A Mathematical Theory of Optimal Milestoning (with a Detour via Exact Milestoning)

TL;DR

This paper introduces optimal and exact milestoning methods that precisely compute mean first passage times in complex systems, enabling efficient kinetic analysis and importance sampling strategies.

Contribution

It provides a rigorous mathematical framework for optimal milestoning, allowing exact calculation of MFPTs and enhancing simulation efficiency compared to previous approaches.

Findings

Optimal milestoning allows exact MFPT computation.

Exact milestoning retains more dynamical information.

Importance sampling strategies improve efficiency.

Abstract

Milestoning is a computational procedure that reduces the dynamics of complex systems to memoryless jumps between intermediates, or milestones, and only retains some information about the probability of these jumps and the time lags between them. Here we analyze a variant of this procedure, termed optimal milestoning, which relies on a specific choice of milestones to capture exactly some kinetic features of the original dynamical system. In particular, we prove that optimal milestoning permits the exact calculation of the mean first passage times (MFPT) between any two milestones. In so doing, we also analyze another variant of the method, called exact milestoning, which also permits the exact calculation of certain MFPTs, but at the price of retaining more information about the original system's dynamics. Finally, we discuss importance sampling strategies based on optimal and exact milestoning that can be used to bypass the simulation of the original system when estimating the statistical quantities used in these methods.