Milestoning without a Reaction Coordinate

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Abstract
Milestoning is a method for calculating kinetics and thermodynamics of long time processes typically not accessible for straightforward Molecular Dynamics (MD) simulation. In the Milestoning approach, the system of interest is partitioned into cells by dividing hypersurfaces (Milestones) and transitions are computed between nearby hypersurfaces. Kinetics and thermodynamics are derived from the statistics of these transitions. The original Milestoning work concentrated on systems in which a one-dimensional reaction coordinate or an order parameter could be identified. In many biomolecular processes, the reaction proceeds via multiple channels or following more than a single-order parameter. A description based on a one-dimensional reaction coordinate may be insufficient. In the present paper, we introduce a variation that overcomes this limitation. Following the ideas of Vanden-Eijnden and Venturoli on Voronoi cells that avoid the use of an order parameter ( J. Chem. Phys. 2009, 130, 194101), we describe another way to “Milestone” systems without a reaction coordinate. We examine the assumptions of the Milestoning calculations of mean first passage times (MFPT) and describe strategies to weaken these assumptions. The method described in this paper, Directional Milestoning, arranges hypersurfaces in higher dimensions that “tag” trajectories such that efficient calculations can be done and at the same time the assumptions required for exact calculations of MFPTs are satisfied approximately. In the original Milestoning papers, trajectories are initiated from an equilibrium set of conformations. Here a more accurate distribution, that mimics the first hitting point distribution, is used. We demonstrate the usage of Directional Milestoning in conformational transitions of alanine dipeptide (in vacuum and in aqueous solution) and compare the correctness, efficiency, and statistical stability of the method with exact MD and with a related method.