3286 J. Chem. Theory Comput., Vol. 6, No. 11, 2010 ensemble averages that are dominated by only a few conﬁgurations with high weight, thereby decreasing the precision of thermodynamic quantities such as free energy changes.22 Here, we propose limiting the acceleration to the degrees of freedom most responsible for conformational changes, thereby reducing the energy added to a system to enhance sampling and resulting in improved reweighting statistics. This work is an extension of a previous study in which aMD was limited to the dihedrals in the backbone of a peptide substrate bound to cyclophilin;23 however, here we demonstrate that accelerated dihedrals may contain atoms which are also contained in nonaccelerated torsions (that is, individual molecules may contain both accelerated and nonaccelerated torsions). Two examples are highlighted. First, we show that even in the case of the model system alanine dipeptide, selectively targeting dihedrals in the molecule’s backbone results in similar acceleration levels while reducing the amount of energy added to the system. Then we turn our attention to the larger problem of calculating the binding energy of the clinically approved drug oseltamivir (marketed as Tamiﬂu by Roche Pharmaceuticals, Basel, Switzerland24) to the N1 ﬂu protein neuraminidase.25,26 Using free energy perturbation with a modiﬁcation to the Bennett acceptance ratio (to account for reweighting),27 we show that the computational cost required to accurately calculate the binding energy may be reduced by as much as 70% while maintaining a similar level of precision. 2. Theory To enhance phase space sampling, the original aMD applies an additional potential only when the potential energy, V(r), is below a speciﬁed criterion E to produce dynamics on the artiﬁcial landscape V*(r) such that V*(r) ) { V(r) if V(r) g E V(r) + ∆ V(r) if V(r) < E The form of the “boost” potential ∆ V(r) is deﬁned as ∆ V(r) ) (E - V(r))2 R + (E - V(r)) (1) (2) The formalism of this boost potential has several practical advantages: it produces a potential energy surface with a smooth ﬁrst derivative, it does not require the deﬁnition of a “reaction coordinate” along which to enhance sampling, it reﬂects the shape of the original potential, and it is relatively simple with only two adjustable parameters (E and R). Simulation results generated with aMD may be reweighted by the exponential of the boost potential, exp( ∆ V(r)), to recover theoretically exact thermodynamic properties for the physically relevant unaccelerated system. In practice, how- ever, the exponential dependence of the reweighting hinders convergence as trajectory averages become dominated by a smaller subset of their snapshots as the range of boost potentials increases. While this does not severely affect small systems such as alanine dipeptide, it does prevent the use of aMD in accurate free energy calculation in larger biomo- lecular systems. To improve the reweighting statistics and enhance sampling, several variants of aMD have been Wereszczynski and McCammon developed including “barrier lowering” aMD,28 replica- exchange aMD,29 and adaptive aMD (personal communica- tion P. Markwick). In this paper we discuss an extension to aMD which may be incorporated into other aMD imple- mentations, which is to selectively accelerate a user-deﬁned subset of dihedrals most pertinent to sampling the relevant degrees of freedom, which we refer to as selectiVe aMD. Selective aMD has the advantage that by only accelerating the degrees of freedom most important to sampling, lower overall boosts may be utilized to achieve a similar accelera- tion level, thus resulting in improved reweighting statistics. The idea of enhancing sampling along a user-deﬁned manifold has been previously shown to improve the calcula- tion of time-correlation functions for kinetics of multidi- mensional systems.30 2.1. Weighted Bennett Acceptance Ratio. Free energy perturbation (FEP) is a well-established technique used in free-energy calculations, speciﬁcally in the case of ligand binding and computational alchemy.31,13 In FEP, a nonphysi- cal energy pathway is constructed between two physical end states, for example, a ligand bound in the active site of an enzyme (which we denote as λ ) 0) and an active site without the ligand (λ ) 1). The path between these two states is divided into a series of “windows” in which the Hamil- tonian is transformed from state 0 to 1. Traditionally, free energy differences between successive windows are esti- mated by exponentially averaging the instantaneous work of going between the states, and the overall free energy is a sum of free energy differences between windows.32 Shirts et al. showed that the Bennett acceptance ratio (BAR) was superior to exponential averaging in producing asymptotically unbiased free energy estimates between two states that could improve precision by an order of magnitude.27,33 For a series of work functions between two states in which individual works do not each have the same weight (as in aMD), the derivation of a weighted BAR follows that in Shirts et al. with the exception that their eqs 5 and 6, the probability of a single measurement of the work Wi for the forward and reverse work functions, are modiﬁed to P(F|W) ) exp( ∆ V(r)) 1 + exp[- (M + Wi - ∆ F)] P(R|W) ) exp( ∆ V(r)) 1 + exp[ (M + Wi - ∆ F)] where the constant M is redeﬁned as M ) kT ln( i)0 nF ∑ exp( ∆ V(r)i) nR ∑ exp( ∆ V(r)j) j)0 ) Therefore, the value of ∆ F that solves (3) (4) (5)