New mean-energy formulae for free energy differences

Abstract

We report a number of new exact expressions, within the realm of classical statistical mechanics, for the solvation contribution ΔA to the free energy change of a solute that undergoes a transformation between two states that differ in the way they interact with the solvent. Our basic result is a coupling-parameter expression that is equivalent to the usual cumulant expansion of ΔA. In our formula, which is derived from a linear coupling-parameter scheme, all the cumulant terms beyond the first one are collected into an integral over the coupling parameter. From this formula, with the help of the mean-value theorems of calculus, we derive exact one- and two-reference-point mean-energy expressions for the solvation free energy change, neither of which is limited to the linear response regime. Both the one- and the two-reference point formulae are given in terms of one distinguished value of the coupling parameter (

and ξ*, respectively) that is not known a priori (except in the linear response regime where we find it exactly). In the case of the two-reference-point formula, we relate ξ* with the cumulative changes of the solvation structure along the intermediate states sampled in the (ξ = 0) → (ξ = 1) transformation of the solute. Finally, and based on these developments, we present an analysis of the Linear Interaction Energy and of the Extended Linear Response semi-empirical computational methods (widely used for the determination of binding constants of ligand/receptor biological systems) in an effort to understand their statistical mechanical foundations as well as their domain of applicability.

Acknowledgement

We dedicate this work to Ben Widom, whose many exact results have been crucial in shaping modern liquid-state thermodynamics. Support for this work from the National Science Foundation, award numbers 0029963 (for FOR) and CHE-0455968 (for DB-A) and the Department of Energy (for GS) is gratefully acknowledged.