The rate constant of the reaction catalyzed by the enzyme carbonic anhydrase II, which removes carbon dioxide from body fluids, is calculated for a model of the active site. The rate-determining step is proton transfer from a zinc-bound water molecule to a histidine residue via a bridge of two or more water molecules. The structure of the active site is known from X-ray studies except for the number and location of the water molecules. Model calculations are reported for a system of 58 atoms including a four-coordinated zinc ion connected to a methylimidazole molecule by a chain of two waters, constrained to reproduce the size of the active site. The structure and vibrational force field are calculated by an approximate density functional treatment of the proton-transfer step at the Self-Consistent-Charge Density Functional Tight Binding (SCC-DFTB) level. A single transition state is found indicating concerted triple proton transfer. Direct-dynamics calculations for proton and deuteron transfer and combinations thereof, based on the Approximate Instanton Method and on Variational Transition State Theory with Tunneling Corrections, are in fair agreement and yield rates that are considerably higher and kinetic isotope effects (KIEs) that are somewhat higher than experiment. Classical rate constants obtained from Transition State Theory are smaller than the quantum values but the corresponding KIEs are five times larger. For multiple proton transfer along water bridges classical KIEs are shown to be generally larger than quantum KIEs, which invalidates the standard method to distinguish tunneling and over-barrier transfer. In the present case, a three-way comparison of classical and quantum results with the observed data is necessary to conclude that proton transfer along the bridge proceeds by tunneling. The results suggest that the two-water bridge is present in low concentrations but makes a substantial contribution to proton transport because of its high efficiency. Bridging structures containing more water molecules may have lower energies but are expected to be less efficient. The observed exponential dependence of the KIEs on the deuterium concentration in H(2)O/D(2)O mixtures implies concerted transfer and thus rules out substantial contributions from structures that lead to stepwise transfer via solvated hydronium ions, which presumably dominate proton transfer in less efficient carbonic anhydrase isozymes.

In this review we discuss a recently introduced method of calculating hydrogen tunnelling rates and tunnelling splittings in medium and large molecules. It is a non-empirical, direct-dynamics method that uses ab initio quantum-chemical output as input data for the calculation of dynam ic properties by means of the instanton approach. This approach is based on the recognition that there is a single path that dominates the tunnelling rate. This so-called instanton trajectory is the path that minimizes the classical action. Although it is very di cult to calculate this trajectory for multidimensional systems, it will be shown that the corresponding instanton action, which is the quantity of practical interest, can be obtained with su cient accuracy to reproduce experimental observations without the explicit evaluation of the instanton trajectory. In this approxim ation schem e the instanton action is calculated from the one-dimensional action through the introduction of appropriate correction terms for all modes coupled to the tunnelling mode. These coupled transverse modes are taken to be harmonic oscillators; the couplings are assumed to be linear and derived from the displacements of the transverse modes between the equilibrium con® guration and the transition state. Nonlinear couplings of large-amplitude transverse modes are also brie¯y discussed. The reaction coordinate is identi® ed with the norm al mode with imaginary frequency in the transition state and not with the minimum-energy path used in variational transition-state theory. The multidimensional potential-energy surface is formulated in terms of the norm al coordinates of the transition state. Form ulas and computer codes are presented which allow direct evaluation of mode-sp eci® c tunnelling splittings as well as of proton transfer rate constants across symmetric or asymmetric barriers as a function of temperature. Results are presented for these rates and splittings that can be critically compared with experimental data. Mode-sp eci® c splittings are discussed for 9-hydroxyphenalenone and tropolone, two large molecules for which excellent experimental data are available. Tunnelling rate constants are discussed for aziridine, oxiranyl and dioxolanyl, three medium -size molecules that undergo inversion by a tunnelling mechanism, and for porphine, a large molecule for which an abundance of high-quality proton-transfer data has been reported. All of these systems are handled successfully by the method. The calculations are performed with the DOIT (dynamics of instanton tunnelling) code, which is available on the internet. This dynam ics code is very e cient compared to other available codes based on transition-state theory with tunnelling corrections and takes only a fraction of the com puter time required for the computation of the quantum-chem ical input data.