Unbiased Rare Event Sampling in Spatial Stochastic Systems Biology Models Using a Weighted Ensemble of Trajectories

Abstract

The long-term goal of connecting scales in biological simulation can be facilitated by scale-agnostic methods. We demonstrate that the weighted ensemble (WE) strategy, initially developed for molecular simulations, applies effectively to spatially resolved cell-scale simulations. The WE approach runs an ensemble of parallel trajectories with assigned weights and uses a statistical resampling strategy of replicating and pruning trajectories to focus computational effort on difficult-to-sample regions. The method can also generate unbiased estimates of non-equilibrium and equilibrium observables, sometimes with significantly less aggregate computing time than would be possible using standard parallelization. Here, we use WE to orchestrate particle-based kinetic Monte Carlo simulations, which include spatial geometry (e.g., of organelles, plasma membrane) and biochemical interactions among mobile molecular species. We study a series of models exhibiting spatial, temporal and biochemical complexity and show that although WE has important limitations, it can achieve performance significantly exceeding standard parallel simulation—by orders of magnitude for some observables.

Weighted Ensemble Simulation: Review of Methodology, Applications, and Software

“…When spatial degrees of freedom are introduced into cellular-scale models, the simulation challenges are still greater. WE was able to sample previously inaccessible low calcium concentrations in a neuromuscular junction model and reproduce experimentally reported power-law behavior (Figure 3), as well as accessing rare events in a complex signaling model embedded in realistic cell and organelle geometry (22). …”

“…In addition, the software can be designed to be interoperable, that is, interfacing with any dynamics engine, because the algorithm does not require under-the-hood modifications of the dynamics engine. To our knowledge, the first freely available, highly scalable, and interoperable WE software package was WESTPA (74), which has been widely applied to a variety of systems ranging in scale from atomistic to cellular (2–4, 21, 22, 59, 62, 63, 75, 76) and interfaced with a diversity of dynamics engines, for example, with GROMACS(36), NAMD(49), OpenMM (23), and AMBER (12) at the atomistic/molecular scale, including GPU versions, and with UIOWA-BD (25, 30), BioNetGen (11, 26), and MCell (39) at the cellular scale. The WESTPA package embodies a wide range of WE capabilities, including the calculation of both equilibrium and kinetic observables using on-the-fly reweighting (10) or a post-analysis procedure (62) as well as plug-ins for using the WE-based string method (3) and WExplore, a recently developed WE strategy that defines sampling regions in a hierarchical fashion (16).…”

“…Several years later, the WE strategy was proven to yield continuous pathways and rate constants in a rigorous manner for any type of stochastic dynamics, including Monte Carlo, BD, and MD, thereby highlighting the generality of the strategy (70). Regardless of the type of dynamics or scale of the system, numerous studies have demonstrated that the WE strategy exhibits significant super-linear scaling in estimating observables; that is, quantities such as rate constants can be estimated with orders of magnitude fewer overall computing resources compared to standard parallelized simulations because WE-spawned trajectories can be focused on sampling bottlenecks (3, 17, 21, 22, 53, 55, 69, 75). Given this efficiency, there has been a resurgence of interest in the WE strategy over the past decade, resulting in the successful generation of pathways and calculation of rate constants for long-timescale processes that would otherwise be infeasible to simulate.…”

“…Here, we review recent advances and applications of the WE path sampling approach, which has enabled a number of otherwise unfeasible applications (2, 17, 19, 22, 55, 59, 76). We present the theoretical basis of WE and also discuss ongoing challenges and future directions for this promising approach.…”

Automatic error control during forward flux sampling of rare events in master equation models

“…Consequently, many varieties of ES can be applied to stochastic biochemical systems with rare event dynamics. Three of the most well known candidates are forward flux sampling (FFS) [32][33][34][35] , nonequilibrium umbrella sampling [36][37][38] , and weighted ensemble [39][40][41][42][43] . Although much discussion has taken place regarding the strengths and weaknesses of each of these methods 44,45 there is no consensus as to an optimal approach.…”