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A final important area is the calculation of free energies with quantum mechanical models [72] or hybrid quantum mechanics/molecular mechanics models (QM/MM) [9]. Such models are being used to simulate enzymatic reactions and calculate activation free energies, providing unique insights into the catalytic efficiency of enzymes. They are reviewed elsewhere in this volume (see Chapter 11).

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10

Reaction Rates and

Transition Pathways

John E. Straub

Boston University, Boston, Massachusetts

I.INTRODUCTION

For 25 years, molecular dynamics simulations of proteins have provided detailed insights into the role of dynamics in biological activity and function [1–3]. The earliest simulations of proteins probed fast vibrational dynamics on a picosecond time scale. Fifteen years later, it proved possible to simulate protein dynamics on a nanosecond time scale. At present it is possible to simulate the dynamics of a solvated protein on the microsecond time scale [4]. These gains have been made through a combination of improved computer processing (Moore’s law) and clever computational algorithms [5].

In spite of these millionfold advances in the length of computer simulation, many dynamic processes remain outside the reach of direct dynamic simulation. Typically these processes involve the crossing of one or more energy barriers that stand between the reactant and product states. To overcome high energy barriers the system must concentrate an amount of energy far greater than the thermal energy in one or a few degrees of freedom. Such fluctuations are highly improbable and are known as ‘‘rare events’’—they occur only infrequently. Processes of this kind include ligand rebinding in heme proteins, proton transfer in enzymatic catalysis, and the configurational reorganization of a folding protein.

A. Defining Reactant and Product ‘‘States’’

The basic chemical description of rare events can be written in terms of a set of phenomenological equations of motion for the time dependence of the populations of the reactant and product species [6–9]. Suppose that we are interested in the dynamics of a conformational rearrangement in a small peptide. The concentration of reactant states at time t is NR(t), and the concentration of product states is NP(t). We assume that we can define the reactants and products as distinct ‘‘macrostates’’ that are separated by a ‘‘transition state’’ dividing surface. The transition state surface is typically the location of a significant energy barrier (see Fig. 1).

For example, when the energy barrier is high compared to the thermal energy, we can assume that when a reactant state is prepared there will be many oscillations in the reactant well before the system concentrates enough energy in the ‘‘reaction coordinate’’

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