Variational implicit solvation with Legendre-transformed Poisson–Boltzmann electrostatics

The variational implicit-solvent model (VISM) is an efficient approach to biomolecular interactions, where electrostatic interactions are crucial. The total VISM free energy of a dielectric boundary (i.e. solute–solvent interface) consists of the interfacial energy, solute–solvent interaction energy and dielectric electrostatic energy. The last part is the maximum value of the classical and concave Poisson–Boltzmann (PB) energy functional of electrostatic potentials, with the maximizer being the equilibrium electrostatic potential governed by the PB equation. For the consistency of energy minimization and computational stability, here we propose alternatively to minimize the convex Legendre-transformed Poisson–Boltzmann (LTPB) electrostatic energy functional of all dielectric displacements constrained by Gauss’ Law in the solute region. Both integrable and discrete solute charge densities are treated, and the duality of the LTPB and PB functionals is established. A penalty method is designed for the constrained minimization of the LTPB functional. In application to biomolecular interactions, we minimize the total VISM free energy iteratively, while in each step of such iteration, minimize the LTPB energy. Convergence of such a min–min algorithm is shown. Our numerical results on the solvation of a single ion indicate that the LTPB performs better than the PB formulation, providing possibilities for efficient biomolecular simulations.