Calculating free energy differences using perturbation theory

Abstract

Thermodynamic perturbation theory represents a powerful tool for evaluating free energy differences in complex molecular assemblies. Like any method, however, FEP has limitations of its own, and particular care should be taken not only when carrying out this type of statistical simulations, but also when interpreting their results. We summarize in a number of guidelines the important concepts and features of FEP calculations developed in this chapter: (a) Formally, FEP is exact for any perturbation. In practice, however, even for moderately large perturbations, the method suffers from convergence issues. It is, therefore, recommended to use a stratification strategy by breaking the reaction pathway into a series of intermediate states through the introduction of an order, or 'coupling' parameter. The choice of the number of intermediate states in a staged FEP calculation should not be dictated by the corresponding change in free energy, but rather by the similarity between the reference and the target ensembles. (b) Although the general FEP theory applies equally to both forward and backward simulations, the efficiencies of these two types of simulations may differ considerably. A properly converged FEP calculation for a 0 → 1 transformation does not necessarily imply that the reverse, 1 → 0, transformation converges equally efficiently to the correct free energy difference. c) The free energy difference between the reference and the target states can be represented as a cumulant expansion. Retaining only the first two terms of this expansion is equivalent to assuming that P0(ΔU) is a Gaussian. Second-order perturbation theory is a very useful tool for analyzing free energy calculations and developing approximate theories. Beyond the second order, however, the cumulant expansion diverges, and should, therefore, be used with extreme care. d) Since free energy is a state function, free energy differences are independent of the pathway chosen for their evaluation. Consequently, 'alchemical transformations,' during which a chemical species is mutated into an alternate one, may be carried out using either a single- or dual-topology paradigm by scaling the non-bonded parameters or the potential energy functions with the order parameter. e) Several techniques are available for improving the efficiency and accuracy of free energy calculations. These techniques require only very modest additional computational effort. Carrying out forward and backward simulations in an appropriate way is one of the more powerful schemes. It is strongly advised that these techniques be used in practice. f) The FEP methodology may be extended to the computation of potential energy, enthalpy and entropy differences. Yet, compared to free energy differences, these quantities are more difficult to estimate with good accuracy, because they inherently depend upon all molecular interactions in the system, and not only on those that are perturbed during the transformation. g) Particular attention should be paid to the interpretation of free energy components obtained by perturbing individual contributions of the potential energy function. These free energy components reflect the pathway defined for their determination, which is not unique. © 2007 Springer-Verlag.