PLoSWiki http://topicpages.ploscompbiol.org/wiki/Main_Page MediaWiki 1.20.4 first-letter Media Special Talk User User talk PLoSWiki PLoSWiki talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_f denotes an apparent dissociation constant, [X] denotes ligand concentration and n_H is the Hill coefficient. If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative negative cooperativity, and if n_{_H}>1, we have positive cooperativity. The total number of ligand binding sites is an upper bound for n_H. The Hill plot, obtained by plotting <math>\log \frac{\bar{Y}}{1-\bar{Y}}</math> versus log([X]) is linear, with slope n_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 2102 2012-12-11T10:33:31Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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]] 2104 2103 2012-12-11T10:40:24Z Nlenovere 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 Nov&egrave;re, N. (2012) Using chemical kinetics to model neuronal signalling pathways. In: Computational Systems Neurobiology, N. Le Nov&egrave;re (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 CO₂ 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^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_H is the Hill coefficient, [X] denotes ligand concentration and K_a denotes an apparent associatin constant (note that the modern form of the equation uses a dissociation constant instead). If n_H<1, the system exhibits negative cooperativity, while if n_{_H}>1, cooperativity is positive. The total number of ligand binding sites is an upper bound for n_H. 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_H 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i. For a reaction of the form <math> P+X \rightarrow PX </math> governed by an association constant K_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_A, 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_i: <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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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_i 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, K₁ is the association constant governing binding of the first ligand molecule, K₂ 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 K₁, K₂, ... 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_i 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2123 2012-12-11T16:16:43Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2139 2138 2012-12-12T14:27:08Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2139 2012-12-12T14:33:03Z Nlenovere 36 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2143 2142 2012-12-12T14:50:49Z Nlenovere 36 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2144 2143 2012-12-12T14:56:09Z Nlenovere 36 /* Conformational Spread */ 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2145 2012-12-12T15:09:55Z Nlenovere 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2148 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt. 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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2151 2150 2012-12-12T17:55:45Z Nlenovere 36 mhanulng38j89cuhgakl5bfl1a709mg {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2152 2151 2012-12-12T17:59:29Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2155 2012-12-13T10:03:44Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2158 2157 2012-12-13T13:51:29Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2159 2158 2012-12-13T14:53:13Z Nlenovere 36 c29z60wcrb33ivqldcjwy7jv86w1bym {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2160 2159 2012-12-14T14:04:11Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2163 2162 2012-12-14T17:08:22Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2163 2012-12-14T17:34:03Z Nlenovere 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2165 2164 2012-12-14T17:48:49Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2203 2012-12-19T11:05:28Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2208 2207 2013-01-03T09:11:00Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2209 2208 2013-01-03T09:23:24Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2226 2225 2013-01-03T22:57:06Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2226 2013-01-03T23:01:47Z 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2228 2227 2013-01-03T23:36:27Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2229 2228 2013-01-03T23:47:11Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2230 2229 2013-01-03T23:49:11Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2231 2230 2013-01-03T23:52:55Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2232 2231 2013-01-03T23:54:43Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2233 2232 2013-01-03T23:56:06Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2234 2233 2013-01-04T00:00:28Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2235 2234 2013-01-04T16:57:24Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2236 2235 2013-01-04T16:58:57Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2237 2236 2013-01-04T17:05:29Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2238 2237 2013-01-04T17:07:00Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2239 2238 2013-01-04T17:27:40Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2244 2239 2013-01-09T22:03:52Z Daniel Mietchen 5 bolding topic upon first use 11fep4d8807zcjbs02y3iqujs9sb19f {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2245 2244 2013-01-09T22:08:58Z Daniel Mietchen 5 attempt at rewriting the introductory definition 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 2246 2245 2013-01-09T22:30:35Z Daniel Mietchen 5 annotations p5haeavhtrl70pzcf90hq13zljk9lmq {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 2246 2013-01-09T22:34:30Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2249 2247 2013-01-09T22:57:29Z Daniel Mietchen 5 , 8sx476nnsuxxnfh5xlz4vp1l04s9qnt {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2250 2249 2013-01-09T22:58:15Z Daniel Mietchen 5 <!--add ref--> 6amxng9gz0y7fgpd8jxmo8k89eqzjac {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2251 2250 2013-01-09T23:01:53Z Daniel Mietchen 5 slight rephrasing 05wt0v8j74su8zmz0oii53cj584oh4q {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2256 2251 2013-01-14T22:39:26Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2270 2256 2013-02-01T11:28:43Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2272 2270 2013-02-01T11:35:46Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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⁺ ([[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]] 2273 2272 2013-02-01T11:37:41Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2275 2274 2013-02-01T11:40:54Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2276 2275 2013-02-01T11:42:35Z Nlenovere 36 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2279 2276 2013-02-01T22:19:13Z Mstefan 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2282 2279 2013-02-05T16:32:29Z Nlenovere 36 kmie6vgrht2prep2f1jane2vru8ts5j {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2510 2282 2013-03-12T01:32:55Z Daniel Mietchen 5 formatting rw0rhb723g92iinooam7wni12xi58lk {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2541 2510 2013-03-25T10:34:28Z Daniel Mietchen 5 formatting 6f5i7g27ywps6a6hps0qfggbyux65m3 {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2542 2541 2013-03-25T10:36:19Z Daniel Mietchen 5 formatting nmrg1hmjqct7vrj4vo0jwc3pdwpgdbj {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂pressure shifted this curve to the right - i.e. higher concentrations of CO₂make 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 CO₂reduces hemoglobin's facility to bind oxygen.) Throughout the 20^thcentury, 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^thcentury, 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 Ca²⁺([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2544 2543 2013-03-25T10:53:16Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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 CO₂ pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2545 2544 2013-03-25T10:56:18Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2546 2545 2013-03-25T11:02:11Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2546 2013-03-25T11:03:24Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2550 2549 2013-03-25T11:10:15Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 5 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2553 2552 2013-03-25T12:27:10Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2554 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2557 2556 2013-03-25T12:37:35Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2558 2557 2013-03-25T12:38:51Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2560 2559 2013-03-25T12:41:53Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2562 2013-03-25T12:47:50Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2563 2013-03-25T12:52:16Z Daniel Mietchen 5 /* The MWC model */ 81xk5anns5ddf9ak8h699kv92yydn08 {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2564 2013-03-25T12:53:42Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2565 2013-03-25T12:57:34Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2567 2566 2013-03-25T12:59:42Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2569 2568 2013-03-25T13:02:51Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2570 2569 2013-03-25T13:04:56Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2570 2013-03-25T13:06:05Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2572 2571 2013-03-25T13:08:00Z Daniel Mietchen 5 /* Transcription factors */ ikbsce1dob6ys9xvv1nndwojp5hp09r {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2573 2572 2013-03-25T13:11:19Z Daniel Mietchen 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 5 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2577 2575 2013-03-26T16:09:02Z Nlenovere 36 /* The Klotz equation */ p6qp5o2t6fo3wgaabqy498xtc0beaei {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2584 2577 2013-03-28T15:41:44Z Daniel Mietchen 5 removed {{citation needed}} as per talk page r1f0f9y1lh63vvw0glvzp3r8hhu9qm8 {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2603 2584 2013-04-12T19:09:52Z Mstefan 25 /* The Hill equation */ - fixed a typo ho8amzbchtawbd8zcm97v2cike5giqs {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 2603 2013-04-12T19:26:45Z Mstefan 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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 25 /* 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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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 Ca²⁺ ([[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]] 2607 2606 2013-04-17T09:33:20Z Nlenovere 36 /* Examples of cooperative binding */ l6rrhrz6utr75q6m28smahyc3i3f0y1 {{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 Kohlens&auml;urespannung des Blutes auf dessen Sauerstoffbindung &uuml;bt.''' ''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|CO₂]] pressure shifted this curve to the right - i.e. higher concentrations of CO₂ 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 CO₂ reduces hemoglobin's facility to bind oxygen.) Throughout the 20^th 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^th 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]] 10 46 203 2011-11-24T02:47:20Z Spencer Bliven 1 Created page with "|" 7boi6rb359a8bssoe6mvr1x1c03ftug | 10 45 194 2011-11-24T02:01:51Z Spencer Bliven 1 Initial Authors template 68jmwnk9kms5epeenfwqf34w56m8o9n <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 }}</nowiki></code> ==Parameters== {| ! Parameter !! Description |- | first || first name or initials |- | last || last name |- | department || Department |- | institution || University or corporation |- | username || The user page of this author, including the User: prefix |} </noinclude><includeonly> <div class="authors">{{{first}}} {{{last}}}¹ {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department}}}, {{{institution}}}, {{{address}}}. |} </div> 196 194 2011-11-24T02:14:03Z Spencer Bliven 1 pdl6dkn3serrpd6y6rpuk8crxsl7c3l <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 }}</nowiki></code> ==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">{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}¹ {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department1|{{{department}}} }}}, {{{institution1|{{{institution}}} }}}, {{{address1|{{{address}}} }}}. |} </div> </includeonly> 197 196 2011-11-24T02:22:12Z Spencer Bliven 1 m6a08w6cyh64ccfnwy2fu1rg8hhd15i <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"> Username:[[{{{username}}}]] {{#if: {{{username1|{{{username|}}} }}} | had a username | didn't have a username }} {{#if: {{{username1|{{{username|}}} }}} | [[{{{username1|{{{username}}} }}}|{{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}}]] | {{{first1|{{{first}}} }}} {{{last1|{{{last}}} }}} }}¹ {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department1|{{{department}}} }}}, {{{institution1|{{{institution}}} }}}, {{{address1|{{{address}}} }}}. |} </div> </includeonly> 199 197 2011-11-24T02:23:09Z Spencer Bliven 1 rlewjkhv08tgcc9wodqhm62ok5h65vw <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}}} }}} }}¹ {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department1|{{{department}}} }}}, {{{institution1|{{{institution}}} }}}, {{{address1|{{{address}}} }}}. |} </div> </includeonly> 200 199 2011-11-24T02:27:05Z Spencer Bliven 1 p4bku5l9gnt3w10p2mzwb8vgkpiphcb <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}}} }}} }}¹ {{#if: {{{last2|}}} | {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}² |}} {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | |- |²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} |} </div> </includeonly> 201 200 2011-11-24T02:29:00Z Spencer Bliven 1 8hgtbgpfl2is6kz6ip624es9gyzr4bs <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 !! 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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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}² |}} {| class="collapsible collapsed" ! Author Affiliations |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} |} </div> </includeonly> 304 202 2012-01-03T10:40:02Z Daniel Mietchen 5 &nbsp; ap0kk1dw38308c7385y6uld8e0nhn11 <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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}² |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{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}}} }}} }}¹ {{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}² |}} {{#if: {{{last3|}}} |, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³ |}} {{#if: {{{last4|}}} |, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴ |}} {{#if: {{{last5|}}} |, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵ |}} {{#if: {{{last6|}}} |, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶ |}} {{#if: {{{last7|}}} |, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷ |}} {{#if: {{{last8|}}} |, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸ |}} {{#if: {{{last9|}}} |, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}² |}} {{#if: {{{last3|}}} |, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³ |}} {{#if: {{{last4|}}} |, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴ |}} {{#if: {{{last5|}}} |, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵ |}} {{#if: {{{last6|}}} |, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶ |}} {{#if: {{{last7|}}} |, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷ |}} {{#if: {{{last8|}}} |, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸ |}} {{#if: {{{last9|}}} |, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}²}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³ |}} {{#if: {{{last4|}}} |, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴ |}} {{#if: {{{last5|}}} |, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵ |}} {{#if: {{{last6|}}} |, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶ |}} {{#if: {{{last7|}}} |, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷ |}} {{#if: {{{last8|}}} |, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸ |}} {{#if: {{{last9|}}} |, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}²}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³ }} {{#if: {{{last4|}}} |, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴ |}} {{#if: {{{last5|}}} |, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵ |}} {{#if: {{{last6|}}} |, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶ |}} {{#if: {{{last7|}}} |, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷ |}} {{#if: {{{last8|}}} |, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸ |}} {{#if: {{{last9|}}} |, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}²}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}²}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{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}}} }}} }}¹{{#if: {{{last2|}}} |, {{#if: {{{username2| }}} | [[{{{username2| }}}|{{{first2}}} {{{last2}}}]] | {{{first2}}} {{{last2}}} }}²}} {{#if: {{{last3|}}}|, {{#if: {{{username3| }}} | [[{{{username3| }}}|{{{first3}}} {{{last3}}}]] | {{{first3}}} {{{last3}}} }}³}} {{#if: {{{last4|}}}|, {{#if: {{{username4| }}} | [[{{{username4| }}}|{{{first4}}} {{{last4}}}]] | {{{first4}}} {{{last4}}} }}⁴}} {{#if: {{{last5|}}}|, {{#if: {{{username5| }}} | [[{{{username5| }}}|{{{first5}}} {{{last5}}}]] | {{{first5}}} {{{last5}}} }}⁵}} {{#if: {{{last6|}}}|, {{#if: {{{username6| }}} | [[{{{username6| }}}|{{{first6}}} {{{last6}}}]] | {{{first6}}} {{{last6}}} }}⁶}} {{#if: {{{last7|}}}|, {{#if: {{{username7| }}} | [[{{{username7| }}}|{{{first7}}} {{{last7}}}]] | {{{first7}}} {{{last7}}} }}⁷}} {{#if: {{{last8|}}}|, {{#if: {{{username8| }}} | [[{{{username8| }}}|{{{first8}}} {{{last8}}}]] | {{{first8}}} {{{last8}}} }}⁸}} {{#if: {{{last9|}}}|, {{#if: {{{username9| }}} | [[{{{username9| }}}|{{{first9}}} {{{last9}}}]] | {{{first9}}} {{{last9}}} }}⁹ |}} {| class="collapsible collapsed" ! Author Affiliations&nbsp; |- |¹{{{department1|{{{department|}}} }}}, {{{institution1|{{{institution|}}} }}}, {{{address1|{{{address|}}} }}}. {{#if: {{{last2|}}} | {{!}}- {{!}}²{{{department2|}}}, {{{institution2|}}}, {{{address2|}}}. |}} {{#if: {{{last3|}}} | {{!}}- {{!}}³{{{department3|}}}, {{{institution3|}}}, {{{address3|}}}. |}} {{#if: {{{last4|}}} | {{!}}- {{!}}⁴{{{department4|}}}, {{{institution4|}}}, {{{address4|}}}. |}} {{#if: {{{last5|}}} | {{!}}- {{!}}⁵{{{department5|}}}, {{{institution5|}}}, {{{address5|}}}. |}} {{#if: {{{last6|}}} | {{!}}- {{!}}⁶{{{department6|}}}, {{{institution6|}}}, {{{address6|}}}. |}} {{#if: {{{last7|}}} | {{!}}- {{!}}⁷{{{department7|}}}, {{{institution7|}}}, {{{address7|}}}. |}} {{#if: {{{last8|}}} | {{!}}- {{!}}⁸{{{department8|}}}, {{{institution8|}}}, {{{address8|}}}. |}} {{#if: {{{last9|}}} | {{!}}- {{!}}⁹{{{department9|}}}, {{{institution9|}}}, {{{address9|}}}. |}} |} </div> </includeonly> 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> 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>''' 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>