Correlation between Solvation Free Energy and Solute-Solvent Interaction Energy in Energy Representation Theory

The energy representation (ER) theory is a type of classical density-functional theory that provides a framework for calculating solvation free energy (SFE) using molecular simulations. A key feature is the use of solute-solvent pair-energy distribution functions and approximate functionals, which eliminate the need for intermediate-state simulations. This enables the application of the ER theory to diverse systems, from small molecules to proteins, polymers, interfaces, and membranes. A strong correlation between SFE and solute-solvent interaction energy (SSIE) has been observed empirically in ER calculations, particularly for proteins with multiple conformations, while the linear interaction energy (LIE) method analytically derives the same relationship for the electrostatic component, both with a coefficient of 1/2. In this study, we derived the same coefficient within the ER framework, where SSIE includes both electrostatic and van der Waals (vdW) contributions. Using 159 conformations of 20-residue Trp-cage protein, we numerically examined the relationship between SFE, SSIE, components of SSIE, and excluded volume. The vdW contribution was much smaller than the electrostatic one, yet its inclusion improved correlations within groups of structurally similar conformations. Moreover, we showed that while the electrostatic component achieves sufficient accuracy with a single reference point, the SSIE can be further improved by introducing multiple reference points. These results clarify the theoretical basis of the SFE-SSIE relationship in the ER theory and highlight its utility for analyzing biomolecular solvation, particularly in systems where direct SFE calculations are computationally demanding.