Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics

Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. Local kinetics between milestones are aggregated to compute the flux through the entire reaction space. In addition to the accuracy of force fields, the accuracy of Milestoning crucially depends on two factors: the initial distribution of the short trajectory ensemble and statistical adequacy of trajectory sampling. The latter can be improved by increasing the number of trajectories while the true initial distribution, i.e., first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we propose two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD for systems at equilibrium condition. Starting from equilibrium Boltzmann distribution on milestones, we calculate the proper weighting factor for equilibrium FHPD approximation and consequently for trajectory flux. The methods are tested on two model examples for illustration purpose. Both methods improve significantly over the widely used classical Milestoning method in terms of the accuracy of mean first passage time (MFPT). In particular, BI-M covers the directional Milestoning method as a special case in the deterministic Hamiltonian dynamics. LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. Furthermore, a locally iterative correction algorithm for FHPD is proposed for exact MFPT calculation on the basis of LPT-M/BI-M, which is much cheaper than the exact Milestoning method.