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Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves.

ABSTRACT: Although knowledge of the pKa values and charge states of individual residues is critical to understanding the role of electrostatic effects in protein structure and function, calculating these quantities is challenging because of the sensitivity of these parameters to the position and distribution of charges. Values for many different proteins which agree well with experimental results have been obtained with modified Tanford-Kirkwood theory in which the protein is modeled as a sphere (reviewed in Ref. 1); however, convergence is more difficult to achieve with finite difference methods, in which the protein is mapped onto a grid and derivatives of the potential function are calculated as differences between the values of the function at grid points (reviewed in Ref. 6). Multigrid methods, in which the size of the grid is varied from fine to coarse in several cycles, decrease computational time, increase rates of convergence, and improve agreement with experiment. Both the accuracy and computational advantage of the multigrid approach increase with grid size, because the time required to achieve a solution increases slowly with grid size. We have implemented a multigrid procedure for solving the nonlinear Poisson-Boltzmann equation, and, using lysozyme as a test case, compared calculations for several crystal forms, different refinement procedures, and different charge assignment schemes. The root mean square difference between calculated and experimental pKa values for the crystal structure which yields best agreement with experiment (1LZT) is 1.1 pH units, with the differences in calculated and experimental pK values being less than 0.6 pH units for 16 out of 21 residues. The calculated titration curves of several residues are biphasic.

SUBMITTER: Oberoi H

PROVIDER: S-EPMC1225699 | biostudies-literature | 1993 Jul

REPOSITORIES: biostudies-literature

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Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves.

Oberoi H H Allewell N M NM

Biophysical journal 19930701 1

Although knowledge of the pKa values and charge states of individual residues is critical to understanding the role of electrostatic effects in protein structure and function, calculating these quantities is challenging because of the sensitivity of these parameters to the position and distribution of charges. Values for many different proteins which agree well with experimental results have been obtained with modified Tanford-Kirkwood theory in which the protein is modeled as a sphere (reviewed ...[more]

PMID: 8369451

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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.

Project description:We present a boundary-element method (BEM) implementation for accurately solving problems in biomolecular electrostatics using the linearized Poisson-Boltzmann equation. Motivating this implementation is the desire to create a solver capable of precisely describing the geometries and topologies prevalent in continuum models of biological molecules. This implementation is enabled by the synthesis of four technologies developed or implemented specifically for this work. First, molecular and accessible surfaces used to describe dielectric and ion-exclusion boundaries were discretized with curved boundary elements that faithfully reproduce molecular geometries. Second, we avoided explicitly forming the dense BEM matrices and instead solved the linear systems with a preconditioned iterative method (GMRES), using a matrix compression algorithm (FFTSVD) to accelerate matrix-vector multiplication. Third, robust numerical integration methods were employed to accurately evaluate singular and near-singular integrals over the curved boundary elements. Fourth, we present a general boundary-integral approach capable of modeling an arbitrary number of embedded homogeneous dielectric regions with differing dielectric constants, possible salt treatment, and point charges. A comparison of the presented BEM implementation and standard finite-difference techniques demonstrates that for certain classes of electrostatic calculations, such as determining absolute electrostatic solvation and rigid-binding free energies, the improved convergence properties of the BEM approach can have a significant impact on computed energetics. We also demonstrate that the improved accuracy offered by the curved-element BEM is important when more sophisticated techniques, such as nonrigid-binding models, are used to compute the relative electrostatic effects of molecular modifications. In addition, we show that electrostatic calculations requiring multiple solves using the same molecular geometry, such as charge optimization or component analysis, can be computed to high accuracy using the presented BEM approach, in compute times comparable to traditional finite-difference methods.

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Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves.

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Multigrid solution of the nonlinear Poisson-Boltzmann equation and calculation of titration curves.

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