3288 J. Chem. Theory Comput., Vol. 6, No. 11, 2010 Wereszczynski and McCammon Figure 2. Free energy landscapes for alanine dipeptide. (a) Metadynamics results representing the expected results for inﬁnite sampling of the system. Classical MD simulations visited all three energy wells on the 10 ns time scale (b); however, the statistics are poor (as shown in Table 1). Sampling is improved on the 50 ns time scale (c). (d) All dihedral aMD, showing that a sampling of all three energy wells is improved relative to the cMD results. (e) Selective dihedral aMD, showing the best agreement to the metadynamics results on the 10 ns time scale. (f) Wells labeled for discussion purposes. Table 1. Free Energy Statistics for Alanine Dipeptide from cMD, All-Dihedral aMD, and Selective aMD Simulations Compared to the Metadynamics Results Which Effectively Represent the Inﬁnite Sampling Limita well 1 well 2 well 3 time 10 ns 50 ns 250 ns simulation type metadynamics classical all dihedral aMD selective aMD classical all dihedral aMD selective aMD classical energy 0.065 0.485 (+.421) 0.000 (-0.065) 0.008 (-0.057) 0.350 (+.285) 0.053 (-0.012) 0.137 (+.072) 0.232 (+.168) population 0.406 0.249 (61%) 0.337 (83%) 0.422 (104%) 0.328 (81%) 0.372 (92%) 0.381 (94%) 0.357 (88%) energy 0.000 0.000 (+.000) 0.085 (+.085) 0.000 (+.000) 0.000 (+.000) 0.000 (+.000) 0.000 (+.000) 0.000 (+.000) population 0.429 0.427 (99%) 0.415 (97%) 0.369 (86%) 0.475 (111%) 0.426 (99%) 0.422 (98%) 0.446 (104%) energy 1.010 0.301 (-0.706) 0.767 (-0.240) 1.010 (-0.004) 1.004 (-0.002) 1.003 (-0.004) 1.038 (+.033) 0.994 (-0.012) population 0.049 0.191 (392%) 0.078 (160%) 0.041 (84%) 0.058 (120%) 0.052 (106%) 0.048 (98%) 0.055 (112%) a For each of the three wells (Figure 2f) the minimum energy relative to the global minimum is calculated, as is the population of states within 1.8 kcal/mol of that minimum. Results show that both aMD forms converge on the order of ﬁve times as fast as the cMD simulations. 10 windows followed by the Lennard-Jones interactions decoupled over 10 windows with a softcore potential using R ) 0.545 (as in a previous study46). Free energies were calculated using BAR, with the modiﬁed BAR formulas described above used in the aMD calculations. A positional restraint of .8 kcal/mol was placed on a central carbon atom in oseltamivir to prevent the ligand from sampling nonactive site portions of the simulation box,47 and calculations were performed for cMD, all-dihedral aMD (with E ) 2600 and R ) 400), and selective aMD (with E ) 13 and R ) 2). For cMD calculations, three sets of windows were run (as has been shown to improve calculated free energies48), each with the same initial coordinates but different velocities for 5 ns per window, whereas for the selective aMD the same three sets of windows were run with 200 ps of all-dihedral aMD (to quickly equilibrate the whole protein) followed by 1.5 ns of selective aMD per window. Note that the times indicated in the text include the simulation time spent in all- dihedral aMD; thus, a time of 500 ps represents 200 ps of all-dihedral and 300 ps of selective aMD. One set of FEP calculations was performed for the all-dihedral case for 1.75 ns window to illustrate the futility of using standard aMD in large-scale biomolecular FEP calculations. The choice of dihedrals to accelerate was based on previous work in which the tetramer was simulated for 100 ns.46 Acceleration was applied to those dihedrals which contained only heavy atoms, were in residues that had a heavy atom within 5 Å of the oseltamivir in the crystal structure, and had a multivariate distribution for a total of 29 dihedrals. Work functions were decorrelated based on the statistical inefﬁciency using code provided by Shirts and Chodera.49 For each BAR calculation, a bootstrap analysis was per- formed (with 50 independent calculations) for an error σB, which was combined with the variance of the three means (σV) to calculate an overall error estimate for the free energy by