Phase-field approach to implicit solvation of biomolecules with Coulomb-field approximation

A phase-field variational implicit-solvent approach is developed for the solvation of charged molecules. The starting point of such an approach is the representation of a solute-solvent interface by a phase field that takes one value in the solute region and another in the solvent region, with a smooth transition from one to the other on a small transition layer. The minimization of an effective free-energy functional of all possible phase fields determines the equilibrium conformations and free energies of an underlying molecular system. All the surface energy, the solute-solvent van der Waals interaction, and the electrostatic interaction are coupled together self-consistently through a phase field. The surface energy results from the minimization of a double-well potential and the gradient of a field. The electrostatic interaction is described by the Coulomb-field approximation. Accurate and efficient methods are designed and implemented to numerically relax an underlying charged molecular system. Applications to single ions, a two-plate system, and a two-domain protein reveal that the new theory and methods can capture capillary evaporation in hydrophobic confinement and corresponding multiple equilibrium states as found in molecular dynamics simulations. Comparisons of the phase-field and the original sharp-interface variational approaches are discussed.