Abstract
Milestoning is a method used to calculate the kinetics and thermodynamics of molecular processes occurring on time scales that are not accessible to brute force molecular dynamics (MD). In milestoning, the conformation space of the system is sectioned by hypersurfaces (milestones), an ensemble of trajectories is initialized on each milestone, and MD simulations are performed to calculate transitions between milestones. The transition probabilities and transition time distributions are then used to model the dynamics of the system with a Markov renewal process, wherein a long trajectory of the system is approximated as a succession of independent transitions between milestones. This approximation is justified if the transition probabilities and transition times are statistically independent. In practice, this amounts to a requirement that milestones are spaced such that trajectories lose position and velocity memory between subsequent transitions. Unfortunately, limiting the number of milestones limits both the resolution at which a system's properties can be analyzed, and the computational speedup achieved by the method. We propose a generalized milestoning procedure, milestoning with transition memory (MTM), which accounts for memory of previous transitions made by the system. When a reaction coordinate is used to define the milestones, the MTM procedure can be carried out at no significant additional expense as compared to conventional milestoning. To test MTM, we have applied its version that allows for the memory of the previous step to the toy model of a polymer chain undergoing Langevin dynamics in solution. We have computed the mean first passage time for the chain to attain a cyclic conformation and found that the number of milestones that can be used, without incurring significant errors in the first passage time is at least 8 times that permitted by conventional milestoning. We further demonstrate that, unlike conventional milestoning, MTM permits milestones to be spaced such that trajectories do not have enough time to lose their velocity memory between successively crossed milestones.
We are indebted to Ron Elber, Ernst-Ludwig Florin, and Serdal Kirmizialtin for advice and many discussions. This work was supported by the National Science Foundation through grants CHE 0848571 and PHY05-51164.
I. INTRODUCTION
II. THE M1TM METHOD
A. Conventional milestoning
B. Milestoning with memory
C. Relationship between M1TM and milestoning
D. Computation of and π_{ ijk }(τ)
III. APPLICATION OF M1TM TO A MODEL SYSTEM
IV. CONCLUDING REMARKS
Key Topics
- Polymers
- 17.0
- Markov processes
- 8.0
- Molecular dynamics
- 8.0
- Brownian dynamics
- 4.0
- Eigenvalues
- 4.0
Figures
A schematic representation of a trajectory x(t) that crosses milestones M _{1}, M _{2}, M _{3}, M _{4}. The piecewise continuous function i(t) tracks the index of the milestone the trajectory last crossed, which indicates the state of the molecular system in the reduced description of milestoning. The ordered pair above each continuous portion of i(t) defines the M1TM state of the molecular system.
A schematic representation of a trajectory x(t) that crosses milestones M _{1}, M _{2}, M _{3}, M _{4}. The piecewise continuous function i(t) tracks the index of the milestone the trajectory last crossed, which indicates the state of the molecular system in the reduced description of milestoning. The ordered pair above each continuous portion of i(t) defines the M1TM state of the molecular system.
A schematic representation of a trajectory confined between milestones j − 1 and j + 1 by Eqs. (9a) and (9b). We illustrate how a single transition from one M1TM state to another may be observed. The trajectory starts by colliding with milestone j − 1 and is reflected. It eventually crosses milestone j, at a point labeled with an “X,” to denote the crossing point as a first hitting point of the M1TM state (j – 1, j). The trajectory then goes on to collide with j + 1, signifying a transition to (j + 1, j). The time for this transition to occur provides a sample of the distribution π_{ ijk }(t), and the values of n _{ ijk } and n _{ jk } may be updated accordingly.
A schematic representation of a trajectory confined between milestones j − 1 and j + 1 by Eqs. (9a) and (9b). We illustrate how a single transition from one M1TM state to another may be observed. The trajectory starts by colliding with milestone j − 1 and is reflected. It eventually crosses milestone j, at a point labeled with an “X,” to denote the crossing point as a first hitting point of the M1TM state (j – 1, j). The trajectory then goes on to collide with j + 1, signifying a transition to (j + 1, j). The time for this transition to occur provides a sample of the distribution π_{ ijk }(t), and the values of n _{ ijk } and n _{ jk } may be updated accordingly.
For the purpose of this paper, polymer cyclization is a process, in which the polymer starts with a predefined end-to-end distance R _{ i } and eventually acquires a cyclic configuration, where the end-to-end distance becomes equal to a capture radius R _{ f }.
For the purpose of this paper, polymer cyclization is a process, in which the polymer starts with a predefined end-to-end distance R _{ i } and eventually acquires a cyclic configuration, where the end-to-end distance becomes equal to a capture radius R _{ f }.
The predicted MFPT of cyclization vs the number of milestones used for (a) M1TM and (b) conventional milestoning, for chains with L = 21, 41, and 61 beads. The capture radius for each chain was selected such that MFPT of cyclization predicted by brute force is τ_{ c }/τ_{ R } = 38 or τ_{ c }/τ_{ R } = 10. In both calculations the data points corresponding to 2 milestones represent the exact result (i.e., no intermediate milestones were used).
The predicted MFPT of cyclization vs the number of milestones used for (a) M1TM and (b) conventional milestoning, for chains with L = 21, 41, and 61 beads. The capture radius for each chain was selected such that MFPT of cyclization predicted by brute force is τ_{ c }/τ_{ R } = 38 or τ_{ c }/τ_{ R } = 10. In both calculations the data points corresponding to 2 milestones represent the exact result (i.e., no intermediate milestones were used).
Autocorrelation function of the relative velocity of the polymer ends.
Autocorrelation function of the relative velocity of the polymer ends.
The relative statistical error of a calculation of the mean first passage time τ_{ c } for a 21 bead polymer to form a cyclic conformation as a function of the minimum number of trajectories n _{ m } sampled for each possible M1TM transition (see Sec. II, Computation of and π_{ ijk }(τ), for the definition of n _{ m }). The calculations were performed for the case of 16 milestones and the value of the capture radius was adjusted such that τ_{ c }/τ_{ R } = 38. In each calculation of the statistical error, we performed 180 independent calculations of the MFPT, and took the statistical error to be the standard deviation of the sample divided by the average of the sample. The results of milestoning calculations are given by circles and the results of M1TM calculations are given by squares.
The relative statistical error of a calculation of the mean first passage time τ_{ c } for a 21 bead polymer to form a cyclic conformation as a function of the minimum number of trajectories n _{ m } sampled for each possible M1TM transition (see Sec. II, Computation of and π_{ ijk }(τ), for the definition of n _{ m }). The calculations were performed for the case of 16 milestones and the value of the capture radius was adjusted such that τ_{ c }/τ_{ R } = 38. In each calculation of the statistical error, we performed 180 independent calculations of the MFPT, and took the statistical error to be the standard deviation of the sample divided by the average of the sample. The results of milestoning calculations are given by circles and the results of M1TM calculations are given by squares.
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