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 .
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).
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 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 .
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).
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.
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