At the Dawn of the 21st Century: Is Dynamics the Missing Link for Understanding Enzyme Catalysis?

Keywords: Dynamics, Enzyme Catalysis, Catalytic Landscape, Tunneling, Promoting Motions, Single Molecule Experiments, Coherence Effects, Polar Preorganization

“Enzyme Dynamics During Catalysis”, Science 295 (2002), 1520–1523.

Although classical enzymology together with structural biology have provided profound insights into the chemical mechanisms of many enzymes, enzyme dynamics and their relation to catalytic function remain poorly characterized. Because many enzymatic reactions occur on time scales of micro- to milliseconds, it is anticipated that the conformational dynamics of the enzyme on these time scales might be linked to its catalytic action.

“Linkage Between Dynamics and Catalysis in a Thermophilic-Mesophilic Enzyme Pair”, Nat. Struct. Mol. Biol. 11 (2004), 945 – 949.

A fundamental question is how enzymes can accelerate chemical reactions. Catalysis is not only defined by actual chemical steps, but also by enzyme structure and dynamics. To investigate the role of protein dynamics in enzymatic turnover, we measured residue-specific protein dynamics in hyperthermophilic and mesophilic homologs of adenylate kinase during catalysis. A dynamic process, the opening of the nucleotide-binding lids, was found to be rate-limiting for both enzymes as measured by NMR relaxation. Moreover, we found that the reduced catalytic activity of the hyperthermophilic enzyme at ambient temperatures is caused solely by a slower lid-opening rate. This comparative and quantitative study of activity, structure and dynamics revealed a close link between protein dynamics and catalytic turnover.

“Intrinsic Dynamics of an Enzyme Underlies Catalysis”, Nature 438 (2005), 117–121.

… This correlation suggests that the protein motions necessary for catalysis are an intrinsic property of the enzyme and may even limit the overall turnover rate. Motion is localized not only to the active site but also to a wider dynamic network. Whereas coupled networks in proteins have been proposed previously, we experimentally measured the collective nature of motions with the use of mutant forms of CypA. We propose that the pre-existence of collective dynamics in enzymes before catalysis is a common feature of biocatalysts and that proteins have evolved under synergistic pressure between structure and dynamics.

“Instrinsic Motions Along an Enzymatic Reaction Trajectory”, Nature 450 (2007), 838–844.

The mechanisms by which enzymes achieve extraordinary rate acceleration and specificity have long been of key interest in biochemistry. It is generally recognized that substrate binding coupled to conformational changes of the substrate-enzyme complex aligns the reactive groups in an optimal environment for efficient chemistry. Although chemical mechanisms have been elucidated for many enzymes, the question of how enzymes achieve the catalytically competent state has only recently become approachable by experiment and computation. Here we show crystallographic evidence for conformational substates along the trajectory towards the catalytically competent ‘closed’ state in the ligand-free form of the enzyme adenylate kinase. Molecular dynamics simulations indicate that these partially closed conformations are sampled in nanoseconds, whereas nuclear magnetic resonance and single-molecule fluorescence resonance energy transfer reveal rare sampling of a fully closed conformation occurring on the microsecond-to-millisecond timescale. Thus, the larger-scale motions in substrate-free adenylate kinase are not random, but preferentially follow the pathways that create the configuration capable of proficient chemistry. Such preferred directionality, encoded in the fold, may contribute to catalysis in many enzymes.

“A Perspective on Enzyme Catalysis”, Science 301 (2003), 1196–1202.

The exploration for links between protein structure, movement, and catalysis will be expanded by the advent of new methods. We anticipate additional examples of enzymes for which motion is important and other cases for which it is not. Particularly attractive are techniques that permit the observation of single molecules on the same millisecond-to-second time scale on which enzymatic reactions normally occur. Such kinetics provide data for conformational changes during enzymatic turnover that may be masked in ensemble-averaged studies. Ensemble studies featuring isotopic editing of specific regions of the protein coupled with temperature jump relaxation also show considerable promise in detecting motions of mobile loops and active-site residues important to catalysis. Similarly, the extensive isotopic labeling of the substrate coupled with kinetic isotope effect analysis can provide structural information for reaction coordinate motions from the vantage point of the substrate. Isotopic editing, of course, remains the basis of Raman, infrared, and NMR studies of protein dynamics.

5.1. Ultrafast Photoisomerization and Electron Transport Reactions

Studying frictional effects in excited state ultrafast reactions has been a field of general interest, for instance in the case of stilbene^46,47, where it was found that frictional effects are generally small when the barrier is large. A related study is one of the femotosecond photoisomerization of bacteriorhodopsin (bR)⁴⁸, which was pumped at either 550 or 405nm and probed between 640 and 950nm. This study found the shape of the simulated emission spectra to be time independent (which is inconsistent with more conventional models that invoke low-frequency motion out of the Franck-Condon region following excitation). This result is similar to the photoisomerization of cis-stilbene, and demonstrates that there exists potentially interesting dynamics in these ultrafast reactions.

Simulations of ultrafast photoisomerization have a long history, dating as far back as the simulations of the primary step in the vision process⁴⁹ in the 1970s. Significant progress has been made on this front since then (e.g. Refs. ^50–54) and, particularly, recent theoretical and experimental studies strongly suggest that subpicosecond isomerization^50,54,55 can have significant dynamical effects.

Another clear case of dynamical effects is provided by the primary electron-transfer step in photosynthetic bacterial reaction centers. In the photosynthetic case, an ensemble of molecules can be coherently excited by use of a short pulse of light. In this system, electron transfer from the excited state occurs on the same timescale as relaxation among the solvent modes that are coupled to the reaction. Vibrational coherence can result in oscillatory kinetics, and deviations from the predictions of Marcus theory^56–59. Related dynamical effects have also been observed in ground-state organic reactions that proceed from an instantaneously generated intermediate⁶⁰.

5.2 Electrostatic Fluctuations and Solvent Relaxation Times

Theoretical studies of solvation dynamics in enzymes have been a major part of the analysis of dynamical ideas^25,61,62. Such studies have included our early determination of the autocorrelation of the electrostatic energy gap in enzymes and solutions (e.g Ref. 26), and an analysis of the corresponding relaxation time (similar simulations have been recently conducted with Hynes’s frictional approach⁶², which is equivalent to the energy gap approach). This energy gap can be separated to the contributions from the reacting system (solute) and the surrounding (solvent). The solvent contribution is of special interest, since this is where the reaction in the enzyme and in solution can be different. This contribution is also strongly related to the key parameters in experimental studies of solvation dynamics. For example, Fig. 1 shows the calculated fluctuations of the energy gap between the reactant and product states during MD simulations of haloalkane dehalogenase and the reference system in water⁶³.

As can be seen from Fig. 1, the fluctuations of the solvent coordinates in the enzyme and solution are quite similar. For a more definitive analysis, one can use the autocorrelation function of the energy gap to compare the transmission coefficients of the enzymatic and solution reactions. Fig. 2 shows the autocorrelation functions of the energy gap in the region of the TS for haloalkane dehalogenase as well as for the reference reaction. This figure presents the autocorrelation functions of both the total energy gap, C(t), as well as the electrostatic component, which we take as the solvent coordinate, C_el(t). It includes the results from two MD simulations of each system, in order to show the variability of the results. Although the results depend somewhat on the initial positions and velocities in the trajectories, the decay kinetics of the autocorrelation function are very similar in the enzyme and water, indicating that the transmission coefficients are not significantly different in the two systems. In both cases, the system relaxes in about 1 ps. Direct simulations of the actual relaxation from the TS to the product state^59,64 give similar results to those obtained from the energy gap, and these simulations give no indication that the enzymatic catalysis depends strongly on dynamical effects.

At this point, it may be useful for the reader to expand the discussion about the dehalogenase test case, which involves the nucleophilic attack of a carboxylate group on the carbon of chloroethane, via an S_N2 mechanism. Here, it has been demonstrated that the fluctuating dipoles of the solvent or protein are capable of either stabilizing or destabilizing the product state relative to the reactant state, thus modulating the chance that the solute moves to the product state²⁴ (a point that has been illustrated for many other systems⁶⁵ as well). The same system has also been examined by Nam et al.¹⁹, who used a QM/MM molecular orbital approach that focuses on the force autocorrelation approach (i.e. C_F(t)), which is a valid but less direct measure of the solvation dynamics of the autocorrelation function of the energy gap (C(t)). Here, Nam et al. found that C_F(t) decays more rapidly in the enzyme than in water, and that in the enzyme, C_F(t) has some oscillatory components that were not seen in water. The finding that C_F(t) can be somewhat different in the enzyme and in water was already described in an earlier study of alcohol dehydrogenase⁶⁵, although the solvation dynamics in the two systems was found to be similar. Additionally, Ref. 19 does not provide a separate analysis for the solute and solvent coordinates, which is difficult to do in standard QM/MM studies. However, the solute contribution cannot be reliably obtained by simply omitting the solvent’s electrostatic contribution from the QM/MM Hamiltonian, as this reproduces the gas-phase results, which are generally quite different from the behavior of the solute in solution (this issue has been discussed at length in Refs. ^61,66). In comparison, we have examined not only the haloalkane DhlA reaction, but also the reference system in water⁶³ (see Figs. 1 and 2), and have used the autocorrelation of the energy gap in order to compare the transmission coefficients in both the enzymatic and in the solution reactions. Our studies demonstrated that even though the precise results obtained depend somewhat on the initial positions and velocities in the trajectories, the decay kinetics of the autocorrelation functions are very similar both in the enzyme and in water, indicating that the two systems have very similar transmission coefficients. It should be pointed out that it has also been possible to examine this system by direct simulations of the actual relaxation from the TS to the product state^59,66. However, overall, none of these simulations have given any indication that enzymatic catalysis is dependent on dynamical effects.

The possible relevance of solvation dynamics to enzyme catalysis involved experimental examination of the solvation dynamics at the active site of an enzyme, glutaminyl-tRNA synthetase (GlnRS), was studied using a fluorescence probe, acrylodan, site-specifically attached at cysteine residue C229 near the active site⁶⁷. The picosecond time-dependent fluorescence Stokes shift indicates that in the absence of any substrate, there is slow solvation dynamics at the active site of the enzyme. This serves as a strong argument against the idea that the dynamics of the enzyme is faster than that of the solvent in the corresponding reference reaction⁶⁸. Here, we see that the dynamics in the enzyme is, if anything, slower rather than faster than the corresponding solvent dynamics, a point which had already been put forward by earlier theoretical studies (e.g. Ref. 63).

6.1 The Experimental Search for the Dynamical Coupling Between the Conformational and Chemical Coordinates

In recent years, nuclear magnetic resonance spectroscopy has been utilized to study the motions of enzymes over a wide range of timescales^15,71–73. An interesting such case is that of the enzyme cyclophilin A (CypA), which catalyzes the cis-trans isomerization of peptidyl proline bonds^15,73–75. Interestingly, the transverse relaxation of Arg55 (which is hydrogen bonded to the substrate and is essential for catalysis) has been demonstrated to accelerate in the presence of the substrate at a rate that approximately matches the dynamics for forming the TS, suggesting that the motions of this residue might play a dynamical role in the catalytic mechanism¹⁵. When considering this proposal, it is important to bear in mind that if Arg55 or any other residue moves along the reaction coordinate, and if its position changes in the TS, it necessarily moves on the same timescale as in the reaction. For example, the solvent molecules in a reaction in solution must rearrange during the reaction, and so must move at more or less the same rate as the solute atoms. Most of the reorganization of the environment will occur on the same timescale as that of the reaction, and the motions of the protein residues near the reacting substrate are not fundamentally different in this regard to the motions of the solvent molecules in solution. Furthermore, as long as the motions of the protein residues follow Boltzmann’s law, they simply reflect probabilistic effects, and not bona fide dynamical effects (see also Ref. 75).

The dynamics of the reaction of CypA was also examined in both its substrate-free state and during catalysis⁷⁶, and it was argued for a dynamic network of coupled motions along the protein, even in the absence of the substrate, that they claimed is essential for catalysis. More specifically, it has been suggested¹⁵ that “the characteristic enzyme motions during catalysis are already present in the free enzyme with frequencies corresponding to the catalytic turnover rate” and that “this correlation suggests that the protein motions necessary for catalysis are an intrinsic property of the enzyme”. The above statement has very problematic implications. That is, if motion along the relevant chemical reaction coordinate in the absence of the substrate is as slow as the turnover time, it would mean that we already have a large barrier along the direction of the chemical reaction coordinate even in the absence of the chemical barrier and thus the sum of chemical barrier and the protein reorganization barrier must be larger than the actual turnover barrier, which is of course not the case. In other words, this time cannot be the same as the chemical reaction time since the barrier along the reaction coordinate is not the same in the absence and the presence of the substrate (note that the above statement implies that the protein contribution is the same in both cases). Obviously, it is hard to understand why motions between two states that are involved in a reaction should contribute to catalysis if such motions occur already relatively slowly in the absence of the substrate (the low speed implies a pre-existing barrier for the reaction). A truly catalytic apo-enzyme should, in the absence of the substrate, have free energy minima corresponding to the reactant and product states that are close together along the reaction coordinate in order to minimize the reorganization energy (see Fig. 5). Already having slow motions (on a scale that is close to the chemical reaction time) along the reaction coordinate in the absence of the substrate means that the reorganization energy was not minimized, and thus that the apo-enzyme has not optimized the reaction speed (which is the opposite of the catalytic implications). Of course, the resolution is simple, and what is probably similar with and without the substrate is the barrier to the conformational change, which has very little to do with the chemical barrier. It is likely that either (a) the NMR measurement with the apo-enzymes does not provide the relevant information about the rate of movement in the direction that will become the chemical reaction coordinate, but rather only along the conformational coordinate, or (b) the protein contribution along the chemical barrier in the apo-enzyme has nothing to do with the corresponding barrier at the ES complex.

Benkovic, Wright and coworkers^9,77,78 have studied the reaction of dihydrofolate reductase by NMR. They found that site-directed mutations of residues in a loop that undergoes relatively large backbone motions had detrimental effects on catalysis, and they suggested that the dynamics of these residues could be important for catalysis. This suggestion was supported by Brooks and coworkers^12,79, who carried out MD simulations of three ternary complexes of the enzyme. Motions of some residues were strongly correlated, and were different in the enzyme-substrate and enzyme-product complexes. Some of these motions were modified in simulations of mutant enzymes with diminished activity. However, these studies did not examine any of the transition states in the reaction or demonstrate any dynamical effects on the rate constant. The different motions of the ES and EP complexes could just reflect the coupling of enzyme-substrate interactions to interactions of various groups in the protein, which is common to all enzymes.

NMR studies have also been used in order to attempt to examine the dynamic energy landscape of enzymes such as dihydrofolate reductase. For instance, Wright et al. used NMR⁷⁰ in order to characterize higher energy conformational substrates of E-coli dihydrofolate reductase. Each intermediate in the catalytic cycle was found to sample low-lying excited states with conformations that resemble the ground-state structures of preceding and subsequent intermediates, with substrate and cofactor exchange occurring through these excited substrates, suggesting that the maximum hydride transfer and steady-state turnover rates are governed by the dynamics of the transitions between the ground and excited states of the intermediates. Thus, the authors have argued that the bound ligands modulate the free energy landscape in order to funnel the enzyme through its reaction cycle along a preferred kinetic path. As stated above, the modulation of the landscape is not a dynamical effect, and claiming that ligand binding changes the landscape (via an induced fit model) does not provide any insight into catalysis, since the important question is how the barrier in the ES complex is reduced relative to that of the reacting complex in water. The implications of the arguments about funnel effects with respect to the landscape will be analyzed in more detail in Section 8.

Finally, in order to complete this section, we would like to focus on the adenylate kinase system, as this enzyme has been the subject of extensive experimental studies of this proposal^29,76,80. For instance, several workers have recently simulated or observed thermally driven dynamics on different timescales, and on this basis have argued for a link between μs to ms domain motions and enzymatic function^34,69,81 (despite the fact that not much is understood about the connection between such motions and local atomic fluctuations, which tend to be much faster). As an example of this, Kern and coworkers²⁸ have studied fluctuations in the hinge regions of adenylate kinase and have demonstrated that while ps to ns timescale atomic fluctuations in this region are quite different between a mesophilic and a hyperthermophilic adenylate kinase, they are very similar at temperatures where the enzymatic activity and folding energy are matched. Thus, based on this, the authors have argued for a connection between the different timescales and the corresponding amplitudes of motions in adeylate kinase in a so-called “hierarchy of timescales”, and believe that this hierarchy and its linkage to catalytic function is likely to be a general characteristic of protein function.

It should be noted that while these studies are extremely valuable, they have not actually established whether or not the conformational motions can transfer energy to the chemical coordinate. More specifically, despite major experimental effort and the publication of multiple articles in high profile journals, there is no single study that actually establishes any connection between the conformational dynamics and the change in the rate of the chemical step. At most, we have studies that have found a similar time range for the chemical and conformational steps, but this does not provide any proof that the two are correlated. Thus, we must explore this issue by theoretical approaches and such studies will be considered in detail in Section 6.3.

6.2 Single Molecule Experiments

In recent years, advances in room-temperature single molecule spectroscopy have allowed for the dynamic behavior of individual molecules to be recorded in real time^82–86. In turn, this has resulted in an increase in interest in the relationship between single-molecule experiments and the nature of protein landscapes. For instance, Xie and coworkers⁸⁷ have examined the enzymatic turnovers of single cholesterol oxidase molecules in real time by monitoring emission from the enzyme’s fluorescent active site, flavin adenine dinucleotide (FAD), and noticed significant and slow fluctuation in the rate of cholesterol oxidation by FAD. They then used single-molecule approaches to examine the static and dynamic disorder of the reaction rates, and used this to argue for a molecular memory phenomenon in which an enzymatic turnover was not independent of its previous turnovers due to a slow fluctuation of protein conformations. Thus, this⁸⁷ and subsequent single-molecule studies (e.g. Refs.^88–90) have proposed that a single enzyme molecule will exhibit large temporal fluctuates of the turnover rate constant over a wide (1ms – 100s) range of timescales. This phenomenon has been broadly referred to as “dynamic disorder”^91,92. It has been argued that the existence of such slow conformational changes on the same timescale of enzymatic reactions makes the application of transition state theory inadequate⁹³ (even though transition state theory has actually been demonstrated to provide an excellent tool for studies of enzymatic catalysis⁹⁴).

Xie and coworkers⁹⁵ have also used fluorescence assays in order to examine conformational changes within bacteriophage T7 DNA polymerase ternary complex upon the binding of a dNTP substrate. The authors found that the binding-induced conformational change is much slower in the case of the wrong base pair (W) than for the right (R) base pair, and they therefore suggested that the conformational change plays an important role in controlling the fidelity. They also observed that the bimolecular association rate constant (k_on) is correlated with the binding energy. The finding in this paper is very important and adds crucial information to the overall picture of DNA fidelity. However, we find some aspects of the authors’ interpretation of their results slightly problematic. That is, they suggest that the overall barrier that corresponds to k_cat/K_M determines the fidelity, and if the barrier that corresponds to the rate of the binding step (k_on) is smaller than the overall barrier, then k_on cannot contribute to the fidelity. In fact, the important finding that k_on is correlated to the binding energy leads to another interesting option, i.e. that the fidelity is most probably controlled by the difference between the overall barriers for R and W. This barrier is determined by two contributions - the binding free energy (rather than the barrier in the binding step) and the activation barrier for the chemical step (see Ref. 96). In order to achieve high fidelity, it is important for the enzyme to decrease the binding free energy (i.e. make it less negative) as well as to increase the chemical barrier upon going from R to W. The decrease in the binding energy leads to an increase in the binding barrier, as a result of the correlation discovered in the present work. The increase in K_M for W is part of the effect that is needed to achieve high fidelity. Thus, the increase in the binding barrier is a result of the factors that control fidelity, rather than the reason for fidelity.

Based on these observations, there have been attempts to test the Michaelis-Menten equation at the single-molecule level^93,97, and a recent work constructed a two-dimensional multisurface reaction free energy description of the catalytic cycle that explicitly connects multi-timescale conformational dynamics and dispersed enzymatic kinetics to the classical Michaelis-Menten equation⁹³ (though this work did not make any novel contribution). Here, the models presented the surface function of a slow conformational motion on a collective enzyme coordinate, Q, that facilitates the catalytic reaction along the intrinsic reaction coordinate, X, thus providing a dynamic realization of transition-state stabilization. The catalytic cycle was subsequently modeled as transitions between multiple displaced harmonic wells in the XQ space (representing different states of the cycle), which was then constructed according to the free energy driving force of the cycle. Based on such a setup, the authors observed that the enzyme-substrate complex under strain will exhibit a non-equilibrium relaxation towards a new conformation that lowers the activation energy of the reaction, thus enslaving the chemical reaction in X to the down hill slow motion on the Q surface. Unfortunately the hypothetical surface was not based on any actual enzyme landscape and had an insignificant barrier along the chemical coordinate. In fact as was pointed out in Ref. 98, there is no evolutionary pressure to reduce the chemical barrier for an enzyme-catalyzed reaction much below the diffusion controlled limit. As a result, it is unlikely that the chemical barrier will ever fall significantly below the binding barrier, and thus even a hypothetical situation in which the chemical barrier is much smaller than the binding barrier⁹³ is highly unrealistic (see Section 6.3).

6.3 Consistent Theoretical Studies Have Established That There Is No Dynamical Coupling Between the Chemical and Conformational Coordinates

As was clarified in the discussion above, there is no current experimental finding that shows a clear relationship between the conformational dynamics and the dynamics of the motion along the chemical coordinate. In the absence of experimental evidence, the only way to actually examine the dynamical hypothesis is to use theoretical studies that simulate the long timescale dynamics along the conformational and chemical coordinates. Such studies are extremely challenging however, and we have only recently started to make progress on this front.

We took a major step in this direction in a recent simulation study⁹⁹ that could actually explore the dynamical idea, by bridging the necessary timescale for examining the dynamical coupling between the conformational and chemical motions. This was achieved by means of a multiscale approach, which allowed for the exploration of the dynamical nature of enzyme catalysis in the ms timescale, making it the first realistic simulation-based analysis of the proposed dynamical coupling on such a long timescale. Our approach is based on renormalizing lower-dimensionality models in such a way that they capture the energetics and dynamics of the full reacting protein system by transforming the energetics and dynamics of the explicit all-atom enzyme to an equivalent low dimensional system. Some elements of this approach have been introduced in our early works (e.g. in Ref. 100), but this is the first work that uses this approach for simulating the long timescale behaviour of enzymatic reactions. The full technical details of this approach are presented in Ref. 99, and here we will only be discussing the major findings of the study of Ref. 99.

In Ref. 99, we used the catalytic reaction of adenylate kinase as a benchmark, though we also demonstrated that our approach can be applied to a general protein model with different types of conformational motions. Our preliminary study showed that the kinetic energy of the conformational motion is completely dissipated during the opening and closing of the active site lid of adenylate kinase, and thus that this cannot affect the time of the chemical process (a situation that holds for as long as the chemical barrier is higher than a few kcal/mol). That is, we found that there is no significant dynamical coupling between the chemical and conformational trajectories, with the inertial part of the conformational motion decaying very rapidly (i.e. on a timescale of far less than a nanosecond), and the remainder of the process being completely guided by the free energy surface and the corresponding Boltzmann probability. Finally, the chemical process was shown to have no significant “memory” of the dynamics of the conformational motion. This crucial finding is illustrated in Fig. 6. We must also clarify that Ref. 99 not only explored the dynamical proposal with the renormalized 2-D model, but also with the full CG protein-substrate model, where it was demonstrated that adding very large excess kinetic energy to the conformational motion (resembling the extreme possible effect of ligand binding) has no significant effect on the rate constant of the chemical step.

Obviously, we are not in any way claiming that the experimental studies discussed above are wrong, but merely that their dynamical interpretation is problematic. It could be argued that our simulation study⁹⁹ is the first to provide a molecular basis of the multitude of NMR and FRET observations, at least as far as coupling between the conformational changes and catalysis is concerned. On the basis of the data shown in Ref. 99, as well as the current available experimental and theoretical studies discussed above, we believe that it is unlikely that enzyme dynamics makes a significant contribution to catalysis.

As an aside, since the issue of semantics is being discussed at length in this paper, it is worth mentioning theoretical studies such as that of Ref. 27, which bring up such terminology as “the directionality of fluctuations” and purports to “show that pico- to nano-second timescale atomic fluctuations in hinge regions of adenylate kinase facilitate the large-scale, slower lid motions that produce a catalytically competent state”. The above study of adenylate kinase shows that this is in fact not the case. However, the more important issue is that whilst such statements are indeed quite elegant, it is not the beauty of the hypothesis, but rather the scientific facts backing it up which are important, otherwise such statements only serve to create even more confusion in a field that is already mired down in soft definitions and vague statements.

7.1 Vibrational Frequencies of Catalytically Important Modes

It is tempting to suggest that such special vibrational motions help in enzyme catalysis. An example of this is a recent study of human aldose reductase¹⁰¹ that uses a vibrational spectroscopy technique that measures the change in electric field at a specific site of a protein as shifts in frequency (Stark shifts) of a calibrated nitrile vibration. It is suggested that such shifts can be used to yield quantitative information on electric fields that can be directly compared with electrostatics calculations. To this end, the authors also perform molecular dynamics simulations in order to reproduce the observed changes in the field. However, at present, it would seem that only the vibrations of the probes in proteins were observed. That is, all other vibrations are implicit, as was, for instance, the case in resonance raman studies of the photoisomerization of bacteriorhodopsin (e.g. Ref. 102–104). Thus, such studies have not actually elucidated any relationship between special vibrational motions and catalysis, nor can these be coherently excited in thermal processes. Also, no relationship can be found between special vibrational motions and enzyme catalysis from simulation studies that have observed overdumped motions in proteins⁵⁰.

Now, in the case of a proton/hydride transfer reaction, the donor-acceptor distance not only influences the height and width of the reaction barrier, but also affects the tunneling coefficient¹⁰⁵. Thus, the rate of proton/hydride transfer is very sensitive to the donor-acceptor vibrational dynamics. Normal-mode analysis (NMA) provides a useful approach to study vibrational motions computationally^106,107. Go and coworkers¹⁰⁸ have used this approach to evaluate the Franck-Condon factors for electron-transfer reactions of cytochrome c. Additionally, Cui and Karplus^109,110 have used an NMA to examine the projections of some of the modes of the protein-substrate system on the reaction coordinate in triosephosphate isomerase. They found that modes that symmetrically raise the energies of the reactant and product (“promoting modes”) favor the reaction, while modes that affect the energies asymmetrically (“demoting modes”) oppose it. Cui and Karplus¹¹¹ also found that crossing the barrier takes only about 30 fs for the proton-transfer step in triosephosphate isomerase, whereas the full redistribution of vibrational energy would take much longer. Therefore, they suggested that these non-equilibrium vibrational modes could influence the transmission coefficient through a dynamical effect. They nevertheless agree that any dynamical effect is likely to be minor, since the transmission coefficient is probably at least 0.5.

The limitation of NMA is that the harmonic approximation is not always valid due to the anharmonic nature of the proteins on one hand and the reaction on the other hand. However, it has been suggested in the above-mentioned studies that NMA is an efficient way to acquire the intrinsic features of the collective motions of proteins, with qualitatively reasonable results. In addition, NMA consumes significantly less computational resources than MD simulations.

A general analysis of the coupling between the protein (or solvent) vibrations and the chemical process have been introduced by Warshel and coworkers in terms of the dispersed polaron (DP) spin boson treatment, which was first used in studies of electron transfer reactions and then in modeling enzymatic reactions^{25,63,112,113}. This approach, which is based on the Fourier transform of the energy gap, provides the projections of the protein modes along the action coordinate and also allows one to explore nuclear quantum mechanical (NQM) effects in an approximated way²⁵. This is achieved by relating the fluctuations of Δε₁₂ during an MD trajectory to the fluctuations of an equivalent harmonic system. Here, we start with the autocorrelation function of the total energy gap:

where u(t)=Δε₁₂(t) − 〈Δε₁₂〉. The power spectrum of the fluctuations in a given diabatic state, J (ω), can be obtained from the Fourier transform of the autocorrelation function:

J (ω) has peaks at the frequencies of the modes that are coupled to the reaction (ω_j), and in the high-temperature limit, the amplitudes of these peaks are proportional to the square of the displacement of the corresponding coordinate in ω₂ relative to ω₁:

where index j now runs over the normal modes of both the solute and the solvent. The Fourier magnitudes obtained from Eq. 8 can be scaled by relating the area under the spectral density function to the overall reorganization energy (λ) as in Eq. 9:

Hynes^{114, 115} had previously used a one-dimensional coordinate to describe the protein environment, and the promoting vibration was introduced to modulate the tunneling splitting. The studies on the rate-promoting motions were subsequently advanced by Schwartz and coworkers, whose analysis¹¹⁶ suggested that promoting vibrations only play a role in the proton/hydride transfer if the frequency of the promoting vibrations is much lower than that of the barrier. Schwartz and coworkers used their approach in studies of several different systems^117–119 and, in all cases, argued for the importance of rate-promoting motions in catalysis. Unfortunately, none of these studies actually demonstrates that the promoting modes account for a catalytic effect (either by showing a rate enhancement relative to the reference reaction in water, or relative to different mutants). Other fundamental problems with this interpretation will also be discussed in greater detail in Section 7.2.

7.2 Networks of Correlated Motions, and Related Issues

Hammes-Schiffer and her coworkers^79,120,121 identified a network of correlated conformational changes with projections on the reaction path in simulations using the MDQT approach, but suggested that these reflect equilibrium structural effects rather than dynamical effects. QM/MM simulations described by Garcia-Vilocoa et al.¹²² also appear to be in accord with this view. It is important to emphasize that in general, the identification of correlated motions does not provide a new view of enzyme catalysis, because reorganization of the solvent along the reaction path in solution also involves highly correlated motions^24,123. Correlated motions of an enzyme do not necessarily contribute to catalysis, and indeed could be detrimental if they increase the reorganization energy of the reaction. The EVB and dispersed-polaron approaches consider the enzyme reorganization explicitly and automatically assess the complete structural changes along the reaction coordinates. As pointed out above, a dispersed-polaron analysis can for example provide information about the projection of the protein motion on the reaction coordinate and provides a basis for a quantitative comparison with a reference reaction in solution. Additionally, the coupling of protein motions to a reaction in an enzyme involves fluctuating electrostatic interactions of the solute with charged or polar residues and bound water molecules. In solution, it involves the reorientation of the solvation shells. Clearly, the reaction coordinate in both cases will involve components along the environmental (solvent) coordinate. The real difference is the amplitude of the change in the solvent coordinates during the reaction, which determines the reorganization energy and generally is smaller in the enzyme because of pre-organization of the active site.

Some of the misunderstandings with regards to the role of coupled motions may be associated with the implications that the chemical step in enzymes is “facilitated by a network of coupled motions that bring the donor and acceptor to the correct distance and orientation and the correct electrostatic environment”¹²⁴. However, this interesting and appealing idea is still problematic, as the minimum of the free energy surface of the reaction in the folded enzyme (which determines the positions of the reacting fragments as well as the degree of electrostatic preorganization and the potential surface) also determines the nature of the reactive modes, which are in fact simply modes that have significant projections onto the reaction path. The reverse, however, is not true: that is, the reactive motions do not bring the system to the pre-organized configuration but rather start from these configurations (at least in the native enzyme). Apparently, the folded enzyme establishes the free energy surface for the catalytic reaction, and the motions on this surface are merely a reflection of the Boltzmann probability of finding the system at different points on the surface. This point is illustrated in Fig. 7, and has been discussed in Refs. ¹²⁵ and ¹²⁶.

Schwartz and coworkers¹¹⁹ have suggested that the catalytic reaction of purine nucleoside phoshorylase involves protein modes that reduce the barrier height by 20% (which is an enormous effect) by compressing the reacting fragments. To evaluate the contribution of such modes to catalysis, one must evaluate the barrier height from the minimum in the ground state, while taking into account the energy associated with the compression. Without such an analysis, one may find that a sufficiently strong compression would eliminate the barrier completely, resulting in a reduction of the barrier by 100%. Additionally, it is important to bear in mind that compression modes similar to those that occur in the protein also occur in the reference solution reaction. In the cases that we have studied, the cost of bringing the reactants to the same distance are similar in solution and in the enzyme, which means that the compression does not contribute significantly to catalysis (see the related discussion of near-attack conformations in Refs. ^127,128). Thus, the problem is that there is simply no attempt to determine the rate enhancement (or barrier reduction relative to any reference) by the enzyme, which is essential when attempting to explore catalytic effects. Furthermore, there is no attempt to reproduce the mutational effect on the barrier. In fact, there is no attempt to calculate or formulate the magnitude of the rate enhancement associated with the dynamical effect of the protein-promoting vibrations. In particular, while the study presents the difference in the power spectra of the H257G and H257A mutants, it has not provided the corresponding difference in activation free energy (Δg^≠) between theses mutants. Claiming that there is a change in the projections of different normal models upon mutation does not provide any evidence of a dynamical effect since this is simply the trivial result of the change of the potential energy surface. Here, one must define what the dynamical effect is, and then convert it to a change in rate constant. Not doing so may amount to semantic excess and potentially to a misleading message to these who confuse such studies with a proper analysis of a catalytic effect where the proposed effect is actually calculated. This might also reflect the belief in some circles that no enzymatic rate can be actually calculated, and thus simulating changes in some other property is equivalent to simulating the assumed resulting changes in the rate constant.

7.3 Flexibility and Catalysis

It has been frequently implied that flexibility helps enzyme catalysis (e.g. Refs. ^6,129–133, amongst others), and this seemingly appealing idea requires some critical examination. That is, the idea that motions are needed for catalysis could, in principle, yield the conclusion that the catalytic power of enzymes is connected to their flexibility¹³⁴. In this vein, several workers have used studies of the thermal adaptation of enzymes to argue for dynamical contributions to catalysis^6,129–131. More specifically, it has been argued that thermophilic (Tm) enzymes (that have evolved to function at highly elevated temperatures) should at most temperatures be more stable than the corresponding mesophilic (Ms) enzymes (that have evolved to function at room temperatures). It is known that Tm enzymes have lower catalytic power than Ms enzymes at the same low temperature, and, based on this, it has been argued that the low catalytic power is presumably due to a decrease in dynamical motion^129,135,136. The possible relationship between the thermal stability and catalytic power of enzymes (and the related implications that the reduced dynamics of Tm enzymes is the root cause for their reduced catalytic power) was recently examined in the specific case of DHFR by means of detailed simulations¹³⁷. This study found that while the Tm enzymes do indeed have restricted motions in the direction of the folding coordinate, this is not relevant to the chemistry of these enzymes, as the motions along the reaction coordinate are perpendicular to the folding coordinate. Additionally, this study demonstrated that the rate of the chemical reaction is not determined by dynamics or flexibility in the ground state, but rather, it is determined by the activation barrier, which is in turned determined by the corresponding reorganization energy, and the displacement along the reaction coordinate is in fact larger in the case of the Tm enzyme than in the case of the Ms, which is the opposite of the trend along the folding coordinate (and is of course contradictory to the flexibility proposal). This suggests that the general trend in enzyme catalysis is that the best catalyst requires less motion during the reaction than other less optimal catalysis, and that to obtain a small electrostatic reorganization energy, some of the folding energy has to be invested into the overall pre-organization process, leading to less stable optimal catalysts.

9.1. NQM Effects Do Occur in Enzymes, But the Same Effects Also Occur in the Corresponding Solution Reactions

Studies of isotope effects in some enzymatic reactions have pointed to nuclear tunneling and other nuclear quantum mechanical effects, such as the zero-point energy contributions of the participating vibrational modes¹⁵⁹. For example, Klinman and coworkers have provided clear evidences for nuclear tunneling in alcohol dehydrogenases^6,160,161, serum amine oxidase¹⁶², lipoxygenase¹⁶³ and glucose oxidase¹⁶⁴. These findings have frequently been interpreted as evidence for dynamical effects. That is, is has been suggested that an enzyme may, for example, exploit a particular vibrational mode that modulates the thickness of the barrier through which an atom can tunnel. The argument that NQM effects contribute to enzyme catalysis is based on the observation of large isotope effects in enzymatic reactions as well as other indications of nuclear tunneling (e.g. Refs. ^159,165). These findings are both very reasonable and very important. They are also fairly consistent with the existence of nuclear tunneling, and are reproducible by simulation studies^166,167. The problem is, however, that such contributions are not necessarily catalytic, since they must be quantified relative to the corresponding reference solution reactions. However, as we have already pointed out in several of our papers^{37,94,166,168,169}, the same NQM effects that occur in the enzyme also occur in the reference reaction in solution, and contribute in a similar way.

Similar findings have also been made in the several experimental studies in which the reference reaction was experimentally observed^36,170. We should nevertheless point out that in many cases it can be hard to measure the relevant reference reaction, and computer simulations thus offer a reasonable way to explore the reference reaction^171–175. However, such studies are only meaningful if they can reproduce the observed catalytic effects, and, fortunately, our EVB approach (which is calibrated on ab initio solution studies) provides the needed reliability. Now, EVB studies coupled with quantum classical path (QCP) calculations¹⁶⁷ of isotope effects have reproduced both the observed catalysis as well as the observed isotope effects in any systems studied by us. The same calculations also found that the isotope effects are similar in the enzymatic and solution reactions.

The most basic argument about the role of tunneling in enzyme catalysis starts with the idea that the enzyme compresses the distance between the donor and acceptor, thus leading to a narrower potential and to greater tunneling^{161,165,176–179}. This idea, that now appears to be taught in standard undergraduate biochemistry courses, is very appealing. It has also been used to rationalize the temperature dependence of observed KIEs¹⁷⁸. However, both us and others have recently found^167,180,181 that in fact even in the vibronic formulation, the isotope effect increases due to the sharp distance dependence of the zero-zero vibrational overlap (see Ref. 126 for a clear analysis). That is, in the more recent view, the NQM effects decrease rather than increase upon compression. This effect is due to the fact that the tunneling in proton and hydride transfer reactions depends on the overlap between the vibrational wave functions of the reactant and product states, and this overlap in turn depends on the distance between the corresponding minima (see Ref. 167). In fact, when the donor and acceptor are pushed to a short enough distance, the mixing between the two states makes the adiabatic surface very flat, and the tunneling effect disappears. While the above claims may sound strange to some, the simplest way to convince the reader is to point out that all key workers in this field (e.g. Refs. ^180,181) now essentially obtain this result.

The above analysis has, of course, major implications with regards to the idea that NQM effects make a significant contribution to enzyme catalysis. In fact, the effects that lead to an increase in the NQM contributions appear to be anticatalytic. This is mainly due to the fact that the rate constant is smaller for larger donor-acceptor distances. Since the fact that the observation of a large KIE reflects an increase (rather than decrease) in the distance between the donor and acceptor is a seemingly counterintuitive point, it still causes significant confusion (see e.g. Ref. 126 for further discussion of this issue).

9.2 The Temperature Dependence of Isotope Effects Does Not Provide Support to the Dynamical Idea

The temperature dependence of the NQM effects as manifested in the corresponding KIE has been a topic of significant current interest^{159,166,180,182}. Part of this interest stems from the hope that this temperature dependence can provide useful information about possible dynamical contributions to enzymatic reactions^159,183.

As far as tunneling is concerned, there is a tendency to argue that the observed temperature dependence of KIEs (e.g. ^184,185) is clear evidence for thermally activated tunneling, and thus, this can be considered support for the idea that tunneling contributes to catalysis. We have no problems with agreeing that tunneling in enzymes can be thermally activated. In fact, our simulations and formulations have long been consistent with this view^{37,94,166,168,169}. We have also recently succeeded in reproducing the observed temperature effect¹⁶⁷. However, our simulations have led to the conclusion that NQM effects do not help in catalysis, since the same thermally activated tunneling also exists in solution.

With the above findings in mind, we can ask what the actual meaning of the observed temperature effect is, and what it tells us about catalysis. Apparently, the studies of DHFR¹⁶⁷ and lipoxygenase¹⁶⁶ indicate that the KIE increases when the distance between the donor and acceptor increases (see the previous section). At any rate, the temperature dependence of the KIE appears to mainly reflect the temperature dependence of the distance between the donor and acceptor. Thus, the temperature dependence actually indicates that tunneling is anti-catalytic (see also our recent review¹²⁶).

Kohen et al.⁶ made the interesting observation that the activation enthalpy (ΔH^‡) for the reaction of the thermophilic alcohol dehydrogenase decreased from 23.6 kcal/mol at low temperatures (0–30° C) to 14.6 kcal/mol at higher temperatures (30–65° C). They interpreted this observation as supporting a contribution to k_cat from vibrationally-enhanced tunneling at higher temperatures. The activation free energy, however, remained essentially constant. In terms of TST, this means that a compensating increase in −TΔS^‡ accompanies the decrease in ΔH^‡ as the temperature is raised.

Altough Kohen et al.⁶ attributed their finding the dynamical effects, in that they believed that their findings on hydrogen transfer under physiological conditions could not “be explained without invoking both quantum mechanics and enzyme dynamics”, Warshel and coworkers have noted (e.g. Refs. ^25,61,153) that the entire effect is in fact a simple entropic effect, and the observed decrease in ΔS^≠ with temperature in the alcohol-dehydrogenase (ADH) reaction can be rationalized by considering the expected interactions of the solute with its surroundings. As the reaction in the direction considered by Kohen et al.⁶ proceeds from a polar ion-pair through to a less polar transition state to a non-polar product, the motions of the surroundings are expected to be less restricted in the transition state than in the reactant state, which contributes a positive term to ΔS^‡. Raising the temperature will release some of the motions that are frozen in the reactant state, which should make ΔS^‡ less positive.

It should be noted that Klinman, who originally supported the dynamical explanation, now agrees with the importance of entropic effects¹⁴⁸, though she attributes this to the new concept of a funnel (see Section 8). This presentation is then used to account for the TΔS^‡ temperature dependence of thermophilic ADH (which was one of the central issues of our previous analysis). This “freezing out” of protein flexibility, which is supposed to accompany a reduction in temperature for the thermophilic proteins, is argued¹⁴⁸ to be representative of a more restrictive conformational space, occurring further down the funnel, where it becomes necessary to increase protein disorder in such a way that the protein moves into the range required for optimal catalytic conditions. As discussed in Section 8, such a proposal has not yet been formulated in such a way that can be analyzed, and our attempt to define such proposal resulted in models that are not likely to be supported by any calculations or conceptual considerations. The proposal also overlooks that fact that in most enzymatic reactions, −TΔS^≠ is very close to zero (see Refs. ^25,151). Furthermore, this has little to do with catalysis, since ΔG^≠ is practically constant and temperature independent in the cases being considered. Finally, even if in some unclear way it is be found by means of calculations that the entropic effect in ADH is due to some changes occurring far from the active site for case A in Figure 10 (which is far less likely than the simple local electrostatic idea that was proposed by us²⁵), it will have no dynamical implications since it is a simple issue of the configurational space. We must clarify here that Case A in Fig. 10 has probably nothing to do with the entropy funnel concept.

9.3 “High Percentage Tunneling” and Other Considerations

At this point we would like to comment on the fact that there have been some recent studies which have classified reactions with a relatively small tunneling contribution^186–188 (i.e. a reduction of 4% in the barriers) as proceeding predominantly by tunneling. This view is based on the fact that the NQM contribution to the rate constant is more than 50% (in one case, it is even claimed that the reaction only proceeds 1% by the classical way, i.e. 99% by tunneling¹⁸⁷). This classification seems to reflect a major misunderstanding, and is actually misleading. This point is discussed clearly in Ref. 126, but here we will clarify that the reaction barrier changes only from e.g. 15 to 14 kcal/mol¹⁸⁶ when we add tunneling and zero point energies, were the 1 kcal/mol is trivial as compared to the actual classical effect of 15 kcal/mol. Perhaps the best way to see the falacy of the definition above is to realize that in order to distinguish between a classical mechanism with proceeds over the barrier to a tunneling mechanism which proceeds through the barrier, one must be able to quantify what fraction of the molecules are reacting via tunneling, and how many are reacting classically. We do not believe that there is any logical way to relate a trivial reduction in a rate constant by a factor of ~20 to a change in the reaction from 100% of the molecules reacting classically to a situation where 95% are reacting through tunneling (i.e. the only point that was established in the works mentioned above is that the rate changes, not that the fraction of molecules tunneling through the barrier changes by that percentage). Now, if there is no possible way to define such a relationship, then such a relationship most likely does not exist.

In summary, a careful analysis of the available experimental studies concluded that there exists no single piece of consistent experimental evidence that demonstrates that tunneling makes a significant contribution to catalysis, whereas all computational studies that actually explore this issue have consistently have found that the catalytic effect of NQM is very small (if there is any at all). Once again, we refer the reader to our recent review¹²⁶ for further discussion.

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