Boundary element solution of the linear Poisson–Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution

Abstract

Abstract:The Poisson-Boltzmann equation is widely used to describe the electrostatic potential of molecules in an ionic solution that is treated as a continuous dielectric medium. The linearized form of this equation, applicable to many biologic macromolecules, may be solved using the boundary element method. A single-layer formulation of the boundary element method, which yields simpler integral equations than the direct formulations previously discussed in the literature, is given. It is shown that the electrostatic force and torque on a molecule may be calculated using its boundary element representation and also the polarization charge for two rigid molecules may be rapidly calculated using a noniterative scheme. An algorithm based on a fast adaptive multipole method is introduced to further increase the speed of the calculation. This method is particularly suited for Brownian dynamics or molecular dynamics simulations of large molecules, in which the electrostatic forces must be calculated for many different relative positions and orientations of the molecules. It has been implemented as a set of programs in Cϩϩ, which are used to study the accuracy and speed of this method for two actin monomers.

Computational Methods for Biomolecular Electrostatics

“…The most common numerical techniques for solving the PB equation are based on discretization of the domain of interest into small regions. Those methods include finite difference (Davis and McCammon, 1989;Nicholls and Honig, 1991;Holst and Saied, 1993;Holst and Saied, 1995;Baker et al, 2001), finite element Friesner, 1997a, 1997b;Baker et al, 2000;Holst et al, 2000;Baker et al, 2001;Dyshlovenko, 2002), and boundary element methods (Zauhar and Morgan, 1988;Juffer et al, 1991;Allison and Huber, 1995;Bordner and Huber, 2003;Boschitsch and Fenley, 2004), all of which continue to be developed to further improve the accuracy and efficiency of electrostatics calculations in the numerous biomolecular applications described below. The major software packages that can be used to solve the PB equation are listed in Table 1.…”

Computation of electrostatic forces between solvated molecules determined by the Poisson–Boltzmann equation using a boundary element method

“…15 The hypersingularity comes from the derivative of the Green's function associated with the double-layer part in the BEM. In the single-layer formulation of the BEM, the hypersingular problem in force and torque calculations can be avoided, such as in the work of Bordner and Huber, 16 who used the "surface charge density" to calculate the force. This "polarized charge" method was described by Zauhar,17 in which the force included both qE forces and boundary pressures, and this was also incorporated into the BEM Langevin dynamics simulations.…”

Efficient solution technique for solving the Poisson–Boltzmann equation

“…We implemented the above scheme for solving the PB equation. The first step in (15) only involves the diffusion operator in the x-direction. Therefore, the resulting set of algebraic equations is tridiagonal.…”

“…After solving for the x-direction in (15), the results for nϩ1/3 are stored in a dummy array. The second step in (15) does not involve derivatives of nϩ1/3 . Therefore, nϩ2/3 is overwritten on the same dummy array.…”

“…The boundary element method utilizes analytical solutions obtained in terms of Green's functions and discretization on the domain surface (molecular surface) are used to compute the potential in the whole domain volume. 14,15 One of the most common approaches used to solve the linear and the nonlinear PB is the finite difference formulation where spatial derivatives are approximated using neighboring points. 16 -18 A successive overrelaxation method has been used to get rapid convergence in solving the linear systems obtained from the finite difference discretization.…”