A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

Experimental evidence for allosteric modifier saturation as predicted by the bi-substrate Hill equation

The cooperative enzyme reaction rates predicted by the bi-substrate Hill equation and the bi-substrate Monod-Wyman-Changeux (MWC) equation when allosterically inhibited are compared in silico. Theoretically, the Hill equation predicts that when the maximum inhibitory effect at a certain substrate condition has been reached, an increase in allosteric inhibitor concentration will have no effect on reaction rate, that is the Hill equation shows allosteric inhibitor saturation. This saturating inhibitory effect is not present in the MWC equation. Experimental in vitro data for pyruvate kinase, a bi-substrate cooperative enzyme that is allosterically inhibited, are presented. This enzyme also shows inhibitor saturation, and therefore serves as experimental evidence that the bi-substrate Hill equation predicts more realistic allosteric inhibitor behaviour than the bi-substrate MWC equation.     

The Hill model for binding myosin S1 to regulated actin is not equivalent to the McKillop-Geeves model

The Hill two-state cooperativity model and the McKillop-Geeves (McK-G) three-state model predict very similar binding traces of myosin subfragment 1 (S1) binding to regulated actin filaments in the presence and absence of calcium, and both fit the experimental data reasonably well [Chen et al., Biophys. J., 80, 2338-2349]. Here, we compared the Hill model and the McK-G model for binding myosin S1 to regulated actin against three sets of experimental data: the titration of regulated actin with S1 and the kinetics of S1 binding of regulated actin with either excess S1 to actin or excess actin to S1. Each data set was collected for a wide range of specified calcium concentrations. Both models were able to generate reasonable fits to the time course data and to titration data. The McK-G model can fit all three data sets with the same calcium-concentration-sensitive parameters. Only K(B) and K(T) show significant calcium dependence, and the parameters have a classic pCa curve. A unique set of the Hill model parameters was extremely difficult to estimate from the best fits of multiple sets of data. In summary, the McK-G cooperativity model more uniquely resolves parameters estimated from kinetic and titration data than the Hill model, predicts a sigmoidal dependence of key parameters with calcium concentration, and is simpler and more suitable for practical use.     

Stabilization of the T-state of hemoglobin

The effect of inositol hexaphosphate and bezafibrate on binding of O2 and CO to HbAO at high concentrations (1 mM) has been evaluated using thin layer optical techniques. Data analysis shows 1) the occurrence of greatly reduced ligand dependent cooperativity (Hill slope of 2.23 for CO and 1.51 for O2), and 2) the presence of significant triply ligated species. The data fits a nested allosteric two-state MWC model in which the T state consists of two allosteric substrates, Tt and Tr, where Tt binds only to the alpha chains and Tr binds to both alpha and beta chains. The model indicates that the triply ligated species consists of a predominant amount of T form, agreeing with kinetic observations of CO ligated hemoglobin. The maximum amount of triply ligated R molecules (CO or O2) implicated is less than 1%, a result similar to that found previously for binding studies made in the absence of BZF and IHP.     

Acetylcholine and epibatidine binding to muscle acetylcholine receptors distinguish between concerted and uncoupled models.

The muscle acetylcholine receptor (AChR) has served as a prototype for understanding allosteric mechanisms of neurotransmitter-gated ion channels. The phenomenon of cooperative agonist binding is described by the model of Monod et al. (Monod, J., Wyman, J., and Changeux, J. P. (1965) J. Mol. Biol. 12, 88-118; MWC model), which requires concerted switching of the two binding sites between low and high affinity states. The present study examines binding of acetylcholine (ACh) and epibatidine, agonists with opposite selectivity for the two binding sites of mouse muscle AChRs. We expressed either fetal or adult AChRs in 293 HEK cells and measured agonist binding by competition against the initial rate of 125I-alpha-bungarotoxin binding. We fit predictions of the MWC model to epibatidine and ACh binding data simultaneously, taking as constants previously determined parameters for agonist binding and channel gating steps, and varying the agonist-independent parameters. We find that the MWC model describes the apparent dissociation constants for both agonists but predicts Hill coefficients that are far too steep. An Uncoupled model, which relaxes the requirement of concerted state transitions, accurately describes binding of both ACh and epibatidine and provides parameters for agonist-independent steps consistent with known aspects of AChR function.     

Analysis of oxygen equilibria of the giant hemoglobin from the earthworm Eisenia foetida using the Adair model

Oxygen equilibrium curves of the giant hemoglobin from the earthworm Eisenia foetida were determined at various concentrations of cations. Using the Adair model of 12 oxygenation steps, we succeeded in fitting the data better than the simple concerted model (MWC model). Analysis of the Adair constants (K1 to K12) indicated that the increase in oxygen affinity occurs in the last six steps (K7 to K12) of the oxygen binding and that it is enhanced by increase in Ca2+ concentration. The Hill coefficient (nmax) at pH 7.5 attained a maximum value of 9.76 at 20 mM CaCl2. In the presence of physiological levels of Ca2+ (5 mM), the Bohr effect was similar to that seen in vertebrates. The data were consistent with the release of two Bohr protons being accompanied by the oxygen-linked binding of one Ca2+. Mg2+ and Na+ exerted a similar effect on the hemoglobin, though to a lesser extent. The stoichiometry of Ca2+ binding of the hemoglobin revealed the presence of two classes of binding sites, of which the affinities are high (Ka = 8.8 x 10(3) +/- 103 M-1) and low. The number of high affinity sites per heme was found to be 0.3, comparable to the number of oxygen-linked Ca2+ binding sites.     

Thermodynamic analysis of carbon monoxide binding by hemoglobin trout I.

Calorimetric measurements at 25 degrees of the differential heat of CO binding by hemoglobin trout I have been examined together with the CO binding isotherms for the protein at 4 degrees and 20 degrees. Simultaneous treatment of these data sets by a statistically rigorous technique permits evaluation of all the thermodynamic parameters for both the Adair and the Monod, Wyman, Changeux (MWC) models. The results show the details of the unusual temperature dependent cooperativity which this hemoglobin exhibits. In the Adair formalism the increasingly favorable free energy change for successive steps of ligand binding are nearly linearly paralleled by increasingly negative enthalpy changes for these steps. This causes the enhanced cooperativity observed as the temperature is decreased. For the MWC case, lowering the temperature increases the stability of the unligated T state relative to the unligated R state since the enthalpy of the T leads to R transition is 29.4 kcal mol-1. Simultaneously, the favorability of ligating R forms relative to T is enhanced since R form ligation is 14.1 kcal (mol CO)-1 more exothermic than that of T. The balance between these opposing effects is to increase ligand binding cooperativity at low temperatures. The predicted temperature dependence of the Hill coefficient for the MWC and Adair models is identical at low and intermediate temperatures, but, interestingly, would show a strong divergence at high temperatures where negative cooperativity is suggested for the Adair case and positive cooperativity for the MWC case.     

Cooperative free energies for nested allosteric models as applied to human hemoglobin.

A model is developed for ligand binding to human hemoglobin that describes the detailed cooperative free-energies for each of the ten different ligated (cyanomet) species as observed by Smith and Ackers (Smith, F.R., and G.K. Ackers. 1985. Proc. Natl. Acad. Sci. USA.82:5347-5351). The approach taken here is an application of the general principle of hierarchical levels of allosteric control, or nesting, as suggested by Wyman (Wyman, J. 1972. Curr. Top. Cell. Reg. 6:207-223). The model is an extension of the simple two-state MWC model (Monod, J., J. Wyman, and J.P. Changeux. 1965. J. Mol. Biol. 12:88-118) using the idea of cooperative binding within the T (deoxy) form of the macromolecule, and has recently been described as a "cooperon" model (Di Cera, E. 1985. Ph.D. thesis). The T-state cooperative binding is described using simple interaction rules first devised by Pauling (Pauling, L. 1935. Proc. Natl. Acad. Sci. USA. 21:186-191). In this application three parameters suffice to describe the cooperative free-energies of the 10 ligated species of cyanomet hemoglobin. The redox process in the presence of cyanide, represented as a Hill plot, is simulated from Smith and Ackers' cooperative free-energies and is compared with available electrochemical binding measurements.     

Nested MWC model describes hydrolysis of GroEL without assuming negative cooperativity in binding

Folding assistance and ATPase activity of GroEL are based on the existence of different conformations. In order to characterise these conformations, published data on steady state ATPase activity in the absence of GroES were reanalysed simultaneously in terms of the Nested MWC model. This model is a hierarchical extension of the symmetry-model of Monod et al. [J. Mol. Biol. 12 (1965) 88]. An unique set of GroEL specific parameters was obtained. This set was supported by comparison of predictions arising from this set of values with experimental data for hydrolysis of ATP in the presence of ADP and ATPgammaS, binding of ATPgammaS and ADP to GroEL in the absence of ATP, and binding of ATP as monitored by fluorescence labelling. Thus, for the first time, multiple data sets for the interaction of nucleotides with GroEL are described quantitatively by an allosteric model. A noteworthy feature of our model is that no negative cooperativity in ATP binding occurs in accordance to experimental observations. Furthermore, the model also includes the existence of a conformation with very high ATPase activity. Such a conformation might be of importance at a certain stage in the folding cycle.     

Calcium currents in the A7r5 smooth muscle-derived cell line. An allosteric model for calcium channel activation and dihydropyridine agonist action

We have investigated the gating kinetics of calcium channels in the A7r5 cell line at the level of single channels and whole cell currents, in the absence and presence of dihydropyridine (DHP) calcium channel agonists. Although latencies to first opening and macroscopic currents are strongly voltage dependent, analysis of amplitude histograms indicates that the primary open-closed transition is voltage independent. This suggests that the molecular mechanisms for voltage sensing and channel opening are distinct, but coupled. We propose a modified Monod-Wyman-Changeux (MWC) model for channel activation, where movement of a voltage sensor is analogous to ligand binding, and the closed and open channels correspond to inactive (T) and active (R) states. This model can account for the activation kinetics of the calcium channel, and is consistent with the existence of four homologous domains in the main subunit of the calcium channel protein. DHP agonists slow deactivation kinetics, shift the activation curve to more negative potentials with an increase in slope, induce intermingled fast and slow channel openings, and reduce the latency to first opening. These effects are predicted by the MWC model if we make the simple assumption that DHP agonists act as allosteric effectors to stabilize the open states of the channel.     

A two-gate model for the ryanodine receptor with allosteric modulation by caffeine and quercetin

We have developed a model of the tetrameric ryanodine receptor--the calcium channel of the sarcoplasmic reticulum. The model accurately describes published experimental data on channel activity at various concentrations of Ca2+, caffeine and quercetin. The proposed mechanisms involve allosteric regulation of Ca2+ affinity by both caffeine and quercetin, and the existence of two independent, A- and I-gates controlled by Ca2+ binding to an activating and an inhibitory module of the receptor. There are four different configurations of the receptor that affect ligand binding to the activation module, but not to the inhibition module. Consequently, there are four kinetic modes for the A-gate and one mode for the I-gate. At a certain moment, the receptor can be in any of the four possible conformations with equal probability. By fitting the data we are able to derive ligand affinities and Hill coefficients, to describe the observation that quercetin is an activating agent stronger than caffeine, and that caffeine and quercetin activate the channel at very low Ca2+ concentration (approximately 10(-11) M). We predict that the activation regime at saturating caffeine or quercetin should present four distinct regions at increasing Ca2+, corresponding to the four different gating modes. Another interesting prediction is the enlargement of the activity domain toward higher Ca2+ concentrations in the presence of caffeine or quercetin.     

The Hill analysis and co-ion–driven transporter kinetics

ConclusionIn mechanistic models for binding of multiple ligands to a biological unit, the saturation behavior depends on the experimental readout. In the binding model, the Hill analysis does not provide information on the number of binding sites N. In contrast, the Hill analysis of the response model does contain this information, in which case the slope of the curve in the lower concentration range corresponds to the number of binding sites. In neither model does the Hill coefficient—defined as the slope of the curve at half-maximal saturation—report this number. In the binding model, the Hill coefficient varies between a value of 1 in the absence of interaction and a value of N in case of extremely strong interaction. In the response model, it varies between a number larger than 1 and N. In both models, the derived Hill coefficient is a measure of the cooperativity and sets a lowest possible number of sites.Ion-coupled transporters are of the response model type, and the saturation behavior of the rate with the co-ion in the lower concentration limit contains the information on the number of cotransported ions. Additionally, in the case of an ordered-binding mechanism, in which the co-ions bind before the transported substrate, the Hill coefficient of co-ion binding is a function of the substrate concentration. The apparent interaction between the co-ion sites increases with the substrate concentration and, consequently, the Hill coefficient extrapolates to the number of co-ions. Measurements of the Hill coefficient over the entire range of substrate concentrations provide information on both the extent of interaction between the sites and the number of sites.

Online supplemental material

Five supplemental texts accompany this review: (1) derivation of the equations for the binding and response models; (2) derivation of the equations for the ordered-binding transporter mechanism; (3) derivation of the equations for the Hill analysis of the saturation level functions; (4) derivation of the equations for substrate-dependent kinetics of mechanistic transporter models; (5) data analysis by curve fitting. The online supplemental material is available at http://www.jgp.org/cgi/content/full/jgp.201411332/DC1.     

Description of hemoglobin oxygenation under universal solution conditions by a global allostery model with a single adjustable parameter

The Monod-Wyman-Changeux allosteric model parameters evaluated from accurate oxygen equilibrium curves (OECs) of hemoglobin that were measured in an extremely wide range of structural constraints, imposed by allosteric effectors, yielded a closed circle when log K(T) and log K(R) were plotted against log L(0) and log L(4), respectively, showing novel phenomena that L(0) and L(4) have a maximal value and a minimal value, respectively, and K(T) and K(R) vary by more than three orders of magnitude. These phenomena were successfully described by a global allostery model, which mathematically keeps the frame work of the MWC model, but allows that K(T) under a set of solution conditions becomes larger than K(R) under another set of solution conditions and postulates that a representative allosteric effector binds to both the T and R states with a lower affinity but with a larger stoichiometry for the R state than for the T state. Thus, this global model can describe any given OEC measured under universal solution conditions with the single adjustable parameter, the concentration of the representative effector.     

Analyses of dose-response curves to compare the antimicrobial activity of model cationic alpha-helical peptides highlights the necessity for a minimum of two activity parameters

To assess and compare different model Leu-Lys-containing cationic alpha-helical peptides, their antimicrobial activities were tested against Escherichia coli as target organism over a broad peptide concentration range. The natural cationic alpha-helical peptides magainin 2 and PGLa and the cyclic cationic peptide gramicidin S were also tested between comparison. The dose-response curves differed widely for these peptides, making it difficult to rank them into an activity order over the whole concentration range. We therefore compared five different inhibition parameters from dose-response curves: IC(min) (lowest concentration leading to growth inhibition), IC(50) (concentration that gives 50% growth inhibition), IC(max) (related to minimum inhibition concentration and minimum bactericidal concentration), inhibition concentration factor (IC(F); describing the increase in concentration of the peptide between minimum and maximum inhibition), and activity slope (A(S); related to the Hill coefficient). We found that these parameters were covariant: two of them sufficed to characterize the dose dependence and hence the activity of the peptides. This was corroborated by showing that the dose dependences followed the Hill equation, with a small, constant aberration. We propose that the activity of antimicrobial peptides can readily be characterized by both IC(50) and IC(F) (or A(S)) rather than by a single parameter and discuss how this may relate to investigations into their mechanisms of action.     

The model of Monod, Wyman, and Changeux generates two sets of parameters when applied to oxygen binding in hemoglobin

The model of Monod, Wyman and Changeux is applied to binding phenomena where the Mass Law and its expansion are employed. In this communication the model of Monod, Wyman and Changeux (MWC) is applied to analyze the oxygen binding reaction in hemoglobin. The symmetrical structure of the MWC model with its three parameters is such that two sets of these parameters, rather than one, fit experimental data for the binding of oxygen to hemoglobin.     

Folding energetics of ligand binding proteins. I. Theoretical model

Heat capacity curves as obtained from differential scanning calorimetry are an outstanding source for molecular information on protein folding and ligand-binding energetics. However, deconvolution of C(p) data of proteins in the presence of ligands can be compromised by indeterminacies concerning the correct choice of the statistical thermodynamic ensemble. By convent, the assumption of constant free ligand concentration has been used to derive formulae for the enthalpy. Unless the ligand occurs at large excess, this assumption is incorrect. Still the relevant ensemble is the grand canonical ensemble. We derive formulae for both constraints, constancy of total or free ligand concentration and illustrate the equations by application to the typical equilibrium Nx <=> N + x <=> D + x. It is demonstrated that as long as the thermodynamic properties of the ligand can be completely corrected for by performing a reference measurement, the grand canonical approach provides the proper and mathematically significantly simpler choice. We demonstrate on the two cases of sequential or independent ligand-binding the fact, that similar binding mechanisms result in different and distinguishable heat capacity equations. Finally, we propose adequate strategies for DSC experiments as well as for obtaining first estimates of the characteristic thermodynamic parameters, which can be used as starting values in a global fit of DSC data.     

Theoretical kinetic studies of models for binding myosin subfragment-1 to regulated actin: Hill model versus Geeves model

It was previously shown that a one-dimensional Ising model could successfully simulate the equilibrium binding of myosin S1 to regulated actin filaments (T. L. Hill, E. Eisenberg and L. Greene, Proc. Natl. Acad. Sci. U.S.A. 77:3186-3190, 1980). However, the time course of myosin S1 binding to regulated actin was thought to be incompatible with this model, and a three-state model was subsequently developed (D. F. McKillop and M. A. Geeves, Biophys. J. 65:693-701, 1993). A quantitative analysis of the predicted time course of myosin S1 binding to regulated actin, however, was never done for either model. Here we present the procedure for the theoretical evaluation of the time course of myosin S1 binding for both models and then show that 1) the Hill model can predict the "lag" in the binding of myosin S1 to regulated actin that is observed in the absence of Ca++ when S1 is in excess of actin, and 2) both models generate very similar families of binding curves when [S1]/[actin] is varied. This result shows that, just based on the equilibrium and pre-steady-state kinetic binding data alone, it is not possible to differentiate between the two models. Thus, the model of Hill et al. cannot be ruled out on the basis of existing pre-steady-state and equilibrium binding data. Physical mechanisms underlying the generation of the lag in the Hill model are discussed.     

Ligand competition curves as a diagnostic tool for delineating the nature of site-site interactions: theory

A few molecular models have been developed in recent years to explain the mechanism of cooperative ligand binding. The concerted model of Monod, Wyman and Changeux and the sequential model of Koshland, Némethy and Filmer were formulated to account for positively cooperative binding. The pre-existent asymmetry model and the sequential model can account for negatively cooperative ligand binding. In most cases, however, it is virtually impossible to deduce the molecular mechanism of ligand binding solely from the shape of the binding isotherm. In the present study we suggest a new strategy for delineating the molecular mechanism responsible for cooperative ligand binding from binding isotherms. In this approach one examines the effect of one ligand on the cooperativity observed in the binding of another ligand, where the two ligands compete for the same set of binding sites. It is demonstrated that the cooperativity of ligand binding can be modulated when a competitive ligand is present in the protein-ligand binding mixture. A general mathematical formulation of this modulation is presented in thermodynamic terms, using model-independent parameters. The relation between the Hill coefficient at 50% ligand saturation with respect to ligand X in the absence, h(x), and in the presence of a competing ligand Z, h(x,z), is expressed in terms of the thermodynamic parameters characterizing the binding of the two ligands. Then the relationship between h(x) and h(x,z), in terms of the molecular parameters of the different allosteric models, is explored. This analysis reveals that the different allosteric models predict different relationships between h(x,z) and h(x). These differences are especially focused when Z binds non-cooperatively. Thus, it becomes possible, on the basis of ligand binding experiments alone, to decide which of the allosteric models best fits a set of experimental data.     

Conformational spread in a ring of proteins: a stochastic approach to allostery

We recently suggested that the sensitivity and range of a cluster of membrane receptors in bacteria would be enhanced by cooperative interactions between neighbouring proteins. Here, we examine the consequences of this "conformational spread" mechanism for an idealised one-dimensional system comprising a closed ring of identical allosteric protomers (protein molecules, or a group of protein domains operating as a unit). We show analytically and by means of Monte Carlo simulations that a ring of allosteric protomers can exhibit a switch-like response to changes in ligand concentration. We derive expressions for the sensitivity and cooperativity of switching and show that the maximum sensitivity is proportional to the number of protomers in the ring. A ring of this kind can reproduce the sensitivity and kinetics of the switch complex of a bacterial flagellar motor, for example, which is based on a ring of 34 FliM proteins. We also compare smaller rings of conformationally coupled protomers to classical allosteric proteins such as haemoglobin and show that the canonical MWC and KNF models arise naturally as limiting cases. Conformational spread appears to be a natural extension of the familiar mechanism of allostery: a physically realistic mechanism that should apply widely to many structures built from protein molecules.