Unpicking allosteric mechanisms of homo-oligomeric proteins by determining their successive ligand binding constants

Abstract

Advances in native mass spectrometry and single-molecule techniques have made it possible in recent years to determine the values of successive ligand binding constants for large multi-subunit proteins. Given these values, it is possible to distinguish between different allosteric mechanisms and, thus, obtain insights into how various bio-molecular machines work. Here, we describe for ring-shaped homo-oligomers, in particular, how the relationship between the values of successive ligand binding constants is diagnostic for concerted, sequential and probabilistic allosteric mechanisms.

This article is part of a discussion meeting issue ‘Allostery and molecular machines’.

1. Background

The allosteric regulation of multi-subunit proteins is often manifested in sigmoidal plots of initial enzyme velocity or fractional saturation as a function of ligand (or substrate) concentration [1]. Such sigmoidal plots usually indicate positive cooperativity in ligand binding, i.e. binding of a ligand to one site favours subsequent ligand binding to other sites. Two classical models that can account for such cooperativity have been put forward: (i) the Monod–Wyman–Changeux (MWC) model [2], in which conformational changes take place in a concerted manner and symmetry is conserved; and (ii) the Koshland–Némethy–Filmer (KNF) model [3], in which conformational changes occur in a sequential manner and symmetry is broken. Distinguishing between these models has, however, proven to be very difficult because sigmoidal plots are very insensitive to the presence of different ligation intermediates. Early studies of allosteric regulation focused on haemoglobin and proteins involved in metabolic control [4], for which discriminating between these models was of less importance because their function is affected mostly by the properties of their end states. More recently, however, it has become clear that allostery is also crucial for the function of molecular machines for which such discrimination is important because their efficiencies are path-dependent. For example, it has been suggested that eukaryotic chaperonins (e.g. CCT/TRiC) undergo ATP-promoted sequential allosteric changes that facilitate domain-by-domain protein substrate release and folding whereas prokaryotic chaperonins (e.g. GroEL) undergo concerted allosteric changes that result in all-or-none release and folding [5]. In support of this suggestion, a GroEL mutant that undergoes ATP-promoted sequential intra-ring conformational changes [6] was indeed shown to facilitate domain-by-domain release and folding of designed bi-domain substrates [7].

Fits of plots of the average initial enzyme velocity or fractional saturation as a function of ligand (or substrate) concentration to the Hill equation can indicate whether and to what extent cooperativity in ligand binding is positive, negative (assuming no site heterogeneity) or absent. These plots, however, rarely provide further insights into the allosteric mechanisms because it is usually not possible to extract from them the values of the individual binding constants corresponding to the successive ligation steps. This problem is particularly difficult in cases of large multi-subunit assemblies such as the chaperonins [8] and other ring-shaped ATP-dependent machines [9]. Owing to advances in native mass spectrometry [10,11] and single-molecule techniques [12] it has become possible, however, in recent years to determine the values of all the successive ligand binding constants for large multi-subunit assemblies and, thus, infer their allosteric mechanisms. In what follows, we show how the relationship between the values of successive ligand binding constants is diagnostic for the allosteric mechanisms of homo-oligomeric proteins, in general, and ring-shaped assemblies in particular. Other allosteric properties of oligomeric rings, such as how the sensitivity of conformational switching to changes in ligand concentration depends on the number of subunits in the ring, have been examined elsewhere [13]. In addition, it should be noted that dynamic allostery [14–16], in the absence of a conformational change in the mean structure, can also be displayed by oligomeric rings. An example for such a system is the trp RNA-binding attenuation protein, which forms an eleven-membered ring [17]. This type of allostery is, however, not discussed here because molecular machines usually undergo conformational changes triggered by ligand (e.g. ATP) binding.

2. Concerted allosteric mechanisms

In the concerted MWC model [2], cooperativity in ligand binding is due to an equilibrium between two unligated states: a tense (T) state with relatively low affinity for the ligand, which is the predominant form in the absence of ligand, and a relaxed (R) state with a higher affinity for the ligand. In this model, the extent of cooperativity is determined by the equilibrium constant L (=[T]/[R]) and by the relative affinities of the ligand for the T and R states (c = K_R/K_T). The MWC model is relatively simple since it involves only two parameters, L and c, but it has the limitation that it cannot account for negative cooperativity, which the more complex KNF model [3] is able to explain. Given that the values of the successive binding constants, in the case of the MWC model, do not depend on the symmetry type of the complex, we will consider here, for simplicity, a tetramer with C4 symmetry (figure 1).

Figure 1.

Scheme describing the concerted (MWC) allosteric model for a tetramer. According to the concerted model, the oligomer in its apo state is in equilibrium between low (T) and high (R) affinity states for the ligand (L = [T]/[R]) designated by square- and circle-shaped subunits, respectively. Ligand-bound and unbound subunits are designated in cyan and dark blue, respectively. In the case of the concerted model with exclusive binding of the ligand to the R state, the values of the intrinsic binding constants of all the successive binding steps will be the same except for the one corresponding to the first site, which will differ by a factor of 1/(1 + L). In the case of the concerted model with non-exclusive binding of the ligand to both the T and R states, the values of all the intrinsic binding constants corresponding to the successive binding steps will differ from each other but will form a series that can be expressed as a function of L and the respective ligand binding constants of the T and R states, K_T and K_R. (Online version in colour.)

In the case of exclusive binding of the ligand (substrate), S, to the R state (i.e. the T state has negligible affinity) of such a tetramer, E, the total concentrations of the different ligand-bound complexes are given by:

2.1

2.2

2.3

2.4

Inspection of equations (2.1)–(2.4) shows that the intrinsic binding constant for the first site is equal to K_R/(L + 1) whereas those for the other three sites are equal to K_R. It should be noted that apparent binding constants can be converted into intrinsic binding constants by applying a statistical correction that accounts for the number of ways the ith ligand molecule can bind or dissociate from the protein when i − 1 molecules are already bound. In general, therefore, finding that all the sites have an intrinsic association constant with the same value, except the first site which has one with a lower value, is diagnostic for the MWC model in the case of exclusive binding to the R state.

In the case of non-exclusive binding of the ligand to a tetramer, the total concentrations of the different ligand-bound complexes are given by:

2.5

2.6

2.7

2.8

It can be seen from inspection of equations (2.5) and (2.6) that the intrinsic association constant, K₂, for the second site is equal to

2.9

More generally, inspecting equations (2.5)–(2.8) shows that the intrinsic association constant for the ith site, K_i, is given by:

2.10

Hence, in the case of the MWC model when binding to the R state is non-exclusive, the intrinsic association constants of all the different sites differ from each other. They can all be expressed, however, as a function of K_R, K_T and L, thereby providing a test for this model for assemblies with a sufficiently large number of sites. In other words, a plot of K_i as a function of i can be fitted in such a case to equation (2.10) as demonstrated previously for GroEL [10].

An extension of the MWC model was proposed to account for observations that allosteric effectors can affect the oxygen affinity of haemoglobin without altering its quaternary equilibrium [18]. In this extended model, each subunit of a protein in the T and R quaternary states is also in equilibrium between two tertiary conformations, t and r. In the case of this model, each ligand binding constant is, therefore, a function of five parameters: the quaternary allosteric constant, a tertiary allosteric constant for each of the T and R states and the ligand affinities of the t and r tertiary states.

3. Sequential allosteric mechanisms

In this section, homo-hexameric rings will be considered since they are ubiquitous in Nature and sufficiently large to allow discrimination between different allosteric models. In the first model considered here, the conformational change propagates either clockwise or anti-clockwise around the ring in a sequential manner and cooperativity is due solely to changes in interactions between adjacent subunits (figure 2). Designating the respective ligand-free and ligand-bound conformations by A and B, it is possible to express the intrinsic association constant for the first site as follows:

3.1

where K_S is the binding constant of a ligand to a subunit in the A state, K_A→B is the equilibrium constant for the conformational switch of a subunit from A to B, and K_AA, K_BA and K_AB are the respective interaction constants for two neighbouring subunits in the A state and for one subunit in the B state and its neighbour in the clockwise or anti-clockwise directions, respectively, in the A state. This model describes induced fit [3] since the A → B conformational change is triggered by ligand binding to a subunit in the A state. The intrinsic association constants for the other five subunits can, therefore, be expressed as follows:

3.2

3.3

3.4

3.5

3.6

Figure 2.

Scheme describing a sequential (KNF) allosteric model for a hexameric ring. In this model, ligand binding induces a conformational change from an A to a B state designated by the change in colour from white to cyan. The intrinsic binding constants corresponding to the binding steps in this scheme are designated by K₁ to K₆. They are a function of the free energy changes associated with: (i) ligand binding to the A state; (ii) the ligand binding-induced conformational switch of a subunit from A to B; and (iii) the interactions between subunits in different states, i.e. AA, AB, BA or BB. (Online version in colour.)

It may be seen from inspection of equations (3.1)–(3.6) that a unidirectional sequential allosteric mechanism in which cooperativity is due to changes in interactions between neighbouring subunits will be reflected in association constants with three different values corresponding to the first, last and other four binding events, respectively. It can be easily verified that a similar result is reached in the case of six subunits in a row if the conformational wave begins at one of the ends. This analysis shows, therefore, that a unidirectional sequential allosteric mechanism can be distinguished from concerted mechanisms. Resolving the direction, clockwise or anti-clockwise, cannot be achieved using this type of analysis but may be addressed for hetero-oligomeric rings [19].

Next, we consider a unidirectional sequential allosteric mechanism in which binding of a ligand to one subunit affects the binding and conformational switching properties of its neighbouring subunit (in the clockwise or anti-clockwise direction) in addition to its interactions with both neighbouring subunits. In such a case, the intrinsic association constant for the first site can be expressed as follows:

3.7

where K_A→A′ is the equilibrium constant for the conformational switch of a subunit from A to A′ and K_BA′ and K_A′A are the interaction constants for neighbouring subunits in the B and A′ states or A and A′ states, respectively, defined as above. The remaining intrinsic association constants can, therefore, be expressed as follows:

3.8

3.9

3.10

3.11

and

3.12

where is the binding constant to a subunit in the A′ state. It may be seen from inspecting equations (3.7)–(3.12) that this allosteric mechanism would be reflected in association constants with four different values corresponding to the first, one before last, last and the remaining three binding events, respectively. Hence, a unidirectional sequential allosteric mechanism in which ligand binding to one subunit affects the binding and conformational switching properties of its neighbouring subunit (in either the clockwise or anti-clockwise direction) can be distinguished from a sequential mechanism in which ligand binding only affects inter-subunit interactions.

4. Probabilistic allosteric mechanisms

In this section, we consider a model in which binding-induced conformational changes need not take place in a sequential manner around the ring but can occur in any order of subunits (figure 3). For simplicity, we will assume here that cooperative effects arise only from binding-induced changes in inter-subunit interactions. Hence, equation (3.1) describes the intrinsic association constant for the first site as before. The expression for the intrinsic association constant for the second binding needs to account for three different ways by which the second ligand can bind and is as follows:

4.1

4.2

4.3

4.4

4.5

Figure 3.

Scheme describing a probabilistic allosteric model for a hexameric ring. In this model, ligand binding can occur in any order of subunits and induces a conformational change from an A to B state. The intrinsic binding constants corresponding to the successive binding steps, designated by K₁ to K₆, take into account all the different ways by which the ligand can bind the oligomer. The other notations are as in figure 2. (Online version in colour.)

Inspection of equations (3.1) and (4.1)–(4.5) shows that, in the case of a probabilistic allosteric mechanism, there is no fixed relationship between the successive intrinsic allosteric constants and that they all have different values.

5. Concluding remarks

Advances in native mass spectrometry [10,11] have made it possible in recent years to determine the values of successive ligand binding constants for multi-subunit proteins. Given these values, it is possible to distinguish between different allosteric mechanisms such as concerted, sequential or probabilistic and, thus, obtain insights into how various bio-molecular machines work. Use of the approach described here is currently limited to homo-oligomers containing at least four subunits that undergo binding-induced conformational changes (i.e. so called K-systems), which are very common in Nature. Its application has revealed that the intra-ring allosteric transitions of GroEL are concerted [10], in agreement with previous computational work [20]. Other approaches based, for example, on single-molecule techniques [12] will be required for resolving the allosteric mechanisms of hetero-oligomers or of assemblies that display cooperativity due to kinetic (V-system) or ligand-promoted assembly effects.

Data accessibility

This article has no additional data.

Authors' contributions

A.H. conceived the study and drafted the manuscript. R.G. helped draft the manuscript. All authors gave final approval for publication.

Competing interests

We declare we have no competing interest.

Funding

This work was supported by grant no. 2015170 from the US-Israel Bi-national Science Foundation. A.H. is an incumbent of the Carl and Dorothy Bennett Professorial Chair in Biochemistry.