Weighted ensemble: Recent mathematical developments

Abstract

The weighted ensemble (WE) method is an enhanced sampling strategy based on periodically replicating and pruning trajectories generated in parallel. WE has grown increasingly popular for computational biochemistry problems, due in part to improved hardware and accessible software implementations. Algorithmic and analytical improvements have also played an important role, and progress has accelerated in recent years.Here, we discuss and elaborate on the WE method from a mathematical perspective, highlighting recent results which enhance the computational efficiency.The mathematical theory reveals a new strategy for optimizing trajectory management that approaches the best possible variance while generalizing to systems of arbitrary dimension.

Adaptive Sampling Methods for Molecular Dynamics in the Era of Machine Learning

“…The last approach that will be described differs from the previous ones because it was formulated to work under weighted ensembles rather than Markov state models. 16 Similar to the approach described before, the goal is to reduce the variance in a metric of importance to the kinetic model. In this case, rather than minimizing the variance of the first nontrivial eigenvalue, the goal is to reduce the variance in the estimated MFPT from the source state to a sink state under recycling boundary conditions (trajectories that enter the sink are immediately restarted from the source).…”

“…Following this allocation rule results in a minimization of the variance of the flux (and therefore the variance on the estimated MFPT). In comparison with the same number of "brute-force" parallel trajectories, we obtain a maximum ratio of variance reduction given by 16 = ( )…”

“…In particular, this result was used to show that in the low temperature limit the advantage with respect to brute-force simulations is exponential in the largest energy barrier. 16 To summarize, in this section, we introduced four theoretical approaches to dissect the advantage of adaptive sampling methods. As we moved from simple parallelization to sophisticated selective seeding criteria, we discussed the known theoretical results associated with each approach and their underlying assumptions.…”

Active Learning of the Conformational Ensemble of Proteins Using Maximum Entropy VAMPNets

“…In these methods, weights for each trajectory are tallied; splitting reduces the weight, and merging increases it. These weights are introduced with the goal of recovering statistically unbiased observables . On the other hand, adaptive sampling techniques tend to prioritize the exploration of a diverse set of molecular conformations by iteratively restarting short simulations from poorly sampled states.…”

“…These weights are introduced with the goal of recovering statistically unbiased observables. 23 On the other hand, adaptive sampling techniques tend to prioritize the exploration of a diverse set of molecular conformations by iteratively restarting short simulations from poorly sampled states. In past approaches, these states are obtained by using some discretization of the conformational space, such as clustering.…”