Partitioning the Hill coefficient into contributions from ligand‐promoted conformational changes and subunit heterogeneity

Abstract

Heterooligomers that undergo ligand‐promoted conformational changes are ubiquitous in nature and involved in many essential processes. Conformational switching often leads to positive cooperativity in ligand binding that is reflected in a Hill coefficient with a value greater than one. The subunits comprising heterooligomers can differ, however, in their affinity for the ligand. Such so‐called site heterogeneity results in apparent negative cooperativity that is reflected by a Hill coefficient with a value less than one. Consequently, positive cooperativity due to the ligand‐promoted allosteric switch can be masked, in cases of such heterooligomers, by apparent negative cooperativity owing to site heterogeneity. Here, we derived expressions for the Hill coefficient, in the case of a heterodimer, in which the contributions from the ligand‐promoted allosteric switch and site heterogeneity are separated. Using these equations and simulations for higher order oligomers, we show under which conditions site heterogeneity can significantly mask the extent of observed positive cooperativity.

1. INTRODUCTION

The term allostery refers to cases when ligand binding (or some other perturbation) at one site affects the activity or function (e.g., ligand binding) of another distant site in the same molecule. Allostery in the function of multi‐subunit proteins is often reflected in sigmoidal plots of fractional saturation of ligand binding sites as a function of ligand concentration. Such data can be fitted to the Hill equation ¹ :

$$\overline{Y}=\frac{K{\left[S\right]}^n}{1+K{\left[S\right]}^n}$$ (1)

where $\overline{Y}$ and K designate the fractional saturation and apparent binding constant, respectively; [S] is the ligand concentration; and n is the Hill coefficient. The Hill equation was originally derived by assuming that binding takes place in an all‐or‐none fashion, although this occurs rarely. Nevertheless, the Hill coefficient provides a model‐independent measure for the extent of cooperativity in a system under equilibrium and steady‐state (when $\overline{Y}$ is replaced by initial enzyme velocity divided by the maximal initial velocity, V/V _max) conditions or in transient kinetic experiments. ²

Cooperativity in ligand binding can be either positive or negative. Positive cooperativity, that is, when ligand binding at one site favors binding at other sites, is reflected in values of n greater than one. Positive cooperativity can arise, for example, when the equilibrium of a protein between states with low‐ and high‐affinity for the ligand is shifted toward the high‐affinity state upon ligand binding. ³ Values of n less than one can be observed in cases of negative cooperativity, that is, when ligand binding at one site disfavors binding at other sites, or when site heterogeneity exists. In the latter case, when sites with different affinities for the ligand preexist, a value of n less than one will be observed even when there is no interaction between the sites because the high‐affinity sites will become bound first.

There are many proteins with site heterogeneity, which also undergo ligand binding‐promoted switching between low‐ and high‐affinity states. A classic example is hemoglobin, which is comprised of α and β subunits with different affinities for oxygen, that undergoes oxygen‐dependent allosteric switching. ⁴ Another example is the chaperonin containing t‐complex polypeptide 1 (CCT/TRiC), which contains eight distinct subunits with different affinities for ATP ⁵ and undergoes a large ATP‐promoted conformational change reflected in n > 2. ⁶ A third example is the human nicotinamide adenine dinucleotide‐dependent isocitrate dehydrogenase hetero‐tetramer, which contains at least two different binding sites for isocitrate and displays positive cooperativity with n > 2, with respect to isocitrate. ⁷ A final example is the hetero‐pentameric α₂βγδ nicotinic receptor, which contains two binding sites with distinct affinities for agonists and antagonists ⁸ and undergoes a conformational change that can be described ⁹ by the Monod–Wyman–Changeux (MWC) model. ³ There are numerous other examples, such as heteromeric G‐protein coupled receptors, ¹⁰ although the biochemical data for many of them regarding site‐specific affinities and cooperativity are often not available. In all these cases, positive cooperativity due to the ligand‐promoted allosteric switch, as reflected by n > 1, will be masked to some extent by the apparent negative cooperativity owing to site heterogeneity. Here, we derived expressions for the Hill coefficient, in the case of a dimer, in which the contributions from the ligand‐promoted allosteric switch and site heterogeneity are separated. Using these equations and simulations, we can determine the conditions under which site heterogeneity can significantly mask the extent of observed positive cooperativity.

2. RESULTS AND DISCUSSION

We first consider cases of dimers for which analytical expressions can be derived. The relationship between the Hill coefficient and the ligand binding constants of a dimer has been derived before ¹¹ as follows. Rearranging Equation (1) yields the following expression for the Hill coefficient:

$$n=\frac{d\log \frac{\overline{Y}}{\left(1-\overline{Y}\right)}}{d\log \left[S\right]}=\frac{\left[S\right]}{\overline{Y}\left(1-\overline{Y}\right)}\frac{d\overline{Y}}{d\left[S\right]}$$ (2)

It follows from Equation (2) that the Hill coefficient at 50% saturation, $n_{0.5}$, is given by:

$$n_{0.5}=4{\left[S\right]}_{0.5}\frac{d\overline{Y}}{d\left[S\right]}{\mid }_{\left[S\right]={\left[S\right]}_{0.5}}$$ (3)

where ${\left[S\right]}_{0.5}$ is the substrate concentration at the midpoint of the binding curve. The fractional saturation, in the case of a dimer, is given by:

$$\overline{Y}=\frac{K_1\left[S\right]+2K_1{K}_2{\left[S\right]}^2}{2\left(1+K_1\left[S\right]+K_1{K}_2{\left[S\right]}^2\right)}$$ (4)

where K ₁ and K ₂ are the apparent association constants for the respective first and second binding events. Hence, ${\left[S\right]}_{0.5}=1/\sqrt[2]{K_1{K}_2}$ (i.e., when $\overline{Y}=0.5).$ Combining this relationship with Equations (3) and (4) results in the following expression for the Hill coefficient ¹⁰ :

$$n_{0.5}=\frac{4}{2+\sqrt[2]{\frac{K_1}{K_2}}}$$ (5)

It can be seen that n _0.5 = 1 for a symmetric dimer when the intrinsic equilibrium constants, K _1,i  = 0.5K ₁ and K _2,i  = 2K ₂, are equal to each other, that is, in the absence of cooperativity. In the presence of strong positive cooperativity (K _2,i  >> K _1,i ) or strong negative cooperativity (K _2,i  << K _1,i ), n _0.5 → 2 and n _0.5 → 0, respectively.

In the case of concerted conformational changes, which are described by the MWC model, ³ cooperativity is due to an equilibrium between two unliganded states: a tense (T) state with low affinity for the ligand, which is the predominant form in the absence of ligand, and a relaxed (R) state with high affinity for the ligand. The extent of cooperativity according to this model 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 _T/K _R). In the case of the MWC model for a symmetric dimer, it is straightforward to show that $K_{1,i}=\frac{K_{\mathrm{R}}+L{\mathrm{K}}_{\mathrm{T}}}{L+1}$ and $K_{2,i}=\frac{K_{\mathrm{R}}^2+L{K}_{\mathrm{T}}^2}{K_{\mathrm{R}}+L{K}_{\mathrm{T}}}$. The Hill coefficient for this case is, therefore, given by:

$$n_{0.5}=\frac{2}{1+\frac{1+\mathit{cL}}{\sqrt[2]{\left(L+1\right)\left(1+c^{2}L\right)}}}$$ (6)

2.1. Site heterogeneity in the concerted case

We now consider an asymmetric dimer AB that switches in a concerted fashion from a low affinity T state (A_TB_T) to a high affinity R state (A_RB_R) where L = [A_TB_T]/[A_RB_R]. In such a case, the apparent binding constants are given by:

$$K_1=\frac{K_{\mathrm{AR}}+K_{\mathrm{BR}}+L\left(K_{\mathrm{AT}}+K_{\mathrm{BT}}\right)}{L+1}$$ (7)

$$K_2=\frac{K_{\mathrm{AR}}{K}_{\mathrm{BR}}+L{K}_{\mathrm{AT}}{K}_{\mathrm{BT}}}{K_{\mathrm{AR}}+K_{\mathrm{BR}}+L\left(K_{\mathrm{AT}}+K_{\mathrm{BT}}\right)}$$ (8)

where K _AR and K _BR are the ligand binding constants of the A and B subunits in the R state and K _AT and K _BT are those for the T state. Combining Equations (5), (7), and (8) and assuming, for simplicity, that α = K _AR/K _BR = K _AT/K _BT, one obtains:

$$n_{0.5}=\frac{4}{2+\frac{\left(1+\alpha \right)}{\sqrt[2]{\alpha }}\frac{\left(1+\mathit{cL}\right)}{\sqrt[2]{\left(L+1\right)\left(1+c^{2}L\right)}}}$$ (9)

The contributions of the ligand‐promoted conformational switch and site heterogeneity to the observed Hill coefficient are separated in Equation (9), thereby allowing to assess their relative contributions. The term $\left(1+\alpha \right)/\sqrt[2]{\alpha }$ represents the contribution of the site heterogeneity. It may be seen that when α = 1, that is, when the two sites are identical, $\left(\left(1+\alpha \right)/\sqrt[2]{\alpha }\right)=2$ and Equation (9) reduces to Equation (6) for a symmetric dimer. It can also be seen that it is always true that $\left(\left(1+\alpha \right)/\sqrt[2]{\alpha }\right)\ge 2$ and $\left(\left(1+\mathit{cL}\right)/\sqrt[2]{\left(L+1\right)\left(1+c^{2}L\right)}\right)\le 1$. The tradeoff between these two respective terms for site heterogeneity and ligand‐promoted conformational switch, therefore, determines the extent of cooperativity and whether it is positive or negative. Finally, it should be noted that this analysis does not depend on whether α is defined as above or by the inverse ratios since $\left(\left(1+\alpha \right)/\sqrt[2]{\alpha }\right)=\left(\left(1+1/\alpha \right)/\sqrt[2]{1/\alpha }\right)$.

Next, we consider the more general case in which α = K _AR/K _BR and β = K _AT/K _BT, that is, the ratios of affinities of the two sites in the T and R states are not equal. In such a case:

$$K_1=\frac{\left(1+\alpha \right)K_{\mathrm{R}}+L\left(1+\beta \right)K_{\mathrm{T}}}{L+1}$$ (10)

$$K_2=\frac{\alpha K_{\mathrm{R}}^2+\mathit{L\beta }{K}_{\mathrm{T}}^2}{\left(1+\alpha \right)K_{\mathrm{R}}+L\left(1+\beta \right)K_{\mathrm{T}}}$$ (11)

The Hill coefficient for this case is, therefore, given by:

$$n_{0.5}=\frac{4}{2+\frac{\left(1+\alpha \right)K_{\mathrm{R}}+L\left(1+\beta \right)K_{\mathrm{T}}}{\sqrt[2]{\left(L+1\right)\left({\mathit{\alpha K}}_{\mathrm{R}}^2+{\mathit{\beta K}}_{\mathrm{T}}^2\right)}}}$$ (12)

To determine how the relative values of α and β affect the contribution of site heterogeneity to the observed cooperativity, we need to solve the inequality:

$$\frac{\left(1+\alpha \right)K_{\mathrm{R}}+L\left(1+\beta \right)K_{\mathrm{T}}}{\sqrt[2]{\left(L+1\right)\left({\mathit{\alpha K}}_{\mathrm{R}}^2+{\mathit{\beta K}}_{\mathrm{T}}^2\right)}}>\frac{\left(1+\alpha \right)}{\sqrt[2]{\alpha }}\frac{\left(K_{\mathrm{R}}+L{K}_{\mathrm{T}}\right)}{\sqrt[2]{\left(L+1\right)\left({\mathit{LK}}_{\mathrm{T}}^2+K_{\mathrm{R}}^2\right)}}$$ (13)

The contribution of site heterogeneity to the observed cooperativity is largest under conditions when this inequality is satisfied. Rearranging and simplifying the above inequality yields:

$${\left(\frac{1+\mathit{Lc}\frac{\beta +1}{\alpha +1}}{1+\mathit{Lc}}\right)}^2>\frac{\left(1+L{c}^2\frac{\beta }{\alpha }\right)}{\left(1+L{c}^2\right)}$$ (14)

Assuming that Lc ² >> 1, which is generally the case, one obtains:

$${\left(\frac{1+\beta }{1+\alpha }\right)}^2>\frac{\beta }{\alpha }$$ (15)

This inequality holds for α > 1 when β > α or β < 1/α and for α < 1 when β > 1/α. These three conditions are equivalent in the sense that they are met when the differences in affinities of the A and B sites in the T state are greater than those in the R state. In other words, site heterogeneity in the T state has a greater masking effect than that in the R state.

2.2. Site heterogeneity in the sequential case

In the Koshland–Némethy–Filmer (KNF) sequential model, ¹² ligand binding induces a conformational change only in the ligand‐bound subunit and symmetry is not conserved. In other words, the ligand‐promoted conformational switch in a multi‐subunit protein does not take place in an all‐or‐none fashion as in the MWC model. In the case of a symmetric dimer that undergoes binding‐induced sequential conformational changes, the apparent binding constants are given by:

$$K_1=K_{\mathrm{b}}{K}_{\mathrm{c}}\frac{K_{\mathrm{tr}}}{K_{\mathrm{tt}}}$$ (16)

$$K_2=K_{\mathrm{b}}{K}_{\mathrm{c}}\frac{K_{\mathrm{rr}}}{K_{\mathrm{tr}}}$$ (17)

where K _b is the intrinsic binding constant, K _c is the equilibrium constant for the conformational switch of an individual subunit, t and r designate the conformations of free and ligand‐bound subunits and K _tt, K _tr, and K _rr are the respective interaction constants for neighboring subunits in the tt, tr, and rr states. The Hill coefficient in this case is given by:

$$n_{0.5}=\frac{2}{1+\sqrt[2]{\frac{K_{\mathrm{tr}}}{K_{\mathrm{tt}}}\frac{K_{\mathrm{tr}}}{K_{\mathrm{rr}}}}}$$ (18)

We now consider an asymmetric dimer AB in which the subunits switch in a sequential fashion from a t state to an r state. In such a case, the apparent binding constants are given by:

$$K_1=K_{\mathrm{b}}^{\mathrm{A}}{K}_{\mathrm{c}}^{\mathrm{A}}\frac{K_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{t}}}}{K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{t}}}}+K_{\mathrm{b}}^{\mathrm{B}}{K}_{\mathrm{c}}^{\mathrm{B}}\frac{K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}}{K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{t}}}}$$ (19)

$$K_2=\frac{K_{\mathrm{b}}^{\mathrm{A}}{K}_{\mathrm{c}}^{\mathrm{A}}{K}_{\mathrm{b}}^{\mathrm{B}}{K}_{\mathrm{c}}^{\mathrm{B}}{K}_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{r}}}}{K_{\mathrm{b}}^{\mathrm{A}}{K}_{\mathrm{c}}^{\mathrm{A}}{K}_{{\mathrm{A}}^{\mathrm{r}}{B}^{\mathrm{t}}}+K_{\mathrm{b}}^{\mathrm{B}}{K}_{\mathrm{c}}^{\mathrm{B}}{K}_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}}$$ (20)

where $K_{\mathrm{b}}^{\mathrm{x}}$ and $K_{\mathrm{c}}^{\mathrm{x}}$ are defined as above for subunit x (x = A or B) and $K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{t}}},K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}$, $K_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{t}}},$ and $K_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{r}}}$ are the respective interaction constants for neighboring subunits A and B in the designated t and r states. Combining Equations (5), (19), and (20), therefore, yields:

$$n_{0.5}=\frac{4}{2+\sqrt[2]{\frac{K_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{t}}}{K}_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}}{K_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{t}}}{K}_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{r}}}}}\left(\sqrt[2]{\frac{K_{\mathrm{b}}^{\mathrm{A}}{K}_{\mathrm{c}}^{\mathrm{A}}{K}_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{t}}}}{K_{\mathrm{b}}^{\mathrm{B}}{K}_{\mathrm{c}}^{\mathrm{B}}{K}_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}}}+\sqrt[2]{\frac{K_{\mathrm{b}}^{\mathrm{B}}{K}_{\mathrm{c}}^{\mathrm{B}}{K}_{{\mathrm{A}}^{\mathrm{t}}{\mathrm{B}}^{\mathrm{r}}}}{K_{\mathrm{b}}^{\mathrm{A}}{K}_{\mathrm{c}}^{\mathrm{A}}{K}_{{\mathrm{A}}^{\mathrm{r}}{\mathrm{B}}^{\mathrm{t}}}}}\right)}$$ (21)

Inspection of Equation (21) shows that the contributions of the ligand‐promoted conformational switch and site heterogeneity to the observed Hill coefficient can be separated also in the sequential case. The term $\sqrt[2]{\left(K_{A^r{B}^t}{K}_{A^t{B}^r}\right)/\left(K_{A^t{B}^t}{K}_{A^r{B}^r}\right)}$, which represents the contribution of the ligand‐promoted conformational switch, is a measure of how the free energy of the transition of one subunit from t to r depends on whether the other subunit is in the t or r conformations. A similar term determines the extent of cooperativity in the sequential case when site heterogeneity is absent. The other two terms correspond to the contribution of the site heterogeneity to the observed cooperativity. It can be seen that both of these terms are equal to one when the subunits are identical in which case Equation (21) simplifies to Equation (18) when site heterogeneity is absent.

2.3. Extending the analysis for higher order oligomers by simulation

Given that analytical expressions can be derived only for dimers, we carried out simulations to explore the effect of site heterogeneity on the Hill coefficient values of higher order heterooligomers. For simplicity, the simulations were carried out assuming concerted conformational changes and that site heterogeneity can be described by a single parameter α (i.e., for N sites K _N,R = α ^N−1 K _1,R and K _N,T = α ^N−1 K _1,T). The results of the simulations show that the value of the Hill coefficient decreases with an increasing value of α, which is a measure of the extent of site heterogeneity (Figure 1a,b). Simulations were also performed for tetramers comprising two subunits of one type and two of a second type as in the case of hemoglobin. The simulations were carried out using different values of L and constant values of c = 0.01 or c = 0.001 (Figure 1c,d). Values of c = 0.01 and L = 10⁵ have been reported for hemoglobin, ¹³ but there is uncertainty in these values owing to a dependency between L and c in data fitting. ¹⁴ , ¹⁵ , ¹⁶ It may be seen that the value of the Hill coefficient is again found to decrease with an increasing value of α. The increase in the value of the Hill coefficient with increasing values of L for c = 0.001 (Figure 1c) and the converse for c = 0.01 (Figure 1d) is due to the bell‐shaped dependence of the value of the Hill coefficient on L when ligand binding is not exclusively to the R state. ¹⁷ Assuming the values of c = 0.01 and L = 10⁵ for hemoglobin, ¹³ the value of the Hill coefficient decreases from about 2.8 for α = 1 to 2.6 for α = 5 (Figure 1d), which was reported to be an upper limit estimate of the nonequivalence between the alpha and beta subunits in hemoglobin. ⁴ The extent of decrease depends on the precise values of L and c and on whether α is the same for the T and R states.

FIGURE 1.

The effect of site heterogeneity on the observed cooperativity of heterooligomers. The values of the Hill coefficient for heterooligomers (dimers, tetramers, or octamers) are plotted against the value of α, which is a measure of the extent of site heterogeneity. The values of the Hill coefficient were calculated using Equation (3) in which the fractional saturation ($\overline{Y})$ was expressed according to the Monod–Wyman–Changeux (MWC) model with (a) L = 1,000 and c = 0.1 or (b) L = 100,000 and c = 0.001. The simulations were performed in a similar manner also for a tetramer comprising two subunits of one type and two of a second type as in the case of hemoglobin. These simulations were carried out using different values of L and constant values of (c) c = 0.01 or (d) c = 0.001. The substrate concentration at half‐saturation ([S]_0.5), that is, when $\overline{Y}$ = 0.5, was determined numerically for each value of α using a python script (Pycharm 3.1). The derivative of $\overline{Y}$ at $\overline{Y}$= 0.5 was determined by setting a small deviation from [S]_0.5 (Δ[S] = 0.00001) and calculating $\frac{\overline{Y}\left({\left[S\right]}_{0.5}+\mathrm{\Delta }\left[S\right]\right)-\overline{Y}\left({\left[S\right]}_{0.5}-\mathrm{\Delta }\left[S\right]\right)}{2\mathrm{\Delta }\left[S\right]}$

3. CONCLUDING REMARKS

The analysis here shows how site heterogeneity can reduce the observed positive cooperativity associated with ligand‐promoted conformational changes. Consequently, heterooligomers with a Hill coefficient of one for binding a certain ligand may in fact be binding it with positive cooperativity. In other words, the measured Hill coefficients for systems with site heterogeneity, therefore, provide only a lower limit estimate of the positive cooperativity that may exist. The simulation results show that the effect of site heterogeneity on the observed cooperativity increases for higher order oligomers (Figure 1a,b). Our analysis shows that the effect of site heterogeneity can also be larger when it cannot be described by a single parameter. The analysis here can account for the fact that ligand binding by large systems with site heterogeneity, such as the chaperonin CCT/TRiC that comprises eight different subunits in a ring, ⁶ can have surprisingly low Hill coefficients.

AUTHOR CONTRIBUTIONS

Mousam Roy: Data curation (equal); formal analysis (supporting); investigation (supporting); methodology (equal); writing – review and editing (supporting). Amnon Horovitz: Conceptualization (lead); data curation (equal); formal analysis (lead); funding acquisition (lead); investigation (equal); methodology (equal); project administration (lead); supervision (lead); writing – original draft (lead); writing – review and editing (lead).

Roy M, Horovitz A. Partitioning the Hill coefficient into contributions from ligand‐promoted conformational changes and subunit heterogeneity. Protein Science. 2022;31(5):e4298. 10.1002/pro.4298