We show how accurate kinetic information, such as the rates of protein folding and unfolding, can be extracted from replica-exchange molecular dynamics (REMD) simulations. From the brief and continuous trajectory segments between replica exchanges, we estimate short-time propagators in conformation space and use them to construct a master equation. For a helical peptide in explicit water, we determine the rates of transitions both locally between microscopic conformational states and globally for folding and unfolding. We show that accurate rates in the ~1/(100 ns) to ~1/(1 ns) range can be obtained from REMD with exchange times of 5 ps, in excellent agreement with results from long equilibrium molecular dynamics.
Replica exchange molecular dynamics (REMD) [1,2] is a powerful method to enhance the conformational sampling, addressing a serious challenge in molecular simulations [3]. Multiple non-interacting copies (or "replicas") of the system are simulated in parallel, each at a different temperature. To transfer the barrier-crossing efficiency from runs at high temperature to those at low temperature, configuration exchanges are attempted periodically (e.g., at time intervals δt REMD ) between replicas at different temperatures (T i and T j ). Those exchange attempts are accepted with a Metropolis probability P REMD (i ↔ j) = min{1, exp[(β j -β i )(U j -U i )]} that enforces detailed balance and maintains canonical distributions at each temperature [with U i the potential energy of the i-th replica, β i = 1/(k B T i ), and k B the Boltzmann constant]. After an accepted exchange, the particle velocities are appropriately re-scaled to the new temperature, or redrawn from respective Maxwell-Boltzmann distributions. Through a series of exchanges, high-temperature conformations are transferred occasionally to low temperature runs, facilitating the exploration of new configurationspace regions.
While enhancing the exploration of conformation space, REMD apparently does not permit the extraction of useful kinetic information. Conformation exchanges result in discontinuous trajectories, precluding the calculation of equilibrium time correlation functions for times longer than the exchange time δt REMD . To improve the sampling efficiency of REMD, the shortest possible δt REMD should be used [4]. With δt REMD much shorter than the time scales of slow conformational changes, the rates of conformational changes appear inaccessible to REMD simulations. To overcome this problem, at least for the special case of a two-state system, an indirect method has recently been proposed in which the two rate coefficients describing the assumed folding/unfolding dynamics are assumed to obey an Arrhenius temperature dependence [5]. However, the protein-folding rate often exhibits non-Arrhenius temperature dependence [6], and folding intermediates are common. To avoid the resulting problems, master-equation approaches have been described by Levy and co-workers [7] in a qualitative, yet insightful analysis. As a quantitative alternative, REMD has recently been used to estimate the local drift and diffusion coefficients [8] within the framework of coarse diffusion equations [9,10,11].
Here we show how one can efficiently extract accurate transition rates from REMD simulations, both locally between microscopic conformational states and globally between folded and unfolded conformations (and possible intermediates), without the assumption of a certain temperature dependence of the underlying kinetics. In fact, our method can be used to investigate the Arrhenius or non-Arrhenius character of a particular system. We determine short-time propagators in conformation space to overcome the problems arising from the intrinsically discontinuous character of REMD trajectories [12,13].
We first realize that REMD permits the accurate (and formally exact) calculation of short-time correlation functions. The initial configurations after a replica exchange (with appropriate velocity assignment) constitute valid representatives of the equilibrium phase-space distributions at the respective temperatures. From the subsequent Hamiltonian dynamics until the next exchange, we can obtain exact correlation functions. The maximum time scale will be a few δt REMD , given by the longest time between accepted replica exchanges.
Specifically, we here determine the frequency of transitions between conformational states. From the observed molecular transitions, we construct a master equation describing the dynamics in a conformation space divided into N distinct states. We later verify that the dynamics in the resulting projected space is captured by a master equation,
, where P i (t) is the population in state i, and k ij ≥ 0 is the transition rate from j to i = j. In vector-matrix notation, we have dP (t)/dt = KP (t), where the N × N rate matrix K has off-diagonal elements k ij and diagonal elements k ii =j =i k ji < 0. The propagators, defined as the probability of being in state j at time t given that the system was in state i at time 0, can be written in terms of the matrix exponential, p(j, t|i, 0) = [exp(Kt)] ji . To estimate the elements of the rate matrix K from either long equilibrium simulations or REMD, we use a maximum-likelihood procedure. We first determine the number N ji of transitions from state i to state j within a time interval ∆t, irrespective of intermediate states. The loglikelihood of observing transition numbers N ji is [12,13] ln
To obtain the rate coefficients of the master equation (with upper and lower diagonal elements related by detailed balance), we maximize ln L with respect to the k ij [12,13].
Effects of non-Markovian dynamics not c
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