## Abstract

One of the most controversial questions in enzymology today is whether protein dynamics are significant in enzyme catalysis. A particular issue in these debates is the unusual temperature-dependence of some kinetic isotope effects for enzyme-catalysed reactions. In the present paper, we review our recent model [Glowacki, Harvey and Mulholland (2012) Nat. Chem. **4**, 169–176] that is capable of reproducing intriguing temperature-dependences of enzyme reactions involving significant quantum tunnelling. This model relies on treating multiple conformations of the enzyme–substrate complex. The results show that direct ‘driving’ motions of proteins are not necessary to explain experimental observations, and show that enzyme reactivity can be understood and accounted for in the framework of transition state theory.

- enzyme catalysis
- kinetic isotope effect (KIE)
- protein dynamics
- quantum tunnelling
- transition state theory (TST)

## Introduction

Ever since their catalytic properties were discovered, the question of exactly why enzymes are such outstanding catalysts has intrigued biochemists, and controversy persists about exactly what it is that makes them such good catalysts [1,2]. There has been encouraging recent progress in enzyme design [3], based on the principle of transition state stabilization, but designed or engineered enzymes, e.g. catalytic antibodies, are generally far less effective catalysts than natural enzymes. This has led to suggestions that existing models are inadequate and that new concepts are needed for understanding enzyme action [1,4–6]. Debates have arisen concerning the role of protein dynamics, in particular for enzymes which catalyse reactions involving quantum tunnelling. Experiments and biomolecular simulations have shown [e.g. by large KIEs (kinetic isotope effects)] that tunnelling is involved in the transfer of hydrogen nuclei and atoms in enzymes [7,8]. Simple models (e.g. based on a constant energy barrier) cannot explain the unusual temperature-dependence of KIEs found for some enzymes reactions involving tunnelling [7–10], including SLO-1 (soya bean lipoxygenase 1) [11], AADH (aromatic amine dehydrogenase) [7], MADH (methylamine dehydrogenase) [9] and DHFR (dihydrofolate reductase) [12]. To what extent, if at all, do these findings indicate that new theories are needed to explain enzyme catalysis?

Conformational changes are certainly central to enzyme function. Smaller structural changes have important functional roles even in simple processes such as the binding of oxygen or carbon monoxide to myoglobin [13]. It is well known that many enzymes change conformation significantly as an intrinsic part of their catalytic cycles [14–16]. It is clear that in many enzymes, e.g. motor proteins and other biological ‘molecular machines’, chemical changes are associated with large conformational changes, and understanding how they are coupled will be essential to understanding their mechanisms of action at a molecular level [17].

Molecular simulations and experiments show that many enzymes exist in multiple conformations [18,19]. The most important reactive conformation (i.e. the conformation responsible for most reactions) may well not predominate [20,21]. At an extreme, some proteins are intrinsically disordered, and only become structured upon binding [22]. Also, functionally important dynamics [23] (such as co-operative motions in some homodimers) seem to be built in to the structure of some enzymes [24]. Enzymes need to fluctuate to function, e.g. many enzymes open to release products, and such conformational changes may be the slowest step for many enzymes [25].

These features of enzymes are fascinating, but are not really controversial. There have been contentious suggestions, however, that protein dynamics directly drive reactions in enzymes (e.g. by compressing interatomic distances on the timescale of the reaction, in order to modulate barrier height and width). Proposals of this sort often identify the dynamics of enzymes as the most important source of their catalytic activity. Analysing this controversial question requires accurate molecular simulations to interpret experimental results and test hypotheses [26–32]. It is also very important to be precise in definitions: strictly speaking, catalysis is the acceleration of a reaction by a catalyst, so the common convenient shorthand of referring to ‘the process of catalysis in an enzyme’ can be misleading. To analyse catalysis, the reaction in an enzyme should be compared with an equivalent uncatalysed or ‘reference’ reaction. Comparison with a less efficient enzyme can also be useful. The first stage in understanding enzyme catalysis, however, is understanding the mechanisms and features of enzyme-catalysed reactions, and to examine whether they show significantly different behaviour from that of other chemical reactions.

## Transition state theory

TST (transition state theory) has proved extremely useful in understanding chemical reactivity. Much of the recent controversy has centred on its applicability to enzyme-catalysed reactions. TST allows reactivity (e.g. rate constants) to be understood and predicted without having to worry about all the details of atomic motion. Insofar as it is based on statistical mechanics, it is applicable to an enzyme system as long as the system is ergodic, i.e. motion of the atoms in the system becomes randomized faster than the reaction occurs [33].

If we take an enzyme–substrate complex to be like any other chemical reactant, R, which is converted into a product, P, R→P, then TST applies as long as the pool of R re-equilibrates faster than the timescale for its conversion into P [34]. In these terms, the movement of enzymes through a range of different conformers is not a ‘dynamical’ property and chemical change of R into P may be treated as decorrelated ‘hopping’ events.

TST gives the rate coefficient as:
(1)where the transmission coefficient, κ, accounts for quantum tunnelling and ballistic re-crossing through the transition state (κ is usually close to 1, except for reactions in which quantum tunnelling is significant) [30]; *k*_{B}, *h* and *R* are the Boltzmann, Planck and gas constants respectively; ϵ is the barrier height (including zero-point energy); *T* is the absolute temperature; and *Q*^{TS} and *Q*^{R} are the partition functions of the transition state and reactant respectively. For small molecules, TST often gives good agreement with experiment, especially when (i) tunnelling is included, (ii) the transition state position is optimized to minimize the rate constant expression and barrier re-crossing, and (iii) the barrier, ϵ, is known accurately. Often, good results are found even without treatment of these factors, and with partition functions calculated from expressions for simple harmonic oscillators and rigid rotors.

For chemical and biochemical reactions, molecular simulations provide a microscopic level of detail, beyond that possible from experiments alone. Highly accurate quantum mechanical electronic structure methods allow calculations of rate constants for reactions of small molecules in the gas phase that can be as accurate as experiments [35]. KIEs can also be calculated accurately, e.g. recent work has given quantitatively accurate predictions of temperature-dependent KIEs for the prototypical reaction of H+H_{2}, for a wide range of isotopes [36]. Accurate potential energy surfaces also allow analysis of reaction dynamics, and often show that reaction dynamics of small molecules in the gas phase are conformationally complex [37]. Investigations of even the simplest gas-phase chemical reactions, e.g. H+H_{2}, are often surprising, as revealed by high-resolution experiments and calculations [38]. Reactions in solution and other condensed phases show even more complexity, as experiments and simulation are now showing [39,40]. The microscopic details of enzyme reaction dynamics of reactions in enzymes will certainly be at least as complex [41].

TST is an approximate theory, and so does have its limitations. Experimental and computational studies of the reactions of small molecules have provided significant insight into these limitations and an understanding of when and how dynamical effects cause deviations from kinetic predictions based on TST. Often, such effects occur because the initial conditions in either energy [42] or configuration space [40,43] are significantly different from a standard thermal distribution (e.g. due to laser excitation) or because the system cannot relax rapidly [40].

## Simulations of enzyme-catalysed reactions

Understanding enzyme reactions similarly requires both experiments and simulation. Several different types of molecular modelling and simulation methods can provide useful insights into enzyme-catalysed reactions. EVB (empirical valence bond) simulations have provided important insights into enzyme catalysis and reaction dynamics [44]. Mechanisms of enzyme-catalysed reactions can be investigated by QM (quantum mechanics) electronic structure calculations, of the type mentioned above, on small models of active sites. Such QM methods can also be used to investigate reactions in larger systems (e.g. whole enzymes) in combined QM/MM (molecular mechanics) simulations. QM/MM calculations on enzyme reactions can now be carried out with highly accurate levels of QM theory [45]; for enzymes such as PHBH (*p*-hydroxybenzoate hydroxylase) and chorismate mutase, such calculations give energy barriers that agree with experiment almost within ‘chemical accuracy’ (1 kcal·mol^{−1}, where 1 kcal=4.184 kJ). TST underlies this comparison, indicating that it is a good basis for understanding these enzyme-catalysed reactions, and that dynamical factors have only a small effect in determining their rates [27,45]. For chorismate mutase, QM/MM and EVB calculations account for the barrier lowering (ΔΔ*G*^{‡}) by this enzyme compared with the reaction in solution, showing that simulations based on a TST framework not only can predict rates of enzyme-catalysed reactions, but also can account for the catalytic rate acceleration by the enzyme [45,46]. Catalysis in chorismate mutase is due to the active site being specifically structured to stabilize the transition state by electrostatic interactions; low reorganization energy is probably a generally important factor in enzyme catalysis [47].

Calculations with lower levels of QM/MM theory allow more extensive simulations (e.g. molecular dynamics simulations of enzymes are possible with semi-empirical QM/MM methods), but are less accurate (e.g. semi-empirical methods typically do not provide accurate barriers, but relative barriers can still be useful). They can provide useful insight, an example being the correlation found between experimental data (ln *k*_{cat}) and QM/MM potential energy barriers for a series of alternative substrates in PHBH and phenol hydroxylase. This supports the applicability of TST and indicates that dynamical and entropic contributions are similar, and relatively small for the whole series of substrates [27].

Molecular simulations have revealed conformational effects in enzyme-catalysed reactions, e.g. identifying minor conformations which may dominate reactivity [21,48]. Reaction barriers in enzymes are affected by many factors, including even small structural changes within the active site [49,50]. Simulations can also analyse the effects of quantum tunnelling, e.g. QM/MM calculations on AADH give a KIE in good agreement with experiment (55±6) [7] and show a significant tunnelling contribution. Tunnelling [51] (and large isotope effects [52,53]) is similarly found in equivalent non-enzymatic reactions, and so the catalytic contribution of tunnelling is probably small [54–56]. For the AADH reaction, shown in Figure 1, simulations indicate that: (i) it is not necessary to invoke ‘driving’ motions to show significant tunnelling and large KIEs, and (ii) small structural changes may be significant, and may favour proton transfer to one carboxylate oxygen or other of the catalytic base; these two reaction paths have very different barriers and tunnelling probabilities [57]. It is important to consider the effects of these small structural variations for a full picture.

These and other simulations [49,58,59] do not support simple models that require very short interatomic distances (driven by supposed ‘barrier squeezing’ motions) to explain observed KIEs [60], suggesting that models that include only a few distances are not useful or physically reasonable. The origin of the observed temperature-dependence of enzyme KIEs thus remains uncertain. Despite encouraging progress, direct calculation of these effects by quantum dynamical molecular simulations remains challenging [61], so simple models can be useful.

## A model for conformational effects and KIEs in enzyme reactions

So, in the light of the apparently surprising temperature-dependence of a number of enzymatic KIEs, the question remains: does TST fail for enzymes? Before addressing this question, it is important to point out that, for reactions of small molecules (when there is only one reactant) [62], the exponential factor in eqn (1) often dominates the temperature-dependence, because the temperature-dependence of the ratio of reactant and transition state partition functions is often smooth and small [63]. Therefore an Arrhenius or Eyring logarithmic plot of ln *k* against 1/*T* gives a reasonably straight line, with some possible curvature at low *T* due to tunnelling. The activation entropy or pre-exponential factor for protium and deuterium transfer reactions will be similar, but the activation energy for hydrogen transfer will be smaller than for deuterium transfer because of zero-point effects, so at higher temperatures the logarithmic plots of *k*_{H} and *k*_{D} against 1/*T* will often be two converging straight lines [1,63]. This simple behaviour of *k*_{H} and *k*_{D} is not always observed, and is not a direct consequence of TST, but instead is due to the fact that, for simple molecules, the temperature-dependence of the partition functions of reactants and transition states is usually fairly simple and monotonous. For enzymes, the existence of multiple conformers means that their partition functions change in a much less predictable way with temperature [64], so it is not surprising that their Arrhenius or Eyring plots often show much more complicated behaviour.

Recently [65], we have presented a simple two-conformation model (Figure 2) that reproduces the measured rate constants for protium and deuterium transfer in SLO-1 [11], AADH [7], MADH [9] and DHFR [12] as a function of temperature. There are, of course, many mathematical expressions that can be used to ‘fit’ the observed experimental data, particularly as these data typically only cover a fairly limited temperature range for enzymes. Hence a successful fit does not guarantee that the model reflects the underlying kinetics exactly. However, provided the model is plausible, a good fit is obtained, and ‘sensible’ (physically realistic) parameters are found, then this can be suggestive of a correct model. And if a particular model provides only poor fits, that model is almost certainly not correct. For all of the enzymes considered, the two-state model gives a very good fit, with physically reasonable values for all parameters.

A crucial feature of this model is that it involves multiple conformations of the enzyme–substrate complex. At an extreme, one could assume an arbitrarily large number of conformers, each with different reactivity behaviour, although this could easily lead to overfitting. A minimal parsimonious model assumes just two reactive conformations, R1 and R2, both of which may lead to a product complex, P, with rate coefficients *k*_{1} and *k*_{2} respectively. R1 and R2 are assumed to interconvert faster than they react, with rate coefficients *k*_{f} and *k*_{r}, and an overall equilibrium constant *K*_{eq}=*k*_{f}/*k*_{r}:

We define [R′]=[R1]+[R2]. The concentrations [R1] and [R2] can be expressed as follows: (2) (3)

The rate of loss of R1 and R2 can then be combined to yield an overall rate:
(4)where *k*′ is the phenomenological loss rate coefficient for [R′]. The equilibrium constant *K*_{eq} can be written in terms of the free energy difference between conformers R1 and R2, so that *k*′ in eqn (4) may be rewritten (in the case of both protium and deuterium transfers) as:
(5)

Using a TST-based expression for *k*_{1} and *k*_{2}, the rate coefficients for the protiated (H) and deuterated (D) substrates are:
(6)
(7)

For both eqns (6) and (7): (8)

The transmission coefficient, κ_{i}^{ℓ}, representing quantum tunnelling is given the form [66]:
(9)where *i* indicates the conformation (1 or 2), ℓ is the isotope (protium or deuterium), β=(*kT*)^{−1}, *V _{i}*

^{TS}is the transition state energy (without zero-point energy), α

_{i}^{ℓ}=2π/(ℏω

_{i}^{ℓ}), ω

_{i}^{ℓ}is the absolute frequency for the barrier, and corresponds to its curvature. Eqns (6) and (7) assume that, at high temperatures, the activated contribution is the same for protium and deuterium transfer, i.e.

*C*

_{1}and

*C*

_{2}are identical for both isotopomeric rate constants.

The fitting of eqns (5)–(7) to the experimental data was constrained so that ϵ_{1}^{H}≤ϵ_{1}^{D} and ϵ_{2}^{H}≤ϵ_{2}^{D} (i.e. a larger barrier for deuterium than for protium transfer), and κ_{1}^{H}(*T*)=κ_{1}^{D}(*T*)=1 (i.e. no tunnelling for conformation R1). For R2, we constrained the transition state frequency, ω_{H}^{TS2}, to lie between reasonable values of 2200 and 3400 cm^{−1}, with ω_{D}^{TS2}=ω_{H}^{TS2}/ (based on the ratio of deuterium and protium frequencies for a harmonic oscillator). The free energy difference, Δ*G*, between the two conformations was constrained to lie between 0 and 9.0 kcal·mol^{−1}. The resulting parameters, which are shown in Table 1, give the fits shown in Figure 3. It can be seen that the model gives good fits to the experimental data.

The model is capable of reproducing KIEs that are constant or decrease with temperature. For SLO-1, the fit only required a single conformation, but all other enzymes required two conformations. These results show that a TST-based model can reproduce the experimental data, as long as sufficient conformational complexity is included.

## Conclusions

A simple model, using physically realistic parameters, accounts for experimentally observed enzyme kinetics. At the heart of this model is the need to include multiple conformations of the enzyme–substrate complex with different reactivity and tunnelling behaviour. Two conformations are sufficient to account for the complex temperature-dependence of all of the enzyme-catalysed reactions considered in the present paper. Real enzymes have very large numbers of conformations, and these do have different reactivity properties, as simulations and, e.g., single-molecule experiments show. Our model is based on TST and, together with molecular simulations, supports TST as an appropriate basis for understanding enzyme reactions and, by implication, catalysis. Altogether, this indicates that there is no direct role for protein dynamics in ‘driving’ enzyme reactions. It is, however, essential to consider multiple enzyme conformations for a full understanding of enzyme-catalysed reactions.

## Funding

We thank the Engineering and Physical Sciences Research Council (EPSRC) for funding. A.J.M. is an EPSRC Leadership Fellow [grant number EP/G007705/1]. J.N.H. and D.R.G. thank the Engineering and Physical Sciences Research Council for support [programme grant number EP/G00224X].

## Footnotes

Frontiers in Biological Catalysis: Biochemical Society Annual Symposium No. 79 held at Robinson College, Cambridge, U.K., 10–12 January 2012. Organized and Edited by David Leys (Manchester, U.K.), Andrew Munro (Manchester, U.K.), Emma Raven (Leicester, U.K.) and Martin Warren (University of Kent, U.K.).

**Abbreviations:**
AADH, aromatic amine dehydrogenase;
DHFR, dihydrofolate reductase;
EVB, empirical valence bond;
KIE, kinetic isotope effect;
MADH, methylamine dehydrogenase;
MM, molecular mechanics;
PHBH, p-hydroxybenzoate hydroxylase;
QM, quantum mechanics;
SLO-1, soya bean lipoxygenase 1;
TST, transition state theory

- © The Authors Journal compilation © 2012 Biochemical Society