Determining the contributions of constraints in free energy calculations: Development, characterization, and recommendations

Abstract

We develop and/or characterize three methods for determining the contributions of constraints in free energy calculations. A new method for determining such contributions in thermodynamic integration (TI) calculations, the potential forces (PF) method, is developed and compared to a second method, the constraint forces (CF) method. Both methods are also compared to a previously described technique for calculating such contributions in free energy perturbation (FEP) simulations. We find that the TI/PF protocol is considerably more efficient than the TI/CF method, and is preferred except in cases where the constraints contributing to the free energy are part of a closed ring. Compared to TI/PF, the FEP method is shown to be relatively poor for generating potential of mean force (PMF) curves, though the FEP method is suitable for determining the ‘‘PMF bond contribution’’ in compositional free energy changes. PMF curves for a system of two neon atoms in a periodic box of water have been derived. The convergence behavior of the free energy derivative ∂G/∂R, where R is a distance constraint, has been examined in detail for this system. As much as a nanosecond of molecular dynamics sampling can be required to derive a fully converged value for this derivative at a single (λ) point. We have determined the sampling ratio for ∂G/∂R as a function of Ne–Ne distance for a 295 water (21 Å/side) periodic water box, and conclude that for free energy changes where long-distance constraint contributions are being determined, a minimum of 0.7 ps of sampling should be performed per window. When constraint contributions arise from short distances (such as when the PMF bond contribution is being calculated), correlation in the constraint derivative series dies out relatively quickly and the minimum sampling—then dictated by correlation in the nonbonded series—should be about 0.6 ps.