Convergence Properties of Free Energy Calculations: .alpha.-Cyclodextrin Complexes as a Case Study

By considering all possible mutations among four para-substituted phenols, p-chlorophenol, p-methylphenol, p-cyanophenol, and p-methoxyphenol, which bind as inclusion compounds in alpha-cyclodextrin, the convergence properties of thermodynamic integration free energy calculations using slow growth as compared to numerical quadrature are investigated and interpreted in terms of structural and dynamical properties of the molecular system. It is shown that a systematic increase in the calculated hysteresis can be expected with increasing simulation time in slow-growth calculations if the system is perturbed faster than the rate at which the various states that make up the equilibrium ensemble are sampled. Using numerical quadrature the effects of nonequilibrium can be largely separated from the effects of insufficient sampling. It is shown, however, that the apparent degree of convergence when using numerical quadrature does not necessarily reflect the accuracy of the calculation. The utility of formulating closed cycles in both the bound and unbound states as a means of determining the minimum error in a given calculation is demonstrated. The effects of the choice of pathway and of the choice of integration scheme on convergence within closed cycles are also discussed. Finally, the quality of the force field used and the relative importance of the force field as opposed to sampling considerations are assessed by comparing the estimated free energy differences to experimental data. It is shown that a meaningful appraisal of a specific force field cannot be made independent of sampling considerations. A modification to the GROMOS force field that improved the agreement between the calculated and experimental free energies for the mutation of p-chlorophenol to p-methylphenol is also proposed.