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.
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).
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.
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