The two-dimensional periodic potential surface for α = 1.125, with the converged path of 20 centers (black dots) and corresponding Voronoi cells. Contour lines are separated by kT, and the corresponding color scale is shown in the same units. In each Voronoi cell, a random sample of the instantaneous positions visited by the replicas is shown in alternating light and dark gray dots for clarity.
Projections of the steady-state distributions for the two-dimensional periodic potential onto the y axis. Solid lines show the distributions for the common and rare parameter sets obtained using conventional sampling.
Convergence of the steady-state distributions projected onto the y axis for both conventional and WE simulations for the two-dimensional periodic potential where α = 1.125 (top) and α = 2.25 (bottom). For both the conventional and WE simulations, the errors are averages over individual error curves calculated for ten independent simulations using Eqs. (13) and (14) . The distributions for the WE simulations do not include statistics from the first 50τ. The target distribution for the shallow potential (top) is calculated from a single long conventional simulation 4.0 × 109 steps in length. For the deep potential (bottom), the target distribution was obtained by averaging ten conventional simulations of 2.0 × 1010 steps. In each case, separate simulations were used to calculate the error and the target distributions. The solid circle marks the average time at which all histogram windows were populated for the ten simulations.
The two-dimensional ring potential with two pathways. (Left panel) An instantaneous snapshot of all of the active replicas during a representative iteration, where replicas that have last visited A are shown in red, and those that last visited B shown in blue. The circles, centered at (−3,0) and (3,0), show the boundaries of the states A and B. (Center and right panels). The center and right panels show the converged strings with the corresponding Voronoi cells for the A → B, and B → A transitions, respectively. The circles delineate A and B as in the left panel. Contour lines are separated by kT, and the corresponding color scale is show in the same units.
Projections of the steady-state distribution for the two-dimensional ring potential onto θ for β = 1.0, 1.5, 2.0, and 2.5, obtained using conventional (lines) and weighted ensemble (circles) simulations. The probability of finding a particle in either of the two metastable intermediates at θ = −π/2 or π/2 decreases with increasing β.
Convergence of the A → B rate constant for the two-dimensional ring potential where β = 1.5 (top) and 2.5 (bottom). For both conventional and WE simulations, the errors are calculated using Eq. (16) , and are the averages of the error curves of ten independent simulations. For the conventional simulations, an estimate of the error cannot be obtained until the first transition from A to B is observed.
Efficiency of the WE method for the two-dimensional ring potential as a function of inverse temperature, β. Efficiency is defined as the ratio of the total aggregate simulation time, T X , to the system's natural MFPT: T X /MFPT, where T X is the time required to achieve a desired error, X. Thus, when the efficiency is 1 the total simulation time is the time required for the MFPT, which may be extremely long. We carried out this analysis for an order of magnitude error estimate (101) of the rate T 1 and a factor of three error estimate (100.5∼3) of the rate T 0.5.
The inactive (left) and active (right) conformations of the NtrC r receiver domain, generated from PDB IDs 1DC7 and 1DC8, respectively. (Center) Ten conformations taken from the Voronoi bin corresponding most closely to the q + = 0.5 (bin index 18). Each conformation is the average of 5 randomly selected snapshots. Coloring is based on residue indices starting from blue at the N-terminus and ending with red at the C-terminus.
Free energy G α associated with each Voronoi tessellation along the string for the elastic network model of NtrC r . The dark shaded region denotes the 95% confidence interval (two standard errors) for the free energy averaged over the last 3000τ of the WE simulation.
Committor probability along the string for the elastic network model of NtrC r . The average committor probability of reaching state B before state A, q +, calculated from an ensemble of 500 conformations in each bin is shown as a solid line, with error bars equal to the standard deviation of the sample. Committors calculated from the WE simulation using Eq. (22) are shown as open circles.
Parameters used in the WE simulations for the periodic two-dimensional potential.
Parameters used in the WE simulations for the two-dimensional ring potential.
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