A minimizing principle for the Poisson-Boltzmann equation

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Abstract

The Poisson-Boltzmann equation is often presented via a variational formulation based on the electrostatic potential. However, the functional has the defect of being non-convex. It can not be used as a local minimization principle while coupled to other dynamic degrees of freedom. We formulate a convex dual functional which is numerically equivalent at its minimum and which is more suited to local optimization.

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Most cited references 13

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Classical electrostatics in biology and chemistry

B. Honig, A. Nicholls (1995)

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Accelerated Poisson-Boltzmann calculations for static and dynamic systems.

Laurent David, Michael Gilson, Ray Luo (2002)

We report here an efficient implementation of the finite difference Poisson-Boltzmann solvent model based on the Modified Incomplete Cholsky Conjugate Gradient algorithm, which gives rather impressive performance for both static and dynamic systems. This is achieved by implementing the algorithm with Eisenstat's two optimizations, utilizing the electrostatic update in simulations, and applying prudent approximations, including: relaxing the convergence criterion, not updating Poisson-Boltzmann-related forces every step, and using electrostatic focusing. It is also possible to markedly accelerate the supporting routines that are used to set up the calculations and to obtain energies and forces. The resulting finite difference Poisson-Boltzmann method delivers efficiency comparable to the distance-dependent dielectric model for a system tested, HIV Protease, making it a strong candidate for solution-phase molecular dynamics simulations. Further, the finite difference method includes all intrasolute electrostatic interactions, whereas the distance dependent dielectric calculations use a 15-A cutoff. The speed of our numerical finite difference method is comparable to that of the pair-wise Generalized Born approximation to the Poisson-Boltzmann method. Copyright 2002 Wiley Periodicals, Inc.

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Beyond Poisson-Boltzmann: Fluctuations and Correlations

Roland R Netz, Henri Orland (1999)

We formulate the non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point, and the effects of fluctuations and correlations are included by a loop-wise expansion around this saddle point. We show that the Poisson equation is obeyed at each order in the loop expansion and explicitly give the expansion of the Gibbs potential up to two loops. We then apply our formalism to the case of an impenetrable, charged wall, and obtain the fluctuation corrections to the electrostatic potential and counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. We also calculate effective interactions between charged particles, which reflect counter-ion correlation effects.