^{1,a)}, Maddalena Venturoli

^{1,b)}, Giovanni Ciccotti

^{2,c)}and Ron Elber

^{3,d)}

### Abstract

Milestoning is a procedure to compute the time evolution of complicated processes such as barrier crossing events or long diffusive transitions between predefined states. Milestoning reduces the dynamics to transition events between intermediates (the milestones) and computes the local kinetic information to describe these transitions via short molecular dynamics (MD) runs between the milestones. The procedure relies on the ability to reinitialize MD trajectories on the milestones to get the right kinetic information about the transitions. It also rests on the assumptions that the transition events between successive milestones and the time lags between these transitions are statistically independent. In this paper, we analyze the validity of these assumptions. We show that sets of optimal milestones exist, i.e., sets such that successive transitions are indeed statistically independent. The proof of this claim relies on the results of transition path theory and uses the isocommittor surfaces of the reaction as milestones. For systems in the overdamped limit, we also obtain the probability distribution to reinitialize the MD trajectories on the milestones, and we discuss why this distribution is not available in closed form for systems with inertia. We explain why the time lags between transitions are not statistically independent even for optimal milestones, but we show that working with such milestones allows one to compute mean first passage times between milestones exactly. Finally, we discuss some practical implications of our results and we compare milestoning with Markov state models in view of our findings.

We are grateful to Philipp Metzner for providing us with the committor function data in the examples of Sec. III. We also thank the referees for their thoughtful comments that led us to rethink the results of this paper and enlarge its scope. Part of this work was performed while the authors were visiting the Erwin Schrödinger Institute (ESI) in Vienna whose support is gratefully acknowledged. This work was also partially supported by NIH under Grant No. GM59796, NSF under Grant Nos. DMS02-09959, DMS02-39625, and DMS07-08140, and ONR under Grant No. N00014-04-1-0565.

I. INTRODUCTION

II. OPTIMAL MILESTONING IN THE OVERDAMPED LIMIT

A. Isocommittor surfaces as optimal milestones

B. First hitting point density and the issue of reinitialization

C. Exact calculation of mean first passage times

III. ILLUSTRATIVE EXAMPLES

IV. OPTIMAL MILESTONING IN SYSTEMS WITH INERTIA

A. Isocommittor surfaces as optimal milestones

B. First hitting point density and the issue of reinitialization

C. Exact calculation of mean first passage times

V. SOME PRACTICAL CONSIDERATIONS

VI. CONCLUDING REMARKS

A. Milestoning versus MSM

B. Milestoning versus TIS and FFS

### Key Topics

- Statistical properties
- 17.0
- Probability theory
- 14.0
- Markov processes
- 9.0
- Molecular dynamics
- 6.0
- Sequence analysis
- 6.0

## Figures

Schematic of a piece of a long ergodic trajectory crossing a set of three milestones: , , and . In this example, , , , and . The part of the trajectory highlighted in bold contributes to one event counted in and the time contributes to the statistics of . The figure also shows the previous transition event from to , which contributes to with the time contributing to , and the next one from to , which contributes to with the time contributing to .

Schematic of a piece of a long ergodic trajectory crossing a set of three milestones: , , and . In this example, , , , and . The part of the trajectory highlighted in bold contributes to one event counted in and the time contributes to the statistics of . The figure also shows the previous transition event from to , which contributes to with the time contributing to , and the next one from to , which contributes to with the time contributing to .

Contour plot of the potential (23) with the three milestones , , and ( and are the boundaries of the reactant set and product set where and , respectively, and is the isocommittor surface for this reaction). The minimum energy path is also shown (dot-dashed line). The gray dots are snapshots every along trajectories starting from points distributed on according to Eq. (10), and the predicted density (10) of first hitting points on (thick black line) is compared to the equilibrium density on (dashed line).

Contour plot of the potential (23) with the three milestones , , and ( and are the boundaries of the reactant set and product set where and , respectively, and is the isocommittor surface for this reaction). The minimum energy path is also shown (dot-dashed line). The gray dots are snapshots every along trajectories starting from points distributed on according to Eq. (10), and the predicted density (10) of first hitting points on (thick black line) is compared to the equilibrium density on (dashed line).

Comparison of the probability density of first hitting points obtained by binning the location where trajectories started from points in distributed according to Eq. (10) hit (black solid curve) with the density [Eq. (10)] (black dashed curve) and with the equilibrium density (gray dot-dashed curve). We also computed the first hitting point density using a long unbiased trajectory and, up to statistical errors, it coincides with the solid curve shown in the figure.

Comparison of the probability density of first hitting points obtained by binning the location where trajectories started from points in distributed according to Eq. (10) hit (black solid curve) with the density [Eq. (10)] (black dashed curve) and with the equilibrium density (gray dot-dashed curve). We also computed the first hitting point density using a long unbiased trajectory and, up to statistical errors, it coincides with the solid curve shown in the figure.

Contour plot of the potential (23) with superimposed isocommittor surfaces used as milestones: from left to right, these surfaces are , , , , , , , , and .

Contour plot of the potential (23) with superimposed isocommittor surfaces used as milestones: from left to right, these surfaces are , , , , , , , , and .

Comparison of the probability density (10) (solid line) on the four milestones shown as thick lines in Fig. 4 with the equilibrium probability density (dashed line). The probability densities are plotted as functions of the arc length along the milestones.

Comparison of the probability density (10) (solid line) on the four milestones shown as thick lines in Fig. 4 with the equilibrium probability density (dashed line). The probability densities are plotted as functions of the arc length along the milestones.

Contour plot of the three-hole potential. We use three milestones, shown as vertical lines and corresponding to (from left to right) , , and . The density (10) (thick solid line) on the surface shows that the lower channel is the preferred one, even though the equilibrium density (thick dashed line) is peaked in the upper channel.

Contour plot of the three-hole potential. We use three milestones, shown as vertical lines and corresponding to (from left to right) , , and . The density (10) (thick solid line) on the surface shows that the lower channel is the preferred one, even though the equilibrium density (thick dashed line) is peaked in the upper channel.

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