A constrained variational model of biomolecular solvation and its numerical implementation

Variational based solvation models of biomolecules with smooth interface have drawn attentions in the past decade since they have been developed as an efficient and reliable representation of solute-solvent interfaces in the framework of implicit solvent models. This work aims at providing solid mathematical supports for a promising geometric flow based computational solvation model with smooth interface (GFBSS) and its involved computational treatments. For this purpose, we improve the GFBSS model by explicitly including two physical constraints: (1) a novel experimental based domain decomposition, and (2) a two-sided obstacle for the characteristic function describing the optimal diffuse solute-solvent boundary. It is shown that the resulting constrained model is mathematically well-posed. Further, to overcome the challenges arising from including these constraints, we propose a family of generalized constrained energy functionals whose variations satisfy a q-Laplacian type equation for nonpolar molecules. The solvation free energies predicted by the generalized models converge to that of the proposed constrained one. Most importantly, the numerical difference between the generalized models and the previous unconstrained GFBSS model is negligible. It implies that the newly proposed constrained solvation model and the previous unconstrained one are equivalent to each other in terms of the solvation free energy calculation and prediction. Our model validation, its numerical implementation, and solvation energy convergence have been demonstrated using several common biomolecular modeling tasks.