The weighted ensemble (WE) path sampling approach orchestrates an ensemble of parallel calculations with intermittent communication to enhance the sampling of rare events, such as molecular associations or conformational changes in proteins or peptides. Trajectories are replicated and pruned in a way that focuses computational effort on under-explored regions of configuration space while maintaining rigorous kinetics. To enable the simulation of rare events at any scale (e.g. atomistic, cellular), we have developed an open-source, interoperable, and highly scalable software package for the execution and analysis of WE simulations: WESTPA (The Weighted Ensemble Simulation Toolkit with Parallelization and Analysis). WESTPA scales to thousands of CPU cores and includes a suite of analysis tools that have been implemented in a massively parallel fashion. The software has been designed to interface conveniently with any dynamics engine and has already been used with a variety of molecular dynamics (e.g. GROMACS, NAMD, OpenMM, AMBER) and cell-modeling packages (e.g. BioNetGen, MCell). WESTPA has been in production use for over a year, and its utility has been demonstrated for a broad set of problems, ranging from atomically detailed host-guest associations to non-spatial chemical kinetics of cellular signaling networks. The following describes the design and features of WESTPA, including the facilities it provides for running WE simulations, storing and analyzing WE simulation data, as well as examples of input and output.

Equilibrium formally can be represented as an ensemble of uncoupled systems undergoing unbiased dynamics in which detailed balance is maintained. Many nonequilibrium 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, 70, 97–110] to generate equilibrium trajectory ensembles and extract nonequilibrium 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.

Because standard molecular dynamics (MD) simulations are unable to access time scales of interest in complex biomolecular systems, it is common to “stitch together” information from multiple shorter trajectories using approximate Markov state model (MSM) analysis. However, MSMs may require significant tuning and can yield biased results. Here, by analyzing some of the longest protein MD data sets available (>100 μs per protein), we show that estimators constructed based on exact non-Markovian (NM) principles can yield significantly improved mean first-passage times (MFPTs) for protein folding and unfolding. In some cases, MSM bias of more than an order of magnitude can be corrected when identical trajectory data are reanalyzed by non-Markovian approaches. The NM analysis includes “history” information, higher order time correlations compared to MSMs, that is available in every MD trajectory. The NM strategy is insensitive to fine details of the states used and works well when a fine time-discretization (i.e., small “lag time”) is used.

As shown by previous theoretical and computational work, absolute entropies of small molecules that populate different conformers can be predicted accurately on the basis of the partitioning of the intramolecular entropy into vibrational and conformational contributions. Herein, we further elaborate on this idea and propose a protocol for entropy calculations of single molecules that combines the rigid rotor harmonic oscillator (RRHO) entropies with the direct sampling of the molecular conformational space by means of classical molecular dynamics simulations. In this approach, the conformational states are characterized by discretizing the time evolution of internal rotations about single bonds, and subsequently, the mutual information expansion (MIE) is used to approach the full conformational entropy from the converged probability density functions of the individual torsion angles, pairs of torsions, triads, and so on. This RRHO&MIE protocol could have broad applicability, as suggested by our test calculations on systems ranging from hydrocarbon molecules in the gas phase to a polypeptide molecule in aqueous solution. For the hydrocarbon molecules, the ability of the RRHO&MIE protocol to predict absolute entropies is assessed by carefully comparing theoretical and experimental values in the gas phase. For the rest of the test systems, we analyze the advantages and limitations of the RRHO&MIE approach in order to capture high order correlation effects and yield converged conformational entropies within a reasonable simulation time. Altogether, our results suggest that the RRHO&MIE strategy could be useful for estimating absolute and/or relative entropies of single molecules either in the gas phase or in solution.

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.

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Multibody type expansions like mutual information expansions are widely used for computing or analyzing properties of large composite systems. The power of such expansions stems from their generality. Their weaknesses, however, are the large computational cost of including high order terms due to the combinatorial explosion and the fact that truncation errors do not decrease strictly with the expansion order. Herein, we take advantage of the redundancy of multibody expansions in order to derive an efficient reformulation that captures implicitly all-order correlation effects within a given cutoff, avoiding the combinatory explosion. This approach, which is cutoff dependent rather than order dependent, keeps the generality of the original expansions and simultaneously mitigates their limitations provided that a reasonable cutoff can be used. An application of particular interest can be the computation of the conformational entropy of flexible peptide molecules from molecular dynamics trajectories. By combining the multibody local estimations of conformational entropy with average values of the rigid-rotor and harmonic-oscillator entropic contributions, we obtain by far a tighter upper bound of the absolute entropy than the one obtained by the broadly used quasi-harmonic method.

First-passage times (FPTs) are widely used to characterize stochastic processes such as chemical reactions, protein folding, diffusion processes or triggering a stock option. In previous work (Suarez et al., JCTC 2014;10:2658-2667, we demonstrated a non-Markovian analysis approach that, with a sufficient subset of history information, yields unbiased mean first-passage times from weighted-ensemble (WE) simulations. The estimation of the distribution of the first-passage times is, however, a more ambitious goal since it cannot be obtained by direct observation in WE trajectories. Likewise, a large number of events would be required to make a good estimation of the distribution from a regular "brute force" simulation. Here, we show how the previously developed non-Markovian analysis can generate approximate, but highly accurate, FPT distributions from WE data. The analysis can also be applied to any other unbiased trajectories, such as from standard molecular dynamics simulations. The present study employs a range of systems with independent verification of the distributions to demonstrate the success and limitations of the approach. By comparison to a standard Markov analysis, the non-Markovian approach is less sensitive to the user-defined discretization of configuration space.