Error and efficiency of replica exchange molecular dynamics simulations

Abstract

We derive simple analytical expressions for the error and computational efficiency of replica exchange molecular dynamics ͑REMD͒ simulations ͑and by analogy replica exchange Monte Carlo simulations͒. 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 ͑MD͒ 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.

Error and efficiency of simulated tempering simulations

“…Even though we explicitly consider only the case of two-state protein folding, the results apply generally to systems with two metastable states that interconvert slowly compared to the relaxation processes within each state. Importantly, 16 in the two-state case the relative computational efficiencies will not depend on the particular property A. In such systems, A will quickly relax to the average values ͗A͘ F and ͗A͘ U in folded and unfolded states F and U, respectively.…”

“…We accordingly define the computational efficiency of the simulations as the rate with which the statistical error in the estimate of the property of interest decreases with the simulation time. Following our earlier work on the efficiency of REMD simulations, 16 we note that for simulation times t sim long compared to the overall longest relaxation time of the system, the central limit theorem implies that the error in the estimate Ā of the exact mean ͗A͘ of any property A decreases as var͑Ā ͒ = c / t sim , where var͑Ā ͒ indicates the variance about the true mean in multiple simulations of the same duration t sim . The constant c depends on the simulation method ͑MD versus ST͒.…”

“…Similar master-equation descriptions resulting in simple first-order kinetic models have previously been used to model replica exchange simulations. 16,[18][19][20][21] …”

“…5,6,[11][12][13][14][15] Key questions in ST, REMD, REMC, or other extended ensemble simulations are as follows: ͑1͒ How large is the possible gain in computational efficiency, ͑2͒ for which simulation parameters can this maximum gain be realized, and ͑3͒ how do the different methods compare with respect to their efficiency gains? We have previously developed a simple analytical formula for the efficiency gain in REMD and REMC simulations 16 for the important case that the slow dynamics of the system of interest can be described by twostate transitions. Here, we derive a formula for the error and computational efficiency of ST simulations.…”

Deterministic replica-exchange method without pseudo random numbers for simulations of complex systems

“…Abraham and Gready introduced some measurement and compared the results [21]. Rosta and Hummer [22] evaluated the practical efficiency of REM simulation for protein folding with a two-state model. However, the examination of the condition for convergence of REM is difficult partly because the mixing of temperature in REM is determined by pseudo random numbers with Metropolis criteria.…”