980>{{cite pmid|3990807}}</ref> each one binding one calcium ion. Interestingly, the molecule does not display a square or tetrahedron structure, but is formed of two lobes, each carrying two EF-hand domains. [[File:CalmodulinConformation.png |thumb|right|{{Figure|7}} Cartoon representation of the protein Calmodulin in its two conformation: "closed" on the left (derived from [[wp:Protein Data Bank|PDB]] id: 1CFD) and "open" on the right (derived from [[wp:Protein Data Bank|PDB]] id: 3CLN). The open conformation is represented bound with 4 calcium ions (orange spheres).]]
=== Transcription factors ===
Cooperative binding of proteins onto nucleic acids has also been shown. A classical example is the binding of the [[wp:Lambda_phage|lambda phage]] repressor to its operators, which occurs cooperatively.<ref name=Ptashne1980>{{cite pmid|6444544}}</ref><ref name=Ackers1982>{{cite pmid|6461856}}</ref> Other examples of transcription factors exhibit positive cooperativity when binding their target, such as the repressor of the TtgABC pumps<ref name=Krell2007>{{cite pmid|17498746}}</ref> (n=1.6).
Conversely, examples of negative cooperativity for the binding of transcription factors were also documented, as for the homodimeric repressor of the ''[[wp:Pseudomonas putida|Pseudomonas putida]]'' [[wp:Cytochrome P450|cytochrome P450cam]] hydroxylase operon<ref name=Arakami2011>{{cite pmid|22093184}}</ref> (n=0.56).
=== Conformational spread and binding cooperativity ===
Early on, it has been argued that some proteins, especially those consisting of many subunits, could be regulated by a generalized MWC mechanism, in which the transition between R and T state is not necessarily synchronized across the entire protein.<ref name=Changeux1967>{{cite pmid|16591474}}</ref> In 1969, Wyman <ref name=Wyman1969>{{cite pmid|5357210}}</ref> proposed such a model with "mixed conformations" (i.e. some protomers in the R state, some in the T state) for respiratory proteins in invertebrates.
Following a similar idea, the conformational spread model by Duke and colleagues<ref name=Duke2001>{{cite pmid|11327786}}</ref> subsumes both the KNF and the MWC model as special cases. In this model, a subunit does not automatically change conformation upon ligand binding (as in the KNF model), nor do all subunits in a complex change conformations together (as in the MWC model). Conformational changes are stochastic with the likelihood of a subunit switching states depending on whether or not it is ligand bound and on the conformational state of neighbouring subunits. Thus, conformational states can "spread" around the entire complex.
== References ==
<references />
[[Category:PLoS Computational Biology drafts]]
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Which produces:
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This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
|}
</div>
</includeonly>
796
304
2012-05-09T20:09:57Z
Cdessimoz
12
je9l7shhii7dgnr2qzgp1a1ht8i71v9
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>
{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>
|}}
{{#if: {{{last3|}}}
|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>
|}}
{{#if: {{{last4|}}}
|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>
|}}
{{#if: {{{last5|}}}
|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>
|}}
{{#if: {{{last6|}}}
|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>
|}}
{{#if: {{{last7|}}}
|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>
|}}
{{#if: {{{last8|}}}
|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>
|}}
{{#if: {{{last9|}}}
|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
797
796
2012-05-09T20:17:57Z
Cdessimoz
12
dvnhv07d10bprk6juziyh46diroob08
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>
|}}
{{#if: {{{last3|}}}
|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>
|}}
{{#if: {{{last4|}}}
|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>
|}}
{{#if: {{{last5|}}}
|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>
|}}
{{#if: {{{last6|}}}
|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>
|}}
{{#if: {{{last7|}}}
|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>
|}}
{{#if: {{{last8|}}}
|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>
|}}
{{#if: {{{last9|}}}
|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
799
797
2012-05-09T20:38:48Z
Cdessimoz
12
eb1kq0om80qxx43a4rr1qphjl8mcgny
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>
|}}
{{#if: {{{last4|}}}
|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>
|}}
{{#if: {{{last5|}}}
|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>
|}}
{{#if: {{{last6|}}}
|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>
|}}
{{#if: {{{last7|}}}
|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>
|}}
{{#if: {{{last8|}}}
|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>
|}}
{{#if: {{{last9|}}}
|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
800
799
2012-05-09T20:40:15Z
Cdessimoz
12
rof8glv24da2ake4myjlvg9ds6pvjz7
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>
}}
{{#if: {{{last4|}}}
|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>
|}}
{{#if: {{{last5|}}}
|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>
|}}
{{#if: {{{last6|}}}
|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>
|}}
{{#if: {{{last7|}}}
|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>
|}}
{{#if: {{{last8|}}}
|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>
|}}
{{#if: {{{last9|}}}
|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
801
800
2012-05-09T20:43:42Z
Cdessimoz
12
pgy30shh4z08d3qbsidmhdjrsfxdkke
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
802
801
2012-05-09T20:46:13Z
Cdessimoz
12
01i0ejmxdyplox2er2uj9plzexebzuj
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}} {{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}} {{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}} {{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}} {{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}} {{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}} {{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}} {{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
805
802
2012-05-09T20:51:08Z
Cdessimoz
12
Undo revision 802 by [[Special:Contributions/Cdessimoz|Cdessimoz]] ([[User talk:Cdessimoz|talk]])
pgy30shh4z08d3qbsidmhdjrsfxdkke
<noinclude>
This template defines the authors of an article, as is typical for a journal article. Note that additional users may have contributed to the finished product. All changes can be viewed in the article history, and the cited authors may choose to include an acknowledgements section separate from this template.
==Usage==
<code><nowiki>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector | username=User:James T Kirk }}</nowiki></code>
Which produces:
<blockquote>{{author| first=James | last=Kirk | department=Starfleet | institution=United Federation of Planets | address=001 Bridge, Starship Enterprise, Deep Space Sector |username=User:James T Kirk}}</blockquote>
==Parameters==
This template allows up to 9 authors. All authors have the following parameters, where 'X' can be a number 1-9.
{|
! Parameter !! Description
|-
| firstX || first name or initials
|-
| lastX || last name
|-
| departmentX || Department
|-
| institutionX || University or corporation
|-
| usernameX || The user page of this author, including the User: prefix
|}
</noinclude><includeonly>
<div class="authors">{{#if: {{{username1|{{{username|}}} }}}
| [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]]
| {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}
}}<sup>1</sup>{{#if: {{{last2|}}}
|, {{#if: {{{username2| }}}
| [[{{{username2| }}}|{{{first2}}} {{{last2}}}]]
| {{{first2}}} {{{last2}}}
}}<sup>2</sup>}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}}
| [[{{{username3| }}}|{{{first3}}} {{{last3}}}]]
| {{{first3}}} {{{last3}}}
}}<sup>3</sup>}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}}
| [[{{{username4| }}}|{{{first4}}} {{{last4}}}]]
| {{{first4}}} {{{last4}}}
}}<sup>4</sup>}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}}
| [[{{{username5| }}}|{{{first5}}} {{{last5}}}]]
| {{{first5}}} {{{last5}}}
}}<sup>5</sup>}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}}
| [[{{{username6| }}}|{{{first6}}} {{{last6}}}]]
| {{{first6}}} {{{last6}}}
}}<sup>6</sup>}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}}
| [[{{{username7| }}}|{{{first7}}} {{{last7}}}]]
| {{{first7}}} {{{last7}}}
}}<sup>7</sup>}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}}
| [[{{{username8| }}}|{{{first8}}} {{{last8}}}]]
| {{{first8}}} {{{last8}}}
}}<sup>8</sup>}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}}
| [[{{{username9| }}}|{{{first9}}} {{{last9}}}]]
| {{{first9}}} {{{last9}}}
}}<sup>9</sup>
|}}
{| class="collapsible collapsed"
! Author Affiliations
|-
|<sup>1</sup>{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}.
{{#if: {{{last2|}}} |
{{!}}-
{{!}}<sup>2</sup>{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}.
|}}
{{#if: {{{last3|}}} |
{{!}}-
{{!}}<sup>3</sup>{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}.
|}}
{{#if: {{{last4|}}} |
{{!}}-
{{!}}<sup>4</sup>{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}.
|}}
{{#if: {{{last5|}}} |
{{!}}-
{{!}}<sup>5</sup>{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}.
|}}
{{#if: {{{last6|}}} |
{{!}}-
{{!}}<sup>6</sup>{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}.
|}}
{{#if: {{{last7|}}} |
{{!}}-
{{!}}<sup>7</sup>{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}.
|}}
{{#if: {{{last8|}}} |
{{!}}-
{{!}}<sup>8</sup>{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}.
|}}
{{#if: {{{last9|}}} |
{{!}}-
{{!}}<sup>9</sup>{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}.
|}}
|}
</div>
</includeonly>
Template:Cite pmid
10
55
373
2012-01-24T23:20:19Z
Spencer Bliven
1
Wrapper for Pubmed citations
jfjewhvo1hr979clgnkhmuvikuou2iq
{{#if: {{{1|}}}|<pubmed>{{{1}}}</pubmed>|<strong class="error">No PMID specified</strong>}}
374
373
2012-01-24T23:29:59Z
Spencer Bliven
1
sxxxmnj93kyxeybxuj9m6xa6snrglaq
{{#if: {{{1|}}}|pubmed {{{1}}}|<strong class="error">No PMID specified</strong>}}
375
374
2012-01-24T23:39:23Z
Spencer Bliven
1
Trying some things to keep the pubmed parser from parsing the template prematurely
i1uj5rmtathjis2w4oarnhop484n8y2
{{#if: {{{1|}}}|<nowiki><pubmed>{{{1}}}</pubmed></nowiki>|<strong class="error">No PMID specified</strong>}}
376
375
2012-01-24T23:41:20Z
Spencer Bliven
1
5cavz9qeejnt68y0eak0i342gterk4g
{{#if: {{{1|}}}|{{tag|pubmed}}{{{1}}}{{tag|/pubmed}}|<strong class="error">No PMID specified</strong>}}
378
376
2012-01-25T00:04:04Z
Spencer Bliven
1
jm41aqrhcttiz5dyihzx1tuuxiq9qto
<includeonly>{{#if: {{{1|}}}|{{#tag:pubmed|{{{1}}}}}|<strong class="error">No PMID specified</strong>}}</includeonly><noinclude>{{Cite pmid|16592676}}
</noinclude>
379
378
2012-01-25T00:20:45Z
Spencer Bliven
1
Added optional layoutfile parameter
s95dlq6jbe8uh2k0d3w1ykydwlpzo0u
<includeonly>{{#if: {{{1|}}}|{{#if: {{{layoutfile|}}}|{{#tag:pubmed|{{{1}}}|layoutfile={{{layoutfile}}}}}|{{#tag:pubmed|{{{1}}}}}}}|<strong class="error">No PMID specified</strong>}}</includeonly><noinclude>{{Cite pmid|16592676}}
</noinclude>
Template:Figure
10
53
334
2012-01-24T00:11:44Z
Spencer Bliven
1
Created page with "'''<span id={{#if {{{2|}}}|{{{2}}}|fig{{{1|}}} }}Figure {{{1|?}}}.'''"
09aqfeftxszwienjuihcbmdoeps8eh7
'''<span id={{#if {{{2|}}}|{{{2}}}|fig{{{1|}}} }}Figure {{{1|?}}}.'''
335
334
2012-01-24T00:12:11Z
Spencer Bliven
1
q2o4kyj7dmk4gomo3w08s3t50v9t6jc
'''<span id={{#if {{{2|}}}|{{{2}}}|fig{{{1|}}} }}>Figure {{{1|?}}}.</span>'''
354
335
2012-01-24T01:11:04Z
Spencer Bliven
1
tl1e85mdww9tsoi7vwtovirahapugvd
'''<span id="fig{{{1|}}}">Figure {{{1|?}}}.</span>'''
370
354
2012-01-24T01:47:53Z
Spencer Bliven
1
ggnkn329ft0o5lre91qmr6kxldaxnfz
'''<span id="fig{{{1|}}}">{{#if: {{{2|}}}|{{{2}}}|Figure {{{1|?}}}}}.</span>'''
Template:See Figure
10
54
337
2012-01-24T00:22:54Z
Spencer Bliven
1
Created page with "{{#if hi | true | false }}"
ra2gnmbkdkq5srt38011bhkgldb2ise
{{#if hi | true | false }}
338
337
2012-01-24T00:27:04Z
Spencer Bliven
1
nn9l20uii5lwpxwwrxfbuwxuwtrwpk3
{{#ifexpr {{{1}}} }}
<noinclude>{{User:Spencer_Bliven/Figures}}</noinclude>
339
338
2012-01-24T00:30:52Z
Spencer Bliven
1
6bheidc65plewhljlx7r5jd6g7slzoi
figure {{#ifexpr {{{1|X}}}|NUMBER|ID }}
340
339
2012-01-24T00:31:29Z
Spencer Bliven
1
d8n7amby4z38kc9zgq1krg7qpkvywm6
figure {{#ifexpr {{{1|X}}} | NUMBER | ID }}
341
340
2012-01-24T00:32:34Z
Spencer Bliven
1
r1mwbylaibgafg9l9yje29wj1zn6y92
figure {{#expr {{{1|X}}} }}
342
341
2012-01-24T00:58:47Z
Spencer Bliven
1
qx23lvikn6ty15q8ufgdhb6m55u0vs3
figure '''{{#expr: 1 and -1 }}'''
343
342
2012-01-24T00:59:06Z
Spencer Bliven
1
f3hpf3x3aen0fxq38xx4ect6sja8cfm
figure '''{{#expr: 2 }}'''
344
343
2012-01-24T00:59:21Z
Spencer Bliven
1
q0ccf2wjslj8k4omdne3lt8v9yhgq5i
figure '''{{#expr: fig2 }}'''
345
344
2012-01-24T00:59:53Z
Spencer Bliven
1
4vsa07493vff700hi0y73xtvtuzbaj7
figure '''{{#iferror {{#expr: fig2 }} | error | no error }}'''
346
345
2012-01-24T01:01:22Z
Spencer Bliven
1
1818u8t6suyrjh5d8qfqxf9qq77o9ne
figure '''{{#expr: 2 }}'''
'''{{#ifexpr: | yes | no}}'''
347
346
2012-01-24T01:01:32Z
Spencer Bliven
1
sndb3vbm5mlogywkn9a31m6l65ij0f9
figure '''{{#expr: 2 }}'''
'''{{#ifexpr: 2| yes | no}}'''
348
347
2012-01-24T01:01:42Z
Spencer Bliven
1
bp2kuwa6gxa3mhfz0o5hy597qmxok2r
figure '''{{#expr: 2 }}'''
'''{{#ifexpr: foo| yes | no}}'''
349
348
2012-01-24T01:02:17Z
Spencer Bliven
1
r5jy4uokfp17u0nf7amwrw7znda6onh
{{#iferror: {{#expr: 1 + X }} | error | correct }}
350
349
2012-01-24T01:04:27Z
Spencer Bliven
1
8bghnydmu1nh85xp6ffn0xw5p5j0evp
{{#iferror: {{#expr: {{{1|fig}}} }} | error - ID | correct - number }}
351
350
2012-01-24T01:05:27Z
Spencer Bliven
1
7es707hmoem144pcpaycjsu3ehdq4w9
{{#iferror: {{#expr: {{{1|fig}}} }} | {{{1|fig}}} | fig{{{1}}} }}
352
351
2012-01-24T01:06:18Z
Spencer Bliven
1
feka48ihdsfczc642lnrep57y9a8lv8
[[#{{#iferror: {{#expr: {{{1|fig}}} }} | {{{1|fig}}} | fig{{{1}}} }} | {{1|?}} ]]
353
352
2012-01-24T01:08:54Z
Spencer Bliven
1
ejr8117oolxhkd61qpg5a5t612cd604
[[#fig{{{1|}}} | {{{1|?}}} ]]
355
353
2012-01-24T01:12:36Z
Spencer Bliven
1
oezs9e7tm58mmj9rpl35r7okiqrpniq
[[#fig{{{1|}}} | {{#if {{{2|}}} | {{{2}}} | {{{1|?}}} }}]]
356
355
2012-01-24T01:13:24Z
Spencer Bliven
1
qdkbpgk45gya7g695mt79dlac5j7piw
[[#fig{{{1|}}} | {{#if {{{2|}}} | {{{2}}} | {{{1|?}}} }} ]]
357
356
2012-01-24T01:13:52Z
Spencer Bliven
1
t6uvg8psuobkx0upm215lofp67qqq7y
[[#fig{{{1|}}} | {{#if: {{{2|}}} | yes | no}} ]]
358
357
2012-01-24T01:14:10Z
Spencer Bliven
1
jlv3itbct5b1vimdm1juxb7agpy5xk6
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}} | no}} ]]
362
358
2012-01-24T01:21:02Z
Spencer Bliven
1
npukyoihn1sdp0e8jyqjjhzb8srl6f7
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}}| no}} ]]
363
362
2012-01-24T01:22:14Z
Spencer Bliven
1
c0o0h899ufxoqluygjz86nh5b8dz3t6
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}}| figure {{{1|?}}} }} ]]<includeonly>
</noinclude>
364
363
2012-01-24T01:22:37Z
Spencer Bliven
1
sbp8gwn5p6pn5o5vlekno3mst97nzu3
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}}| figure {{{1|?}}} }} ]]<includeonly>
</includeonly>
365
364
2012-01-24T01:22:58Z
Spencer Bliven
1
6y99wk146vsf4lns3dd8oll6m75xfus
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}}| figure {{{1|?}}} }} ]]<noinclude>
</noinclude>
366
365
2012-01-24T01:23:35Z
Spencer Bliven
1
2til7jdlp339ti7gmedg83z987ev8pt
[[#fig{{{1|}}} | {{#if: {{{2|}}} | {{{2}}}| figure {{{1|?}}} }}]]<noinclude>
</noinclude>