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Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.

ABSTRACT: Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ??Gel, and solvation, ?Gel, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ??Gel are more significant than those in ?Gel, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ??Gel, even though its predictions of ?Gel are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ??Gel.

SUBMITTER: Harris RC

PROVIDER: S-EPMC3756143 | biostudies-literature | 2013 Aug

REPOSITORIES: biostudies-literature

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Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.

Harris Robert C RC Boschitsch Alexander H AH Fenley Marcia O MO

Journal of chemical theory and computation 20130801 8

Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔGel, and solvation, ΔGel, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔGel are more significant than those in ΔGel, and can be divided into two parts: (i) errors associated with the ...[more]

PMID: 23997692

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Project description:We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L(∞) estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Project description:The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Dataset Information

Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.

Publications

Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation.

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