Volume 3, Issue 5
The Geometry Behind Numerical Solvers of the Poisson-Boltzmann Equation

Xinwei Shi and Patrice Koehl

Commun. Comput. Phys., 3 (2008), pp. 1032-1050.

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  • Abstract

Electrostatics interactions play a major role in the stabilization of biomolecules: as such, they remain a major focus of theoretical and computational studies in biophysics. Electrostatics in solution is strongly dependent on the nature of the solvent and on the ions it contains. While methods that treat the solvent and ions explicitly provide an accurate estimate of these interactions, they are usually computationally too demanding to study large macromolecular systems. Implicit solvent methods provide a viable alternative, especially those based on Poisson theory. The Poisson-Boltzmann equation (PBE) treats the system in a mean field approximation, providing reasonableestimates of electrostatics interactions in a solvent treatedas continuum. In the first part of this paper, we review the theory behind the PBE, including recent improvement in which ions size and dipolar features of solvent molecules are taken into account explicitly. The PBE is a non linear second order differential equation with discontinuous coefficients, for which no analytical solution is available for large molecular systems. Many numerical solvers have been developed that solve a discretized version of the PBE on a mesh, either using finite difference, finite element, or boundary element methods. The accuracyof the solutions providedby these solvers highly depend on the geometry of their underlying meshes, as well as on the method used to embed the physical system on the mesh. In the second part of the paper, we describe a new geometric approach for generating unstructured tetrahedral meshes as well as simplifications of these meshes that are well fitted for solving the PBE equation using multigrid approaches.

  • History

Published online: 2008-03

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