Studies on Poisson–Nernst–Planck Systems with Large Permanent Charges Under Relaxed Neutral Boundary Conditions

Modeling ion transport through membrane channels is crucial for understanding cellular processes, and Poisson–Nernst–Planck (PNP) equations provide a fundamental continuum framework for such ionic fluxes. We investigate a quasi-one-dimensional steady-state PNP system for two oppositely charged ion species, focusing on how large permanent charges within the channel and realistic boundary conditions impact ion transport. In contrast to classical models that impose ideal electroneutrality at the channel ends (a simplification that eliminates boundary layers near the membrane interfaces), we adopt relaxed neutral boundary conditions that allow small charge imbalances at the boundaries. Using asymptotic analysis treating the large permanent charge as a singular perturbation, we derive explicit first-order expansions for each ionic flux, incorporating boundary layer parameters ( σ , ρ ) to quantify slight deviations from electroneutrality. This analysis enables a qualitative characterization of individual cation and anion flux behaviors. Notably, we identify two critical transmembrane potentials, V 1 c and V 2 c , at which the cation and anion fluxes, respectively, vanish, signifying flux-reversal thresholds that delineate distinct monotonic regimes in the flux-voltage response; these critical values depend on the permanent charge magnitude and the boundary layer parameters. We further show that both ionic fluxes exhibit saturation: as the applied voltage becomes extreme, each flux approaches a finite limiting value, with the saturation level modulated by the degree of boundary charge imbalance. Moreover, allowing even small boundary charge deviations reveals non-intuitive discrepancies in flux behavior relative to the ideal electroneutral case. For example, in certain parameter regimes, a large permanent charge that enhances an ionic current under strict electroneutral conditions will instead suppress that current under relaxed-neutral conditions (and vice versa). This new analytical framework exposes subtle yet essential nonlinear dynamics that classical electroneutral assumptions would otherwise obscure. It provides deeper insight into the interplay between large fixed charges and boundary-layer effects, emphasizing the importance of incorporating such realistic boundary conditions to ensure accurate modeling of ion transport through membrane channels. Numerical simulations are performed to provide more intuitive illustrations of our analytical results.