sing Selectively Applied Accelerated Molecular Dynamics J. Chem. Theory Comput., Vol. 6, No. 11, 2010 3289 σ ) 3 σV + ∑ σB/3 i)1 (8) Results Alanine Dipeptide. The free energy landscape of alanine dipeptide can be described by the rotation of two torsional angles, φ and ψ, making it an ideal model system for methodological development that has been extensively studied (Figure 1). The metadynamics, or “inﬁnite sampling limit”, results (Figure 2a) show three distinct energy wells which we label for further discussion in Figure 2f. The energy barrier between wells 1 and 2 is relatively low, and classical MD (cMD) simulations sample both sets of conﬁgurations on the 10 ns time scale (Figure 2b). Well 3 is substantially oversampled, which we attribute to the system becoming trapped in this state due to the higher energy barrier, thereby discouraging transitions to and from well 3. With 50 ns of simulation the sampling of all three wells are improved (Figure 2c). Further detail is shown in Table 1, which compares the minimum energy of each well and probability of all states within 1.8 kcal/mol of that minimum to the theoretically exact answer derived from metadynamics. The 10 ns cMD shows good agreement for well 2 (it is identiﬁed as the global minimum and the population at 3kT is nearly identical to the metadynamics results); however, well 1 is undersampled by 39% whereas well 3 is oversampled by 292%, and the minimum energies are incorrect by .4 and .7 kcal/mol, respectively. With 50 ns of simulation time the sampling improves such that errors in the populations range from 11% to 20%, and by 250 ns of sampling the statistics agree much better with the metadynamics results for all three wells, although the populations of wells 1 and 3 are still off by >10%. For comparison, the all-dihedral and selective aMD simulations sampled all three energy wells on the 10 ns time scale (Figure 2d and 2e). Free energy statistics indicate a maximum error in the minimum well energy estimate of 0.24 kcal/mol in the all-dihedral case and 0.06 kcal/mol in the selective aMD simulation, whereas the greatest disagreement in well populations was an undersampling of well 1 by 17% in the all-dihedral aMD and an undersampling of well 3 by 16% in the selective aMD. Extension of the simulations to 50 ns results in further improved statistics of the well populations, with well 1 only being undersampled by 8% in the all-dihedral simulation and by 6% in the selective aMD. To further examine the convergence of the free energy statistics we deﬁned the parameter to quantitate the difference in the two-dimensional energy proﬁles (as dis- cussed in the methods) which has the property of a value of 1 representing ideal sampling of the wells as compared to the metadynamics results. In Figure 3 we compare the time course of this parameter between the cMD and the aMD simulations (note the different time scales for the two sets of simulations). The scores for both aMD are similar to those for cMD simulations of ﬁve times the length, with cMD simulations requiring 200 ns before consistently having values above 0.9, whereas the aMD simulations pass this value at 44 and 35 ns for all-dihedral and selective aMD, respectively. Figure 3. Comparison of the time evolution of our order parameter . Classical MD results (shown in black) correspond to the lower x axis, which is ﬁve times longer than that of the upper x axis, which corresponds to both aMD simulations. Despite this 5-fold difference in time scales, all three simula- tions show similar convergence patterns, with the lower weights in the selective dihedral slightly improving the statistics relative to the all-dihedral aMD. A comparison of the boosts applied throughout the aMD simulations (Figure 4a) shows that higher boost potentials are applied throughout the all-dihedral simulation than in the selective aMD. For the all-dihedral case, ∆ V varies from 0 to 4.2 kcal/mol, with an average value of 0.45 kcal/mol, whereas the selective aMD has a range of 0 to 0.54 kcal/ mol with an average of 0.11 kcal/mol. This decrease in the applied boosts has a signiﬁcant impact on reweighting as the maximum weight relative to an unaccelerated state is reduced from 1097 to 2.46. In Figure 4b the amount of total weight recovered from a simulation is plotted against the percentage of frames that contribute to that weight. In the case of all-dihedral aMD, very few frames contribute a substantial portion of the reweighting, 50% of the total weight comes from 4.8% of the trajectory, whereas 90% of the weight comes from 53.7% of the trajectory. This results in almost one-half the sampling (46.3%) contributing very little (less than 10%) to the calculated ensemble averages. In the selective aMD case the lower boosts result in more uniform weights, 50% of the weight comes from 37.9% of the trajectory and 90% from 87.6% of the trajectory (for comparison, in cMD conﬁgurations in the trajectory are uniformly weighted, so 50% of the weight comes from 50% of the trajectory). This increased reweighting efﬁciency improves the recovered statistics as high-boost conﬁgurations tend to dominate the ensemble average. For example, wells 1 and 2 appear signiﬁcantly smoother in the energy landscape of the selective aMD relative to the all-dihedral aMD (Figure 2e and 2d) and is consistently higher in the selective case, both of which can be attributed to smoother statistics in the reweighted energy proﬁles. N1 Neuraminidase. The binding of oseltamivir to neuraminidase is an example of how highly accurate free energy calculations may be employed in the study and development of novel pharmaceutical compounds. In order to validate our aMD simulations of the decoupling of oseltamivir in the N1 active site, we performed extensive sampling at each step of the alchemical transformation with 5 ns of cMD simulation for each of the 21 λ values, which