Image Approximations to Electrostatic Potentials in Layered Electrolytes/Dielectrics and an Ion-Channel Model

Image charge approximations are developed for electric potentials in the Poisson-Boltzmann theory in inhomogeneous media consisting of dielectrics or electrolyte solutions such as the layered structure in a membrane or cylindrical ion-channels. The image charges are obtained either by a least square fitting between the potential of unknown images and the exact reaction potential (for the layered media or cylindrical region) or by a Prony fitting to the Fourier transform of the exact potential (layered media only) and a Sommerfeld-type identity, which yields the locations and strengths of the image charges. Next, combining the results for the two geometries, the image charge approximation for the reaction potential, due to a charge inside an ion-channel model, is obtained, which accounts for the polarization of the region outside the ion-channel (consisting of a membrane and electrolyte solutions below and above). Such an approximation to the reaction field in the ion-channel model can be used for an explicit/implicit hybrid treatment of electrostatics interaction in modeling ion-channels. Numerical tests show that the proposed method has an attractive performance in computing electrostatic interactions of source charges inside the ion-channel model via a simple summation of pairwise interactions among source and image charges.