Canonical variational theory for enzyme kinetics with the protein mean force and multidimensional quantum mechanical tunneling dynamics. Theory and application to liver alcohol dehydrogenase

We present a theoretical framework for the calculation of rate constants of enzyme-catalyzed reactions that combines variational optimization of the d ynamical bottleneck for overbarrier reactive events and multidimensional qu antum mechanical tunneling dynamics for through-barrier reactive events, bo th in the presence of the protein environment. The theory features a two-zo ne, three-stage procedure called ensemble-averaged variational transition s tate theory with multidimensional tunneling (EA-VTST/MT) with the transmiss ion coefficient based on the equilibrium secondary-zone (ESZ) approximation for including the effects of the protein on a catalytic reaction center, c alled the primary zone. The dynamics is calculated by canonical variational theory with optimized multidimensional tunneling contributions, and the fo rmalism allows for Boltzmann averaging over an ensemble of reactant and tra nsition state conformations. In the first stage of the calculations, we ass ume that the generalized transition states can be well described by a singl e progress coordinate expressed in primary-zone internal coordinates; in su bsequent steps, the transmission coefficient is averaged over a set of prim ary-zone reaction paths that depend on the protein configuration, and each reaction path has its own reaction coordinate and optimized tunneling path. We also present a simpler approximation to the transmission coefficient th at is called the static secondary-zone (SSZ) approximation. We illustrate b oth versions of this method by carrying out calculations of the reaction ra te constants and kinetic isotope effects for oxidation of benzyl alcoholate to benzaldehyde by horse liver alcohol dehydrogenase. The potential energy surface is modeled by a combined generalized hybrid orbital/quantum mechan ical/ molecular mechanical/semiempirical valence bond (GHO-QM/MM/SEVB) meth od. The multidimensional tunneling calculations are inicrocanonically optim ized by employing both the small-curvature tunneling approximation and vers ion 4 of the large-curvature tunneling approximation. We find that the vari ation of the protein mean force as a function of reaction coordinate is qua ntitatively significant, but it does not change the qualitative conclusions for the present reaction. We obtain good agreement with experiment for bot h kinetic isotope effects and Swain-Schaad exponents.