Exact milestoning

Abstract

A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied.

Extracting the diffusion tensor from molecular dynamics simulation with Milestoning

“…16,50 Recently, an extension to Milestoning was introduced that makes this method formally exact (provided that the statistics is sufficient). 52 4. We assume that the approaches to place milestones in one and higher dimensions allow a good estimate of the components of diffusion tensor.…”

“…The assessment is possible since Milestoning is derived from atomically detailed trajectories, and can be made exact. 52 Furthermore, the high numerical efficiency of Milestoning makes it attractive for the construction of CV-based models since statistical convergence at relevant time scales can be achieved. Finally since Milestoning is based on the GME and not the ME, extensions to time dependent friction kernel are possible and are topic for future research.…”

Markovian and Non-Markovian Modeling of Membrane Dynamics with Milestoning

“…(8) (or Eq. (3)) can be solved with the help of Laplace transforms (the details of the derivation can be read somewhere else 20,23,24,34 ). The stationary flux vector (where the vector components are the stationary flux through each milestone) is computed as the solution of the linear equation ${\mathbf{q}}_{\mathrm{stat}}=\langle {\mathbf{q}}_{\mathrm{stat}}\mathrm{boldK}\rangle$…”

“…If the phase space points at the interface are also kept in the kernel (i.e. we have $K_{\alpha \beta }false(x_{\alpha },x_{\beta },tfalse)$ where $x_{\alpha }$ is a phase space point at interface $\alpha$), then the algorithm is exact 20 for time homogeneous processes. The trajectory sampling in the exact version of Milestoning is equivalent to the non-equilibrium umbrella sampling of Dinner 21 and trajectory tilting of vanden Eijnden.…”

Systematic Dissociation Pathway Searches Guided by Principal Component Modes

“…43–47 Because sampling a long and continuous trajectory for investigating complex conformational changes and binding kinetics may still be excessively time-consuming or not practical even when using enhanced MD or MC techniques, methods have been developed to run many short time-length MD simulations for several microstates and using a Markovian or non-Markovian model to describe the transitions among these microstates. 48–51 However, it is not always easy to define microstates and accurately model a continuous long binding/unbinding trajectory from multiple short MD simulations.…”