Is cooperative oxygen binding by hemoglobin really understood?

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1 review Is cooperative oxygen binding by hemoglobin really understood? William A. Eaton 1, Eric. R. Henry 1, James Hofrichter 1 and Andrea Mozzarelli 2 The enormous success of structural biology challenges the physical scientist. Can biophysical studies provide a truly deeper understanding of how a protein works than can be obtained from static structures and qualitative analysis of biochemical data? We address this question in a case study by presenting the key concepts and experimental results that have led to our current understanding of cooperative oxygen binding by hemoglobin, the paradigm of structure function relations in multisubunit proteins. We conclude that the underlying simplicity of the two-state allosteric mechanism could not have been demonstrated without novel physical experiments and a rigorous quantitative analysis. ing process and a quantitative explanation of the experimental data in terms of a (statistical mechanical) model. It is, however, an interesting challenge for the physical scientist to demonstrate that such studies provide a truly deeper understanding of protein function than is already apparent from the beautiful, static color pictures of the structural biologist and qualitative analysis of biochemical data. We address this issue by describing the key findings in the evolution of our understanding of cooperative ligand binding by a single protein, hemoglobin (Hb), the paradigm of structure function relations in multisubunit proteins (Fig. 1). Even though this history spans almost a century 1, it is quite relevant to current protein research. With today s technology comparable results on a newly discovered protein would be obtained in a much shorter time, but the conceptual approach would be very similar. From Bohr to Perutz The physiologically important and physically interesting property of hemoglobin and other so-called allosteric proteins is that they exhibit cooperative interactions in binding ligands to sites that are distant from each other in the structure. The initial critical observation was made by the physiologist, Christian Bohr, father of the atomic physicist Niels Bohr 1,2. Bohr s careful measurements of hemoglobin oxygenation showed that the binding deoxy, T oxy, R Fig. 1 Schematic structures of hemoglobin (adapted from ref. 42). The two views on the right are looking down the pseudo two-fold x axis. The quaternary conformational change consists of a rotation of the symmetrically related αβ dimers by ~15 relative to each other and a translation of ~0.1 nm along the rotation axis. 1 Laboratory of Chemical Physics, Building 5, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland , USA. 2 Institute of Biochemical Sciences and National Institute of the Physics of Matter, University of Parma, Parma, Italy. Correspondence should be addressed to W.A.E. eaton@helix.nih.gov There are many levels at which a scientific question can be answered. An answer that is quite satisfying to a scientist from one discipline may be totally unsatisfactory to a scientist from another. This has become an increasingly important issue in research on biological problems as scientists from different disciplines ask what appear to be the very same questions. For proteins, the most common questions are: what does a protein look like, what does it do, and how does it do it? Although a threedimensional structure at atomic resolution provides a clear answer to the first, the latter questions concerning protein function may be quite problematic. Part of the problem is describing just what is meant by understanding protein function. A genome scientist, faced with trying to determine the role of tens of thousands of proteins, may consider understanding a protein s function to be a brief descriptor of what it does in a cell, such as bind ligand x, or oxidize protein y, or phosphorylate protein z. To a molecular biologist the problem of understanding the ligandbinding function of a protein, for example, may be solved once the binding site has been identified in the three-dimensional structure. To a physical scientist, solution of the structure and the identification of a binding site mark just the first, albeit absolutely essential, step in understanding the function of this protein. At the very least the physical scientist would like to have a complete thermodynamic and kinetic description of the bindnature structural biology volume 6 number 4 april

2 review a fractional saturation veins observed oxygen pressure (torr) arteries Fig. 2 MWC description of hemoglobin oxygenation. a, The observed binding curve (green) and the corresponding hyperbolic (noncooperative) binding curves for the R and T quaternary structures (red and blue, respectively). Binding begins at low oxygen pressure along the T-state binding curve and ends at high pressure along the R-state binding curve. As successive oxygen molecules bind, the equilibrium between the two quaternary structures shifts toward R. The sigmoidal shape of the observed curve and its shift to the right by acid and carbon dioxide facilitate loading of oxygen in the lungs and unloading in the tissues. b, Population of 10 tetrameric species as a function of ligand saturation for a two-state MWC allosteric model with four identical binding sites 40. Each curve is labeled by quaternary state (T or R) with a subscript to indicate the number of ligands bound. The populations of T 2, T 3, T 4, R 0, R 1, and R 2 are not visible on this scale. curve is sigmoidal instead of hyperbolic, as would occur with a simple Hb + O 2 HbO 2 equilibrium. The binding of oxygen by hemoglobin is therefore cooperative. That is, as the number of bound oxygen molecules increases in the association reaction, the apparent binding affinity increases (Fig. 2). In the same study Bohr also discovered that carbon dioxide lowers the oxygen affinity. (The necessary reciprocal effect of increased affinity for carbon dioxide and protons as the oxygen pressure is lowered was measured much later.) These properties make hemoglobin an efficient transporter of oxygen from the lungs to the tissues and of carbon dioxide from the tissues to the lungs (Fig. 2a). By the year 1904 both the protein and its physiological function had been clearly identified, and the functional significance of cooperative binding had been explained. How much more would a genome scientist want to know? We are of course at the very beginning of the subject, and a fundamental question remains: how does hemoglobin exhibit this cooperative behavior? Another physiologist, Gilbert Adair, took the next major step. He made extremely precise osmotic pressure measurements from which he determined the molecular weight of hemoglobin to be 67,000 g mol 1 (refs. 1,3). At that time it was known that ~17,000 g of protein contained one mole of iron atoms, so Adair had discovered that hemoglobin contained four binding sites. He then formulated binding in terms of four successive binding constants, which increased with the addition of each oxygen molecule, the fourth oxygen molecule binding with an affinity much greater than the first. The legendary chemist Linus Pauling suggested the first structural explanation of cooperative binding 4. He discovered that he could reproduce the oxygen-binding curve with a two-parameter model in which binding oxygen to one heme caused an interaction with neighboring hemes b population saturation that increased their affinity. One parameter was taken as the interaction parameter and the other as the intrinsic binding constant. Since fitting the binding curve required only a single interaction parameter Pauling reasoned that hemoglobin possesses either a tetrahedral or square symmetric arrangement of the hemes. He preferred the latter, because he could then explain the change in affinity as somehow resulting from a resonance electronic interaction between adjacent hemes. (For reasons that are not clear, Pauling assumed that the hemes were on the surface of the molecule and with this arrangement tetrahedral symmetry would place them too far away for the direct interaction he thought necessary.) It was Hemoglobin was known to contain four hemes that serve as binding sites for oxygen; its cooperative behavior could be explained by a sequential model in which binding to one heme raises the affinity of its neighbors through a direct heme heme interaction. Would a modern-day biologist want more of an explanation? The answer is almost certainly yes, because the biologist would like to see hemoglobin. A three-dimensional structure would be essential to explain the alteration in hemoglobin s function produced by mutations, particularly those that cause disease, such as sickle Fig. 3 Simplified schematic of the MWC/Perutz mechanism 7,9,14. Open symbols designate unliganded subunits and filled symbols liganded subunits. Arrows connecting subunits represent salt bridges the quaternary bonds of MWC theory that constrain and stabilize the T quaternary structure. There are actually six intersubunit salt bridges in the T quaternary structure (each arrow between α subunits represents a pair of salt bridges) that are not present in the R quaternary structure and two intrasubunit salt bridges (not shown) in the β subunits. Four of the eight salt bridges contain ionizable groups, and their breakage produces a change in pk of the participating residues that leads to a net release of protons. Proton release upon binding to the T quaternary structure is called the tertiary Bohr effect, while proton release upon the change of quaternary structure is called the quaternary Bohr effect. Ligand binding to a subunit in the T quaternary structure breaks the salt bridge originating from that subunit, while the change from the T to R quaternary structure at any ligation state breaks all four salt bridges. Ligand binding to the T quaternary structure occurs with an association constant (K T ) that is simply the R-state association constant (K R ) reduced by a factor that is proportional to the strength of the salt bridge (c). The quaternary equilibrium constant between the zero-liganded tetramers is defined as L. 352 nature structural biology volume 6 number 4 april 1999

3 review a c log [y/(1 y)] optical density b y d wavelength (nm) Fig. 4 Oxygen binding to a single crystal of hemoglobin in the T quaternary structure. a, Projection of four hemes onto the (ac) crystal face of the optical measurements. b, Absorption spectra of a crystal at nine oxygen pressures between 0 and 760 torr with light linearly polarized parallel to either the a (blue) or c (red) crystal axes. c, Hill plot of fractional saturation with oxygen (y) versus oxygen pressure from spectra measured with linearly polarized light parallel to the a crystal axis. The circles are for increasing oxygen pressure, while the squares are for decreasing oxygen pressure. Binding is perfectly reversible and noncooperative. The p50 (oxygen pressure at 50% saturation) measured from the a-axis data is 155 ± 1 torr and 141 ± 1 torr from the c-axis data; absorption of light polarized parallel to the c-axis has a greater contribution from the α-hemes, indicating an intrinsically higher affinity for the α-subunits 20. d, Binding curves for the separate α and β subunits calculated using the polarized absorption spectra and the heme orientations derived from the X-ray structures 28. log oxygen pressure (torr) oxygen pressure (torr) cally, so that the intrinsic oxygen affinity of each heme in either R or T is independent of the number of oxygen molecules already bound (Fig. 2). MWC called cooperative interaction between identical ligands homotropic, while the cooperative interaction between unlike ligands, such as oxygen and protons (or substrate and allosteric inhibitor in the case of enzymes) was called heterotropic. What was the relation between the MWC model and the structure of hemoglobin? Perutz took a bold approach that went far beyond simply identifying the R and T quaternary structures of the MWC model with the structures of oxy- and deoxyhemoglobin, and describing them in detail (Fig. 1). In a tour de force he saw through the complexity of his atomic models and developed a structural explanation for exactly how hemoglobin worked his stereochemical mechanism 9,10. The key finding was a set of salt bridges at the subunit interfaces that are present in the T quaternary structure but absent in R. Perutz described how oxygen binding to the heme in the T quaternary structure with its associated iron displacement could move a helix, break a salt bridge, release a proton, and destabilize the structure at the subunit interface between αβ dimers of the tetramer, thereby biasing the quaternary equilibrium toward the R state (Figs 1, 3). In Perutz s mechanism the salt bridges play three roles: they stabilize the T quaternary structure relative to R, lower the oxygen affinity in T because of the energy required to break them upon oxygen binding, and release protons upon breakage, explaining why the overall affinity is lowered when the ph decreases (the Bohr effect). He viewed the mechanism as a combination of the MWC and KNF models, because in the KNF model ligand binding induces conformational changes in the protein (the KNF model ignores cell anemia (the first example of a molecular disease, also discovered by Pauling 5 ). The problem was taken up by Max Perutz. After almost three decades of pioneering work on developing the X-ray crystallographic method for proteins, Perutz solved the three-dimensional structure of hemoglobin. A comparison of the structures of oxy- and deoxyhemoglobin at low resolution produced one of his first major results: the β subunits move closer together upon oxygenation 6. This experimental fact helped motivate a completely different and powerful theory on the origin of cooperative interactions, not only for hemoglobin oxygenation but for multisubunit proteins in general. Jacques Monod and Jean-Pierre Changeux observed that many enzymes are activated and inhibited by substrates and ligands in a cooperative fashion, and that such enzymes contain more than one protein subunit. They saw the analogy to hemoglobin (where the substrate is now oxygen), and together with Jeffries Wyman they developed a theoretical model that would apply to all types of multisubunit proteins 7. Their model was very different from Pauling s, later elaborated by Daniel Koshland and coworkers (KNF) 8. In the MWC model, cooperativity arises from an equilibrium between two structures having different arrangements of the subunits, so-called quaternary structures (Fig.1). The tense or T quaternary structure has a low affinity for ligands, while the relaxed or R quaternary structure has a high affinity for ligands. Cooperativity in the MWC model arises from a shift in the population from the low-affinity T quaternary structure to the high-affinity R quaternary structure with increasing oxygen pressure, as required by Le Chatelier s principle (Fig. 2a). Each quaternary structure, however, binds statistinature structural biology volume 6 number 4 april

4 review a ligand binding b conformational changes fraction deoxyhemes geminate R bimolecular fraction spectral change tertiary quaternary T bimolecular log time (s) log time (s) Fig. 5 Kinetics of hemoglobin following nanosecond photodissociation of carbon monoxide complex 40. a, Ligand rebinding kinetics obtained from the average deoxy minus carbonmonoxy difference spectrum (inset). Geminate rebinding (rebinding of dissociated ligands before they escape from the protein) to R is followed by bimolecular rebinding to R and then to molecules that have switched from R to T. b, Protein conformational changes obtained from the change in the deoxyhemoglobin spectrum (inset). In both (a) and (b) the points are experimental, and the dotted curves are calculated from the extended MWC model of Henry et al. 40. Because the deoxyheme spectral change is more than 10-fold smaller than the spectral change due to ligand binding (as indicated by the relative amplitudes in the insets), the deviations between the data and the fit in (b) represent less than 1% of the total spectral amplitude, and may not be significant. the important structural finding of two quaternary structures). Although oxygen binding results in conformational changes, there appeared to be no structural mechanism for transmitting these changes to the heme of the neighboring subunits other than through a change in quaternary structure. For homotropic effects Perutz s stereochemical mechanism appeared, therefore, to be pure MWC in that the intrinsic affinity of a subunit is solely determined by the quaternary structure. Perutz s structure and mechanism were also successful in qualitatively rationalizing the altered behavior of mutant hemoglobins. The site of the sickle cell mutation (β6 Glu to Val), for example, is found on the molecular surface, creating a sticky hydrophobic patch that causes polymerization to form a solid fiber of 14 intertwined helical strands, a structure solved by Stuart Edelstein and coworkers 11. Moreover, fiber formation in T, but not in R, explains sickling of red cells by deoxygenation in the tissues and unsickling by oxygenation in the lungs 12. The structure and mechanism also elegantly explain how substitutions at the subunit interface distant from the heme produce changes in oxygen affinity by altering the quaternary equilibrium 13. Overall, Perutz s analysis of hemoglobin could be viewed as a turning point in the history of proteins. It represented the beginning of an era in which structural changes at the atomic level were being used to explain how a protein functioned and how mutations in a protein caused disease. It was To structural biologists the hemoglobin problem was solved, and it was time to move on to other proteins. Would further inquiry by physical scientists lead to a truly deeper understanding of how hemoglobin functions? After the structure Perutz s work aroused the interest of many physical scientists, both theorists and experimentalists, and stimulated an enormous amount of research. His mechanism qualitatively explained a vast array of experimental facts, but could it also provide a quantitative explanation? Monod, Wyman, and Changeux had invented a very general model with little structural detail, while Perutz had proposed a detailed structural mechanism which appeared consistent with their model, but contained no prescription for making it quantitative. This important step was taken by Attila Szabo and Martin Karplus, who developed a statistical mechanical formulation of Perutz s structural mechanism (Fig. 3) 14. Szabo and Karplus showed that the Perutz mechanism was consistent with the MWC model for homotropic effects and could indeed provide a quantitative explanation of equilibrium experimental data. They also recognized that the MWC model had to be modified to account for heterotropic effects, particularly the influence of ph on the T-state affinity. This is, however, not a serious criticism of the MWC model. Monod, Wyman, and Changeux clearly recognized that it was unrealistic to expect ph changes to affect the affinity only by altering the quaternary equilibrium, and not also to affect the affinity of R or T for oxygen 7. The Szabo and Karplus analysis of heterotropic effects made the interesting prediction that once the salt bridges are completely broken, as in the fully oxygenated molecule, the quaternary equilibrium constant (Lc 4, Fig. 3) would be relatively insensitive to solution conditions. Their prediction of Lc 4 constant was dramatically confirmed a decade later 15, a fact that has remained largely unrecognized. The most important idea of MWC that the intrinsic affinity depends on the quaternary structure alone, and not on the number of ligands bound per se, was strongly supported by a series of novel spectroscopic studies and insightful analyses by Robert Shulman, John Hopfield, and Seiji Ogawa 16. Nevertheless, the applicability of the MWC model remained controversial for several reasons. First, subsequent papers by Perutz and others appeared to contradict the conclusion that his stereochemical mechanism was consistent with the MWC model 17. Second, in an investigation of individual ligation intermediates using the cyanide complex of oxidized hemoglobin as an analog of oxyhemoglobin, Gary Ackers and coworkers reported a very large cooperative interaction in ligand binding to the T-state 18. Finally, studies of hemoglobin kinetics by Quentin Gibson also appeared inconsistent with the MWC model 19. A reinvestigation of the problem was clearly required. To do so required new kinds of experiments. 354 nature structural biology volume 6 number 4 april 1999

5 review Hemoglobin in the R quaternary structure binds oxygen with very nearly the same high affinity as the αβ dimer (that is, the half-molecule; Fig. 1), as well as the isolated α and β chains. The question of the relative contribution to cooperativity from the quaternary change versus direct subunit subunit interaction therefore centers on the binding properties of the low-affinity T quaternary structure. Does cooperativity arise solely from the T to R transition, as in the MWC model, or is there significant cooperativity in binding to T, as in a Pauling/KNF sequential model? To address this question directly, Rivetti et al. 20 measured oxygen binding to single crystals of hemoglobin known to remain in the T quaternary structure by X-ray crystallography 21. There was now no ambiguity about the quaternary structure, which has always been a source of uncertainty in solution studies, and possible differences between crystal and solution were not an issue. The crystal binding curve was found to be noncooperative, confirming the essential point of the MWC model that the T quaternary structure binds oxygen noncooperatively (Fig. 4). However, a small amount of cooperative binding was detected, which is masked by unequal binding to the α and β subunits (Fig. 4) (see below). Since the X-ray results showed that oxygenation does not break the salt bridges 21, the crystal-binding studies also supported a major element of Perutz s mechanism. According to the Perutz mechanism there should be no ph dependence to the T-state affinity unless the salt bridges break, and none was observed in the crystal 20. (The contribution of the salt bridges to the Bohr effect in solution has been studied in detail using nuclear magnetic resonance by Chien Ho and coworkers 22.) Removal of two of the six salt bridges, moreover, raises the oxygen affinity of the crystal T-state, but only about threefold 23. Factors in addition to the constraints of the salt bridges must therefore contribute to the low affinity of the T-state. One obvious criticism of the crystal studies is that the lattice forces may prevent hemoglobin from undergoing the conformational relaxation that would lead to more significant cooperative binding within the T quaternary structure and a ph-dependent affinity. Strong evidence against this argument has recently been obtained by showing that deoxyhemoglobin encapsulated in a silica gel is trapped in the T quaternary structure, binds oxygen noncooperatively 24, has the same affinity as T-state hemoglobin in solution (about fourfold higher than the crystal) 25, and exhibits most of the solution T-state Bohr effect 25. An important question regarding equilibrium properties remains. How can the MWC model be reconciled with the results of Ackers? Using chemical analogs to investigate the properties of singly, doubly, and triply liganded molecules, normally present at very low population (Fig. 2b), Ackers and coworkers made an extensive series of measurements of the free energy of dissociation of the tetramer into two αβ dimers for all 10 ligation microstates 26. The new information on ligand binding to the tetramer from this approach is based on thermodynamic linkage, namely that the difference in the tetramer-todimer dissociation free energy for ligation microstates is the same as the difference in the free energy of binding ligands to the tetramers compared to the dissociated dimers. Since the free dimers bind oxygen noncooperatively and have nearly the same affinity as the R-state tetramer, these free energy differences are measures of cooperative interactions associated with liganding each microstate, and are called cooperative free energies. Using primarily the results from a zinc for iron substitution, Ackers has recently derived these energies for the 10 distinct ligation microstates of unmodified hemoglobin (Table 1) 26. Although the discrepancy is now much less than that found earlier using hν or kt geminate escape entry conformational change exp{ (kt) β } Fig. 6 Schematic for ligand rebinding to a single subunit in the R quaternary structure 40. A ligand dissociated from the heme or entering the heme pocket from the solvent may either bind or escape into the solvent. The rate of ligand entry and escape is independent of quaternary or tertiary structure, while the bond making and breaking steps at the heme depend on both the tertiary and quaternary structure. Following dissociation the subunit conformation relaxes from r*, the liganded conformation existing before dissociation, to r, the liganded conformation in the relaxed equilibrium R quaternary structure. The rate of the R to T quaternary transition now depends both on the number of ligands bound and the tertiary conformation of the four subunits. An analogous scheme applies to subunits in the T quaternary structure, with the subunits labeled t* and t. the cyanometheme substitution, one of the 10 microstates is still inconsistent with a perfect MWC model. The discrepant microstate is the doubly liganded tetramer with both ligands on the same αβ dimer (Table 1). The most straightforward way of explaining this result is to simply add intradimer cooperativity in the T-state to the MWC model, as was done by Gill et al. 27 That is, the second ligand binds to the αβ dimer with a (δ -fold) higher intrinsic affinity than the first, instead of with the same affinity as required by the MWC model. This modified MWC model explains noncooperative binding in the crystal (Hill n = 1.0) as resulting from intradimer cooperativity (1 < δ < 2.5, corresponding to 1 < n < 1.2), exactly compensating for the 2 5 fold higher affinity of the α subunit compared to the β subunit (0.97 > n > 0.85) 28. A comparable value of δ = 4 is consistent with the cooperative free energies observed for the microstates (Table 1). Although there is clearly some cooperative binding within the T quaternary structure, it is very small compared to the ~1,000-fold increase in affinity accompanying the change in quaternary structure from T to R (the MWC parameter c in Table 1). The surprise is that this small deviation from the MWC model does not result from a cooperative interaction across the interface between αβ dimers that is known to change with quaternary structure and thereby affect the oxygen affinity. Instead it results from cooperative interaction between subunits of the same αβ dimer. X-ray crystallography detects no change at the interface between these dimers upon either ligand binding or quaternary transition 10, so there is as yet no structural explanation for either intradimer cooperativity or why it may be considerably exaggerated in some chemical analogs. The large cooperativity in the T-state found with the cyanometheme substitution (δ 170) (a result that has recently been challenged on technical grounds 10,26,29 ) prompted Ackers to discard the MWC model prematurely 18. With the new results from the crystal, gel, and tetramer dimer dissociation studies, almost every major element of the equilibrium formulation of the MWC model and Perutz stereochemical mechanism for homotropic effects has been subjected to critical tests, with no major discrepancies between theory and experiments. For nature structural biology volume 6 number 4 april

6 review Table 1 Comparison of cooperative free energies obtained from tetramer dimer dissociation experiments with predictions of MWC and modified MWC models Ligation microstate 1 Cooperative free energy (kcal mol 1 ) 2 Expt. 3 MWC 4 Modified MWC 5 x in [-RT ln (x/1+l)] 6 α 1 α 2 β 1 β L α 1 O 2 α 2 β 1 β ± ε (1 + Lc) α 1 α 2 β 1 O 2 β ± ε (1 + Lc) α 1 O 2 α 2 β 1 O 2 β ± ε 2 (1 + Lc 2 δ) α 1 O 2 α 2 β 1 β 2 O ± ε 2 (1 + Lc 2 ) α 1 O 2 α 2 O 2 β 1 β ± ε 2 (1 + Lc 2 ) α 1 α 2 β 1 O 2 β 2 O ± ε 2 (1 + Lc 2 ) α 1 O 2 α 2 β 1 O 2 β 2 O ± ε 3 (1 + Lc 3 δ) α 1 O 2 α 2 O 2 β 1 O 2 β ± ε 3 (1 + Lc 3 δ) α 1 O 2 α 2 O 2 β 1 O 2 β 2 O ± ε 4 (1 + Lc 4 δ 2 ) 1 There are 16 ligation microstates. Because of the two-fold rotation axis of symmetry that interchanges the α 1 β 1 dimer with the α 2 β 2 dimer (Fig. 1), 10 of the 16 are distinct and are shown here. 2 Defined as the free energy change upon dissociating the fully deoxygenated tetramer into two αβ dimers minus the free energy of dissociation for the microstate (omitting the statistical factors). 3 From Table 1 of Ackers 26. Experimental uncertainties from Huang et al Calculated from -RT ln (x/1+l) with δ = 1 (that is, no cooperative binding to the T-state), ε = 3.1 (the ratio of the intrinsic R-state affinity to the dimer affinity; Doyle et al. 44 reported ε = 4 (+4,-2)) and MWC parameters (L = 4.4 x 10 6, c = K T /K R = , and K T = 1/(77 torr)) which simultaneously fit the cooperative free energies (weighted by the uncertainties) and the oxygen-binding curve under identical solution conditions (Fig. 17 in ref. 26). 5 Same as footnote 4, except δ fixed at 4 (that is, cooperative binding to the αβ dimer in the T-state tetramer), ε = 3.4, L = 6.4 x 10 6, c = 0.001, K T = 1/(109 torr). Allowing δ to vary in the minimization yields cooperative free energies of 0, 3.3, 3.3, 4.9, 6.6, 6.6, 6.6, 7.0, 7.0, 6.3 with δ = 21, ε = 3.7, L = 8.8 x 10 6, c =9.2 x 10 4, K T = 1/(110 torr), but decreases the sum of squares by only 15%. However, such a large δ is not possible. For equal α and β affinities it would result in a Hill n of 1.6, compared to the value of n < 1 observed in gels 25. Furthermore, to produce a Hill n of 1.0 consistent with δ = 21, the ratio of α to β affinities would be greater than 80 (for n = 1, δ = (q+1) 2 /4q, where q K α /K β ) 20, inconsistent with the approximately equal affinities observed for the microstates. 6 These expressions are the same as those in Gill et al. 27, except that they contain the quaternary enhancement factor ε not considered by Gill et al. They are also the same as those given in Table 4 of Ackers 45 with c α = c β, δ α = δ β ε, and δ αβ δ. One difference is that Ackers expressions 45 assume that R and T exhibit identical intradimer cooperativity. The expressions above can be obtained from the partition functions: with K Rα = K Rβ, and K Tα = K Tβ. 7 The value reported for this ligation microstate with cyanide bound to the two oxidized hemes as analogs for oxyhemes is 3.1 kcal mol 1 yielding δ 170, but as mentioned in the text this value could be the result of experimental artifacts 10,26,29. most biochemists cooperative oxygen binding by hemoglobin is not only well understood, but far better understood than any other multisubunit protein. There was, however, a large body of work on the complex kinetics of hemoglobin to be explained. Kinetics make far greater demands on mechanism, and are particularly important for the hemoglobin mechanism where there is a large number of intermediate species with low equilibrium populations (Figs 2b and 3). In kinetic experiments large populations of these intermediates can be generated transiently and interrogated. Hemoglobin kinetics Gibson made the critical observation for understanding cooperativity in hemoglobin kinetics 30. Gibson discovered that following photodissociation by an intense light pulse the bimolecular rate of carbon monoxide rebinding is more than 20 times faster than the initial rate obtained by mixing deoxyhemoglobin with carbon monoxide. His result suggested that the rate of ligand binding, and therefore the ligand affinity, is not determined by the number of ligands already bound. This is consistent with the MWC model (formulated several years later) but not with a Pauling/KNF sequential model. Gibson also discovered that the deoxyhemoglobin optical spectrum produced immediately after the flash is different from that of deoxyhemoglobin at equilibrium, and suggested that this fastreacting deoxyhemoglobin has a different structure. This work was followed by a series of innovative kinetic experiments using rapid mixing and flash photolysis methods by Gibson, Eraldo Antonini and Maurizio Brunori, summarized in the now classic book by Antonini and Brunori 31. The connection of the kinetics to the MWC model was made by Hopfield, Shulman, and Ogawa 32. They identified fast-reacting hemoglobin with hemoglobin that had not yet switched from the R to T quaternary structure after ligand photodissociation. These investigators also provided at least a semiquantitative explanation of almost all of the kinetic experiments on both ligand association and dissociation with a kinetic version of the MWC model. In this model there are only two pairs of binding and dissociation rates, one for T and one for R, and all quaternary rates are assumed to be fast relative to ligand binding or dissociation. There was still a crucial missing piece to the story. In spite of the importance of the quaternary conformational changes, their kinetics had not yet been observed. This prompted Gibson to take advantage of the deoxyheme spectral changes to measure the microsecond-millisecond kinetics of conformational changes in photolysis experiments 19. He was, however, unable to use an MWC model to fit the ligand rebinding kinetics at neutral ph simultaneously with the conformational kinetics, which he attributed to R T quaternary structural changes. Gibson concluded that the experimental results are inconsistent with the MWC model 19. This of course presented a serious problem, and, as mentioned earlier, was a major source of the controversy concerning the applicability of the MWC model. Several key ingredients went into solving this problem. First, nanosecond-resolved spectroscopy showed that the spectral changes of the deoxyhemes observed by Gibson begin much earlier than a microsecond, and extend from less than a nanosecond to milliseconds (Fig. 5) 33. Conformational changes before ~1 µs were shown to be purely tertiary, while those occurring later include quaternary changes as well. A second key element was the realization that the tertiary relaxation is a single extended process. This view was motivated by the experiments of Philip Anfinrud on myoglobin 34 which showed that, as in glassy systems 35, the time course of conformational relaxation could be closely approximated by a stretched exponential function (that is, exp[- (kt) β ], β 0.1) extending from hundreds of femtoseconds to almost a microsecond. A third ingredient was the idea of Noam Agmon and Hopfield that conformational relaxation in myoglobin slows geminate rebinding (that is, rebinding of dissociated ligands before they escape from the protein) of carbon monoxide 36. This proposal was most directly confirmed in experiments 356 nature structural biology volume 6 number 4 april 1999

7 review on myoglobin embedded in a glass of the sugar trehalose at room temperature 37. As in a low-temperature (<180 K) glycerol/water glass 38, the high viscosity of the room-temperature trehalose glass traps the substrates in the liganded conformation, prevents the ligand from escaping the protein and results in a large increase in the geminate rebinding rate compared to the conformationally relaxed protein 37. The final important element was the simplification of the description of the quaternary rates. In the MWC model there are 10 such rates, defining R T equilibrium constants at the 5 ligation states (Fig. 3). The observation of a linear free energy relation, however, between quaternary rates and equilibrium constants (k(r i T i ) = γ(lc i ) α, i = 0,1,2,3,4) reduced the description of the quaternary rates to the two MWC parameters, L and c (Fig. 3), and two new kinetic parameters, α which is a measure of the distance along a reaction coordinate between R and T, and γ which sets the absolute time 39. These four new findings motivated a reinvestigation of the kinetic version of the MWC model, with the addition of geminate rebinding, the change in tertiary conformation before the quaternary change, and a linear free energy relation between quaternary rates and equilibria (Fig. 6) 40. The tertiary conformational change was treated as a stretched exponential process that slows the geminate rebinding rate. This model simultaneously fits the ligand rebinding kinetics and the kinetics of tertiary and quaternary conformational changes (Fig. 5), as well as the carbon monoxide binding curve and population of ligation intermediates as a function of carbon monoxide saturation determined in the low-temperature electrophoresis experiments of Michele Perrella and coworkers 41. The early attempt by Gibson 19 was unsuccessful because he did not have the necessary time resolution and therefore did not consider the possibility that tertiary conformational changes beginning before the quaternary change contributed to the spectral kinetics. Conclusion The MWC two-state allosteric model and the Perutz stereochemical mechanism have survived the demanding test of quantitatively explaining both the equilibrium and complex kinetic behavior of hemoglobin. At this point physical scientists would argue that at least the homotropic part of cooperative oxygen binding to hemoglobin is well understood and might treat it as a closed subject. But is it? Fundamental questions remain. Foremost among these is a quantitative structural explanation for the low affinity of the T-state. The difference in binding free energy of R and T is never more than ~4 kcal mol 1. Is it possible in such a complex machine to explain quantitatively such a small free energy difference? Finally, we return to the original question. Have physical scientists produced a truly deeper understanding of hemoglobin function than Perutz had already obtained from static structural models and qualitative analysis? We believe the answer is yes. First, Shulman, Hopfield, and Ogawa showed that a vast array of equilibrium, spectroscopic, and complex kinetic behavior is consistent with the two-state allosteric model of Monod, Wyman, and Changeux. Szabo and Karplus clarified the relation between the Perutz mechanism and the MWC model, and showed how to make quantitative predictions from a complicated structural mechanism. The single-crystal and gel studies provided the convincing experimental evidence in support of the MWC model and Perutz s mechanism for describing cooperative effects in equilibrium experiments. Finally, time-resolved spectroscopy from the picosecond to the millisecond regime, together with an improved understanding of protein physics from studies on myoglobin, led to an interpretation of hemoglobin s complex kinetics in terms of the MWC model. This underlying simplicity in the apparently complex behavior of hemoglobin could not have been demonstrated without these novel physical experiments and a rigorous quantitative analysis. Acknowledgments We thank A. Szabo for numerous helpful discussions on the hemoglobin mechanism and for his comments on the manuscript. We also thank M. Brunori, P. Wolynes, and R. Zwanzig for helpful discussions, and G.L. Rossi for his generous support and collaboration in the single-crystal studies. This work was supported by a NATO Collaborative Research grant. This work was presented by W.A.E. at the Dahlem Workshop on Simplicity and Complexity in Proteins and Nucleic Acids, Berlin, Germany, May 17 22, 1998 (eds Frauenfelder, H., Deisenhofer, J. & Wolynes, P.G.) Dahlem University Press (in the press). Received 21 August,1998; accepted 16 December, 1998 nature structural biology volume 6 number 4 april

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