^{1,a)}

### Abstract

Milestoning is a method used to calculate the kinetics of molecular processes occurring on timescales inaccessible to traditional molecular dynamics (MD) simulations. In the method, the phase space of the system is partitioned by milestones (hypersurfaces), trajectories are initialized on each milestone, and short MD simulations are performed to calculate transitions between neighboring milestones. Long trajectories of the system are then reconstructed with a semi-Markov process from the observed statistics of transition. The procedure is typically justified by the assumption that trajectories lose memory between crossing successive milestones. Here we present Milestoning with Coarse Memory (MCM), a generalization of Milestoning that relaxes the memory loss assumption of conventional Milestoning. In the method, milestones are defined and sample transitions are calculated in the standard Milestoning way. Then, after it is clear where trajectories sample milestones, the milestones are broken up into distinct neighborhoods (clusters), and each sample transition is associated with two clusters: the cluster containing the coordinates the trajectory was initialized in, and the cluster (on the terminal milestone) containing trajectory's final coordinates. Long trajectories of the system are then reconstructed with a semi-Markov process in an extended state space built from milestone and cluster indices. To test the method, we apply it to a process that is particularly ill suited for Milestoning: the dynamics of a polymer confined to a narrow cylinder. We show that Milestoning calculations of both the mean first passage time and the mean transit time of reversal—which occurs when the end-to-end vector reverses direction—are significantly improved when MCM is applied. Finally, we note the overhead of performing MCM on top of conventional Milestoning is negligible.

I am indebted to Dmitrii Makarov, Ron Elber, John Straub, Ernst-Ludwig Florin, Serdal Kirmizialtin, Ryan Cheng, Lei Huang, and Sai Konda for providing good advice and engaging in many helpful discussions. I am also thankful to the National Science Foundation (NSF) (Grant No. CHE 0848571) and the Robert A. Welch Foundation (Grant No. F-1514) for providing financial support for this project, and again to Dr. Makarov who procured their support. Computer time was generously provided by Texas Advanced Computing Center.

I. INTRODUCTION

II. MILESTONING WITH COARSE MEMORY

A. Theory

B. Initial configurations and transitions

C. Clustering

1. K-means clustering

2. Transition memory clustering

D. Extracting reaction pathways

E. Optimal number of clusters

III. APPLICATION OF MCM TO A MODEL PROCESS: POLYMER REVERSAL

A. Extracting timescales: The MFPT and MTT

B. Analysis of the statistical noise

IV. CONCLUDING REMARKS

### Key Topics

- Cluster analysis
- 23.0
- Polymers
- 11.0
- Molecular dynamics
- 8.0
- Probability theory
- 7.0
- Reaction mechanisms
- 7.0

## Figures

(a) A schematic representation of a trajectory **x**(*t*) that crosses milestones *H* _{1}, *H* _{2}, *H* _{3}, *H* _{4}. Each milestone is divided into 3 numbered regions (clusters), distinguished by their opacity. (b) The piecewise continuous function *i*(*t*) that tracks the index of the milestone the trajectory in (a) last crossed. Its value is the first index in the ordered pair above each continuous portion of *i*(*t*). The 2nd index of each ordered pair denotes the cluster the trajectory sampled when it transitioned to the current milestone. These ordered pairs define the MCM state of the molecular system. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

(a) A schematic representation of a trajectory **x**(*t*) that crosses milestones *H* _{1}, *H* _{2}, *H* _{3}, *H* _{4}. Each milestone is divided into 3 numbered regions (clusters), distinguished by their opacity. (b) The piecewise continuous function *i*(*t*) that tracks the index of the milestone the trajectory in (a) last crossed. Its value is the first index in the ordered pair above each continuous portion of *i*(*t*). The 2nd index of each ordered pair denotes the cluster the trajectory sampled when it transitioned to the current milestone. These ordered pairs define the MCM state of the molecular system. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

A fixed height histogram in which the set of initial configurations on milestone *i,* {*x* _{ i }}, is binned along *u*(*x*). Each short line is a value of *u*(*x*) assumed by a sample initial configuration (i.e., a first hitting point) at *x*. The height of each bin is fixed at 3 samples, and the intervals between the thick lines define the clusters on the milestone. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

A fixed height histogram in which the set of initial configurations on milestone *i,* {*x* _{ i }}, is binned along *u*(*x*). Each short line is a value of *u*(*x*) assumed by a sample initial configuration (i.e., a first hitting point) at *x*. The height of each bin is fixed at 3 samples, and the intervals between the thick lines define the clusters on the milestone. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

The reversal of a polymer, as monitored by the z coordinate of the polymer's end-to-end vector, *q*(*x*) = *z* _{ N } − *z* _{1}.

The reversal of a polymer, as monitored by the z coordinate of the polymer's end-to-end vector, *q*(*x*) = *z* _{ N } − *z* _{1}.

Figures (a) and (b) give the MFPT, τ, and the MTT, *t* of polymer reversal as a function of the pore radius (in units of bond length, σ), respectively. In MCM calculations, 40 MCM states were defined on milestone 1 (noting that it only has one reachable milestone), and 80 MCM states were defined on each subsequent milestone.

Figures (a) and (b) give the MFPT, τ, and the MTT, *t* of polymer reversal as a function of the pore radius (in units of bond length, σ), respectively. In MCM calculations, 40 MCM states were defined on milestone 1 (noting that it only has one reachable milestone), and 80 MCM states were defined on each subsequent milestone.

The runtime, *t* _{run}, for calculations of the MFPT as a function of pore radius. Circles are the runtimes of Milestoning calculations, and squares are the runtimes of the brute force MD calculations. For the brute force MD calculations, we performed 4 independent trials, in which 162 reversal events were observed in each trial at each radius. 64 processors (16 Opteron quad-core 64 bit processors with core frequency of 2.3 GHz) were run in parallel with a master-slave load balancing algorithm. On the same hardware, the MCM calculations were obtained by requiring that all allowed transitions between the M1TM states were sampled by a minimum of 600 trajectories. Figure 4 was generated from the same data used in these calculations. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

The runtime, *t* _{run}, for calculations of the MFPT as a function of pore radius. Circles are the runtimes of Milestoning calculations, and squares are the runtimes of the brute force MD calculations. For the brute force MD calculations, we performed 4 independent trials, in which 162 reversal events were observed in each trial at each radius. 64 processors (16 Opteron quad-core 64 bit processors with core frequency of 2.3 GHz) were run in parallel with a master-slave load balancing algorithm. On the same hardware, the MCM calculations were obtained by requiring that all allowed transitions between the M1TM states were sampled by a minimum of 600 trajectories. Figure 4 was generated from the same data used in these calculations. This figure has been previously published in a Ph.D. dissertation. ^{ 30 }

(a) The MFPT, τ, of reversal for a 35 bead polymer in a pore of radius 2.46σ, as a function of *n* _{ m }(see main text for definition). The ordinate is here normalized by the MFPT found from MD simulations, τ_{ MD }. (b) The ratio of the standard deviation of the MFPT, to τ_{ MD }, as a function of *n* _{ m }. The standard deviation was computed by 32 independent calculations of the MFPT. A total of 6 milestones were used in all calculations. Blue triangles correspond to conventional Milestoning calculations. For circles, 16 MCM states are defined on milestone 1, 32 MCM states are defined on subsequent milestones. For red squares, 4 MCM states are defined on milestone 1, 8 MCM states are defined on subsequent milestones. This figure has previously been published in a Ph.D. dissertation. ^{ 30 }

(a) The MFPT, τ, of reversal for a 35 bead polymer in a pore of radius 2.46σ, as a function of *n* _{ m }(see main text for definition). The ordinate is here normalized by the MFPT found from MD simulations, τ_{ MD }. (b) The ratio of the standard deviation of the MFPT, to τ_{ MD }, as a function of *n* _{ m }. The standard deviation was computed by 32 independent calculations of the MFPT. A total of 6 milestones were used in all calculations. Blue triangles correspond to conventional Milestoning calculations. For circles, 16 MCM states are defined on milestone 1, 32 MCM states are defined on subsequent milestones. For red squares, 4 MCM states are defined on milestone 1, 8 MCM states are defined on subsequent milestones. This figure has previously been published in a Ph.D. dissertation. ^{ 30 }

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