(Color online) Free energy surface of the alanine dipeptide obtained by multidimensional adaptive umbrella sampling (Ref. 19). All simulations were done using the ACE implicit solvent model (Ref. 23) with a smoothing parameter of 1.6 and a solvent dielectric constant of 78.5, as implemented in the CHARMM program (Ref. 24). The temperature was held constant at with the Berendsen thermostat (Ref. 25); a time step of was used. The angles (deg) and energies (kcal/mol) of the minima (stars) are 0.0, 3.8, 1.1, (53, 53) 4.5. The angles and energies of the saddle points (crosses) are 3.5, 7.1, (9, 89) 7.2, 6.0, 4.4, (63, 127) 5.8, 5.5, (121, 139) 9.3, 10.0. The minimum energy pathway from to leads through , , and .
Schematic representation of the TMD method. The starting configuration is labeled with , the target with , and the dotted semicircles represent the hypersheres with, going from left to right, a rmsd of , , etc., with the target. At each time step, the trajectory is propagated by an unperturbed, Newtonian move (gray), followed by a perturbation (black) that brings the trajectory to the next rmsd hypersphere.
(Color online) TMD simulations of the alanine dipeptide. (a) Averages (circles) ± the fluctuations (vertical lines) of the total perturbation as a function of in TMD simulations of the alanine dipeptide. Each TMD trajectory starts near the conformation; the starting points were generated from constrained equilibration simulations that used different random seeds for the heating procedure. The TMD simulations were stopped when a rmsd of from the conformation was reached. All values are in angstrom. (b)–(e) TMD trajectories projected on the free energy surface. was set to [(b) and (d)] or [(c) and (e)].
(Color online) RP-TMD simulations of the alanine dipeptide. (a)–(d) RP-TMD trajectories projected on the free energy surface. (a) . (b) . (c) . (d) . (e) Rmsd with respect to the target conformer (in angstrom) for the trajectories shown in (a) (black), (b) (dark gray), (c) (gray), and (d) (light gray). (f) Number of simulation steps needed to reach the target. 20 simulations were performed for each value of (in angstrom); the average number of steps is indicated by the bigger dots. The larger graph shows the results for the simulations; the inset shows the results for simulation sets with , , and in going from left to right with (left columns), (middle), and (right) in each set.
Free energy distributions in the TMD and RP-TMD simulations. Shown are the average distributions of 20 simulations each. (a) TMD simulations with (light gray), 0.0001 (gray), 0.000 05 (dark gray), and (black). (b) RP-TMD simulations with (light gray), 0.0004 (gray), 0.0003 (dark gray), and (black). (c) RP-TMD simulations with , and (gray), 0.000 24 (dark gray) and (black).
(Color online) RP-TMD simulations of the alanine dipeptide using directional control. (a) (no directional control). (b) , .
Relative timings of unbiased, TMD, and RP-TMD simulations of uncapped alanine-helices, using the ACE (Ref. 23) implicit solvent model and no cut off. The averages over 3 runs of 10 000 MD steps are shown; targets in the TMD and RP-TMD simulations were the fully extended conformations, and the timing of the unbiased MD simulation of is set equal to unity.
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