NUMERICAL-SOLUTION OF THE NONLINEAR POISSON-BOLTZMANN EQUATION FOR A MEMBRANE-ELECTROLYTE SYSTEM

Two features that characterize the complex nature of a membrane-electrolyte system are the change in dielectric at the lipid-solvent interface and the periodicity of the charge-embedded membrane. The former can be treated within a continuum model, and the planar nature of the membrane can be accounted for through the enforcement of periodic boundary conditions. Here we describe a numerical technique, based on a finite-difference formulation, for solving the full nonlinear Poisson-Boltzmann equation which incorporates the above features of a membrane-electrolyte system. This method is used to calculate the electrostatic potential for a model membrane containing a rectangular array of charges at a variety of lattice spacings and ionic strengths. At sufficiently large distances from the membrane, the results are in good agreement with the Gouy-Chapman theory, which is based on the assumption of a uniform charge density in an infinite plane. Electrostatic potentials are also obtained in the interior of the membrane for tl;e model system. In addition, this method is used to find the potential for a case where a set of dipoles is embedded in a membrane. This procedure can be applied to the investigation of the electrostatic properties of lipid-bound proteins and in other cases where Gouy-Chapman theory is inadequate.