Extraction of kinetics from equilibrium distributions of states using the Metropolis Monte Carlo method
The Metropolis Monte Carlo (MC) method is used to extract reaction kinetics from a given equilibrium distribution of states of a complex system. The approach is illustrated by the folding/unfolding reaction for two proteins - a model beta-hairpin and a-helical protein a3D. For beta-hairpin, the free energy surfaces (FESs) and free energy profiles (FEPs) are employed as the equilibrium distributions of states, playing a role of the potentials of mean force to determine the acceptance probabilities of new states in the MC simulations. Based on the FESs and PESs for a set of temperatures that were simulated with the molecular dynamics (MD) method, the MC simulations are performed to extract folding/ unfolding rates. It has been found that the rate constants and first-passage time (FPT) distributions obtained in the MC simulations change with temperature in good agreement with those from the MD simulations. For a3D protein, whose equilibrium folding/unfoldingwas studied with the single-molecule FRET method (Chung et al., J. Phys. Chem. A, 115, 2011, 3642), the FRET-efficiency histograms at different denaturant concentrations were used as the equilibrium distribution of protein states. It has been found that the rate constants for folding and unfolding obtained in the MC simulations change with denaturant concentration in reasonable agreement with the constants that were extracted from the photon trajectories on the basis of theoretical models. The simulated FPT distributions are single-exponential, which is consistent with the assumption of two-state kinetics that was made in the theoretical models. The promising feature of the present approach is that it is based solely on the equilibrium distributions of states, without introducing any additional parameters to perform simulations, which suggests its applicability to other complex systems.