## Abstract

The physical basis of the catalytic power of enzymes remains contentious despite sustained and intensive research efforts. Knowledge of enzyme catalysis is predominantly descriptive, gained from traditional protein crystallography and solution studies. Our goal is to understand catalysis by developing a complete and quantitative picture of catalytic processes, incorporating dynamic aspects and the role of quantum tunnelling. Embracing ideas that we have spearheaded from our work on quantum mechanical tunnelling effects linked to protein dynamics for H-transfer reactions, we review our recent progress in mapping macroscopic kinetic descriptors to an atomistic understanding of dynamics linked to biological H-tunnelling reactions.

- enzyme mechanism
- H-tunnelling
- kinetic isotope effect
- protein dynamics
- redox catalysis
- temperature-dependence

## The physical basis of enzyme catalysis, tunnelling and dynamics

Enzymes achieve high catalytic rates. However, the mechanism by which this is achieved remains elusive. Rate enhancements of up to 10^{21} have been reported, but the physical basis of this catalytic power remains contentious [1], despite sustained and intensive research efforts [1–7]. The role of protein motions in enzyme catalysis, both in classical and quantum mechanical transfers, is hotly debated [1,7–9]. The very presence and identity of motions, coupled with catalysis, are currently one of the most important unanswered questions in enzyme catalysis, yet one of the most difficult ones to address experimentally. In recent years, we have provided key experimental and computational evidence that supports full tunnelling models for enzyme-catalysed hydrogen transfer (for a review, see [10]) and we have relied on a strong interplay between high-resolution/time-resolved structural analysis, detailed computational simulations, chemical biology and fast reaction kinetics to develop, and provide experimental support for, new theoretical frameworks for enzyme catalysis. This interdisciplinary approach is yielding new and important insight into dynamics coupled with the reaction co-ordinate required to ‘squeeze’ tunnelling barriers and thus increase the probability of H-transfer by quantum mechanical tunnelling.

To what extent dynamic processes have an impact on enzyme catalysis and how these could have evolved is a key question, in particular as to whether enzymes have evolved to use quantum tunnelling to the best advantage. That tunnelling occurs is now widely accepted, with the conceptual frameworks incorporating protein motion into the enzymatic H-tunnelling process [11–13]. More controversial is a role for compressive motion to promote H-tunnelling [8], a special property of enzymes that might be endowed by evolutionary pressure. While an increasing number studies have suggested a role for protein motion in facilitating tunnelling (see e.g. [10,14–17] for recent reviews), others have argued that motions contributing to tunnelling are similar both in enzymes and in free solution and therefore the degree of tunnelling available in the enzyme system is equivalent to that in free solution [18–20]. Although a number of kinetic and computational studies [10,14–17] employing analysis of KIEs (kinetic isotope effects) are in agreement with environmentally coupled models of H-tunnelling [11–13] (see Figure 1 for a theoretical description), some workers have argued that correlated motions do not contribute to catalysis (see e.g. [21]). Environmentally coupled models of H-tunnelling represent a radical departure from physical frameworks based on semi-classical transition-state theory, but atomic-level insight into the tunnelling event and accompanying motions of the enzyme–substrate and enzyme–intermediate complexes can only be obtained from high-resolution protein structures of catalytically competent intermediates combined with computer simulation. Our work with AADH (aromatic amine dehydrogenase) and bacterial MR (morphinone reductase) is beginning to provide unique insight into the importance of dynamics linked to tunnelling reactions.

## Environmentally coupled tunnelling models, promoting vibrations and the origin of the temperature-dependence of KIEs

The theoretical framework of the Kuznetsov and Ulstrup model [11] has been used to rationalize experimental findings, including the temperature-dependence of KIEs, reaction rates and thermodynamic parameters. A more detailed analysis is presented in our previous study [13] and that of Knapp and Klinman [12]. The tunnelling rate constant, *k*_{tunnel}, can be related to both the non-thermally equilibrated (sub-picosecond) and thermally equilibrated motion terms by the following general expression:
(1)
Where const. is an isotope-independent term; the term in square brackets is an environmental energy term relating the driving force of the reaction, Δ*G*°, and the reorganizational energy, λ (*R* is the gas constant and *T* the temperature in kelvin); the F.C. Term is the Franck–Condon nuclear overlap along the hydrogen co-ordinate and arises from the overlap between the initial and the final states of the hydrogen's wavefunction. In the simplest limit, when only the lowest vibrational level is occupied for the nuclear wavefunction of the hydrogen nucleus, the F.C. Term is independent of temperature; otherwise, the F.C. Term will be temperature-dependent. Temperature-dependent ‘gating’ dynamics (motion), which can be likened to a ‘squeezing’ of the potential energy barrier, can modulate the F.C. Term. From eqn (1), the KIE can be expressed as:
(2)
Thus the KIE is temperature-independent if the F.C. Term dominates and temperature-dependent if there is a significant contribution from a gating motion. The trends in KIE are conceptually similar for non-adiabatic (e.g. H radical transfer) and adiabatic (e.g. proton transfer) reactions. For H radical transfer, the KIE (eqn 3) is written as [12]:
(3)
where *k*_{B} is Boltzmann's constant, *r*_{0} and *r*_{1} the equilibrium and final separations of the reactant and product potential wells along the hydrogen co-ordinate, ω_{H} and ω_{D} the frequencies of the reacting bond and *m*_{H} and *m*_{D} the masses of the transferred particle for protium and deuterium respectively. The H/^{2}H transfer distance, *r*_{H}/*r*_{D}, is reduced by the distance the gating unit moves, *r*_{X}(*r*_{H/D}=*r*_{0}−*r*_{X}). The energetic cost of gating (*E*_{X}) is given by:
(4)
where *k*_{HO} is the force constant, and the gating co-ordinate (*X*) is related to the gating oscillation (ω_{X}), the mass of the gating unit (*m*_{X}), and *r*_{X} as follows:
(5)

This model predicts that if the gating term dominates (i.e. ħω_{X}<*k*_{B}*T*), the observed KIE can be temperature-dependent, since this leads to different transfer distances for the heavy and light isotopes. In this regime, the Arrhenius prefactor ratio *A*_{H}/*A*_{D} value is predicted to be less than unity. Alternatively, if the Franck–Condon term dominates (i.e. ħω_{X}>*k*_{B}*T*), the KIE will be temperature-independent. In this scenario, occupation of excited vibrational levels could result in some temperature-dependence. However, the Boltzmann distribution at 298 K suggests that tunnelling should be predominantly from the vibrational ground state of the nuclear wavefunction of hydrogen. In the regime where ħω_{X}≈*k*_{B}*T* gating plays some role in modulating the tunnelling probability, temperature-dependent KIEs are observed and the *A*_{H}/*A*_{D} values decrease (compared with the regime where the Franck–Condon term dominates) and may approach unity.

## Are measurements of the temperature-dependence of KIEs reliable indicators of the presence or absence of promoting motions?

The environmentally coupled H-tunnelling model can be used qualitatively, based on the temperature-dependence of the KIE, to define enzymatic H-tunnelling systems as either requiring (i.e. the observed KIE is temperature-dependent) or not requiring (i.e. the observed KIE is temperature-independent) a promoting vibration (‘gating’ motion) for the tunnelling event to occur. We have used this qualitative approach extensively in our previous studies (examples reviewed in [10]). However, as we seek a more detailed understanding of the factors that drive H-tunnelling, it is essential to move from a qualitative to a quantitative understanding. Recently, we made this important step forward. We have shown that the apparent temperature-independence of the observed KIE for the H-tunnelling event in the AADH/tryptamine system can be accommodated within the framework of the environmentally coupled model of Kuznetsov and Ulstrup [11] for H-tunnelling [2,22] and, moreover, that this approach can reconcile apparent discrepancies that can arise if only the qualitative approach is used. Our study (which also required as input the data from the analysis of our Molecular Dynamics simulations and DFT (density functional theory) calculations [2,22], as well as kinetic data [2]) revealed that, while the observed KIE for this system is not measurably temperature-dependent (suggesting that a promoting vibration is not required for H-tunnelling), a promoting vibration is nevertheless required to explain the observed KIE and the experimentally observed difference in phenomenological enthalpies of activation. Our numerical modelling showed the reaction rates to be consistent with a promoting vibration within the environmentally coupled model when tunnelling proceeds also from vibrationally excited states [22]. A detailed analysis of our numerical modelling, biomolecular simulations and kinetic data can be found elsewhere [2,22], but in summary a localized promoting vibration, with a frequency of ∼165 cm^{−1}, was identified for the proton tunnelling step in the oxidation of tryptamine catalysed by AADH and shown to be consistent with there being a weak temperature-dependence (i.e. not detectable in experimental kinetic studies) of the KIE. This promoting vibration does not require large-scale dynamics of the protein scaffold but is inherent to the proton donor, the catalytic iminoquinone intermediate, as shown by Molecular Dynamics simulations of this intermediate and density functional theory calculations (Figure 2A) [22]. This promoting vibration corresponds to a rotation of the donor C-1/H-1 methylene group, which couples with the acceptor oxygen, O-2, so that these atoms all move towards a configuration from which tunnelling becomes more probable (Figure 2B). In addition to being mechanistically, thermodynamically and kinetically favoured, proton transfer to O-2 is also dynamically favoured. Also, motions other than this promoting vibration (i.e. preorganization) are likely to be involved in moving the system towards a tunnelling-ready configuration, from which the gating motion can take effect. We surmise that the repositioning of O-2 would bring O-2 sufficiently close to H-1 for an electrostatic coupling of their respective vibrations.

## Hydrostatic pressure as a probe of promoting vibrations, numerical modelling and molecular interpretation

Recently, we reported the use of the pressure-dependence of KIEs, coupled with a study of their temperature-dependence, as a probe for promoting motions in enzymatic H-tunnelling reactions [4]. We employed the flavoprotein MR as our model system. The reductive half-reaction of MR involves hydride transfer from the C-4 *R*-hydrogen of β-NADH to the N-5 atom of FMN and this reaction is directly observed in a rapid mixing stopped-flow instrument, is kinetically resolved from steps involving coenzyme binding and formation of an enzyme–NADH CT (charge-transfer) complex, and the observed KIE is essentially the intrinsic KIE [3,4,23]. From these studies, we have demonstrated that the 1° KIE of MR is highly temperature-dependent [23]. These data, coupled with a recent study showing that the 2° KIE is also exalted and consistent with preorganization in MR [3], led us to describe the reaction within the context of modern environmentally coupled models of H-tunnelling with a requirement for a promoting motion to move the nicotinamide C-4-H sufficiently close to the FMN N-5 atom to facilitate tunnelling.

We have measured, using stopped-flow methods, the hydride transfer rate as a function of both hydrostatic pressure and temperature. We demonstrated that increasing the pressure from 1 bar to 2 kbar (1 bar=100 kPa) accelerates the hydride transfer reaction when both protium (from 50 to 161 s^{−1} at 25 °C) and deuterium (from 12 to 31 s^{−1} at 25 °C) are transferred [3]. We also demonstrated that the observed primary KIE increases with pressure (from 4.0 to 5.2 at 25 °C), an observation shown to be incompatible with the Bell correction model for H-tunnelling but consistent with an environmentally coupled model for hydride tunnelling. By numerical modelling, we show that both the pressure-dependence and temperature-dependence of the reaction rates are consistent with the framework of the environmentally coupled tunnelling model of Kuznetsov and Ulstrup [3,11], providing additional support for the role of a promoting motion in the hydride tunnelling reaction in MR (Figure 3). We implemented the environmentally coupled tunnelling model as we also used to describe the H-tunnelling reaction in AADH [22]. Thus the rate of a tunnelling reaction is described by:
(6)
Equation (6) is a more complete rate expression that accounts for tunnelling from and/or to vibrationally excited states where the rate is weighted according to the Boltzmann populations of each vibrationally excited reactant states (*P*_{i}), *V*_{el} is the isotope-independent electronic coupling constant, *E*_{vib} the (isotope-dependent) difference in vibrational energy between the reactants and products and F.C._{i}_{,j} is the F.C. term for tunnelling from the vibrational state *i* to the vibrational state *j*. Note that for simplification, only the first vibrationally excited states are included (the Boltzmann populations of the higher excited states are negligible at physiological temperatures):
(7)
(8)
(9)
(10)
To incorporate gating, and thus a strongly temperature-dependent KIE, a Franck–Condon term incorporating a gating mode is used:
(11)
Δ*r*=(*r*_{0}−*r*_{X}) is the tunnelling distance reduced from an equilibrium separation, *r*_{0}, by the distance of gating, *r*_{X}. Thus, by systematically varying *E*_{X}, *T* and *r*_{0}, while solving for the experimentally determined KIE, temperature-dependence (ΔΔ*H*‡) and *A*′_{H}/*A*′_{D} values at each pressure measured, we can estimate the effect of pressure on gating during this reaction.

Our preliminary X-ray diffraction data of MR with bound tetrahydro-NADH indicates that the coenzyme nicotinamide and MR flavin moieties are coplanar with the coenzyme C-4 and FMN N-5 atoms in close proximity. These data are consistent with the structure of a homologous Old Yellow Enzyme, which was solved with an NADPH analogue that was bound coplanarly to the enzyme-bound flavin. Molecular modelling of the crystal structure co-ordinates of the MR tetrahydro-NADH complex allowed us to determine the position of the transferred hydride in the reactant (NADH C-4-*R*H) and product states (FMN N-5-H). From this model, the equilibrium tunnelling distance, *r*_{0}, was then estimated to be ∼1.7 Å (1 Å=0.1 nm) (Figure 3). This distance would appear to be much too large for tunnelling to occur, as the de Broglie wavelength of hydrogen is only approx. 0.6 Å. This provides a further justification for the use of an environmentally coupled tunnelling model rather than a tunnelling correction model.

## Summary

Studies of either the (i) temperature- or (ii) pressure- dependence of KIEs have proven to be informative in deciphering mechanisms of H-transfer, but in combination additional insight is gained. Our studies of the combined hydrostatic pressure and temperature-dependence of KIEs demonstrate the utility of ‘barrier engineering’ using hydrostatic pressure as a probe of tunnelling regimes in enzyme systems and the approach has provided added and independent support for the requirement for promoting motions in such tunnelling reactions. A combination of experimental studies, numerical modelling and biomolecular simulations is beginning to provide atomistic insight into tunnelling reactions and the role of protein dynamics. Our work highlights the benefit of pursuing interdisciplinary approaches in our quest to investigate the nature of coupled motions in enzyme catalysis.

## Acknowledgments

This work was funded by the U.K. BBSRC (Biotechnology and Biological Sciences Research Council) and the U.K. EPSRC (Engineering and Physical Sciences Research Council). N.S.S. is a BBSRC Professorial Research Fellow. Some of the material presented in this article is based on our previous paper [13] and is included for the sake of completeness.

## Footnotes

Bringing Together Biomolecular Simulation and Experimental Studies: A Biochemical Society Focused Meeting in conjunction with the Molecular Graphics and Modelling Society held at Manchester Interdisciplinary Biocentre, Manchester, U.K., 10–11 September 2007. Organized and Edited by Mike Sutcliffe (Manchester, U.K.).

**Abbreviations:**
AADH, aromatic amine dehydrogenase;
DFT, density functional theory;
KIE, kinetic isotope effect;
MR, morphinone reductase

- © The Authors Journal compilation © 2008 Biochemical Society