A Generalized Theory for the Structural and Spatial Mapping of Energy, Entropy, and Free Energy

Systems in which the free energy density is nonuniform in space are familiar; the surface tension of a water droplet and the surface energy of a solid are good examples. Some such cases can be treated with prior theory, notably inhomogeneous solvation theory (IST), but IST is applicable only to liquids. Despite this limitation, IST has proven useful as a guide to the design of ligands to bind a targeted protein, based on the idea that ligands that displace high free energy water will tend to bind more tightly, other things being equal. Here, we present Generalized Thermodynamic Mapping (GTM) theory, a more general theory that is applicable to the entirety of a biomolecular or other chemical system, and which thus may provide additional guidance for molecular design. For example, it might highlight parts of a ligand whose local free energy rises on binding, thus suggesting where modifications could improve affinity. Starting from classical statistical thermodynamics, we derive structural decompositions that assign energy and entropy to individual atoms or to larger chemical components, such as amino acid residues, and spatial decompositions that define continuously varying thermodynamic densities throughout the system. The potential energy is decomposed via the multibody expansion, and the entropy via the mutual information expansion. The resulting thermodynamic densities satisfy key desiderata: their spatial integrals yield the correct total thermodynamic quantities, the densities vanish where the atomic number density is zero, and all entropy terms above first order go to zero in the absence of correlation. The entropy decomposition in GTM theory is closely related to that of IST, but it is simpler, largely because it expresses the entropy in terms of normalized probability density functions instead of nonnormalized correlation functions. We show that GTM theory is formally exact for any number of particles, N, whereas IST is not. The unified framework presented here enables thermodynamic mapping of solute and solvent alike and is expected to support a range of applications, including in structure-based drug design, protein design, the analysis of allostery, and materials science. The use of GTM theory to gain insight into protein-ligand binding is illustrated here with an initial case study.