A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule

Abstract

The Poisson-Boltzmann equation (PBE) is one widely-used implicit solvent continuum model for calculating electrostatics of ionic solvated biomolecule. In this paper, a new finite element and finite difference hybrid method is presented to solve PBE efficiently based on a special seven-overlapped box partition with one central box containing the solute region and surrounded by six neighboring boxes. In particular, an efficient finite element solver is applied to the central box while a fast preconditioned conjugate gradient method using a multigrid V-cycle preconditioning is constructed for solving a system of finite difference equations defined on a uniform mesh of each neighboring box. Moreover, the PBE domain, the box partition, and an interface fitted tetrahedral mesh of the central box can be generated adaptively for a given PQR file of a biomolecule. This new hybrid PBE solver is programmed in C, Fortran, and Python as a software tool for predicting electrostatics of a biomolecule in a symmetric 1:1 ionic solvent. Numerical results on two test models with analytical solutions and 12 proteins validate this new software tool, and demonstrate its high performance in terms of CPU time and memory usage.

SDPBS Web Server for Calculation of Electrostatics of Ionic Solvated Biomolecules

“…It is developed based on an unstructured interface-fitted tetrahedral mesh of Ω to fully take advantages of the finite element method in handling the complicated surface shape of interface Γ and the interface conditions. The second one is a novel finite element and finite difference hybrid solver of PBE proposed in [65]. It is constructed based on the Schwarz alternating domain decomposition method and a novel seven-boxes partition of Ω for the purpose of taking advantages of both finite element and finite difference methods.…”

“…It has been shown to be able to significantly improve the performance of the PBE finite element solver in both CPU time and computer memory amount, especially for large biomolecule cases. Both PBE solvers can guarantee the numerical convergence and numerical stability [59,65].…”

“…Furthermore, we find that the Newton equation (2.11) can be rewritten in the strong partial derivative form as a linear interface boundary value problem as follows [65]:…”

“…To further improve the accuracy and numerical stability of PBE numerical solutions, we recently developed new numerical PBE solvers using finite element, finite difference, solution decomposition, domain decomposition, and multigrid methods [29,34,35,59,63,65]. A collection of these solvers provides us with a new PBE solver library, which we refer to as SDPBS (Solution Decomposition Poisson-Boltzmann Solvers), since the core parts of these solvers are constructed from a decomposition of PBE solution.…”

“…In fact, when the interface problems of Ψ and Φ are reformulated to variational forms, the interface conditions have been treated naturally without requiring any further numerical approximation. Because of using our new modified Newton minimization scheme, the state-of-the-art finite element library DOLFIN [37] from the FEniCS project, and the scientific computation toolkit PETSc [5,6], our SDPBS solvers have been shown to work efficiently and effectively for various biomolecules [59,65].…”