Analytical solutions of Poisson's equation for realistic geometrical shapes of membrane ion channels

Analytical solutions of Poisson's equations satisfying the Dirichlet boundary conditions for a toroidal dielectric boundary are presented. The electric potential computed anywhere in the toroidal conduit by the analytical method agrees with the value derived from an iterative numerical method. We show that three different channel geometries, namely, bicone, catenary, and toroid, give similar potential profiles as an ion traverses along their central axis. We then examine the effects of dipoles in the toroidal channel wall on the potential profile of ions passing through the channel. The presence of dipoles eliminates the barrier for one polarity of ion, while raising the barrier for ions of the opposite polarity. We also examine how a uniform electric field from an external source is affected by the protein boundary and a mobile charge. The channel distorts the field, reducing it in the vestibules, and enhancing it in the constricted segment. The presence of an ion in one vestibule effectively excludes ions of the same polarity from that vestibule, but has little effect in the other vestibule. Finally, we discuss how the solutions we provide here may be utilized to simulate a system containing a channel and many interacting ions.