Dynamics of Ion Transport in Biological Cells with Ion Fluxes from the Boundaries

We propose to determine solutions of the time-dependent Poisson-Nernst-Planck (PNP) system for systems with arbitrary ion types. Models that involve more than two ions are crucial in many biological applications and systems with three or more ion types are significantly more challenging to analyse than the widely studied two-ion case that arises in semi-conductors. We will consider the case of arbitrary valences and with arbitrary diffusion coefficients for each ionic species. We will make use of the fact that the Debye length is significantly smaller than the domain size in all biological applications. We will hence use asymptotic techniques that will allow us to solve for the dynamics in the Debye layers and electro-neutral interior regions. The full solution will then be obtained by matching the dynamic solutions in the outer regions and the Debye layer. Our preliminary investigation points to the fact that matching techniques that can be applied in the time-independent problem cannot match the boundary layer solutions and the bulk solution in the case of the generic time-dependent problem. We therefore anticipate that complicated intermediate boundary layer structures must exist in the the time-dependent problem. We will determine these structures and discuss the consequences for biological applications.