Relaxed Boundary Conditions in Poisson-Nernst-Planck Models: Identifying Critical Potentials for Multiple Cations.

Ion channels are protein pores that regulate ionic flow across cell membranes, enabling vital processes such as nerve signaling. They often conduct multiple ionic species simultaneously, leading to complex nonlinear transport phenomena. Because experimental techniques provide only indirect measurements of ion channel currents, mathematical models-particularly Poisson-Nernst-Planck (PNP) equations-are indispensable for analyzing the underlying transport mechanisms. In this work, we examine ionic transport through a one-dimensional steady-state PNP model of a narrow membrane channel containing multiple cation species of different valences. The model incorporates a small fixed charge distribution along the channel and imposes relaxed electroneutrality boundary conditions, allowing for a slight charge imbalance in the baths. Using singular perturbation analysis, we first derive approximate solutions that capture the boundary-layer structure at the channel-reservoir interfaces. We then perform a regular perturbation expansion around the neutral reference state (zero fixed charge with electroneutral boundary conditions) to obtain explicit formulas for the steady-state ion fluxes in terms of the system parameters. Through this analytical approach, we identify several critical applied potential values-denoted Vka (for each cation species