Simulating Enzyme Reactivity: Computational Methods in Enzyme Catalysis
- 1.1 Introduction
- 1.2 Defining and Calculating the Catalytic Effect
- 1.2.1 Using a Logical and Useful Definition
- 1.2.2 Evaluating Reliable Activation Free Energies by Computational Approaches
- 1.2.3 Electrostatic Transition State Stabilisation (TSS)
- 1.3 What was Found by Reliable Computational Studies?
- 1.3.1 General Findings
- 1.3.2 Quantifying the Source of Electrostatic Contributions to Catalysis
- 1.4 What are the Problems with Other Proposals?
- 1.4.1 Ground-state Destabilisation by Steric Strain Does Not Provide a Large Catalytic Effect
- 1.4.2 Dynamical Effects Do Not Contribute Significantly to Enzyme Catalysis
- 1.4.3 Correlated Modes Clearly Exist in Proteins, but They Also Exist in Solution
- 1.4.4 Problems with the Generalised Compression Idea
- 1.4.5 RSD by Desolvation Effects Does Not Provide Large Catalytic Effects
- 1.4.6 Entropy Contributions of Bringing the Reactants Together are Unlikely to Account for Large Catalytic Effects
- 1.4.7 Allosteric Control of Catalytic Activity is Also Associated with Electrostatic Effects
- 1.5 Conclusions and Perspectives
Chapter 1: Perspective on Computer Modelling of Enzymatic Reactions
Published:16 Nov 2016
Arieh Warshel, Ram Prasad Bora, 2016. "Perspective on Computer Modelling of Enzymatic Reactions", Simulating Enzyme Reactivity: Computational Methods in Enzyme Catalysis, Inaki Tunon, Vicent Moliner
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Enzymes perform key life professes at time ranges needed for synchronisation between the relevant reaction cycles. This sometimes requires enormous rate acceleration, whose origin presents a major puzzle both on a fundamental and practical level. Several proposals have been put forward to try to rationalise the catalytic power of enzymes but, in our view, resolving the merit of the different options cannot be accomplished without the use of computer simulations. To provide the reader with a perspective of the crucial direction of modelling of enzymatic reactions we consider the advances in the field both in the technical and fundamental directions. In doing so, we describe the development of the QM/MM approaches and the need for careful sampling. We then discuss what careful simulations of enzymatic reactions found; namely that the electrostatic polar preorganisation is the key catalytic factor, and then demonstrate the use of simulations in excluding or finding the problems in problematic catalytic proposals.
Understanding how enzymes work has both fundamental and practical importance, as these remarkable molecules play a key role in controlling and performing most life processes.1,2 In many respects the most crucial issue is understanding the origin of the enormous catalytic power of enzymes. Although some aspects of this puzzle were elucidated by biochemical and structural studies, the source of the catalytic power of enzymes is still controversial (e.g. see references in Warshel,1 Fersht2 and Warshel et al.3 ). The current discussion is sometimes reduced to statements such as, “the enzyme binds the transition state more strongly than the ground state”, without providing any clear idea as to how such extra stabilisation can be provided.
The search for the origin of enzyme catalysis is frequently guided poorly by overlooking the crucial need to select a proper reference state. This issue has been carefully addressed in ref. 3, which defines a reference stat where we have the reaction in a water cage as in the enzyme active site. This unique ‘chemistry-filtered’ reference selection3 allows one to focus on the key issues in enzyme catalysis in a well-defined way, asking what is the real difference in the environment (and its interaction with the reacting substrate) that makes such an enormous difference.
One of the points that we have tried to emphasise is our view that enzyme catalysis is too complex to be resolved experimentally without the use of computer-based models.4 With this view in mind we can move to the pioneering works in this field and the introduction of the quantum mechanical/molecular mechanical (QM/MM) method5 in 1976. We will also discuss some of the advances since the inception of the field, emphasising efficient sampling and outlining related problems.
Our perspective will also address different catalytic proposals, focusing on the validation of the electrostatic preorganisation idea and on exploring less consistent ideas. We will also comment on the problems with the relatively slow current advances in enzyme design.
1.2 Defining and Calculating the Catalytic Effect
1.2.1 Using a Logical and Useful Definition
In analysing the origin of the catalytic effect it is crucial to ask catalysis relative to what? Here, as hinted at in the previous section, we must select a reasonable reference state, and this can be done in several ways. For example, we may start with the scale introduced by Wolfenden and coworkers.6 This scale has established the catalytic power associated with the binding free energy of the transition state (TS) in the enzyme relative to the energy of reaching the TS in the uncatalysed reaction in water, but still left significant misunderstandings about the challenge in rationalising enzyme catalysis. One problem is associated with the fact that the mechanism in the enzyme and in solution can be different (see above) and this difference is a part of the Wolfenden scale. Another serious problem arises from the fact that the real challenge in rationalising enzyme catalysis has not been emphasised by the Wolfenden scale, because it also reflects the binding free energy of the substrate (whose nature is well understood), whereas the real problem is associated with rationalising the large change in free energy upon going from the ES (enzyme–substrate) to the ES‡ (TS corresponding to the ES) states (i.e. the free energy associated with the kcat and the corresponding Δg of Figure 1.1).
More specifically, it is useful to start with the Figure 1.1(A), where the activation barrier, Δg, corresponds to kcat/KM (more precisely, kcat/KD), and Δg corresponds to kcat (or more precisely, to the enzyme rate constant for the chemical step). The energetics of the reaction in the enzyme can now be compared to the corresponding energetics of the reaction in solution (Figure 1.1(B)). In this respect, it is useful to consider the free energy profile or the potential of mean force (PMF) for the reaction in water and in the protein, and then to divide the second-order process into the free energy of bringing the fragments to the same cage and the activation barrier of the first-order reactive event. The comparison of the enzyme and solution reactions can be done by either comparing Δg to Δg, or by comparing Δg to Δg (see ref. 3).
As pointed out above, the comparison of Δg and Δg should reflect the fact that many enzymatic reactions involve mechanisms different from the corresponding solution reactions (see, for example, ref. 3). Fortunately, this effect is well understood and can be determined by using a proper thermodynamic cycle and in recent years by reliable quantum mechanical calculations.7,8
1.2.2 Evaluating Reliable Activation Free Energies by Computational Approaches
Attempts to explore the origin of the catalytic power of enzymes should be based on quantitative methods for calculating the rate constant for reactions starting from the structure of the given enzymes. Obviously the key point in such calculations is the validity of the calculated activation free energies.9 The general QM/MM strategy provides a generic way of obtaining potential surfaces and, in principle, activation free energies of chemical processes in enzymes. This approach5 has gained popularity in recent years and has been used in a variety of forms (for reviews see Kamerlin et al.10 ). However, implementation of rigorous, ab initio QM/MM approaches in quantitative calculations of activation free energies is still extremely challenging. Nevertheless, significant progress has started to be made in recent works.7,8,11–17 Furthermore, semi-empirical QM/MM studies with reasonable PMF calculations, and in some cases even with least energy paths, can be used to assess the validity of some catalytic proposals.18–20
Despite the future promise of well sampled ab initio QM/MM evaluations of activation barriers, we prefer to focus here on the empirical valence bond (EVB) method,1,21,22 because even at this stage it provides what is probably the most effective available way for quantifying the catalytic effect and determining its origin. The EVB method is a QM/MM approach, which describes the system with two or more resonance states (or more precisely, diabatic states) corresponding to classical valence-bond structures (which are basically described as empirical force fields). These diabatic states are allowed to interact with the surroundings through their electrostatic charge distribution and then mixed in an effective Hamiltonian using the same mixing terms (off-diagonal terms) in solution, in the gas phase and in solution (an assumption that has gained major support from our constrained density functional theory calculations (CDFT)).23
The free energy surfaces are described as a function of the diabatic energy gap that is taken as the generalised free energy surface where the overall free energy barrier is obtained by a specialised free energy perturbation umbrella sampling approach (FEP/US).24 This approach allows us to sample the EVB energy surface in an effective way by molecular dynamics (MD) simulations. Now, because trajectories on the reactant surface will reach the TS only rarely, it is usually necessary to run trajectories on a series of potential surfaces (‘mapping’ potentials) that drive the system adiabatically from the reactant to the product state.24 In the simple case of two diabatic states the mapping potential (εm) can be written as a linear combination of the reactant and product potentials and the FEP/US provides a way to obtain the ‘PMF’ (or the free energy functional) along the energy gap coordinate.
The FEP/US approach can also be used to obtain the free energy functional of the individual diabatic states for the reactant and product states (the free energy functional) represent microscopic equivalents of the Marcus parabolas in electron-transfer theory.25 The intersection of this free energy functional provides a quantitative estimate of the reorganisation energy, which will play a key role in our considerations.
The powerful physical picture of intersecting electronic states provided by the EVB treatment is particularly useful for exploring environmental effects on chemical reactions in condensed phases.26 The ground-state charge distribution of the reacting species (solute) polarises the surroundings (solvent), and the charges of each resonance structure of the solute then interact with the polarised solvent.1 For example, if ionic and covalent states are used to describe the solute, preferential stabilisation of the ionic state by the solvent will give the adiabatic ground state more ionic character. This allows us, for example, to obtain a very well-defined separation of covalent (charge transfer) and electrostatic effects and thus, to analyse in a clear way some covalent hypotheses. In addition, the EVB method lends itself to proper configurational sampling and converging free energy calculations, which makes it possible to evaluate non-equilibrium solvation effects.9
The EVB and other QM/MM methods allow us to simulate chemical reactions in enzyme active sites and solution and to reproduce the corresponding changes in activation barriers. The results of the corresponding calculations of the total EVB catalytic effects in some enzymes are given in Figure 1.2.
The results shown are taken from ref. 3, and recent EVB studies gave similarly good agreement (for example, Kamerlin and Warshel,22 as well as very challenging cases that involve evaluation of activation entropy (e.g. Isaksen et al.27 )).
1.2.3 Electrostatic Transition State Stabilisation (TSS)
Our calculations have indicated that in all cases studied the catalytic effect is due to electrostatic effects (see Introduction and ref. 3). With current insight it might be argued that electrostatic effects must have been the most obvious candidates for explaining enzyme catalysis. However, careful studies in the early stages of the field have basically excluded this possibility. That is, early experiments with model compounds in solution (e.g. ref. 28 and 29) that explored the role of electrostatic effects (by introducing charged groups to stabilise the TS charge distribution) concluded that such effects must be small (e.g. see ref. 28 and 29). Similarly, phenomenological attempts to estimate the magnitude of electrostatic contributions to catalysis30 also indicated that such effects are small. Thus it was assumed that more or less uniformly (at least in studies that attempted to quantify the catalytic effect) electrostatic effects do not play a very important role. The problem has, however, been that the physical organic chemistry experiments in solution might be rather irrelevant to an enzyme active site. Overall, phenomenological attempts to estimate the electrostatic effects in enzymes have been very problematic because it is almost impossible to assess the dielectric effects in the protein without a proper computational model. In this respect, it is also important to clarify that the view expressed by the pioneering work of Jencks31 did not consider electrostatic stabilisation of the TS as a major catalytic effect.
1.3 What was Found by Reliable Computational Studies?
1.3.1 General Findings
Before we consider the growing theoretical support for the role of electrostatic stabilisation in catalysis, it is important to comment about the insight that emerged from mutation experiments starting from around 1984 (e.g. ref. 2, 32–37). These mutation experiments have provided major insights, and in many cases pointed towards the importance of electrostatic effects. More recent works have added extensive support to this view (e.g. ref. 38, 39). However, since the catalytic effect reflects the overall effect of the enzyme active site, it has been very hard to reach unique experimental conclusions about the overall electrostatic effect. Furthermore, even in the seemingly unique case in which a mutation of an ionised group to a non-polar group leads to a large reduction in kcat, it has been very hard to determine experimentally whether this is an electrostatic effect or some other factor (an excellent example is the D102N mutation of trypsin,40 discussed in ref. 41).
With the above observation in mind, it seems to us that the use of QM/MM and related approaches provides what is perhaps the best way to convert the structures of enzyme active sites to catalytic contributions. In fact, since 1976, there has been a growing number of molecular orbital–QM/MM (MO–QM/MM) and EVB calculations that identify electrostatic effects as the key factor in enzyme catalysis. This trend has moved the field gradually from a stage of qualitative statements (e.g. see ref. 42) to more quantitative conclusions. Here it is useful to consider the studies summarised in ref. 3. This work only considered studies that were done until the works mentioned in ref. 3, where the actual catalytic effect was reproduced, rather than general QM/MM studies. Full quantitative evaluation of the electrostatic effects has been provided at present mainly by EVB studies, because this requires not only calculations of the activation free energy in enzyme and solution but also evaluation of the electrostatic contribution to the binding free energy of the reactant state (RS) and the TS. However, MO–QM/MM approaches have studied the electrostatic interaction energies (e.g. ref. 19, 43) and even reported systematic progress in evaluating change in electrostatic free energy along the reaction coordinate (e.g. ref. 44). Furthermore, many of the EVB studies evaluated the solvent reorganisation energy and demonstrated that this contribution accounts for a major part of the catalytic effect (see below). There were also some more recent attempts to estimate the reorganisation energies by MO–QM/MM approaches (e.g. ref. 45), but these only considered the change in the environment MM energy rather than systematic calculations using eqn (1.1) (see below) or related linear response approximation (LRA) treatments. Unfortunately, the change in the very large total molecular mechanics (MM) energy during the reaction is a rather unstable quantity, which is hard to evaluate in a quantitative way.
In many cases MO–QM/MM calculations can provide clear indications that the electrostatic effects play a major role in catalysis by simply evaluating the contributions of different residues to the activation barrier. Unfortunately, this type of ‘mutational’ analysis is frequently very qualitative because the simulations do not provide a sufficient dielectric screening. The underestimation of the screening effect is quite problematic, when one deals with ionised protein residues (see discussion in Schutz and Warshel46 ). Moreover, a proper analysis of the catalytic effect should explore the overall electrostatic contribution of the active site rather than just the contribution of some residues. In any case, we also list in ref. 3 studies that explore the electrostatic effects of different residues. In a few cases (e.g. ref. 47) we already have QM/MM studies that gradually excluded the electrostatic effect of the enzyme environment and thus established the importance of the overall electrostatic effect.
Some QM/MM and other related studies (see below) have not supported the idea of electrostatic TSS. However, at present (see also Section 1.4), all studies that did not support the electrostatic idea have involved significant inconsistencies. Key examples are: (a) Works that attributed the catalysis to desolvation or the ground-state electrostatic destabilisation. These works did not consider the actual binding of the TS and RS (e.g. ref. 48, 49) or could not reproduce the actual catalytic effect by the binding calculations (e.g. ref. 20, 49, 50). (b) Works that could not reproduce the catalytic effect without the use of entirely inconsistent entropic cycles and calculations that involved major overestimates based on gas-phase vibrational analysis (e.g. ref. 51, also see discussion in ref. 52). (c) Works that promoted the near attack conformations (NAC) proposal of Bruice and coworkers53 and also were supported by other groups in related forms (e.g. ref. 54, 55) might have seemed reasonable. However, a deeper physically-based analysis has proved that the NAC reflects an incorrect analysis of what is actually an electrostatic TSS effect (this point requires one to follow the discussion below and in ref. 56). (d) Finally, it is important to comment here on the idea that enzyme catalysis is due to RS destabilisation and to the decrease of the enzyme self-energy upon moving to the TS. As will be shown below this idea is also based on inconsistent considerations.
To summarise this section, it seems to us that careful considerations of the works mentioned in ref. 3, as well as consistent attempts to identify the origin of large catalytic effects, point towards the conclusion that electrostatic effects are the key factors in enzyme catalysis (this issue will be emphasised and quantified further in the following section).
1.3.2 Quantifying the Source of Electrostatic Contributions to Catalysis
The studies reported above provide a general support to the electrostatic proposal. A more quantitative analysis is provided in Table 3 of ref. 3. As seen from the table we have clear examples of specific cases where most of the catalytic effect is due to electrostatic interactions. What remains to be established is that these effects are associated with TSS and to examine the reasons as to why the protein is able to provide such large effects. These issues can be explored by using the LRA expression (Lee et al.57 ):
where U is the solute–solvent interaction potential, Q designates the residual charges of the solute atoms where Q‡ indicates the TS charges and 〈ΔU〉Q designates an average over configurations obtained from a MD run with the given solute charge distribution. The quantity λ in eqn (1.2) is the reorganisation energy. The first term in eqn (1.1) is the above-mentioned interaction energy at the TS, where Q=Q‡, which is similar in the enzyme and in solution. The second term expresses the effect of the environment preorganisation. If the environment is randomly oriented towards the TS in the absence of charge (as is the case in water), then the second term is zero and we obtain:
where the electrostatic free energy is half of the average electrostatic potential.58 However, in the preorganised environment of an enzyme, we obtain a significant contribution from the second term and the overall ΔQ (Q‡) is more negative than in water. This extra stabilisation leads to the catalytic effect of the enzyme. Another way to see this effect is to realise that in water, where the solvent dipoles are randomly oriented around the uncharged form of the TS, the activation free energy includes the free energy needed to reorganise the solvent dipoles towards the charged TS. On the other hand, the reaction in the protein costs less reorganisation energy because the active site dipoles (associated with polar groups, charged groups and water molecules) are already partially preorganised towards the TS charge.59 The reorganisation energy is related to the well-known Marcus’ reorganisation energy, but it is not equal to it. More specifically, the Marcus’ reorganisation energy60 is related to the transfer from the reactant to the product state, while here we deal with charging the TS. In other words, the suggestion (e.g. ref. 61, 62) that the reduction in the protein reorganisation energy will result in catalysis according to the Marcus relationship is problematic, because it implies that the reduction in the reorganisation energy is due to the existence of a non-polar active site. Unfortunately, protein active sites are polar (instead of being non-polar) and having a non-polar active site would drastically destabilise rather than stabilise ionic TSs (see discussion of desolvation models in Warshel et al.63 and references given in this paper). In fact, the source of enzyme catalysis is the preorganisation of a very polar environment (see ref. 9 for a detailed discussion and a demonstration in ref. 3).
Regardless of the above clarification, it is almost always true that the catalytic effect is associated with the reduction of the Marcus’ reorganisation energy so that λp≤λw. This point and the related role of preorganisation of the electrostatic environment is demonstrated schematically in Figure 1.3 and ref. 3, and quantified for the case of DhlA64,65 in Figure 10 of ref. 3.
An LRA analysis based on eqn (1.2) is given in Table 3 of ref. 3 for haloalkane dehalogenase (DhlA) and for chorismate mutase (CM) in Table 4 of ref. 3. As seen from these cases and other related studies, the catalytic effect appears to be associated mainly with the electrostatic stabilisation of its TS, and a large part of the effect is associated with the preorganisation contribution. Interestingly, even in the case of peptide bond formation by the ribosome (which constitutes a very early stage in the evolution of biocatalysts) it has been found that the preorganisation effect provides the major catalytic effect.66,67
The correlation between the reorganisation energy and the catalytic effect has been explored (ref. 68) in a study of the effect of mutations in dihydrofolate reductase (DHFR). This is a very interesting benchmark, because the effects of mutations were used as evidence of the catalytic effect of correlated motions, but our calculations68 established that the actual trend reflects a good correlation between the reorganisation energy and the catalytic effect (see Figure 1.4).
Another important issue that may be mentioned at this point is the Jencks’ idea that enzymes use their binding energies to destabilise the substrate and to bring about the positioning of the reacting groups. Actually, a large part of the preorganisation effect is due to the inherent folding energy and not due to the interaction with the substrate. Furthermore, the preorganisation effect results in TSS rather than ground-state destabilisation.
As stated above the catalytic power of enzymes is largely due to the preorganised electrostatic environment of their active sites. Our considerations of the overall energetics of this effect led to the idea that the preorganisation is associated with reduction in the protein folding energy.1,69 This stability/activity idea was also supported by experimental works70,71 and electrostatic modelling.72
Recent works have tried to argue that enzyme structure can tell us about changes in the protein preorganisation upon mutation73 and that with such a concept one can show that the changes in the rates in mutants of DHFR are due to reduced dynamics and not to increases in reorganisation energy. Fortunately, due to our ability to actually calculate reorganisation energy and the activation barrier it was possible to show74 that the rate constant change reflects the change in reorganisation energy. The same conclusions were obtained subsequently by other research groups.75–77
In exploring the robustness of our conclusions it is useful to try to evaluate the observed mutational effect. In this respect we consider our evaluation of mutational effects in CM (Figure 6 of ref. 78) or the results of directed evolution in the design of Kemp elimenase,79 as well as the results of distanced mutations in DHFR,80 as powerful elements in the validation of our computational methods. All the above analysis requires a very significant benchmark that appears to be provided by the EVB.
1.4 What are the Problems with Other Proposals?
Although we have introduced compelling evidence for the overwhelming importance of electrostatic contributions, it is important to consider other proposals. This issue has been discussed extensively elsewhere (e.g. ref. 1, 11, 52, 81, 82), but it seems appropriate to summarise the results of computer modelling of the main alternative proposals.
1.4.1 Ground-state Destabilisation by Steric Strain Does Not Provide a Large Catalytic Effect
The idea that enzyme catalysis is associated with ground-state destabilisation was put forward in the classical studies of lysozyme.83 Later studies that examined the actual amount of energy associated with steric strain found it to be small, due to the inherent flexibility of proteins.1,84,85 Nevertheless, the strain proposal has been invoked in several recent studies, which were shown to be inconsistent with energy considerations (see ref. 3); this will be discussed below.86,87
One instructive example provided by spectroscopic studies was interpreted as a ground-state destabilisation due to electrostatic effects (electrostatic strain-induced mechanism). In particular, this idea was further elaborated by Anderson.88 Unfortunately, the logic of ref. 88 was shown to be very problematic, as shown in ref. 3. At any rate, the main effect of the field from the reorganised active site is to stabilise the TS and not to destabilise the RS, and this fact has been established in many detailed computational studies that actually examined this issue (e.g. see Table 5 in ref. 3).
One of the systems that challenges the idea that strain is not important to catalysis is the action of B12-containing enzyme catalysis; thus it may be useful to consider the Co–C bond cleavage in coenzyme B12 enzymes. This system involves a radical bond breaking process and yet displays a very large catalytic effect of about 12 orders of magnitude.89,90 This catalytic effect has been attributed to reactant state destabilisation (RSD) and, in particular, to the distortion of the corrin ring or other strain effects.49,89–94 In particular, it was suggested that the strain is operated by the so-called mechanochemical trigger mechanism associated with the upward folding of the corrin ring (e.g. ref. 95–97). However, recent theoretical studies show that such compression cannot destabilise the Co–C bond (e.g. ref. 98, 99). A recent QM/MM study93 provides an impressive analysis of the system and reproduces the catalytic effects. The decomposition of the catalytic effect resulted in about 8 kcal mol−1 electrostatic effect (between the protein and the leaving group) and about 15 kcal mol−1 strain in the leaving group. However, decomposition to energy contributions in QM/MM calculations, which do not involve free energy calculations and sufficient sampling and relaxation (e.g. see ref. 100), is extremely challenging and can lead to unstable results. A more recent study101 that used the EVB and a very extensive free energy umbrella sampling calculation found, in agreement with ref. 93, that the catalysis is due to the interaction with the leaving group, but concluded that this effect is almost entirely an electrostatic effect (the catalysis disappears with a hypothetical, fully non-polar leaving group). The study of ref. 101 also used the LRA approach and established that the enzyme does not use RSD and stabilise the substrate more strongly than water does. The enzyme stabilisation of the leaving group increases, however, when the Co–C bond is stretched towards the TS. Interestingly in a more recent work,102 we succeeded in accounting for a very large electrostatic stabilisation due to entropic effects.
1.4.2 Dynamical Effects Do Not Contribute Significantly to Enzyme Catalysis
The proposal that special ‘dynamical’ effects play a major role in enzyme catalysis (e.g. ref. 103, 104) has become popular in recent years (e.g. ref. 105–113). Unfortunately, the only way to explore this proposal is to start with a clear definition of the dynamical effects and then to examine carefully whether the corresponding contributions are different in enzymes and in solution. Although this issue has been analysed in great detail in several recent reviews,9,52,114 we will consider here some key points as well as some recent works that supported the dynamical proposal.
There are several ways to define dynamical effects, and these will be considered below. However, in order to establish a dynamical contribution to catalysis by a given definition, we must obtain (with the specific definition) different magnitudes of dynamical contributions to the rate constant in the enzyme and in water. Now, in considering different definitions, we may start with the transmission factor, because it is agreed in the chemical physics community (see references in Villa and Warshel9 ) that all the dynamical effects are contained in this factor, which corrects the absolute rate theory for recrossing of the reactive trajectories (see ref. 52 for a clear definition). To the best of our knowledge, all the reported simulation studies going back to the earliest analysis115 and to subsequent studies (e.g. ref. 69) found that the transmission factors are similar in the enzyme and in solution and do not differ much more than unity in the enzyme (e.g. ref. 9, 110). Typical values of the transmission factors are 0.8 and 0.6 in enzyme and solution, respectively.116 These values are too similar to each other to be considered a source for any catalytic effect.
One of the best definitions of dynamical effects is to distinguish between inertial models (where coherent models funnel the binding motion to motion over the TS) and motions where the TS theory (with a minor tunnelling correction) is fully valid. The problems with the inertial model were established by the first realistic simulation117 of the long-term behaviour of a reacting enzyme. This simulation has shown that once the barrier for the chemical step is sufficiently high, the system loses all its memory about the movement to the bottom of the reaction barrier. Unfortunately, after the clear demonstration that properly defined dynamical effects do not contribute to catalysis, it was argued by the supporters of the dynamical idea that no dynamical proposal was ever put forward by the experimental community and that the theoreticians are those who reject the ‘active dynamics’ proposal. Thus we wish to clarify that in clear contrast to the implications of ref. 118 and 119 the issues are not semantic at all. That is, a recent review119 argued that the experimental community talks about stochastic dynamical models (which it called passive dynamics), because this is what Marcus-like models imply. Unfortunately, as we illustrated with explicit examples in our previous review,120 it is very clear that the majority of these works explicitly refer to a dynamical contribution to catalysis, and stochastic approaches are not dynamical and thus are not part of a correctly formulated dynamical proposal. This problem extends to the extremely puzzling implications that enzymologists knew this and meant that dynamics is related to statistical models. It is obviously hard to prove what someone knew without providing references, but to attribute knowledge of effect, which is not discussed in the literature, is not an accepted way of establishing knowledge and understanding. Additionally, as far as Marcus-like models are concerned (which we introduced consistently to solution and enzymes in the correct adiabatic limit in 1990121 and 1991122 ), attempts to use such models in analysing tunnelling, in for instance all the arguments and discussions in early and ongoing works,123–126 seem to clearly be an attempt to promote the problematic real dynamical (gating) proposal, as can be established by the reader, and not an attempt to support the established relationship between the activation free energy and the electrostatic preorganisation.3,127 Finally, we would also like to clarify that the nature of the catalytic landscape presents a thermodynamic (including entropic) factor rather than a dynamical effect.
Fortunately, the key issues in elucidating the origin of enzyme catalysis and catalytic contributions have already been proposed, and thus what was meant by the proposal can be frequently established, without being sidetracked by the names that either have been or are now being used to describe such proposals. Of course, to eliminate confusion, it is crucial to have clear logical and scientific definitions of what is meant by a dynamical proposal. Here, we will try to examine the validity of the dynamical proposal in cases where it is fully clear that what is meant is a dynamical contribution to the chemical step of catalysis. These proposals will be examined and analysed within the framework of clear, physically-based definitions of dynamical contributions to catalysis, in order to demonstrate that within rigorous frameworks, there is no need to invoke dynamical contributions to rationalise the observed effects, and of course, for example, conformational sampling gives entropic contributions, but these again have nothing to do with dynamics but rather with the available configurational space. Thus, having presented our arguments, we will leave it to the readers to decide whether, within rigorously defined frameworks, it is still possible to argue that dynamical contributions play a major role in enzyme catalysis. In addition we would like to clarify that in contrast to the implication of ref. 118 and 119 it is not true that the theoretical community objected to the active dynamics proposal. In fact, it was only Warshel who has evaluated the dynamical effects in the early 80s128 and showed that they do not contribute to catalysis, and only recent changes in the common trend led to new identification of the problems associated with the dynamical proposal (see ref. 129–131).
Another definition can imply that dynamical effects are related to the availability of special coherent motions. In this way, the dynamical proposal implies that enzymes ‘activate’ a special type of coherent motions, which are not available in the solution reaction. However, if the difference between the reaction in enzyme and in solution can be accounted for by evaluating the corresponding Δg‡ using non-dynamical Monte Carlo (MC) methods we do not have dynamical contributions. In other words, if the results from MC and MD are identical, then we do not have dynamical contributions to catalysis. Now, careful and systematic studies (e.g. ref. 9, 132) have shown that the reactions in both enzymes and solutions involved large electrostatic fluctuations. However, these fluctuations follow the Boltzmann distribution and, thus, do not provide dynamical contributions to catalysis.
It has been suggested (e.g. ref. 110) that dynamical effects are associated with the so-called non-equilibrium solvation effects. However, the corresponding analysis has been shown to be very problematic (see ref. 9, 52). Furthermore, it has been clearly demonstrated that the difference between the non-equilibrium solvation effects in the enzyme and that in solution is an integral part of the difference between the corresponding activation free energies.
Apparently, there is no single experimental finding that can be used to consistently support the dynamical hypothesis. Most of the experiments that were used to support this proposal have not compared the catalysed and uncatalysed reaction, and thus have not addressed the issue of catalysis (see discussion in ref. 9). Instructive NMR experiments (e.g. ref. 111) demonstrated the involvement of different motions in enzymatic reactions (see also below). The obvious existence of motions that have components along the reaction coordinate does not constitute a dynamical effect unless these motions are shown to be coherent. Probably all the motional effects identified so far are related to entropic factors (i.e. to change in the available configurational space) rather than to real dynamical effects.
Despite our previous reviews of the dynamical proposal (e.g. ref. 114) we find it useful to consider the most recent work that implied or explicitly supported this idea. We start by recognising that the advance in NMR studies (e.g. ref. 111, 133) allows one to probe the interesting nature of the relatively slow protein motions. This, however, does not prove that proteins can “harness thermal motions through specific dynamic networks to enable molecular function” as suggested by ref. 133.
An instructive example of what we see as an over-interpretation of exciting experimental findings is a follow-up134 to the study of ref. 111. That is, study of the action of cyclophilin111 found that the protein motions are correlated with the substrate turnover, while the more recent study of ref. 134 found that the same motions still exist in the absence of the substrate. This led to the interesting proposal that both protein structure and dynamics have co-evolved synergistically and that dynamical pre-sampling is “harvested for catalytic turnover”. Unfortunately, the authors have not addressed the facts that catalysis must be defined relative to a reference reaction in solution, and that the catalytic effect of virtually every enzyme that has been studied consistently has been found to be associated with electrostatic rather than dynamical effects (this is true also in the present case, e.g. ref. 135).
A theoretical work136 that was considered as a support of the finding of ref. 134, has attempted to evaluate the dynamical contribution from the protein's vibrations to the transmission factor of the erection of cyclophilin, and concluded that the dynamical contribution is significant. This study propagated trajectories from the TS, placing different amount of kinetic energy in the protein normal modes. Unfortunately this work had several problems. First, adding arbitrarily non-Boltzmann energy to specific modes at the TS, or any other state, has no relationship to correct rate theories. One has to prove that these vibrations are populated in a non-Boltzmann way and then to use the correct density matrix or an alternative treatment to examine whether there is any validity to such an assumption. In other words, adding arbitrary kinetic energy in the direction of the product will certainly change the recrossing in any model and, thus, cannot serve as a way of examining the contributions of the protein mode; this challenging problem can perhaps be addressed by starting an assumed coherent mode from the ground state and examining whether it retained coherence in the long time that it takes to reach the TS. Second, the same approach, whether justified or not, should have been performed on the reference solution reaction. Such a study would almost certainly reproduce similar effects in solution and thus correspond to little or no catalytic effect.
Another recent theoretical attempt to support the dynamical proposal137 used transition path sampling to explore the catalytic reaction of lactate dehydrogenase (LDH). It was concluded that some trajectories in the TS region move in a concerted way and some in a stepwise path, and this was used to imply that the enzyme dynamics help to catalyse the reaction. However, this study also involved several problems. First, no attempt was made to evaluate the activation free energy and no comparison was made to the uncatalysed reaction, in contrast to earlier studies that actually elucidated the role of the reduction in reorganisation energy in the same enzyme.138 Second, the fact that the reaction path may involve both concerted and stepwise paths has little to do with dynamical effects. It simply reflects the shape of the calculated reaction landscape.
In summary of the discussion, it is useful to recognise that consistent simulation studies found no evidence for dynamical contributions to catalysis. Another related issue is associated with the suggestion that vibrationally enhanced tunnelling (VET) plays a major role in enzyme catalysis (see, for example, ref. 112, 139). Some workers (e.g. ref. 113) assumed that there exists here an entirely new phenomenon that makes transition state theory (TST) inapplicable to enzymatic reactions. However, the VET effect is not new and is common to many chemical reactions in solution.140–142 Moreover, the VET is strongly related to TST. That is, when the solvent fluctuates and changes the energy gap (see ref. 132, 140), the light atom sees a fluctuating barrier that allows in some cases for a greater rate of tunnelling. As shown in ref. 132, these fluctuations are taken into account in the statistical factor of the classical TST and the same is true when quantum effects are taken into account. Thus, the recent finding that the solvent coordinates should be considered in tunnelling studies is not new and does not mean that this effect is important in catalysis.
Hwang et al.143 were the first to calculate the contribution of tunnelling and other nuclear quantum effects to enzyme catalysis. Since then and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical–nuclear effects in enzyme reactions. The approaches used range from the quantised classical path (QCP) (e.g. ref. 9, 144, 145), the centroid path integral approach,146,147 vibrational TST,148 and the molecular dynamic with quantum transition (MDQT) surface hopping method.149 Most studies have not yet examined the reference water reaction and thus, could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g. ref. 9, 114, 144, 145) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis (see also ref. 129, 130).
1.4.3 Correlated Modes Clearly Exist in Proteins, but They Also Exist in Solution
Early studies108,150,151 have explored the reaction of dihydrofolate reductase by NMR. They found that site-directed mutations of the 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 MD simulations107,152 that did not examine, however, any of the TSs in the reaction or demonstrate any dynamical effects on the rate constant.
More recent studies (e.g. ref. 153–155) have led to growing recognition that the mutational effects in DHFR reflect equilibrium structural effects rather than dynamical effects. However, the focus shifted to discussion of correlated motions (e.g. ref. 156, 157) rather than on reorganisation effects identified in ref. 80 and shown in Figure 1.4. This seems to create an impression that here we have a special catalytic effect with new implications beyond the concept of electrostatic TSS. However, the identification of correlated motions does not provide a new view of enzyme catalysis, because the reorganisation of the solvent along the reaction path in solution also involves highly correlated motions.132,143 Correlated motions of an enzyme do not necessarily contribute to catalysis, and indeed could be detrimental if they increase the reorganisation energy of the reaction. Our EVB and dispersed-polaron approaches described elsewhere (e.g. ref. 52) considered the enzyme reorganisation explicitly and automatically assess the complete structural changes along the reaction coordinates. A dispersed-polaron analysis of the type represented in ref. 52, for example, determines the projection of the protein motion on the reaction coordinate and provides a basis for a quantitative comparison with a reference reaction in solution. In other words, our early studies indicated quite clearly that the motions along the reaction coordinate involve many modes in both the enzyme and solution reactions, but could not find any evidence that the existence of coupled modes contribute to catalysis.
One may still wonder about the connection between correlated motions and the effect of mutations on enzyme catalysis. However, the effect of distant mutations in DHFR (see Figure 1.4) is likely to be due to propagation of structural changes to the active site region, as is the case in many allosteric systems (e.g. ref. 158, 159). The new active site configuration is then unable to provide the same preorganised environment as the native enzyme. In other words, the mutation can change the curvature of the reaction coordinate and this change can be described as the effect of coupled modes (although such a description is neither predictive nor particularly useful). However, the issue is not the decomposition of the reaction path to the different protein modes but the height of the activation barrier. This barrier is determined by the reorganisation energy, which depends on the sum of the displacement of the different modes upon motion from the reactant to product state. Apparently, the mutations lead to an increase in the distance between the product and reactant states and in fact to larger displacements of the modes that are projected on the reaction coordinate. This means that the coupled modes reduce rather than increase the catalytic effect.
Perhaps the most effective way to classify and quantify the effect of mutation energy is to use allosteric diagrams of the type discussed in ref. 160. In this case, the focus is on the transfer of information due to energy coupling rather than just the correlation between simulated structural changes and the relationship to the active site preorganisation is clearer.
1.4.4 Problems with the Generalised Compression Idea
Bruice and coworkers have advanced the idea that enzymes catalyse reactions by favouring configurations in which the reactants are pushed to a close interaction distance (e.g. ref. 53). In most cases that we have studied, the energy associated with moving the reacting fragments from their average configuration in water to the average configuration in the enzymes was small, indicating that the corresponding catalytic effect was relatively minor.65,161 In one case, where the NAC effect appeared to be large, it was found that the actual catalytic effect was attributable to electrostatic stabilisation of the TS.56 In other words, the NAC effect evidently has been found to be a consequence rather than the reason for the electrostatic catalytic effect.56
The most notable example is CM, whose RS and TS are illustrated in ref. 56, where both the RS and TS of CM have similar charge distributions, and thus the same preorganisation effects that stabilise the RS also stabilise the TS and lead to an apparent NAC effect by making the RS structure closer to that of the TS. However, this is an inherent result of the TSS rather than being the reason for catalysis (see ref. 56).
Interestingly, despite the fact that the NAC effect has been shown to reflect incorrect energy considerations, it is still assumed to be an important effect and a reflection of compression effect. The inconsistency in the compression idea has been established by careful studies (e.g. ref. 162).
1.4.5 RSD by Desolvation Effects Does Not Provide Large Catalytic Effects
The idea that enzymes reduce the activation barrier by desolvating and destabilising the ground state of their reacting fragments has been put forward by many workers (e.g. ref. 48, 163–165). However, systematic analyses have demonstrated that the TS are solvated much more strongly in many enzymes than in the reference solution system.1,11,64 It is important to note that the only way to test the desolvation proposal computationally is to calculate the actual binding energies of the reactants in the ground and TSs (see, for example, ref. 64). Most of the computational studies that are claimed to favour the desolvation proposal have not included such calculations.
One of the best illustrations of the problem with the RSD proposal has been given in the case of orotidine 5′-monophosphate decarboxylase (ODCase).63 Although this case was discussed extensively, it gained additional importance due to an experiment166 that justifies taking this as a specific general example. Now, the catalytic action of ODCase was first proposed to reflect the desolvation effect.165 This was shown to involve an incorrect thermodynamic cycle (e.g. ref. 63). The elucidation of the structure of this enzyme showed that its active site is extremely polar (highly charged), but this led to a new RSD proposal, where the negatively-charged groups of the protein destabilise the carboxylate of the orotate substrate.20 This proposal was shown to be inconsistent with the fact that a destabilised orotate will accept protons and become stable.63 Furthermore, a careful computational study illustrates that the protein works by TSS and not by RSD (see ref. 63 and discussion below). Finally, studies by Wolfenden and coworkers167,168 have provided strong evidence against the RSD proposal. These studies demonstrated that mutations of Asp96 and other residues that were supposed to destabilise the orotate led to weaker rather than stronger binding. As predicted in ref. 63, this result is inconsistent with the RSD, because destabilisation of the RS should result in a reduction of the binding energy. The strongest support has been provided in a very unique way by the experiments of Amyes et al.,166 once these experiments have been analysed in a physically consistent way as has been done in ref. 3. In fact, the analysis can be used to disprove Jencks’ proposal that enzymes work by using binding energies to destabilise the ground state of the reactive part of the substrate.
More major and fundamental problems with the analysis of ODC and the analysis of Wu et al.20 that serve as a basis for the RSD proposal are provided in ref. 3.
1.4.6 Entropy Contributions of Bringing the Reactants Together are Unlikely to Account for Large Catalytic Effects
The idea that enzyme catalysis is associated with the entropy loss upon substrate binding was advanced in the early work of Jencks and coworkers163,169 and has gained some support in recent computational studies.170,171 However, Villa et al.172 have argued that this proposal is based on an incomplete thermodynamic cycle. The entropic contribution probably cannot be large because the activation entropy in solution is usually much smaller than one might assume. This reflects the fact that the formation of the TS does not lead to loss of many degrees of freedom.172 Problems with the entropic proposal have also emerged from experimental studies of cystidine deaminase by Wolfenden and coworkers,173 as established by Åqvist and coworkers.174
1.4.7 Allosteric Control of Catalytic Activity is Also Associated with Electrostatic Effects
Allosteric effects control many enzymatic processes where interaction with another protein or with effectors drastically changes the catalytic activity of the given enzymes. So far, all the systems that have been explored by consistent simulations are found to be controlled by electrostatic effects.
The activation of Ras by GTPase activating protein provides a general example of a molecular switch that controls cell differentiation (e.g. see discussion in ref. 175). Through our simulation studies of this system, we have shown that the binding of GTPase activating protein leads to a major electrostatic stabilisation of the TS for the guanosine triphosphate (GTP) hydrolysis by both the so-called arginine finger176 and by the transfer to a catalytic configuration, where the p-loop and other dipolar motifs stabilise the product of the hydrolysis reaction (see ref. 159, 175).
Another example is provided by the transition from the inactive chymotrypsinogen to the active chymotrypsin. This transition involves the cutting of the single bond between residues 15 and 16.177 The new amino terminus at Ile-16 then forms a salt bridge with Asp 194, and this leads to a large shift of the main chain dipoles and the formation of the preorganised oxyanion hole.178 The energetics of a related structural change due to the Gly-216/Gly-226 mutation to alanine's was explored by EVB calculations178 and shown to reflect changes in electrostatic TSS.
Even in the case of haemoglobin we were able to show158 that a significant fraction of the allosteric effect is associated with the change in interaction between the charge shift upon oxygen binding and the change in protein tertiary structure (see discussion in ref. 158).
The fidelity of DNA replication by DNA polymerases is controlled by the active site (where the incorporation reaction is catalysed) and by the binding site of the incoming nucleotide that already includes the template base (e.g. see discussion in ref. 179). The high fidelity is guaranteed by the fact that the rate of incorporation of an incoming wrong nucleotide, W, is drastically slower than the corresponding rate of the right nucleotide, R (see ref. 180). Now, the origin of this control can be quantified by considering the interplay between the binding site of the incoming base and the stabilisation of the TS in the chemical site. Our previous studies181,182 already indicated that the binding of the incoming base is determined by the preorganisation energy provided by the base binding site (which includes the template base); now the remaining challenge is to show that the TS stabilisation by the preorganised active site is anti-correlated with the preorganisation in the base binding site. This point has been explored in our preliminary studies, when we generated an ‘interaction matrix’ to describe the interaction between the TS and the protein groups as well as the interaction of the base of the incoming nucleotide with its surroundings.160 Using such diagrams for the R and W systems (at the corresponding relaxed TS structures) provides an instructive decomposition of the allosteric effect that controls replication fidelity. In particular, taking the difference between the R and W matrices helps to identify the residues that are involved in the transfer of information from the base site to the TS site. Without going into the details (which will be addressed elsewhere) we note that the transfer of information between the base site and the chemical active site is controlled by electrostatic energies.
1.5 Conclusions and Perspectives
The exploration of the catalytic power of enzymes requires one to use well-defined references and the most unique reference is our ‘chemistry-filtered’ reference state that involves the same mechanism in the corresponding enzymatic reaction. With this reference we can address the challenge of evaluating the enormous catalytic power of some enzymes and relating it to the nature of the active site environmental effect.
The evaluation of the catalytic effect requires models that convert the active site structure to activation free energy and such models started with our 1976 QM/MM model5 and evolved over the last 40 years. This review describes some of the directions taken in using and developing QM/MM methods. It appears that many studies are still overlooking the need for significant sampling and the need for calibration and validation on reference reactions in solutions. One way to see this point is to examine whether the given model can evaluate pKa in a protein active site, as large errors in such a calculation would also be reflected in calculations of activation barriers.
Using QM/MM methods that give converging and calibrated results, and in particular the EVB method, has indicated that enzyme catalysis is due mainly to electrostatic effects. These effects are due to polar preorganisation of the active site environment that stabilises the TS much more that the corresponding environment in water. We also clarified and demonstrated that the preorganisation effect involves a reduction in the folding energy and results in an inverse activity/stability correlation.
In order to verify the proposal that the catalytic effect is primarily due to electrostatic effects, it is important to demonstrate that the contributions from other factors and proposals are relatively small. This was done in the present review by considering various proposals and summarising studies that established the problems with those proposals. Thus, although it is reasonable to assume that evolution has exploited many possible catalytic effects, it appears that, with the exception of the electrostatic preorganisation effects, most of the mechanisms that have been proposed cannot lead to significant catalytic effects. Of course, our findings cannot be extrapolated to enzymes that have not yet been studied. But the only way to examine the feasibility of a proposed effect is to assess its magnitude in a variety of known enzymes, and the finding that a particular effect is relatively unimportant in all of these test cases indicates that this effect cannot contribute significantly to catalysis.
In summary, our studies and those of others have provided clear support for the view that electrostatic effects are the most important factor in enzyme catalysis.59,183 It also appears that the issue in studies of enzyme catalysis is not the reformulation of TST, but the ability to evaluate the activation free energy in a reliable way including, if needed, quantum corrections. We believe that the accelerated increase in theoretical studies will provide increasing support to the electrostatic proposal and that the ability of such theoretical studies to reproduce experimental observations will lend credibility to their ability to dissect the overall catalytic effects to their key components and thus to establish the origin of enzyme catalysis.
The EVB can be parameterised conveniently in many cases by studying the solution reaction, but in some cases the surface of the solution reaction is very complex and requires careful ab initio calculations of the free energy surface in solution. A good example is GTP hydrolysis in solution, which requires major effort184 in an attempt to distinguish between a mechanism with one nucleophilic water (1W) and with an additional water (2W) where the proton of the nucleophilic water is transferred to a second water and then to the phosphate oxygen. Calibrating the EVB surface on the ab initio surface provides a powerful way to study the reaction in G proteins.8
The advances in QM/MM studies are likely to continue in the near future and are likely to help in rational enzymes, and some progress in this direction has been reported. However, most attempts to use computer-aided enzyme design involved approaches that cannot reproduce the barriers in known cases (see ref. 185) and thus are not expected to provide realistic predictions for effective catalysis. Furthermore, even approaches that reproduce observed activation barriers are still facing a major challenge in taking non-active enzymes and reaching the same enormous catalytic effect presented in enzymes that were developed by evolution. Accomplishing this task would clearly be a way to demonstrate real understanding of enzyme catalysis.
This work was supported by the National Institutes of Health (NIH) grants GM 24492 and U19CA105010, and NSF grant MCB-1243719. We thank the High Performance Computing Center (HPCC) at the University of Southern California (USC) for computer time. We also thank Extreme Science and Engineering Discovery Environment (XSEDE) (which is supported by National Science Foundation grant number ACI-1053575) for computer time.