Multigrid solution of the Poisson—Boltzmann equation

Abstract

A multigrid method is presented for the numerical solution of the linearized Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the finite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for two-dimensional interface problems occurring in reactor physics. A detailed analysis of the resulting method is presented for several computer architectures, including comparisons to diagonally scaled CG, ICCG, vectorized ICCG and MICCG, and to SOR provided with an optimal relaxation parameter. Our results indicate that the multigrid method is superior to the preconditioned CG methods and SOR, and that the advantage of multigrid grows with the problem size.

Development of constant-pH simulation methods in implicit solvent and applications in biomolecular systems

“…It was the beginning of the Poisson-Boltzmann (PB) equation era in biophysics and biochemistry. The pioneering and landmark work of this "new" approach for macromolecules represented at atomistic level and nonuniform dielectrics (in implicit solvent) is due to Warwicker and Watson (1982), which was followed by many others (e.g., Davis and McCammon (1990), Holst (1993), Davis et al (1991), Juffer et al (1991), Juffer (1998), Honig and Nicholls (1995), Bashford et al (1988), Bashford and Karplus (1990), Beroza et al (1991), Warwicker (1999), Baker et al (2001), Li et al (2005), and Anandakrishnan et al (2012)). This approach is indicated in Fig.…”

“…The PB equation is obtained by the combination of fundamental electrostatic equations, the Poisson and the Boltzmann equations (a detailed and mathematically oriented derivation can be found elsewhere (Holst 1993)). The Poisson equation is used to calculate the three-dimensional electric potential (φ) generated by a macromolecule lying in an ionic solvent.…”

“…For example, the "finite element method", (Davis and McCammon 1990;Project 1995;Harvey 1989;Orttung 1977;Terán et al 1989) the "finite difference method", (Davis and McCammon 1990;Project 1995;Harvey 1989;Warwicker and Watson 1982;Holst 1993;Davis et al 1991;Sakalli and Knapp 2015) and the "boundary element method" (Davis and McCammon 1990;Project 1995;Harvey 1989;Juffer 1993Juffer , 1998Juffer et al 1991) are applied to solve the PB equation for biomolecular systems. There are also a number of generalized program packages available to study biomolecular phenomena (Warwicker and Watson 1982;Holst 1993;Davis et al 1991;Bashford and Gerwert 1992;Honig and Nicholls 1995;Juffer 1992) and web-servers (Calixto 2010;Anandakrishnan et al 2012;Smith et al 2012;Wang et al 2016). A quite recent new numerical implementation is the Gaussian-based dielectric function description for the nonlinear PB (NLPB) .…”

Investigation of protein binding affinity and preferred orientations in ion exchange systems using a homologous protein library

“…Electrostatic potential (EP) surfaces of the native protein and the mutants were generated using the adaptive PoissonBoltzmann Solver (APBS) program (Baker et al, 2001;Bank and Holst, 2000;Holst and Saied, 1993;Holst and Saied, 1995). The VMD molecular graphics viewer was used to depict the EP surfaces developed from the APBS solver (Humphrey et al, 1996).…”

Efficient solution technique for solving the Poisson–Boltzmann equation

“…18 Also, regular grids lead to simpler and higher order numerical schemes and they allow implementation of multigrid methods. 20 Tomac and Gräslund 21 demonstrated a multigrid finite volume method for solving a modified PB equation for spherical charged particles. The modified PB equation agrees with the results obtained from Monte Carlo simulations on 1:1 and 2:1 ionic solutions.…”