Solvation is an elementary process in nature and is of paramount importance to many sophisticated chemical, biological and biomolecular processes. The understanding of solvation is anessential prerequisite for the quantitative description and analysis of biomolecular systems.Implicit solvent models, particularly those based on the Poisson-Boltzmann (PB) equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory and have some severe limitations.We have introduced differential geometry based solvation models which allow the solvent-solute interface to be determined by the variation of a total free energy functional. Our models extend the scaled particle theory (SPT) of nonpolar solvation models with a solvent-solute interaction potential. The nonpolar solvation model is completed with a PB theory based polar solvation model. In our Eulerian formation, the differential geometry theory of hypersurface is utilized to define and construct smooth interfaces with good stability and differentiability, for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. Some techniques from the geometric measure theory are employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation, so as to bring all energy terms on an equal footing. In our Lagrangian formulation, the differential geometry theory of surfaces is used toprovide a natural description of solvent-solute interfaces. By optimizing the total free energy functional, we derive a coupling of the generalized Poisson-Boltzmann equation (GPBE) andthe generalized geometric flow equation (GGFE or also called Laplace-Beltrami equation) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. The coupled partial differential equations (PDEs) are solved with iterativeprocedures to reach a steady state, which delivers the desired solvent-solute interface and electrostatic potential for many problems of interest. These quantities are utilized to evaluatethe solvation free energies, protein-protein binding affinities, etc.The above proposed approaches have been extensively validated.Extensive numerical experiments have been designed to validate the present theoretical models, to test the computationalmethods, and to optimize the numerical algorithms. Solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new models and numerical approaches. Comparisonis given to both experimental and theoretical results in the literature.Moreover, to account for the charge rearrangement during the solvation process, we also propose a differential geometry based multiscale solvation model which makes use of electrondensities computed directly from a quantum mechanical approach. We construct a new total energy functional, which consists of not only polar and nonpolar solvation contributions, but also the electronic kinetic and potential energies. We show that the quantum formulation of our solvation model improves the prediction of our earlier models, and outperforms some explicit solvation analysis.
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