Dataset Information

Asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low salt concentration: analytic expressions for surface potential and preferential interaction coefficient.

ABSTRACT: The analytic solution to the nonlinear Poisson-Boltzmann equation describing the ion distributions surrounding a nucleic acid or other cylindrical polyions as a function of polyion structural quantities and salt concentration ([salt]) has been sought for more than 80 years to predict the effect of these quantities on the thermodynamics of polyion processes. Here we report an accurate asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low to moderate concentration of a symmetrical electrolyte (< or = 0.1 M 1:1 salt). The approximate solution for the potential is derived as an asymptotic series in the small parameter var epsilon(-1), where var epsilon identical with kappa(-1)/a, the ratio of the Debye length (kappa(-1)) to the polyion radius (a). From the potential at the polyion surface, we obtain the coulombic contribution to the salt-polyelectrolyte preferential interaction (Donnan) coefficient (Gamma(u)coul) per polyion charge at any reduced axial charge density xi. Gamma(u)coul is the sum of the previously recognized low-salt limiting value and a salt-dependent contribution, analytically derived here in the range of low-salt concentrations. As an example of the application of this solution, we obtain an analytic expression for the derivative of the midpoint temperature of a nucleic acid conformational transition with respect to the logarithm of salt concentration (dT(m)/d ln[salt]) in terms of [salt] and nucleic acid structural quantities. This expression explains the experimental observation that this derivative is relatively independent of salt concentration but deviates significantly from its low-salt limiting value in the range 0.01-0.1 M.

SUBMITTER: Shkel IA

PROVIDER: S-EPMC122393 | biostudies-literature | 2002 Mar

REPOSITORIES: biostudies-literature

ACCESS DATA

Publications

Asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low salt concentration: analytic expressions for surface potential and preferential interaction coefficient.

Shkel Irina A IA Tsodikov Oleg V OV Record M Thomas MT

Proceedings of the National Academy of Sciences of the United States of America 20020301 5

The analytic solution to the nonlinear Poisson-Boltzmann equation describing the ion distributions surrounding a nucleic acid or other cylindrical polyions as a function of polyion structural quantities and salt concentration ([salt]) has been sought for more than 80 years to predict the effect of these quantities on the thermodynamics of polyion processes. Here we report an accurate asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low to moderate concentration of a ...[more]

PMID: 11880615

Similar Datasets

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

Asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low salt concentration: analytic expressions for surface potential and preferential interaction coefficient.

Publications

Asymptotic solution of the cylindrical nonlinear Poisson-Boltzmann equation at low salt concentration: analytic expressions for surface potential and preferential interaction coefficient.

Similar Datasets

OmicsDI is part of the ELIXIR infrastructure

Tweets