The Geometry Behind Numerical Solvers of the Poisson-Boltzmann Equation (Communications in Computational Physics, 2008)
Electrostatics interactions play a major role in the stabilization ofbiomolecules: as such, they remain a major focus of theoretical and computationalstudies in biophysics. Electrostatics in solution is strongly dependent on the nature ofthe solvent and on the ions it contains. While methods that treat the solvent and ionsexplicitly provide an accurate estimate of these interactions, they are usually computationally too demanding to study large macromolecular systems. Implicit solventmethods provide a viable alternative, especially those based on Poisson theory. ThePoisson-Boltzmann equation (PBE) treats the system in a mean field approximation,providing reasonable estimates of electrostatics interactions in a solvent treated as continuum. In the first part of this paper, we review the theory behind the PBE, includingrecent improvement in which ions size and dipolar features of solvent molecules aretaken into account explicitly. The PBE is a non linear second order differential equation with discontinuous coefficients, for which no analytical solution is available forlarge molecular systems. Many numerical solvers have been developed that solve adiscretized version of the PBE on a mesh, either using finite difference, finite element,or boundary element methods. The accuracy of the solutions provided by these solvershighly depend on the geometry of their underlying meshes, as well as on the methodused to embed the physical system on the mesh. In the second part of the paper, wedescribe a new geometric approach for generating unstructured tetrahedral meshes aswell as simplifications of these meshes that are well fitted for solving the PBE equationusing multigrid approaches.