MWC allosteric model explains unusual hemoglobin-oxygen binding curves from sickle cell drug binding

An oxygen-affinity-modifying drug, voxelotor, has very recently been approved by the FDA for treatment of sickle cell disease. The proposed mechanism of action is by preferential binding of the drug to the R quaternary conformation, which cannot copolymerize with the T conformation to form sickle fibers. Here, we report widely different oxygen dissociation and oxygen association curves for normal blood in the presence of voxelotor and interpret the results in terms of the allosteric model of Monod, Wyman, and Changeux with the addition of drug binding. The model does remarkably well in quantitatively explaining a complex data set with just the addition of drug binding and dissociation rates for the R and T conformations. Whereas slow dissociation of the drug from R results in time-independent dissociation curves, the changing association curves result from slow dissociation of the drug from T, as well as extremely slow binding of the drug to T. By calculating true equilibrium curves from the model parameters, we show that there would be a smaller decrease in oxygen delivery from the left shift in the dissociation curve caused by drug binding if drug binding and dissociation for both R and T were rapid. Our application of the Monod, Wyman, and Changeux model demonstrates once more its enormous power in explaining many different kinds of experimental results for hemoglobin. It should also be helpful in analyzing oxygen binding and in vivo delivery in future investigations of oxygen-affinity-modifying drugs for sickle cell disease.     

Comments on Extensions of the Allosteric Model for Haemoglobin

RECENTLY Edelstein¹ has concluded on the basis of a numerical analysis that the sequential model as formulated by Koshland, Nemethy and Filmer (KNF)² describes the oxygen binding curves of a number of species of human haemoglobin less well than does the two-state allosteric model of Monod, Wyman and Changeux (MWC)³. This communication demonstrates that Edelstein's analysis is incomplete and that extension of his analysis reveals that no such conclusion can be drawn from the data considered.     

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.     

An analytic solution to the Monod-Wyman-Changeux model and all parameters in this model.

Starting from the Monod-Wyman-Changeux (MWC) model (Monod, J., J. Wyman, and J. P. Changeux. 1965. J. Mol. Biol. 12:88-118), we obtain an analytical expression for the slope of the Hill plot at any ligand concentration. Furthermore, we derive an equation satisfied by the ligand concentration at the position of maximum slope. From these results, we derive a set of formulas which allow determination of the parameters of the MWC model (kR, C, and L) from the value of the Hill coefficient, nH, the ligand concentration at the position of maximum slope [( A]0), and the value of nu/(n-nu) at this point. We then outline procedures for utilizing these equations to provide a "best fit" of the MWC model to the experimental data, and to obtain a refined set of the parameters. Finally, we demonstrate the applicability of the technique by analysis of oxygen binding data for Octopus hemocyanin.     

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.     

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.     

Oxygen binding and subunit interaction of hemoglobin in relation to the two-state model

Mills and Ackers (Mills, F.C., and Ackers, G.K. (1979) J. Biol. Chem. 254, 2881-2887) have reported the subunit interactions of hemoglobin to decrease on binding of the fourth molecule of oxygen to hemoglobin. This effect, which they called quaternary enhancement, is incompatible with the two-state Monod, Wyman, and Changeux allosteric model. Their free energy of binding of the fourth molecule (-9.3 kcal/mol) has been compared with independent kinetic estimates which give -8.6 kcal/mol. This smaller value is consistent with literature values and allows reasonable representation of the equilibrium curve using the two-state model without invoking quaternary enhancement.     

Binding of oxygen and carbon monoxide to arthropod hemocyanin: an allosteric analysis

The binding of oxygen and carbon monoxide to hemocyanin from the mangrove crab Scylla serrata and the lobster Homarus americanus has been studied by thin-layer optical absorption and front face fluorescence techniques. Three types of experiments were performed on subunit and oligomeric preparations of each hemocyanin: oxygen binding, carbon monoxide binding, and oxygen-carbon monoxide competition studies. The results obtained from the subunit preparations of dissociated oligomers from both hemocyanins show that the binding site can be ligated by either one oxygen or one carbon monoxide. The binding results obtained with the oligomeric samples of hemocyanin from both species cannot be described by the two-state MWC model [Monod, J., Wyman, J., & Changeux, J. P. (1965) J. Mol. Biol. 12, 88-118] since the data from the three types of binding experiments cannot be fit with a single set of binding constants. The MWC model has been extended by including a third allosteric form, and an analysis based on the three-state model is able to fit the data from the three types of experiments with the same set of binding constants. The comparison of the oxygen to carbon monoxide affinity ratios (kO2/kCO) indicates that the structure around the binding site of subunits in the T form oligomer is similar to that of the free subunits. The oligomeric forms of both these hemocyanins bind carbon monoxide with a weak but definite positive cooperativity. An analysis of the affinity ratios for the T, S, and R forms suggests that the high affinity of the R form results from a specific interaction between oxygen and binding site.     

Evolution of allosteric models for hemoglobin

We compare various allosteric models that have been proposed to explain cooperative oxygen binding to hemoglobin, including the two-state allosteric model of Monod, Wyman, and Changeux (MWC), the Cooperon model of Brunori, the model of Szabo and Karplus (SK) based on the stereochemical mechanism of Perutz, the generalization of the SK model by Lee and Karplus (SKL), and the Tertiary Two-State (TTS) model of Henry, Bettati, Hofrichter and Eaton. The preponderance of experimental evidence favors the TTS model which postulates an equilibrium between high (r)- and low (t)-affinity tertiary conformations that are present in both the T and R quaternary structures. Cooperative oxygenation in this model arises from the shift of T to R, as in MWC, but with a significant population of both r and t conformations in the liganded T and in the unliganded R quaternary structures. The TTS model may be considered a combination of the SK and SKL models, and these models provide a framework for a structural interpretation of the TTS parameters. The most compelling evidence in favor of the TTS model is the nanosecond - millisecond carbon monoxide (CO) rebinding kinetics in photodissociation experiments on hemoglobin encapsulated in silica gels. The polymeric network of the gel prevents any tertiary or quaternary conformational changes on the sub-second time scale, thereby permitting the subunit conformations prior to CO photodissociation to be determined from their ligand rebinding kinetics. These experiments show that a large fraction of liganded subunits in the T quaternary structure have the same functional conformation as liganded subunits in the R quaternary structure, an experimental finding inconsistent with the MWC, Cooperon, SK, and SKL models, but readily explained by the TTS model as rebinding to r subunits in T. We propose an additional experiment to test another key prediction of the TTS model, namely that a fraction of subunits in the unliganded R quaternary structure has the same functional conformation (t) as unliganded subunits in the T quaternary structure.     

On the pH-dependence of the reaction of hemoglobin with carbon monoxide.

Flash-photolysis experiments were performed on solutions of carbonmonoxy hemoglobin (human Hb A) as function as pH. The fraction of fast reaction and the amount of photodissociation as produced by a given amount of light quanta has been analyzed in terms of the allosteric model of ligand binding by Monod, Wyman and Changeux. It is shown how the switch-over point of the allosteric transition from the T or R state is controlled by the protons, which act as allosteric effectors.     

Kinetics of ligand binding and quaternary conformational change in the homodimeric hemoglobin from Scapharca inaequivalvis

The kinetics of the reaction with oxygen and carbon monoxide of the homodimeric hemoglobin from the bivalve mollusc Scapharca inaequivalvis has been extensively investigated by flash and dye-laser photolysis, temperature jump relaxation, and stopped flow methods. The results indicate that cooperativity in ligand binding, already observed for oxygen at equilibrium, finds its kinetic counterpart in a large decrease of the oxygen dissociation velocity in the second step of the binding reaction. In the case of carbon monoxide, cooperativity is clearly evident in the increase of the combination velocity constant as the reaction proceeds. Therefore, the ligand-binding kinetics of this dimeric hemoglobin shows the characteristic features of the corresponding reactions of tetrameric hemoglobins. Analysis of the data in terms of the allosteric model proposed by Monod et al. (Monod, J., Wyman, J., and Changeux, J. P. (1965) J. Mol. Biol. 12, 88-118) has shown that the values of the allosteric parameters cannot be fixed uniquely for a dimeric hemoglobin. The rapid changes in absorbance observed at the isosbestic points of unliganded and liganded hemoglobin following laser photolysis provided a value of 7 X 10(4) S-1 at 20 degrees C for the rate of the ligand-free quarternary conformational change, postulated on the basis of cooperative ligand binding. Comparison of the rapid absorbance changes observed during ligand rebinding in this hemoglobin with those observed in tuna hemoglobin indicate that, at full photolysis, binding to the T state is followed by further binding and conversion to the liganded R state; at partial photolysis, population of the liganded T state occurs immediately and is followed by a decay to the liganded R state upon further ligand binding. These new results, in conjunction with previous equilibrium data on the same system, show unequivocally that the presence of two different types of chain is not an absolute prerequisite for cooperativity in hemoglobins, contrary to currently accepted ideas.     

Thermodynamic studies on oxygen binding by human red blood cells

Oxygen equilibrium curves have been measured on human normal red blood cells, at the temperatures of 20, 25, 30, 37 and 41 degrees C, and at pHs ranging from 6.8 to 8.2. The thermodynamical parameters have been determined for the four successive steps of oxygenation and for overall oxygenation, according to the Adair and MWC models [Monod J, Wyman J, Changeux JP. On the nature of allosteric transitions: a plausible model. J Mol Biol 1965;12:88-118]. The heat release appears to be nearly equal for the four steps. At the first three steps, the delta H change is counterbalanced by a nearly equivalent change of delta S, resulting in a rather small delta G value. delta G is greater at the fourth step, because of diminution of this enthalpy-entropy compensation phenomenon. The four steps are both enthalpy and entropy driven. According to the MWC model, the T to R transition is endothermic, and allosteric quaternary transition occurs at binding of the third oxygen. The average heat release increases by 2.8 kcal/mol when pH raises from 7.4 to 8.2, but flattens below pH 7.4. After correction for the heat of solution of oxygen and for the heat of proton release (referred to intracellular pH), an intrinsic heat for oxygenation of the heme of approximately--13 kcal/mol is obtained for the successive steps of oxygenation (at pH 7.4, 37 degrees C). These results are compared with those previously obtained for pigeon and trout red blood cells.     

Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes

In our effort to elucidate the systems biology of the model organism, Escherichia coli, we have developed a mathematical model that simulates the allosteric regulation for threonine biosynthesis pathway starting from aspartate. To achieve this goal, we used kMech, a Cellerator language extension that describes enzyme mechanisms for the mathematical modeling of metabolic pathways. These mechanisms are converted by Cellerator into ordinary differential equations (ODEs) solvable by Mathematica. In this paper, we describe a more flexible model in Cellerator, which generalizes the Monod, Wyman, Changeux (MWC) model for enzyme allosteric regulation to allow for multiple substrate, activator and inhibitor binding sites. Furthermore, we have developed a model that describes the behavior of the bifunctional allosteric enzyme aspartate kinase I-homoserine dehydrogenase I (AKI-HDHI). This model predicts the partition of enzyme activities in the steady state which paves the way for a more generalized prediction of the behavior of bifunctional enzymes.     

Allosteric control in Limulus polyphemus hemocyanin: functional relevance of interactions between hexamers

Hemocyanin of the horseshoe crab Limulus polyphemus is composed of 48 oxygen-binding subunits, which are arranged in eight hexameric building blocks. Allosteric interactions in this oligomeric protein have been examined by measurement of high-precision oxygen-equilibrium curves, using an automated Imai cell. Several models were compared in numerical analysis of the data. A number of conclusions can be drawn with confidence. (1) Oxygen binding by Limulus hemocyanin cannot satisfactorily be described by the two-state MWC model [Monod, J., Wyman, J., & Changeux, J.P. (1965) J. Mol. Biol. 12, 88-118] for allosteric transitions with either the hexamer or dodecamer as the allosteric unit. (2) Of the models tested, the data sets can be best described by an extended MWC model that allows for an equilibrium, within the 48-subunit ensemble, between cooperative hexamers and cooperative dodecamers. The model invokes T and R states for both hexamers (T6 and R6) and dodecamers (T12 and R12). Allosteric effectors modulate oxygen affinity and cooperativity by affecting the R to T equilibria within hexamers and dodecamers and by shifting the equilibria between hexamers and dodecamers. (3) The fitted model parameters show that under most conditions the intersubunit contacts within T-state hexamers are more constrained than those within T-state dodecamers. (4) The oxygen affinities of the hexameric and dodecameric R states are the same, but under all conditions examined the conformation of the fully oxygenated molecule is that of the dodecameric R state. (5) Between pH 7.4 and pH 8.5 the dodecameric T state has a higher affinity for oxygen than the hexameric T state, allowing for "T-state cooperativity".(ABSTRACT TRUNCATED AT 250 WORDS)     

Comparison of the applicability of several allosteric models to the pH and 2,3-bis(phospho)glycerate dependence of oxygen binding by human blood

Oxygen saturation curves of blood were measured by the mixing method at several concentrations of protons and bis(phospho)glycerate. Various theoretical models for the co-operativity of oxygen binding by haemoglobin were then tested for their ability to fit the experimental curves. The effects of pH on oxygen binding could be described by both the Monod, Wyman & Changeux, and the Herzfeld & Stanley models, with most success when protons were assumed to affect oxygen affinity directly rather than through effects on the quaternary-state equilibrium. When the combined effects of pH and bis(phospho)glycerate were considered, however, all the versions of the Monod model that were used, including the three-state version, were unsuccessful. The best fit to the saturation curves was obtained with the Herzfeld & Stanley model, with protons acting as a direct effector of oxygen affinity, and bis(phospho)glycerate acting to lower oxygen affinity as well as influencing the quaternary-state equilibrium.     

High and low oxygen affinity conformations of T state hemoglobin

To understand the interplay between tertiary and quaternary transitions associated with hemoglobin function and regulation, oxygen binding curves were obtained for hemoglobin A fixed in the T quaternary state by encapsulation in wet porous silica gels. At pH 7.0 and 15 degrees C, the oxygen pressure at half saturation (p50) was measured to be 12.4 +/- 0.2 and 139 +/- 4 torr for hemoglobin gels prepared in the absence and presence of the strong allosteric effectors inositol hexaphosphate and bezafibrate, respectively. Both values are in excellent agreement with those found for the binding of the first oxygen to hemoglobin in solution under similar experimental conditions. The corresponding Hill coefficients of hemoglobin gels were 0.94 +/- 0.02 and 0.93 +/- 0.03, indicating, in the frame of the Monod, Wyman, and Changeux model, that high and low oxygen-affinity tertiary T-state conformations have been isolated in a pure form. The values, slightly lower than unity, reflect the different oxygen affinity of alpha- and beta-hemes. Significantly, hemoglobin encapsulated in the presence of the weak effector phosphate led to gels that show intermediate oxygen affinity and Hill coefficients of 0.7 to 0.8. The heterogeneous oxygen binding results from the presence of a mixture of the high and low oxygen-affinity T states. The Bohr effect was measured for hemoglobin gels containing the pure conformations and found to be more pronounced for the high-affinity T state and almost absent for the low-affinity T state. These findings indicate that the functional properties of the T quaternary state result from the contribution of two distinct, interconverting conformations, characterized by a 10-fold difference in oxygen affinity and a different extent of tertiary Bohr effect. The very small degree of T-state cooperativity observed in solution and in the crystalline state might arise from a ligand-induced perturbation of the distribution between the high- and low-affinity T-state conformations.     

An extended Monod-Wyman-Changeaux-model expressed in terms of the Herzfeld-Stanley formalism applied to oxygen and carbonmonoxide binding curves of hemoglobin trout IV.

An extended Monod, Wyman, Changeaux (MWC)-model, the mathematical basis of which had been formulated by Herzfeld and Stanley (J. Mol. Blol. 82:231. 1974.) was used to fit oxygen and CO-binding curves of hemoglobin trout IV measured at different pH-values between pH = 8.0 and 6.0. From this calculation one obtains that even the fully liganded molecule exhibits a R----T quaternary transition upon approaching the acid pH-region. In the case of O2-binding, the cooperativity becomes negative below pH = 6.5. This can be related to the difference between the equilibrium constants of proton binding to the alpha- and beta-subunits. Furthermore, it can be shown that the interaction between the quaternary T----R- and the tertiary t----r-transitions is different for the alpha- and beta-subunits.     

Experiments on Hemoglobin in Single Crystals and Silica Gels Distinguish among Allosteric Models

Trapping quaternary structures of hemoglobin in single crystals or by encapsulation in silica gels has provided a demanding set of data to test statistical mechanical models of allostery. In this work, we compare the results of those experiments with predictions of the four major allosteric models for hemoglobin: the quaternary two-state model of Monod, Wyman, and Changeux; the tertiary two-state model of Henry et al., which is the simplest extension of the Monod-Wyman-Changeux model to include pre-equilibria of tertiary as well as quaternary conformations; the structure-based model of Szabo and Karplus; and the modification of the latter model by Lee and Karplus. We show that only the tertiary two-state model can provide a near quantitative explanation of the single-crystal and gel experimental results.