Qualitative properties of zero-current ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes

We consider a one-dimensional Poisson-Nernst-Planck system with two oppositely charged particles and nonuniform finite ion sizes modeled through a local hard-sphere potential. The existence and local uniqueness result is established under the framework of geometric singular perturbation theory. Treating the fi- nite ion size as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to further study the qualitative properties of zero-current ionic flows, a special state among the range of the value for ionic current, which is significant for physiology. Of particular interest are the effects on the zero-current ionic flows from finite ion sizes, diffusion coefficients and ion valences. Critical potentials are identified and their important roles played in the study of ionic flow properties are characterized. Those non-intuitive observations from mathematical analysis of the system provide better understandings of the mechanism of ionic flows through membrane channels, particularly the internal dynamics of ionic flows, which cannot be detected via current technology. Numerical simulations are performed to provide more intuitive illustrations of the analytical results.