Effects of multi-component small permanent charges on the dynamics of Poisson-Nernst-Planck models

In this paper, the classical Poisson-Nernst-Planck (PNP) model describing ion transport through a membrane channel is used to study the effects of small permanent charges and the structures of ion channels on ionic flows. The model under study includes two oppositely charged ion species, and the permanent charge in this model is a piecewise constant function with two nonzero regions. By rescaling, the classical PNP model can be viewed as a singularly perturbed differential equation system. Therefore, the geometric singular perturbation theory is employed to get a singular orbit. Assuming that the permanent charge density is small, a regular perturbation expansion is used to obtain the first-order approximation of the individual flux, which acts as a basis for our analysis. Then, the effects of small permanent charges on the fluxes and the current-voltage relation, which not only depend on the boundary conditions, but also depend on the structures of ion channels and the ratio between two nonzero permanent charge densities, are analyzed in this paper. Particularly, our results indicate that the geometric structures of three-dimensional ion channels have a short and narrow cross-section, which is explained in [1]. Also, our results indicate that the ratio between two nonzero permanent charge densities can change the position of a short and narrow cross-section in ion channels.