Ke. Forsten et al., NUMERICAL-SOLUTION OF THE NONLINEAR POISSON-BOLTZMANN EQUATION FOR A MEMBRANE-ELECTROLYTE SYSTEM, Journal of physical chemistry, 98(21), 1994, pp. 5580-5586
Two features that characterize the complex nature of a membrane-electr
olyte system are the change in dielectric at the lipid-solvent interfa
ce and the periodicity of the charge-embedded membrane. The former can
be treated within a continuum model, and the planar nature of the mem
brane can be accounted for through the enforcement of periodic boundar
y conditions. Here we describe a numerical technique, based on a finit
e-difference formulation, for solving the full nonlinear Poisson-Boltz
mann equation which incorporates the above features of a membrane-elec
trolyte system. This method is used to calculate the electrostatic pot
ential 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 w
ith the Gouy-Chapman theory, which is based on the assumption of a uni
form 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 wher
e a set of dipoles is embedded in a membrane. This procedure can be ap
plied to the investigation of the electrostatic properties of lipid-bo
und proteins and in other cases where Gouy-Chapman theory is inadequat
e.