sing Selectively Applied Accelerated Molecular Dynamics J. Chem. Theory Comput., Vol. 6, No. 11, 2010 3291 we obtain a free energy value of 67.3 ( 1.5 kcal/mol, identical (within error) to that from the longer cMD calculations. Results from individual aMD runs are shown in Figure S2, Supporting Information. A comparison of the time evolutions of the BAR results shows similar behavior for the aMD and cMD free energies (with the aMD on shorter time scales) for the short, medium, and long equilibration times (Table 2 and Figure S3, Supporting Information). For short equilibration periods (cMD, 1000 ps; aMD, 250 ps) the free energy is initially overestimated, and while it approaches the values obtained for longer equili- bration, the bias introduced in this nonequilibrated period results in free energies ∼1.5 kcal/mol too high. Medium length equilibration periods (cMD, 200 ps; aMD, 500 ps) suffer from this effect as well but are not quite as biased, whereas for long equilibration times (cMD, 3000 ps; aMD, 750 ps) the calculated free energies remain stable (within error) with increased sampling. As a comparison, we also performed BAR calculations on a single set of windows run for 1750 ps with all-dihedral aMD (Figure S4, Supporting Information). The much larger range of weights resulted in very few conﬁgurations con- tributing to the BAR results and poor free energy estimates. For example, with 250 ps of sampling and 1250 ps of equilibration a free energy of -15.0 ( 17.0 kcal/mol was computed, whereas 750 ps of equilibration and 750 ps of sampling resulted in a free energy of decoupling of 36.9 ( 17.3 kcal/mol. Both of these values are wildly inaccurate compared to the values of 66.9 ( 1.2 and 67.3 ( 1.5 kcal · mol obtained from cMD and selectively applied aMD. 5. Concluding Discussions The method of accelerated molecular dynamics has been well established as a means of enhancing phase space sampling with minimal computational cost; however, the exponential re- weighting required for the recovery of ensemble averages in the unaccelerated case introduces excessive noise such that it is often difﬁcult, if not impossible, to recover accurate ensemble averages. Even in the case of the well-studied system alanine dipeptide this becomes evident. For example, in Figure 2d the free energy proﬁle of well 1 appears discontinuous in the region of (φ, ψ) ) (-50,150), due in large part to the fact that several of the conformations visited have weights 1-3 orders of magnitude below that of the maximum weighted conformation. In contrast, by limiting the acceleration to only those dihedrals which are most pertinent to phase space sampling (in this case, φ and ψ) the maximum weight was reduced from 1097 to 2.46, which not only increased the smoothness of the recovered free energy proﬁle (Figure 2e) but also moderately improved the agreement to the inﬁnite sampling limit (as calculated with metadynamics) results in Table 1 and Figure 3. However, both aMD formalisms showed approximately a factor of 5-fold increase in efﬁciency relative to cMD for our order parameter . Extension of this idea to the pharmaceutically relevant case of oseltamivir binding to neuraminidase shows the expensive free energy calculation of the decoupling of the ligand in the protein’s active site may be reduced by up to 70% (from 5 to 1.5 ns/window) without a loss in precision. While this may not be crucial in the case of studying the binding of only a single ligand to a protein, one could imagine that in the lead optimization stage of a drug-design effort, when highly accurate binding energies are necessary, the ability to examine three times the number of possible compounds at little extra computational cost may be highly desirable. In the case presented here, the accelerated dihedrals were chosen based upon extensive prior MD simulations; however, if this data were not available one could choose the accelerated dihedrals by residue type (accelerating dihedrals in residues with highly mobile side chains such as arginine and not accelerating dihedrals in aromatic rings), atom types (non-hydrogen containing), and proximity to the ligand. Additionally, in some cases where the ligand is bulky and has multiple torsions with high-energy barriers between local minima, one could accelerate dihedrals in the ligand mol- ecules themselves, as was done in the case of cyclophilin.23 Selective aMD may easily be incorporated into other free energy algorithms. For example, the methods of one-step perturbation and envelope distribution sampling provide tech- niques for effectively calculating the binding of several ligands to a protein with a single extended MD simulation.51,52 They have, however, not been extensively utilized due to the fact that, depending on the system being studied, they may require simulations on the microseconds time scale.53 Therefore, a reduction of the computational cost of 3- to 5-fold (as observed in this study) could reduce the necessary simulation length from the highly expensive 1-2 µ s range into the manageable 200 ns time scale. These methods highlight that sampling of the partition function is, in general, a slow process requiring extensive calculations; therefore, methods such as selective aMD, which can enhance sampling of the relevant portions of phase space while not introducing excessive noise into the calculations, may prove useful in future applications. Acknowledgment. We extend our thanks to M. Lawrenz, M. Fajer, P. Gasper, and Y. Wang for valuable discussion concerning the work presented here. The project described was supported by Award Number F32GM093581 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the ofﬁcial views of the National Institute of General Medical Sciences or the National Institutes of Health. This research used resources of the National Energy Research Scientiﬁc Computing Center, which is supported by the Ofﬁce of Science of the U.S. Department of Energy under Contract No. DE-AC02- 05CH11231. Additional support has been provided by the NSF, NIH, HHMI, CTBP, NBCR, and the NSF Supercomputer Centers. Supporting Information Available: Plots of the free energies for oseltamivir decoupling as a function of sampling time for individual cMD and aMD simulations; comparisons between these cMD and aMD runs; and free energies calculated from a single all-dihedral aMD simulations. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Karplus, M.; McCammon, J. A. Nat. Struct. Biol. 2002, 9, 646–652.