Geometric Singular Perturbation Analysis of Poisson– Nernst–Planck Models with Large Permanent Charges and Multi-Cation Transport

Assuming the permanent charge density is much larger than the ion concentrations on two boundaries of an open ion channel, this paper studies the impacts of large permanent charges which are positioned along the channel wall and play a significant role in channel functioning. Through this work, the quasi-one-dimensional Poisson–Nernst–Planck (PNP) system is utilized to analyze the electrodiffusion properties of ionic flows through an ion channel. The geometric singular perturbation theory is applied to derive the existence and local uniqueness of solutions to the corresponding PNP system containing large permanent charges and two monovalent cations. The matching asymptotic expansions are conducted to generate the expansions of state variables in terms of the permanent charge density, from which the permanent charge effects on individual fluxes can be discussed by further using the regular perturbation analysis. The mechanisms of the inhibited role played by large permanent charge in the co-ions’ flow are also elucidated from the perspective of internal dynamics. We hope this work could provide more comprehensive information on large permanent charge effects on ion channels and display more realistic interactions between related physical parameters such as channel geometry, diffusion coefficients, boundary potential and boundary ion concentrations.