Simultaneous Computation of Dynamical and Equilibrium Information using a Weighted Ensemble of Trajectories

Abstract

Equilibrium formally can be represented as an ensemble of uncoupled systems undergoing unbiased dynamics in which detailed balance is maintained. Many non-equilibrium processes can be described by suitable subsets of the equilibrium ensemble. Here, we employ the "weighted ensemble" (WE) simulation protocol [Huber and Kim, Biophys. J., 1996] to generate equilibrium trajectory ensembles and extract non-equilibrium subsets for computing kinetic quantities. States do not need to be chosen in advance. The procedure formally allows estimation of kinetic rates between arbitrary states chosen after the simulation, along with their equilibrium populations. We also describe a related history-dependent matrix procedure for estimating equilibrium and nonequilibrium observables when phase space has been divided into arbitrary non-Markovian regions, whether in WE or ordinary simulation. In this proof-of-principle study, these methods are successfully applied and validated on two molecular systems: explicitly solvated methane association and the implicitly solvated Ala4 peptide. We comment on challenges remaining in WE calculations.

Computational estimation of ms-sec atomistic folding times

“…[66][67] We also employed haMSM analysis, which is unbiased for steady-state flux estimation at arbitrary lag times, and small lag times allow fuller use of the extensive WE data. 56,64,68 The approach is of particular interest for Protein G because, in principle, a haMSM can estimate steady-state behavior using trajectories generated in the transient period -i.e., in the approach to steady state. As noted, the flux profile for Protein G indicates those WE simulations clearly remained in the transient regime.…”

“…The novelty of the results is their relatively high precision and unbiased nature due to the theoretical foundations of the WE and haMSM methods. 35,56,64 Hence, although comparison to experimental folding times are shown in Table 1, readers are cautioned that the present study should be considered a first step in assessment of molecular models and initial ensembles. Given these caveats, the rough agreement with experimental values is encouraging but also points to the need for further investigation of solvent modeling and initial ensembles as discussed below.…”

“…[96][97] The continuous trajectories generated by WE allow application of the historyaugmented MSMs at arbitrary lag times, which are unbiased for estimation of steady-state fluxes. 56,64 Using short lag-times for haMSMs in turn allows use of all data generated. Figure 1: The WE procedure and comparison to regular MD simulation.…”

“…Although all these methods are theoretically well-grounded, WE does offer the pragmatic advantage of being fully independent of the dynamics engine employed, which has enabled its application with a wide range of both molecular and cell-scale simulation software. 34,[53][54][55][56][57][58][59] This versatility facilitated the integration of the WESTPA software package 60 with the GPUaccelerated version of the AMBER molecular dynamics package [61][62][63] as employed here. The WE method yields ensembles of fully continuous trajectories from which non-equilibrium observables can be calculated, including kinetic and mechanistic properties.…”

“…Importantly, the continuous WE trajectories spanning from unfolded to folded macrostates enable folding rate estimation using the history-augmented Markov state model (haMSM) formalism which is unbiased at arbitrary lag times. 56,64…”