A new boundary condition for the nonlinear Poisson-Boltzmann equation in electrostatic analysis of proteins

Abstract

As a well-established implicit solvent model, the Poisson-Boltzmann equation (PBE) models the electrostatic interactions between a solute biomolecule and its surrounding solvent environment over an unbounded domain. One numerical challenge in solving the nonlinear PBE lies in the boundary treatment. Physically, the boundary condition of this solute solvent system is defined at infinity where the electrostatic potential decays to zero. Computationally, a finite domain has to be employed in grid-based numerical algorithms. However, the Dirichlet boundary conditions commonly used in protein simulations are known to produce unphysical solutions in some cases. This motivates the development of a few asymptotic conditions in the PBE literature, which are global boundary conditions and have to resort to iterative algorithms for calculating volume integrals from the previous step. To overcome these limitations, a modified Robin condition is proposed in this work as a local boundary condition for the nonlinear PBE, which can be implemented in any finite difference or finite element method. The derivation is based on the facts that away from the biomolecule, the asymptotic decaying pattern of the nonlinear PBE is essentially the same as that of the linearized PBE, and the monopole term will dominate other terms in the multipole expansion. Asymptotic analysis has been carried out to validate the application range and robustness of the proposed Robin condition. Moreover, a second order boundary implementation by means of a matched interface and boundary (MIB) scheme has been constructed for three-dimensional biomolecular simulations. Extensive numerical experiments have been conducted to examine the robustness, accuracy, and efficiency of the new boundary treatment for calculating electrostatic free energies of Kirkwood spheres and various protein systems.