Elsevier

Chemical Physics Letters

Volume 418, Issues 1–3, 25 January 2006, Pages 268-271
Chemical Physics Letters

Calculation of the transition state theory rate constant for a general reaction coordinate: Application to hydride transfer in an enzyme

https://doi.org/10.1016/j.cplett.2005.10.129Get rights and content

Abstract

An expression for the transition state theory rate constant is provided in terms of the potential of mean force for a general reaction coordinate and the mass-weighted gradient of this reaction coordinate. The form of the rate constant enables the straightforward calculation of rates for infrequent events with conventional umbrella sampling and free energy perturbation methods. The approach is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase using a hybrid quantum/classical molecular dynamics method. Inclusion of the nuclear quantum effects of the transferring hydrogen increases the transition state theory rate constant by a factor of 244.

Introduction

Transition state theory (TST) [1], [2] has played a central role in theoretical studies of chemical reactions in solution and proteins. In the framework of TST, the rate constant for a unimolecular chemical reaction is expressed ask=κkTST.In this expression, kTST is the transition state theory rate constant, which is defined to be the equilibrium flux in the forward direction through a dividing surface separating the reactant and product regions of configuration space, and κ is the transmission coefficient, which accounts for dynamical recrossings of the dividing surface. The transition state theory rate constant can be expressed in terms of the free energy of activation ΔG:kTST=1βhe-βΔG,where β = 1/kBT. The transmission coefficient can be calculated with molecular dynamics methods in which trajectories are started at the dividing surface and are propagated backward and forward in time [3], [4], [5], [6], [7].

This Letter focuses on the calculation of kTST for condensed phase reactions. We provide an expression for kTST in terms of the potential of mean force (PMF) for a general reaction coordinate. Although this expression is related to those previously derived in [8], [9], the form of the rate constant presented here enables the straightforward calculation of rates for infrequent events with conventional umbrella sampling and free energy perturbation methods [10], [11], [12], [13]. To illustrate the utility of this approach, we apply this expression, in conjunction with a hybrid quantum/classical molecular dynamics method [14], to hydride transfer in the enzyme dihydrofolate reductase (DHFR). Furthermore, we examine the impact of including the nuclear quantum effects of the transferring hydrogen.

Section snippets

Theory

In the reactive flux formulation of transition state theory [15], [16], [17], [18], the TST rate constant can be expressed in terms of the reactive flux through a dividing surface that separates the reactants and products,kTST=δ(ξ-ξ)ξ˙Θ(ξ˙)Θ(ξ-ξ).Here, ξ is the reaction coordinate, ξ = ξ at the dividing surface, and the reactant and product regions are defined as ξ < ξ and ξ > ξ, respectively. In Eq. (3), the term ξ˙ is the time derivative of the reaction coordinate, δ is the Dirac delta

Computational methods

We applied Eq. (15) to hydride transfer in the enzyme DHFR using the hybrid quantum/classical molecular dynamics methodology developed in our group [14]. Since the details of these specific simulations are given elsewhere [22], [23], here we provide only a brief description of the method and the simulations. The reaction studied is the hydride transfer from the NC4 position of the NADPH (nicotinamide adenine dinucleotide phosphate) cofactor to the C6 position of the protonated dihydrofolate

Results and discussion

We used Eq. (15) to calculate the TST rate constant for hydride transfer in DHFR with the hydride treated both classically and quantum mechanically. The value of the first factor, {[Zξ/(2πβ)]1/2}ξcond, is 1.56 × 1015 (kcal/mol)/s for the classical treatment and 1.00 × 1016 (kcal/mol)/s for the quantum mechanical treatment of the transferring hydrogen nucleus. The value of the second factor, [e-βW(ξ)/-ξdξe-βW(ξ)], is 1.02 × 10−13 (kcal/mol)−1 for the classical treatment and 3.89 × 10−12 (kcal/mol)−1

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