GroEL Allostery Illuminated by a Relationship between the Hill Coefficient and the MWC Model

Abstract

A fundamental problem that has hindered the use of the classic Monod-Wyman-Changuex (MWC) allosteric model since its introduction is that it has been difficult to determine the values of its parameters in a reliable manner because they are correlated with each other and sensitive to the data-fitting method. Consequently, experimental data are often fitted to the Hill equation, which provides a measure of cooperativity but no insights into its origin. In this work, we derived a general relationship between the value of the Hill coefficient and the parameters of the MWC model. It is shown that this relationship can be used to select the best estimate of the true combination of the MWC parameter values from all the possible ones found to fit the data. Here, this approach was applied to fits to the MWC model of curves of the fraction of GroEL molecules in the high-affinity (R) state for ATP as a function of ATP concentration. Such curves were collected at different temperatures, thereby providing insight into the hydrophobic effect associated with the ATP-promoted allosteric switch of GroEL. More generally, the relationship derived here should facilitate future thermodynamic analysis of other MWC-type allosteric systems.

Significance

Many multi-subunit proteins undergo ligand-promoted conformational changes. Such allosteric switching can often be described with a concerted model of conformational changes. Fitting experimental data to this model is, however, often difficult because its parameters are correlated with each other and sensitive to the fitting procedure. Here, we derive a general expression that relates the Hill coefficient, a measure of cooperativity in ligand binding, to the parameters of the concerted model. This expression is used to help extract the parameters of the concerted model for GroEL, a chaperone that assists protein folding. The results shed light on the hydrophobic-to-hydrophilic switch that accompanies the ATP-promoted conformational changes of GroEL, which are crucial for its function.

Introduction

Protein oligomerization confers many selective advantages to organisms and is therefore found to be ubiquitous in nature (1). One advantage is the potential for evolution of allosteric regulation of protein function (2). Such allosteric control of multi-subunit proteins is often achieved through ligand-promoted conformational changes (3) that result in cooperative ligand binding. Two classic models were put forward in the 1960s to describe allostery in multimeric proteins: the concerted Monod-Wyman-Changeux (MWC) model (4, 5, 6) and the sequential Koshland-Némethy-Filmer (KNF) model (7). In the concerted model, the ligand-free protein is assumed to be in equilibrium between two states: a low-affinity “tense” (T) state and a high-affinity “relaxed” (R) state. Ligand binding shifts the equilibrium from T to R, but symmetry is maintained, i.e., all the subunits in a given molecule are either in T or R conformations. According to the MWC model, the extent of cooperativity in ligand binding is determined by the equilibrium in the absence of ligand between the T and R states (L = [T₀]/[R₀]) and the relative association constants of the ligand for the two states (c = K_T/K_R). The MWC model is elegant and relatively simple, but it cannot account for negative cooperativity, i.e., when ligand binding to one site decreases the affinity of other, still-unoccupied sites. In the case of the sequential KNF model, each subunit undergoes a ligand binding-induced conformational change, i.e., symmetry is not maintained. This model involves more parameters, but it can describe negative cooperativity.

Two fundamental problems have hindered the use of the MWC and KNF models ever since their introduction. First, it has been difficult to distinguish between the two models (except in cases of negative cooperativity) because sigmoidal binding curves are insensitive to the presence of different ligation intermediates. More recently, however, it was shown that the values of successive ligand binding constants of a homo-oligomeric protein can be determined by native mass spectrometry, thereby allowing for distinguishing between various allosteric models (8,9). Second, it has been difficult to determine the values of the parameters in the two models (e.g., L, K_R, and K_T in the MWC model) in a reliable manner because they are sensitive to the data-fitting method and correlated with each other (10,11). In the case of hemoglobin, the paradigm for allosteric oligomers (12), strategies for circumventing the difficulty in data fitting to the MWC model have been proposed (10,11), but no general solution is available. Consequently, experimental data are often fitted to the Hill equation:

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

where $\overline{Y}$is the fractional saturation of ligand binding sites, [S] is the ligand concentration, K is the apparent binding constant, and n is the Hill coefficient, which is a measure of the extent of cooperativity. The main drawback of this approach is that, except in special cases such as dimers, it has been difficult to relate the Hill coefficient to the more physically meaningful parameters of the MWC or KNF models.

Given that cooperativity according to the MWC model is due to ligand-promoted conformational changes, it is possible to monitor the fraction of sites in the R state, $\overline{R}$, as a function of substrate concentration instead of monitoring $\overline{Y}$. Here, we show that the Hill coefficient for $\overline{R}$ can be directly related to the parameters L and c of the MWC model. We use this relationship to select the best estimate of the true combination of L and c values from all the possible ones found to fit the experimental data. This strategy was applied to fits to the MWC model of $\overline{R}$ as a function of ATP concentration for the allosteric chaperonin GroEL (13) at different temperatures. The results provide insight into thermodynamic effects that accompany the ATP-promoted allosteric switching of GroEL, which is crucial for its function (14).

Materials and Methods

Single-ring GroEL with the mutations F44W and K327C was generated, purified, and labeled as described (15). The final labeling ratio was ∼0.4 fluorophore molecules per ring. The ATP-promoted T-to-R conformational change of this single-ring variant was monitored by measuring fluorescence emission at 681 nm upon excitation at 635 nm using a Fluorolog-3 fluorimeter (Horiba Jobin Yvon, Edison, NJ). The experiments were carried out using 10 nM of the single-ring mutant in 50 mM HEPES buffer (pH 7.5) containing 10 mM KCl, 10 mM MgCl₂, 1 mM dithiothreitol, and 0.0124% Tween at temperatures ranging from 12 to 29.5°C. ATP was added in a stepwise fashion, and the concentration of ADP was minimized owing to the presence of an ATP-regenerating system containing 250 μM phosphoenolpyruvate and 9.2 U/mL pyruvate kinase. The fluorescence intensity was measured for 1 min after each ATP addition, and the average value was then calculated. The temperature of the sample was monitored inside the cuvette using a thermocouple. Data fitting was carried out using the Levenberg-Marquardt algorithm in OriginPro 8 with a tolerance value of 10⁻⁹.

Results

Theory

The fraction of molecules (also sites) in the R state, $\overline{R}$, is defined in accordance with the MWC model as follows:

$$\overline{R}=\frac{\sum R_i}{\sum T_i+\sum R_i}=\frac{{\left[R_0\right](1+K_R\left[S\right])}^N}{\left[T_0\right]{\left(1+K_T\left[S\right]\right)}^N+\left[R_0\right]{\left(1+K_R\left[S\right]\right)}^N}=\frac{{\left(1+K_R\left[S\right]\right)}^N}{L{\left(1+K_T\left[S\right]\right)}^N+{\left(1+K_R\left[S\right]\right)}^N},$$ (2)

where ΣR_i and ΣT_i designate all the molecules in the R and T states, respectively, with different numbers i of bound sites from 0 to the total number of sites, N. By analogy to the ligand concentration-dependent definition of the Hill coefficient for $\overline{Y}$, the Hill coefficient for $\overline{R}$ was defined in previous work (16) as follows:

$$n_{\overline{R}}=\frac{d\text{log}\frac{\overline{R}}{1-\overline{R}}}{d\text{log}\left[S\right]}.$$ (3)

This definition of the Hill coefficient for $\overline{R}$ implies that $\overline{R}=K{\left[S\right]}^n/(1+K{\left[S\right]}^n)$ and is therefore inconsistent with the assumptions of the MWC model because it follows that $\overline{R}$ = 0 and $\overline{R}$ = 1 at zero and infinite substrate concentrations, respectively. Inspection of Eq. 2 shows, however, that ${\overline{R}}_0$ = 1/(L + 1) and ${\overline{R}}_{\infty }$ = 1/(Lc^N + 1) where ${\overline{R}}_0$ and ${\overline{R}}_{\infty }$ are the fraction of molecules in the R state at zero and infinite substrate concentrations, respectively. The Hill equation for $\overline{R}$ is, therefore, defined in the work here as follows:

$$\overline{R}={\overline{R}}_0+\left({\overline{R}}_{\infty }-{\overline{R}}_0\right)\frac{K{\left[S\right]}^n}{1+K{\left[S\right]}^n}.$$ (4)

Rearranging Eq. 4 leads to

$$\frac{\overline{R}-\overline{R_0}}{\overline{R_{\infty }}-\overline{R}}=K{\left[S\right]}^n.$$ (5)

The Hill coefficient is, therefore, given by

$$n_{\overline{R}}=\frac{d\text{log}\frac{\overline{R}-{\overline{R}}_0}{{\overline{R}}_{\infty }-\overline{R}}}{d\text{log}\left[S\right]}=\frac{\left[S\right]\left({\overline{R}}_{\infty }-{\overline{R}}_0\right)}{\left(\overline{R}-{\overline{R}}_0\right)\left({\overline{R}}_{\infty }-\overline{R}\right)}\frac{d\overline{R}}{d\left[S\right]}.$$ (6)

Given ${\overline{R}}_0$ = 1/(L + 1) and ${\overline{R}}_{\infty }$ = 1/(Lc^N + 1), it is possible to express the Hill coefficient at $\overline{R}$ = 0.5, $n_{\overline{R},0.5}$, as follows:

$$n_{\overline{R},0.5}=\frac{4{\left[S\right]}_{0.5}L\left(c^N-1\right)}{\left(L-1\right)\left(L{c}^N-1\right)}\frac{d\overline{R}}{d\left[S\right]}.$$ (7)

Setting $\overline{R}$ = 0.5 in Eq. 2 leads to the following expression for [S]_0.5:

$${\left[S\right]}_{0.5}=\frac{1-\sqrt[N]{L}}{\sqrt[N]{L}{K}_T-K_R}.$$ (8)

The derivative $d\overline{R}/d\left[S\right]$ is given by

$$\frac{d\overline{R}}{d\left[S\right]}=\frac{NL\left(K_R-K_T\right){\left(1+K_R\left[S\right]\right)}^{N-1}{\left(1+K_T\left[S\right]\right)}^{N-1}}{{\left(L{\left(1+K_T\left[S\right]\right)}^N+{\left(1+K_R\left[S\right]\right)}^N\right)}^2}.$$ (9)

Combining Eqs. 7, 8, and 9, therefore, yields

$$n_{\overline{R},0.5}=\frac{N{L}^{\left(N-1\right)/N}\left(c^N-1\right)\left(\sqrt[N]{L}-1\right)\left(1-c\sqrt[N]{L}\right)}{\left(L-1\right)\left(L{c}^N-1\right)\left(1-c\right)}.$$ (10)

Inspection of Eq. 10 shows that it is possible to relate the Hill coefficient for $\overline{R}$ to the parameters of the MWC model for a homo-oligomer with any number (N) of subunits.

Plots of $n_{\overline{R},0.5}$ as a function of ln(L) using Eq. 10 are bell-shaped and become progressively broader for decreasing values of c (Fig. 1 A). This behavior is similar to previous plots of n_0.5 for $\overline{Y}$as a function of log(L) (17), but the plots here are based on the relationship between n_0.5 for $\overline{R}$and the parameters L and c. Both here and before (17), the bell-shaped behavior reflects the existence of lower cooperativity when either the T or R states are very stable (i.e., at high and low values of L, respectively, when they undergo less allosteric switching), provided that both states are active (i.e., at relatively high values of c). In the absence of an active T state (i.e., at c = 0), allosteric switching is required for activity. Cooperativity, therefore, increases monotonically with increasing values of L until reaching its maximum value of N.

Figure 1.

Characterizing the dependence of the Hill coefficient on the value of the allosteric constant. (A) Plots of the value of the Hill coefficient for $\overline{R}$ as a function of the natural logarithm of L for N = 7 and different fixed values of c. The plots were generated using Eq. 10. (B) The natural logarithm of L at $n_{\overline{R},0.5}^{max}$ is plotted against that of c for curves such as the ones shown in (A). The values of $n_{\overline{R},0.5}^{max}$ were extracted from the plots that were generated using Eq. 10. The data were subjected to a linear fit, and the slope was found to be −3.5, as expected from Eq. 13, which was derived by assuming that Lc^N ≪ 1.

Despite the similarities between the plots of n_0.5 as a function of ln(L) for $\overline{R}$ and $\overline{Y}$ there are also some differences. First, it is important to note that Eq. 10 is not defined for certain values of c and L, namely, c = 1, L = 1 and Lc^N = 1. In the case of c = 1 (i.e., K_T = K_R), it may be seen by inspecting Eq. 2 that $\overline{R}$ = 1/(1 + L), i.e., the value of $\overline{R}$ is a constant, and no ligand-promoted allosteric transition can be observed. In the case of Lc^N = 1, $\overline{R}$ → 0.5 when [S] → ∞, and therefore, n_0.5 cannot be defined. Finally, in the case of L = 1, $\overline{R}$ = 0.5 only when c = 1, in which case, as discussed above, no allosteric transition can be observed. A second difference between n_0.5 for $\overline{R}$ and for $\overline{Y}$ is their ranges of values. Inspection of Eq. 10 shows that the value of n_0.5 for $\overline{R}$ approaches N, the total number of sites, when L → ∞ and c → 0, as in the case of n_0.5 for $\overline{Y}$. By contrast, the value of n_0.5 for $\overline{R}$ can be less than 1 in the case of low cooperativity, whereas the value of n_0.5 for $\overline{Y}$, in the case of positive cooperativity, can never be under 1.

It is also interesting to consider how the maximum value of n_0.5 for $\overline{R}$, $n_{\overline{R},0.5}^{max}$, depends on L and c. Given that L > 1 and that c < 1, and assuming that Lc^N ≪ 1, one can simplify Eq. 10 as follows:

$$n_{\overline{R},0.5}\simeq N\left(\frac{c^N-1}{c-1}\right)\left(1-L^{-\frac{1}{N}}-c{L}^{\frac{1}{N}}+c\right).$$ (11)

The derivative of $n_{\overline{R},0.5}$ with respect to L is therefore given by

$$\frac{d{n}_{\overline{R},0.5}}{dL}=\left(\frac{c^N-1}{c-1}\right)L^{-\frac{N+1}{N}}\left(1-c{L}^{2/N}\right).$$ (12)

Setting $\left(d{n}_{\overline{R},0.5}/dL\right)=0$, one obtains the relationship between the values of L and c at $n_{\overline{R},0.5}^{max}$, which is given by

$$L_{n_{\overline{R},0.5}^{max}}=\frac{1}{c^{N/2}}\text{or}\ln L_{n_{\overline{R},0.5}^{max}}=-\frac{N}{2}\ln c.$$ (13)

A plot of $\ln \left(L_{n_{\overline{R},0.5}^{max}}\right)$ as a function of ln(c) for curves generated for N = 7 (as in single-ring GroEL) using Eq. 10 shows that the slope is indeed −3.5, as expected from Eq. 13 (Fig. 1 B).

Monitoring the T-to-R transition of single-ring GroEL

Two mutations, F44W and K327C, were introduced into single-ring GroEL (18), which contains seven ATP binding sites. The mutant single-ring GroEL was then labeled at position 327 with the oxazine dye Atto 655 and purified as described before (15). The distance in the T state between the C_α atoms of residue 44 in one subunit and residue 327 in the nearest-neighbor subunit in the counterclockwise direction is ∼17 Å (19). In the T state, the Trp44 side chain can therefore interact with the dye molecule and quench its fluorescence. In the R conformation, the distance between Trp44 and the dye increases to ∼29 Å (20), thereby reducing the fluorescence quenching due to their interaction. It was therefore possible to monitor the ATP-promoted T-to-R transition of the labeled single-ring GroEL by measuring the increase in fluorescence intensity as a function of ATP concentration.

Plots of fluorescence, F, as a function of ATP concentration were fitted, in accordance with the Hill equation (Eq. 4), to

$$\frac{F-F_0}{F_{\infty }-F_0}=\frac{K{\left[S\right]}^n}{1+K{\left[S\right]}^n},$$ (14)

where F₀ and F_∞ are the fluorescence intensities at zero and saturating ATP concentrations (Fig. 2 A). The data were also fitted to the MWC model (Eq. 2) using

$$F=F_T+\left(F_R-F_T\right)\frac{{\left(1+K_R\left[S\right]\right)}^7}{L{\left(1+K_T\left[S\right]\right)}^7+{\left(1+K_R\left[S\right]\right)}^7},$$ (15)

where F_T and F_R are the fluorescence intensities of the T and R states, respectively (Fig. 2 B). Three titrations were carried out at each temperature, and the data for each temperature were then subjected to global fitting.

Figure 2.

Monitoring the ATP-promoted allosteric transition of single-ring GroEL at different temperatures. The allosteric switch of single-ring GroEL was monitored at different temperatures by measuring the fluorescence emission at 681 nm upon excitation at 635 nm as a function of ATP concentration. The data were fitted to either (A) the Hill equation using Eq. 14 or (B) the MWC model using Eq. 15. The r² values of the fits to the MWC model range from 0.9992 to 0.9997 and are always higher than the corresponding ones to the Hill equation, which range from 0.9963 to 0.9991. Likewise, the root-mean-square deviation (RMSD) values of the fits of the unnormalized data to the MWC model range from 719 to 1539 and are always lower than the corresponding ones to the Hill equation, which range from 1234 to 3221.

Correlations between L, K_T, and K_R were encountered when Eq. 15 was used to fit the data. In other words, different initial guesses of the parameter values result in different combinations of L and c values that yield good fits of the data (Fig. S1). However, not all the combinations of L and c values that fit the data are consistent with Eq. 10. Eq. 10 was, therefore, used to select the best estimate of the true values of L and c by requiring that the difference between the values of $n_{\overline{R}}$ calculated using the Hill equation (Eq. 14) and from Eq. 10 (using the values of L and c obtained from fitting the data to Eq. 15) is minimal for each temperature. In the case of the data for 26°C, for example, the smallest difference between the values of $n_{\overline{R}}$ calculated using the Hill equation (Eq. 14) and from Eq. 10 was obtained for the combination of L and c values shown in red (Fig. S1). Simulations were carried out to test the validity of this approach to select the best estimate of the true combination of L and c values from all the possible ones found to fit the data. Data with Gaussian noise comparable to that in our experiments were generated using the MWC equation with different known values of L, K_R and K_T similar to those determined here (Figs. S2 and S3; Table S1). It may be seen that using the criterion of smallest difference between the values of $n_{\overline{R}}$, calculated using the Hill equation (Eq. 14) and from Eq. 10, enables estimation of the true values of L, K_R, and K_T (Figs. S2 and S3; Table S1). By contrast, fits with the best r² values often result in wrong estimates of these values (Figs. S2 and S3; Table S1). It is important to note that an assumption involved in applying this criterion is that the value of $n_{\overline{R}}$ calculated using the Hill equation is equal to the value at the substrate concentration at which $\overline{R}$ = 0.5. Difficulties in accurate estimation of the Hill constant from the data can arise, however, when the experimental error is large and when the fit to the log-log transformation of the Hill equation is not linear over a sufficiently wide range of concentrations. In such cases, our approach becomes less effective. This is the case here for the data at 29.5°C, in which there is some uncertainty because the linear range is narrower. Our fitting procedure also involved substituting L^y (values of y = 7 or higher such as 100 were used) instead of L in Eq. 15 because this reduced the errors in L, which were back-calculated from the errors in L^y and found to be acceptable (Figs. 3 and 4). It should be noted that the values of L determined here differ by many orders of magnitude from those determined previously assuming exclusive binding to the R state, but the values of the Hill constant are similar (21). The values of K_T and K_R (and thus c) (Fig. 3; Table 1) are also similar to those reported before for single-ring GroEL (22).

Figure 3.

Plot of the natural logarithm of the ATP binding constants for the T and R states as a function of temperature. The natural logarithms of K_T and K_R are plotted as a function of 1/T in accordance with the van’t Hoff equation, ln(K) = −ΔH^o/RT + ΔS^o/R. The linear plots indicate that $\Delta C_p=\left(\partial \Delta H/\partial T\right)\approx 0$ for ATP binding to the T and R states. Error bars represent standard errors.

Figure 4.

Plot of the natural logarithm of the allosteric constant L as a function of temperature. The data were fitted to Eq. 16, in which the enthalpy and entropy are temperature-dependent, but the heat capacity is not. Error bars represent standard errors.

Table 1.

Parameters of the MWC Model for Single-Ring GroEL

T (°C)	KR (μM−1)	KT (μM−1)	ln(L)
12.0	1.2 ± 0.1	0.15 ± 0.01	10.7 ± 0.4
16.0	1.4 ± 0.1	0.18 ± 0.01	10.7 ± 0.5
18.5	2.8 ± 0.1	0.25 ± 0.02	13.6 ± 0.5
20.5	6.2 ± 0.1	0.32 ± 0.03	17.6 ± 0.5
22.0	7.6 ± 0.2	0.33 ± 0.04	18.8 ± 0.8
26.0	17.1 ± 0.5	0.46 ± 0.01	22.5 ± 0.1
29.5	86.8 ± 2.8	0.56 ± 0.02	32.5 ± 0.1

Temperature dependence of the allosteric transition of single-ring GroEL

Given the values of L at different temperatures obtained from the fits to Eq. 15 (Fig. 2 B), it is possible to determine the thermodynamic parameters of the allosteric transition. A plot of ln(L) as a function of the inverse of the temperature, 1/T, was fitted to (Fig. 4)

$$\text{ln}\left(L\right)=-\frac{\Delta H_0}{RT}+\frac{\Delta S_0}{R}-\frac{\Delta C_p}{R}\left(1-\frac{T_0}{T}\right)+\frac{\Delta C_p}{R}ln\frac{T}{T_0},$$ (16)

where ΔH₀ and ΔS₀ are the enthalpy and entropy changes at the reference temperature T₀ and ΔC_p is the change in heat capacity. The values obtained for ΔH₀, ΔS₀, and ΔC_p at T₀ = 25°C are ∼301 (±31) kcal mol⁻¹, 1.1 (±0.1) kcal mol⁻¹ K⁻¹, and 23 (±6) kcal mol⁻¹ K⁻¹, respectively. These are unusually large values, but it should be borne in mind that the T-to-R switch of single-ring GroEL is accompanied by unusually large changes in nonpolar and polar accessible surface areas. The nonpolar surface area increases by ∼11,900 Å², and the polar surface area decreases by ∼13,400 Å² (these changes in polar and nonpolar surface areas, ΔA_np and ΔA_p, were calculated for single rings using the Protein Data Bank (PDB): 1GRL (19) and 4KI8 (20) structures of the respective T and R states with Naccess 2.1.1 (S. L. Hubbard and J. M. Thornton, University College London) and its default parameters). Given the relationship ΔC_p = 0.45ΔA_np − 0.26ΔA_p (23), one obtains a value for ΔC_p of ∼8.8 kcal mol⁻¹ K⁻¹. Other sources such as hydrogen bonding, electrostatics, and conformational entropy might therefore also contribute to ΔC_p but their magnitude and even sign are uncertain (24,25). Another contribution to ΔC_p, which is potentially more unique to GroEL, might arise from a change in water ordering in the cavity. Given the large changes in surface area associated with the allosteric switch of GroEL, it is not surprising that the values of the thermodynamic parameters determined here are more similar to those observed for the folding of some proteins such as yeast phosphoglycerate kinase, which has a ΔC_p of unfolding of 7.8 kcal mol⁻¹ K⁻¹ (26). In contrast with the large ΔC_p value associated with the T-to-R transition, those for ATP binding to the T and R states are close to zero as indicated by the linear van’t Hoff plots (Fig. 3).

Conclusions

In this work, a general relationship between the value of the Hill coefficient and the parameters of the MWC model was derived. The relationship was used to select the best estimates of the MWC model parameters from all the possible ones found to fit data of the fraction of GroEL molecules in the high-affinity (R) state for ATP as a function of ATP concentration. The data were collected at different fixed temperatures, thereby enabling the determination of the thermodynamic parameters of GroEL’s ATP-promoted intraring allosteric switch. The large positive value of ΔC_p determined here is consistent with the large changes in nonpolar and polar surface area associated with GroEL’s T-to-R allosteric transition. Surprisingly, to the best of our knowledge, there are few available data regarding the thermodynamics of allosteric switching of other systems, and comparisons are therefore not yet possible. The procedures described in this work will facilitate future thermodynamic analysis of GroEL and other MWC-type allosteric systems.

Author Contributions

R.G. performed research and analyzed the data. T.M. analyzed data. A.H. performed research and wrote the manuscript.

Editor: Doug Barrick.