^{1}, Ron Elber

^{2}and David Shalloway

^{3}

### Abstract

A recently introduced computational algorithm to extend time scales of atomically detailed simulations is illustrated. The algorithm, milestoning, is based on partitioning the dynamics to a sequence of trajectories between “milestones” and constructing a non-Markovian model for the motion along a reaction coordinate. The kinetics of a conformational transition in a blocked alanine is computed and shown to be accurate, more efficient than straightforward molecular dynamics by a factor of about 9, and nonexponential. A general scaling argument predicts a linear speedup with the number of milestones for diffusive processes and an exponential speedup for transitions over barriers. The algorithm is also trivial to parallelize. As a side result, milestoning also produces the free energy profile along the reaction coordinate and is able to describe nonequilibrium motions along one (or a few) degrees of freedom.

The authors thank Eric van den Eijnden for many useful discussions, and John Straub and Eric Darve for pointing us to important references. This research was supported by a NIH Grant No. GM59796 to one of the authors (R.E.). The authors acknowledge NIH Grant No. RR020889 for the purchase of a computer cluster on which these calculations were performed.

I. INTRODUCTION

II. THEORY

III. THE COMPUTATIONAL GAIN OF MILESTONING

A. Parallelization

B. Diffusive enhancement

C. Exponential bootstrapping

IV. THE MILESTONING APPROXIMATION

V. METHODS

VI. RESULTS

VII. MARKOVMODEL AND A SINGLE EXPONENTIAL RELAXATION

VIII. DISCUSSION

A. Avoiding a reaction coordinate

B. Generating reaction coordinates

C. Computing rates from reaction coordinate

D. Computing free energies

E. Non-Markovian model

F. Extending variables of milestoning

G. Extending milestoning to more than one dimension

### Key Topics

- Free energy
- 21.0
- Markov processes
- 14.0
- Peptides
- 14.0
- Molecular dynamics
- 11.0
- Trajectory models
- 9.0

## Figures

A schematic representation of the milestoning approach. A set of nonintersecting hypersurfaces establishes “milestones” between the reactant (R) and the product (P). The circles indicate the volume in coordinate space that defines the boundaries of each of the final states. Also shown (dashed line) is a single trajectory initiated at and terminated at . An ensemble of such trajectories is used to compute the first passage time distributions between the milestones (see text for more details).

A schematic representation of the milestoning approach. A set of nonintersecting hypersurfaces establishes “milestones” between the reactant (R) and the product (P). The circles indicate the volume in coordinate space that defines the boundaries of each of the final states. Also shown (dashed line) is a single trajectory initiated at and terminated at . An ensemble of such trajectories is used to compute the first passage time distributions between the milestones (see text for more details).

A ball-and-stick model of blocked alanine, the molecular system whose transition from an helix to a sheet in aqueous solution is studied in the present paper (water molecules not shown). Also shown is the dihedral angle that was used as a reaction coordinate. The conformation shown is .

A ball-and-stick model of blocked alanine, the molecular system whose transition from an helix to a sheet in aqueous solution is studied in the present paper (water molecules not shown). Also shown is the dihedral angle that was used as a reaction coordinate. The conformation shown is .

projection of the OEQ ensembles of the simulation, showing hyperplanes 25, 33, 49, 65, 81, 97, 113, and 129. Note that the hyperplane is defined in Cartesian space (including all the peptide degrees of freedom). The points shown satisfy the Cartesian constraints exactly, so their projection onto the (non-Cartesian) plane is not one dimensional.

projection of the OEQ ensembles of the simulation, showing hyperplanes 25, 33, 49, 65, 81, 97, 113, and 129. Note that the hyperplane is defined in Cartesian space (including all the peptide degrees of freedom). The points shown satisfy the Cartesian constraints exactly, so their projection onto the (non-Cartesian) plane is not one dimensional.

The local first passage time distribution of the population initiated at milestone 6 to milestone 5 (the trajectories are terminated either at milestone 5 or 7). This distribution is taken from the seven-milestone simulations (see Table I).

The local first passage time distribution of the population initiated at milestone 6 to milestone 5 (the trajectories are terminated either at milestone 5 or 7). This distribution is taken from the seven-milestone simulations (see Table I).

Convergence of the local first passage time (between milestones) in picoseconds as a function of the number of trajectories. Although we have used 3000 trajectories, reasonable convergence can be achieved with just a few hundred.

Convergence of the local first passage time (between milestones) in picoseconds as a function of the number of trajectories. Although we have used 3000 trajectories, reasonable convergence can be achieved with just a few hundred.

The rate of product appearance (milestone is absorbing) as a function of the number of milestones. The results for three milestones are exact and calculations with 19 milestones (or less) are accurate. Corresponding overall first passage time are given in parentheses. Note that the calculation of the time course does not assume exponential kinetics. Indeed the early-time regime shows a significant deviation from exponential behavior. See further discussion on the nonexponential kinetics of the transition in alanine dipeptide in Sec. VII.

The rate of product appearance (milestone is absorbing) as a function of the number of milestones. The results for three milestones are exact and calculations with 19 milestones (or less) are accurate. Corresponding overall first passage time are given in parentheses. Note that the calculation of the time course does not assume exponential kinetics. Indeed the early-time regime shows a significant deviation from exponential behavior. See further discussion on the nonexponential kinetics of the transition in alanine dipeptide in Sec. VII.

The dependence of the overall first passage time on the number of milestones. We obviously expect the accuracy of the rate calculations to increase as the number of milestones decreases. However, the plot shows early convergence to the right answer. The calculations are accurate with 19 and fewer milestones. The speedup obtained with 19 milestones compared to straightforward MD trajectories is a factor of 9. See text for the analysis.

The dependence of the overall first passage time on the number of milestones. We obviously expect the accuracy of the rate calculations to increase as the number of milestones decreases. However, the plot shows early convergence to the right answer. The calculations are accurate with 19 and fewer milestones. The speedup obtained with 19 milestones compared to straightforward MD trajectories is a factor of 9. See text for the analysis.

Velocity correlation function for the dihedral angle. A rough estimate for the relaxation time suggests that it is smaller than . Instead of an exact time derivative, we have used the finite difference expression .

Velocity correlation function for the dihedral angle. A rough estimate for the relaxation time suggests that it is smaller than . Instead of an exact time derivative, we have used the finite difference expression .

Computation of free energies with milestoning. Multiple free energy curves as a function of the dihedral angle are shown for a different number of milestones. Exact results (squares) were obtained by a long MD trajectory. (a) includes free energy profiles (going from top to the lower curves) of runs with 144, 74, 73, 37, 19, 11, 7, and 5 milestones, while the lower figure (b) only (, 17, 7, and 5). When the milestoning approximation breaks down, the free energy surface is distorted in values, but it still maintains the correct positions of maxima and minima. Note that for a sufficiently small number of milestones (in which the kinetics are described accurately), the coarse free energy surfaces are similar.

Computation of free energies with milestoning. Multiple free energy curves as a function of the dihedral angle are shown for a different number of milestones. Exact results (squares) were obtained by a long MD trajectory. (a) includes free energy profiles (going from top to the lower curves) of runs with 144, 74, 73, 37, 19, 11, 7, and 5 milestones, while the lower figure (b) only (, 17, 7, and 5). When the milestoning approximation breaks down, the free energy surface is distorted in values, but it still maintains the correct positions of maxima and minima. Note that for a sufficiently small number of milestones (in which the kinetics are described accurately), the coarse free energy surfaces are similar.

A schematic drawing of the termination condition of a milestoning trajectory that follows a circle. The toy model is an illustration of a potential error in determining termination times (see text for more details).

A schematic drawing of the termination condition of a milestoning trajectory that follows a circle. The toy model is an illustration of a potential error in determining termination times (see text for more details).

## Tables

Summary of milestoning runs. A detailed list of all milestoning runs performed in this study. The run with three milestones is exact. The run with 19 milestones or less provides accurate answers.

Summary of milestoning runs. A detailed list of all milestoning runs performed in this study. The run with three milestones is exact. The run with 19 milestones or less provides accurate answers.

The first column is the number of milestones that were used in the rate calculation. The second column shows computed MFPTs of the alanine dipeptide transition, as predicted by Eq. (5). Statistical errors for the first passage time (in brackets) were computed according to the procedure of Ref. 4. The results of the last row are for an exact model. The third column shows the predictions of a single rate constant (exponential rate, see Sec. VII). Note the significant deviation of the MFPT from the prediction of a single, exponential reaction, i.e., the prediction. The fourth column shows the milestone AIT. The last two columns are the standard deviations of the first passage time distribution with the Markovian model (not single exponential!) and with milestoning. The second moment is a measure of the nonexponential behavior of the kinetics. We did not compute the second moments for simulations with because even their first moments are highly inaccurate. Overall, the differences between the Markovian model and milestoning are not large. For the non-Markovian contribution to be important, the incubation kinetics must be nonexponential. This is the case of three milestones.

The first column is the number of milestones that were used in the rate calculation. The second column shows computed MFPTs of the alanine dipeptide transition, as predicted by Eq. (5). Statistical errors for the first passage time (in brackets) were computed according to the procedure of Ref. 4. The results of the last row are for an exact model. The third column shows the predictions of a single rate constant (exponential rate, see Sec. VII). Note the significant deviation of the MFPT from the prediction of a single, exponential reaction, i.e., the prediction. The fourth column shows the milestone AIT. The last two columns are the standard deviations of the first passage time distribution with the Markovian model (not single exponential!) and with milestoning. The second moment is a measure of the nonexponential behavior of the kinetics. We did not compute the second moments for simulations with because even their first moments are highly inaccurate. Overall, the differences between the Markovian model and milestoning are not large. For the non-Markovian contribution to be important, the incubation kinetics must be nonexponential. This is the case of three milestones.

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