A novel method is presented for solving the Poisson-Boltzmann (PB) equation based on a rigorous treatment of geometric singularities of the dielectric interface and a Green's function formulation of charge singularities. Geometric singularities, such as cusps and self-intersecting surfaces, in the dielectric interfaces are bottleneck in developing highly accurate PB solvers. Based on an advanced mathematical technique, the matched interface and boundary (MIB) method, we have recently developed an accurate PB solver, MIBPB-II by rigorously enforcing the flux continuity conditions at the dielectric interface where geometric singularities may occur. However, when mesh size approaches half of the van der Waals radius, MIBPB-II cannot maintain its accuracy because the grid points that carry the interface information overlap with points carrying distributed charges. In the present Green's function formalism, charge singularities are transformed into interface jump conditions, which are treated on an equal footing as the geometric singularities in our MIB framework. The resulting method, denoted as MIBPB-III, is able to provide highly accurate electrostatic potentials at a mesh as coarse as 1.2 angstrom for proteins. The MIBPB-III has been extensively validated by using analytically solvable problems and molecular surfaces of polyatomic systems, and proteins.
An important and direct application of the solution of PBE is to calculate the solvation forces, which are the basis for molecular dynamics simulation. We provide MIB based solvation forces calculation schemes, taking advantage of the accurate potentials output from MIBPB solver and again the incorporation of interface jump conditions. As one of the important components of the solvation forces, the accuracy of dielectric boundary forces, which were traditionally calculated by using smoothed dielectric interface, is improved by resorting to a level set based 1st order surface integral. Several examples and comparisons are presented to validate the schemes.
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