Error and efficiency of replica exchange molecular dynamics simulations.

We derive simple analytical expressions for the error and computational efficiency of replica exchange molecular dynamics (REMD) simulations (and by analogy replica exchange Monte Carlo simulations). The theory applies to the important case of systems whose dynamics at long times is dominated by the slow interconversion between two metastable states. As a specific example, we consider the folding and unfolding of a protein. The efficiency is defined as the rate with which the error in an estimated equilibrium property, as measured by the variance of the estimator over repeated simulations, decreases with simulation time. For two-state systems, this rate is in general independent of the particular property. Our main result is that, with comparable computational resources used, the relative efficiency of REMD and molecular dynamics (MD) simulations is given by the ratio of the number of transitions between the two states averaged over all replicas at the different temperatures, and the number of transitions at the single temperature of the MD run. This formula applies if replica exchange is frequent, as compared to the transition times. High efficiency of REMD is thus achieved by including replica temperatures in which the frequency of transitions is higher than that at the temperature of interest. In tests of the expressions for the error in the estimator, computational efficiency, and the rate of equilibration we find quantitative agreement with the results both from kinetic models of REMD and from actual all-atom simulations of the folding of a peptide in water.