Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes

Cortis, Christian M.; Friesner, Richard A.

doi:10.1002/(sici)1096-987x(199710)18:13<1591::aid-jcc3>3.0.co;2-m

Cited by 176 publications

(179 citation statements)

References 18 publications

(23 reference statements)

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“…14 Finite elements methods have been applied both to the linearized PB equation 67 and to the nonlinear PB equation. 17 Recently, Holst and colleagues 68 established its rigorous solution and approximation theory, resulting in the first rigorous convergence result for any numerical methods applied to PBE.…”

Section: Discussionmentioning

confidence: 99%

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Koehl

Delarue

2010

The Journal of Chemical Physics

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The Poisson-Boltzmann ͑PB͒ 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 ͑DPBL͒ 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 ͑PDE͒. 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 considered͒. 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 ͑1995͔͒ 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.

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Section: Discussionmentioning

confidence: 99%

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Koehl

Delarue

2010

The Journal of Chemical Physics

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“…(5) or (6) can be solved numerically with the finite difference [32][33][34] or finite element methods. 35 In the finite difference method, ionic concentrations, the electrostatic potential, as well as fixed atomic charges are represented in a finite box on a 3D Cartesian grid. Typical grid resolution used in contemporary studies is in the range of 0.2-1.0 Å , and the box extends for a minimum of 10-20 Å in each direction from the molecular surface.…”

Section: Pb Equationmentioning

confidence: 99%

Continuum molecular electrostatics, salt effects, and counterion binding—A review of the Poisson–Boltzmann theory and its modifications

2007

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This work is a review of the Poisson–Boltzmann (PB) continuum electrostatics theory and its modifications, with a focus on salt effects and counterion binding. The PB model is one of the mesoscopic theories that describes the electrostatic potential and equilibrium distribution of mobile ions around molecules in solution. It serves as a tool to characterize electrostatic properties of molecules, counterion association, electrostatic contributions to solvation, and molecular binding free energies. We focus on general formulations which can be applied to large molecules of arbitrary shape in all‐atomic representation, including highly charged biomolecules such as nucleic acids. These molecules present a challenge for theoretical description, because the conventional PB model may become insufficient in those cases. We discuss the conventional PB equation, the corresponding functionals of the electrostatic free energy, including a connection to DFT, simple empirical extensions to this model accounting for finite size of ions, the modified PB theory including ionic correlations and fluctuations, the cell model, and supplementary methods allowing to incorporate site‐bound ions in the PB calculations. © 2007 Wiley Periodicals, Inc. Biopolymers 89: 93–113, 2008.This article was originally published online as an accepted preprint. The “Published Online” date corresponds to the preprint version. You can request a copy of the preprint by emailing the Biopolymers editorial office at biopolymers@wiley.com

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“…In such cases, the PB equation becomes three-dimensional, but its solution is still rather straightforward. [14,15] Sometimes effects of polarization and hydration are included by considering a distant-dependent and field-dependent dielectric constant. The relevant theory to treat this class of models is the modified PB equation.…”

Section: Dna Polyelectrolyte Models and Theoriesmentioning

confidence: 99%

Molecular Simulations of DNA Counterion Distributions

Lyubartsev¹

2004

Dekker Encyclopedia of Nanoscience and Nanotechnology, Second Edition - Six Volume Set (Print Version)

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Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes

Cited by 176 publications

References 18 publications

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

AQUASOL: An efficient solver for the dipolar Poisson–Boltzmann–Langevin equation

Continuum molecular electrostatics, salt effects, and counterion binding—A review of the Poisson–Boltzmann theory and its modifications

Molecular Simulations of DNA Counterion Distributions

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