A new paradigm for electrostatic catalysis of radical reactions in vitamin B12 enzymes

Integrating computational methods to retrofit enzymes to synthetic pathways

Microbial production of desired compounds provides an efficient framework for the development of renewable energy resources. To be competitive to traditional chemistry, one requirement is to utilize the full capacity of the microorganism to produce target compounds with high yields and turnover rates. We use integrated computational methods to generate and quantify the performance of novel biosynthetic routes that contain highly optimized catalysts. Engineering a novel reaction pathway entails addressing feasibility on multiple levels, which involves handling the complexity of large-scale biochemical networks while respecting the critical chemical phenomena at the atomistic scale. To pursue this multi-layer challenge, our strategy merges knowledge-based metabolic engineering methods with computational chemistry methods. By bridging multiple disciplines, we provide an integral computational framework that could accelerate the discovery and implementation of novel biosynthetic production routes. Using this approach, we have identified and optimized a novel biosynthetic route for the production of 3HP from pyruvate.

Characterization of protein contributions to cobalt-carbon bond cleavage catalysis in adenosylcobalamin-dependent ethanolamine ammonia-lyase by using photolysis in the ternary complex

Protein contributions to the substrate-triggered cleavage of the cobalt-carbon (Co-C) bond and formation of the cob(II)alamin-5'-deoxyadenosyl radical pair in the adenosylcobalamin (AdoCbl)-dependent ethanolamine ammonia-lyase (EAL) from Salmonella typhimurium have been studied by using pulsed-laser photolysis of AdoCbl in the EAL-AdoCbl-substrate ternary complex, and time-resolved probing of the photoproduct dynamics by using ultraviolet-visible absorption spectroscopy on the 10(-7)-10(-1) s time scale. Experiments were performed in a fluid dimethylsulfoxide/water cryosolvent system at 240 K, under conditions of kinetic competence for thermal cleavage of the Co-C bond in the ternary complex. The static ultraviolet-visible absorption spectra of holo-EAL and ternary complex are comparable, indicating that the binding of substrate does not labilize the cofactor cobalt-carbon (Co-C) bond by significantly distorting the equilibrium AdoCbl structure. Photolysis of AdoCbl in EAL at 240 K leads to cob(II)alamin-5'-deoxyadenosyl radical pair quantum yields of <0.01 at 10(-6) s in both holo-EAL and ternary complex. Three photoproduct states are populated following a saturating laser pulse, and labeled, P(f), P(s), and P(c). The relative amplitudes and first-order recombination rate constants of P(f) (0.4-0.6; 40-50 s(-1)), P(s) (0.3-0.4; 4 s(-1)), and P(c) (0.1-0.2; 0) are comparable in holo-EAL and in the ternary complex. Time-resolved, full-spectrum electron paramagnetic resonance (EPR) spectroscopy shows that visible irradiation alters neither the kinetics of thermal cob(II)alamin-substrate radical pair formation, nor the equilibrium between ternary complex and cob(II)alamin-substrate radical pair, at 246 K. The results indicate that substrate binding to holo-EAL does not "switch" the protein to a new structural state, which promptly stabilizes the cob(II)alamin-5'-deoxyadenosyl radical pair photoproduct, either through an increased barrier to recombination, a decreased barrier to further radical pair separation, or lowering of the radical pair state free energy, or a combination of these effects. Therefore, we conclude that such a change in protein structure, which is independent of changes in the AdoCbl structure, and specifically the Co-C bond length, is not a basis of Co-C bond cleavage catalysis. The results suggest that, following the substrate trigger, the protein interacts with the cofactor to contiguously guide the cleavage of the Co-C bond, at every step along the cleavage coordinate, starting from the equilibrium configuration of the ternary complex. The cleavage is thus represented by a diagonal trajectory across a free energy surface, that is defined by chemical (Co-C separation) and protein configuration coordinates.

Understanding the determinants of selectivity in drug metabolism through modeling of dextromethorphan oxidation by cytochrome P450

Cytochrome P450 enzymes play key roles in the metabolism of the majority of drugs. Improved models for prediction of likely metabolites will contribute to drug development. In this work, two possible metabolic routes (aromatic carbon oxidation and O-demethylation) of dextromethorphan are compared using molecular dynamics (MD) simulations and density functional theory (DFT). The DFT results on a small active site model suggest that both reactions might occur competitively. Docking and MD studies of dextromethorphan in the active site of P450 2D6 show that the dextromethorphan is located close to heme oxygen in a geometry apparently consistent with competitive metabolism. In contrast, calculations of the reaction path in a large protein model [using a hybrid quantum mechanical-molecular mechanics (QM/MM) method] show a very strong preference for O-demethylation, in accordance with experimental results. The aromatic carbon oxidation reaction is predicted to have a high activation energy, due to the active site preventing formation of a favorable transition-state structure. Hence, the QM/MM calculations demonstrate a crucial role of many active site residues in determining reactivity of dextromethorphan in P450 2D6. Beyond substrate binding orientation and reactivity of Compound I, successful metabolite predictions must take into account the detailed mechanism of oxidation in the protein. These results demonstrate the potential of QM/MM methods to investigate specificity in drug metabolism.

Interheme electron tunneling in cytochrome c oxidase

Cytochrome c oxidase (CcO) is the terminal enzyme of the respiratory chain that catalyzes respiratory reduction of dioxygen (O(2)) to water in all eukaryotes and many aerobic bacteria. CcO, and its homologs among the heme-copper oxidases, has an active site composed of an oxygen-binding heme and a copper center in the vicinity, plus another heme group that donates electrons to this site. In most oxidoreduction enzymes, electron transfer (eT) takes place by quantum-mechanical electron tunneling. Here we show by independent molecular dynamics and quantum-chemical methods that the heme-heme eT in CcO differs from the majority of cases in having an exceptionally low reorganization energy. We show that the rate of interheme eT in CcO may nevertheless be predicted by the Moser-Dutton equation if reinterpreted as the average of the eT rates between all individual atoms of the donor and acceptor weighed by the respective packing densities between them. We argue that this modification may be necessary at short donor/acceptor distances comparable to the donor/acceptor radii.

Structural basis for adenosylcobalamin activation in AdoCbl-dependent ribonucleotide reductases

Class II ribonucleotide reductases (RNR) catalyze the formation of an essential thiyl radical by homolytic cleavage of the Co-C bond in their adenosylcobalamin (AdoCbl) cofactor. Several mechanisms for the dramatic acceleration of Co-C bond cleavage in AdoCbl-dependent enzymes have been advanced, but no consensus yet exists. We present the structure of the class II RNR from Thermotoga maritima in three complexes: (i) with allosteric effector dTTP, substrate GDP, and AdoCbl; (ii) with dTTP and AdoCbl; (iii) with dTTP, GDP, and adenosine. Comparison of these structures gives the deepest structural insights so far into the mechanism of radical generation and transfer for AdoCbl-dependent RNR. AdoCbl binds to the active site pocket, shielding the substrate, transient 5'-deoxyadenosyl radical and nascent thiyl radical from solution. The e-propionamide side chain of AdoCbl forms hydrogen bonds directly to the α-phosphate group of the substrate. This interaction appears to cause a "locking-in" of the cofactor, and it is the first observation of a direct cofactor-substrate interaction in an AdoCbl-dependent enzyme. The structures support an ordered sequential reaction mechanism with release or relaxation of AdoCbl on each catalytic cycle. A conformational change of the AdoCbl adenosyl ribose is required to allow hydrogen transfer to the catalytic thiol group. Previously proposed mechanisms for radical transfer in B12-dependent enzymes cannot fully explain the transfer in class II RNR, suggesting that it may form a separate class that differs from the well-characterized eliminases and mutases.

On catalytic preorganization in oxyanion holes: highlighting the problems with the gas-phase modeling of oxyanion holes and illustrating the need for complete enzyme models

Oxyanion holes play a major role in catalyzing enzymatic reactions, yet the corresponding energetics is frequently misunderstood. The main problem may be associated with the nontrivial nature of the electrostatic preorganization effect, without following the relevant formulation. That is, although the energetics of oxyanion holes have been fully quantified in early studies (which include both the enzymatic and reference solution reactions), the findings of these studies are sometimes overlooked, and, in some cases, it is assumed that gas-phase calculations with a fixed model of an oxyanion hole are sufficient for assessing the corresponding effect in the protein. Herein, we present a systematic analysis of this issue, clarifying the problems associated with modeling oxyanions by means of two fixed water molecules (or related constructs). We then re-emphasize the point that the effect of the oxyanion hole is mainly due to the fact that the relevant dipoles are already set in an orientation that stabilizes the TS charges, whereas the corresponding dipoles in solution are randomly oriented, resulting in the need to pay a very large reorganization energy. Simply calculating interaction energies with relatively fixed species cannot capture this crucial point, and considering it may help in advancing rational enzyme design.

Exploring challenges in rational enzyme design by simulating the catalysis in artificial kemp eliminase

One of the fundamental challenges in biotechnology and in biochemistry is the ability to design effective enzymes. Doing so would be a convincing manifestation of a full understanding of the origin of enzyme catalysis. Despite an impressive progress, most of the advances on this front have been made by placing the reacting fragments in the proper places, rather than by optimizing the environment preorganization, which is the key factor in enzyme catalysis. Rational improvement of the preorganization would require approaches capable of evaluating reliably the actual catalytic effect. This work takes previously designed kemp eliminases as a benchmark for a computer aided enzyme design, using the empirical valence bond as the main screening tool. The observed absolute catalytic effect and the effect of directed evolution are reproduced and analyzed (assuming that the substrate is in the designed site). It is found that, in the case of kemp eliminases, the transition state charge distribution makes it hard to exploit the active site polarity, even with the ability to quantify the effect of different mutations. Unexpectedly, it is found that the directed evolution mutants lead to the reduction of solvation of the reactant state by water molecules rather that to the more common mode of transition state stabilization used by naturally evolved enzymes. Finally it is pointed out that our difficulties in improving Kemp eliminase are not due to overlooking exotic effect, but to the challenge in designing a preorganized environment that would exploit the small change it charge distribution during the formation of the transition state.

Examining the case for the effect of barrier compression on tunneling, vibrationally enhanced catalysis, catalytic entropy and related issues

The idea that tunneling is enhanced by the compression of the donor-acceptor distance has attracted significant interest. In particular, recent studies argued that this proposal is consistent with pressure effects on enzymatic reactions, and that the observed pressure effects support the idea of vibrationally enhanced catalysis. However, a careful analysis of the current works reveals serious inconsistencies in the evidence presented to support these hypotheses. Apparently, tunneling decreases upon compression, and external pressure does not lead to the applicable compression of the free energy surface. Additionally, pressure experiments do not provide actual evidence for vibrationally enhanced catalysis. Finally, the temperature dependence of the entropy change in hydride transfer reactions is shown to reflect simple electrostatic effects.

Continuous wave photolysis magnetic field effect investigations with free and protein-bound alkylcobalamins

The activation of the Co-C bond in adenosylcobalamin-dependent enzymes generates a singlet-born Co(II)-adenosyl radical pair. Two of the salient questions regarding this process are: (1) What is the origin of the considerable homolysis rate enhancement achieved by this class of enzyme? (2) Are the reaction dynamics of the resultant radical pair sensitive to the application of external magnetic fields? Here, we present continuous wave photolysis magnetic field effect (MFE) data that reveal the ethanolamine ammonia lyase (EAL) active site to be an ideal microreactor in which to observe enhanced magnetic field sensitivity in the adenosylcobalamin radical pair. The observed field dependence is in excellent agreement with that calculated from published hyperfine couplings for the constituent radicals, and the magnitude of the MFE (<18%) is almost identical to that observed in a solvent containing 67% glycerol. Similar augmentation is not observed, however, in the equivalent experiments with EAL-bound methylcobalamin, where all field sensitivity observed in the free cofactor is washed out completely. Parallels are drawn between the latter case and the loss of field sensitivity in the EAL holoenzyme upon substrate binding (Jones et al. J. Am. Chem. Soc. 2007, 129, 15718-15727). Both are attributed to the rapid removal of the alkyl radical immediately after homolysis, such that there is inadequate radical pair recombination for the observation of field effects. Taken together, these results support the notion that rapid radical quenching, through the coupling of homolysis and hydrogen abstraction steps, and subsequent radical pair stabilization make a contribution to the observed rate acceleration of Co-C bond homolysis in adenosylcobalamin-dependent enzymes.

The EVB as a quantitative tool for formulating simulations and analyzing biological and chemical reactions

Recent years have seen dramatic improvements in computer power, allowing ever more challenging problems to be approached. In light of this, it is imperative to have a quantitative model for examining chemical reactivity, both in the condensed phase and in solution, as well as to accurately quantify physical organic chemistry (particularly as experimental approaches can often be inconclusive). Similarly, computational approaches allow for great progress in studying enzyme catalysis, as they allow for the separation of the relevant energy contributions to catalysis. Due to the complexity of the problems that need addressing, there is a need for an approach that can combine reliability with an ability to capture complex systems in order to resolve long-standing controversies in a unique way. Herein, we will demonstrate that the empirical valence bond (EVB) approach provides a powerful way to connect the classical concepts of physical organic chemistry to the actual energies of enzymatic reactions by means of computation. Additionally, we will discuss the proliferation of this approach, as well as attempts to capture its basic chemistry and repackage it under different names. We believe that the EVB approach is the most powerful tool that is currently available for studies of chemical processes in the condensed phase in general and enzymes in particular, particularly when trying to explore the different proposals about the origin of the catalytic power of enzymes.

On the importance of ribose orientation in the substrate activation of the coenzyme B12-dependent mutases

The degree to which the corrin ring portion of coenzyme B(12) can facilitate the H-atom-abstraction step in the glutamate mutase (GM)-catalyzed reaction of (S)-glutamate has been investigated with density functional theory. The crystal structure of GM identifies two possible orientations of the ribose portion of coenzyme B(12). In one orientation (A), the OH groups of the ribose extend away from the corrin ring, whereas in the other orientation (B) the OH groups, especially that involving O3', are instead directed towards the corrin ring. Our calculations identify a sizable stabilization amounting to about 30 kJ mol(-1) in the transition structure (TS) complex corresponding to orientation B (TS(B)CorIm). In the TS complex where the ribose instead is positioned in orientation A, no such effect is manifested. The observed stabilization in TS(B)CorIm appears to be the result of favorable interactions involving O3' and the corrin ring, including a C-HO hydrogen bond. We find that the degree of stabilization is not particularly sensitive to the Co-C distance. Our calculations show that any potential stabilization afforded to the H-atom-abstraction step by coenzyme B(12) is sensitive to the orientation of the ribose moiety.

DFT and ONIOM(DFT:MM) studies on Co-C bond cleavage and hydrogen transfer in B12-dependent methylmalonyl-CoA mutase. Stepwise or concerted mechanism?

The considerable protein effect on the homolytic Co-C bond cleavage to form the 5'-deoxyadenosyl (Ado) radical and cob(II)alamin and the subsequent hydrogen transfer from the methylmalonyl-CoA substrate to the Ado radical in the methylmalonyl-CoA mutase (MMCM) have been extensively studied by DFT and ONIOM(DFT/MM) methods. Several quantum models have been used to systematically study the protein effect. The calculations have shown that the Co-C bond dissociation energy is very much reduced in the protein, compared to that in the gas phase. The large protein effect can be decomposed into the cage effect, the effect of coenzyme geometrical distortion, and the protein MM effect. The largest contributor is the MM effect, which mainly consists of the interaction of the QM part of the coenzyme with the MM part of the coenzyme and the surrounding residues. In particular, Glu370 plays an important role in the Co-C bond cleavage process. These effects tremendously enhance the stability of the Co-C bond cleavage state in the protein. The initial Co-C bond cleavage and the subsequent hydrogen transfer were found to occur in a stepwise manner in the protein, although the concerted pathway for the Co-C bond cleavage coupled with the hydrogen transfer is more favored in the gas phase. The assumed concerted transition state in the protein has more deformation of the coenzyme and the substrate and has less interaction with the protein than the stepwise route. Key factors and residues in promoting the enzymatic reaction rate have been discussed in detail.

Toward accurate screening in computer-aided enzyme design

The ability to design effective enzymes is one of the most fundamental challenges in biotechnology and in some respects in biochemistry. In fact, such ability would be one of the most convincing manifestations of a full understanding of the origin of enzyme catalysis. In this work, we explore the reliability of different simulation approaches, in terms of their ability to rank different possible active site constructs. This validation is done by comparing the ability of different approaches to evaluate the catalytic contributions of various residues in chorismate mutase. It is demonstrated that the empirical valence bond (EVB) model can serve as a practical yet accurate tool in the final stages of computer-aided enzyme design (CAED). Other approaches for fast screening are also examined and found to be less accurate and mainly useful for qualitative screening of ionized residues. It is pointed out that accurate ranking of different options for enzyme design cannot be accomplished by approaches that cannot capture the electrostatic preorganization effect. This is in particular true with regard to current design approaches that use gas phase or small cluster calculations and then estimate the interaction between the enzyme and the transition state (TS) model rather than the TS binding free energy or the relevant activation free energy. The ability of the EVB model to provide a tool for quantitative ranking in the final stage of CAED may help in progressing toward the design of enzymes whose catalytic power is closer to that of native enzymes than to that of the current generation of designer enzymes.

Progress in ab initio QM/MM free-energy simulations of electrostatic energies in proteins: accelerated QM/MM studies of pKa, redox reactions and solvation free energies

Hybrid quantum mechanical/molecular mechanical (QM/MM) approaches have been used to provide a general scheme for chemical reactions in proteins. However, such approaches still present a major challenge to computational chemists, not only because of the need for very large computer time in order to evaluate the QM energy but also because of the need for proper computational sampling. This review focuses on the sampling issue in QM/MM evaluations of electrostatic energies in proteins. We chose this example since electrostatic energies play a major role in controlling the function of proteins and are key to the structure-function correlation of biological molecules. Thus, the correct treatment of electrostatics is essential for the accurate simulation of biological systems. Although we will be presenting different types of QM/MM calculations of electrostatic energies (and related properties) here, our focus will be on pKa calculations. This reflects the fact that pKa's of ionizable groups in proteins provide one of the most direct benchmarks for the accuracy of electrostatic models of macromolecules. While pKa calculations by semimacroscopic models have given reasonable results in many cases, existing attempts to perform pKa calculations using QM/MM-FEP have led to discrepancies between calculated and experimental values. In this work, we accelerate our QM/MM calculations using an updated mean charge distribution and a classical reference potential. We examine both a surface residue (Asp3) of the bovine pancreatic trypsin inhibitor and a residue buried in a hydrophobic pocket (Lys102) of the T4-lysozyme mutant. We demonstrate that, by using this approach, we are able to reproduce the relevant side chain pKa's with an accuracy of 3 kcal/mol. This is well within the 7 kcal/mol energy difference observed in studies of enzymatic catalysis, and is thus sufficient accuracy to determine the main contributions to the catalytic energies of enzymes. We also provide an overall perspective of the potential of QM/MM calculations in general evaluations of electrostatic free energies, pointing out that our approach should provide a very powerful and accurate tool to predict the electrostatics of not only solution but also enzymatic reactions, as well as the solvation free energies of even larger systems, such as nucleic acid bases incorporated into DNA.

Computational Studies of Bioorganometallic Enzymes and Cofactors

Because of their complex geometric and electronic structures, the active sites and cofactors of bioorganometallic enzymes, which are characterized by their metal–carbon bonds, pose a major challenge for computational chemists. However, recent progress in computer technology and theoretical chemistry, along with insights gained from mechanistic, spectroscopic, and X-ray crystallographic studies, have established an excellent foundation for the successful completion of computational studies aimed at elucidating the electronic structures and catalytic cycles of these species. This chapter briefly reviews the most popular computational approaches employed in theoretical studies of bioorganometallic species and summarizes important information obtained from computational studies of (i) the enzymatic formation and cleavage of the Co–C bond of coenzyme B12; (ii) the catalytic cycle of methyl-coenzyme M reductase and its nickel-containing cofactor F430; (iii) the polynuclear active-site clusters of the bifunctional enzyme carbon monoxide dehydrogenase/acetyl-coenzyme A synthase; and (iv) the magnetic properties of the active-site cluster of Fe-only hydrogenases.

A quantum mechanical/molecular mechanical study on the catalysis of the pyridoxal 5'-phosphate-dependent enzyme L-serine dehydratase

The catalytic mechanism of a pyridoxal 5'-phosphate-dependent enzyme, l-serine dehydratase, has been investigated using ab initio quantum mechanical/molecular mechanical (QM/MM) methods. New insights into the chemical steps have been obtained, including the chemical role of the substrate carboxyl group in the Schiff base formation step and a proton-relaying mechanism involving the phosphate of the cofactor in the beta-hydroxyl-leaving step. The latter step is of no barrier and follows sequentially after the elimination of the alpha-proton, leading to a single but sequential alpha, beta-elimination step. The rate-limiting transition state is specifically stabilized by the enzyme environment. At this transition state, charges are localized on the substrate carboxyl group, as well as on the amino group of Lys41. Specific interactions of the enzyme environment with these groups are able to lower the activation barrier significantly. One major difficulty associated with studies of complicated enzymatic reactions using ab initio QM/MM models is the appropriate choices of reaction coordinates. In this study, we have made use of efficient semiempirical models and pathway optimization techniques to overcome this difficulty.

The energetics of the primary proton transfer in bacteriorhodopsin revisited: it is a sequential light-induced charge separation after all

The light-induced proton transport in bacteriorhodopsin has been considered as a model for other light-induced proton pumps. However, the exact nature of this process is still unclear. For example, it is not entirely clear what the driving force of the initial proton transfer is and, in particular, whether it reflects electrostatic forces or other effects. The present work simulates the primary proton transfer (PT) by a specialized combination of the EVB and the QCFF/PI methods. This combination allows us to obtain sufficient sampling and a quantitative free energy profile for the PT at different protein configurations. The calculated profiles provide new insight about energetics of the primary PT and its coupling to the protein conformational changes. Our finding confirms the tentative analysis of an earlier work (A. Warshel, Conversion of light energy to electrostatic energy in the proton pump of Halobacterium halobium, Photochem. Photobiol. 30 (1979) 285-290) and determines that the overall PT process is driven by the energetics of the charge separation between the Schiff base and its counterion Asp85. Apparently, the light-induced relaxation of the steric energy of the chromophore leads to an increase in the ion-pair distance, and this drives the PT process. Our use of the linear response approximation allows us to estimate the change in the protein conformational energy and provides the first computational description of the coupling between the protein structural changes and the PT process. It is also found that the PT is not driven by twist-modulated changes of the Schiff base's pKa, changes in the hydrogen bond directionality, or other non-electrostatic effects. Overall, based on a consistent use of structural information as the starting point for converging free energy calculations, we conclude that the primary event should be described as a light-induced formation of an unstable ground state, whose relaxation leads to charge separation and to the destabilization of the ion-pair state. This provides the driving force for the subsequent PT steps.

Magnetic field effect studies indicate reduced geminate recombination of the radical pair in substrate-bound adenosylcobalamin-dependent ethanolamine ammonia lyase

The apparent conflict between literature evidence for (i) radical pair (RP) stabilization in adenosylcobalamin (AdoCbl)-dependent enzymes and (ii) the manifestation of magnetic field sensitivity due to appreciable geminate recombination of the RP has been reconciled by pre-steady-state magnetic field effect (MFE) investigations with ethanolamine ammonia lyase (EAL). We have shown previous stopped-flow MFE studies to be insensitive to magnetically induced changes in the net forward rate of C-Co homolytic bond cleavage. Subsequently, we observed a magnetic-dependence in the continuous-wave C-Co photolysis of free AdoCbl in 75% glycerol but have not done so in the thermal homolysis of this bond in the enzyme-bound cofactor in the presence of substrate. Consequently, in the enzyme-bound state, the RP generated upon homolysis appears to be stabilized against the extent of geminate recombination required to observe an MFE. These findings have strong implications for the mechanism of RP stabilization and the unprecedented catalytic power of this important class of cobalamin-dependent enzymes.

A quantum-chemical picture of hemoglobin affinity

Understanding the molecular mechanism of hemoglobin cooperativity remains an enduring challenge. Protein forces that control ligand affinity are not directly accessible by experiment. We demonstrate that computational quantum mechanics/molecular mechanics methods can provide reasonable values of ligand binding energies in Hb, and of their dependence on allostery. About 40% of the binding energy differences between the relaxed state and tense state quaternary structures result from strain induced in the heme and its ligands, especially in one of the pyrrole rings. The proximal histidine also contributes significantly, in particular, in the alpha-chains. The remaining energy difference resides in protein contacts, involving residues responsible for locking the quaternary changes. In the alpha-chains, the most important contacts involve the FG corner, at the "hinge" region of the alpha(1)beta(2) quaternary interface. The energy differences are spread more evenly among the beta-chain residues, suggesting greater flexibility for the beta- than for the alpha-chains along the quaternary transition. Despite this chain differentiation, the chains contribute equally to the relaxed substitute state energy difference. Thus, nature has evolved a symmetric response to the quaternary structure change, which is a requirement for maximum cooperativity, via different mechanisms for the two kinds of chains.