THE INFLUENCE OF SMALL PERMANENT CHARGES AND PARTIAL BOUNDARY CONDITIONS ON INDIVIDUAL FLUXES VIA POISSON-NERNST-PLANCK SYSTEMS

We investigate ionic flow through membrane channels within a one-dimensional Poisson-Nernst-Planck framework, considering two oppositely charged ion species and incorporating a small but non-zero permanent charge. Building upon prior analysis in [<span class="xref"><a href="#b50" ref-type="bibr">50</a></span>], which was confined to cases with higher left-side concentrations, we relax this constraint to examine a broader and more physiologically relevant range of boundary conditions. Crucially, instead of considering the product of zeroth- and first-order terms in the flux expansion, we focus on the first-order term itself, which directly captures the leading effects of channel geometry and permanent charge. Under partial electroneutral boundary conditions, our analysis results demonstrate that the permanent charge plays a regulatory role: It can either enhance or suppress both ion fluxes simultaneously, or exert opposing effects by increasing anion flux while decreasing cation flux - an outcome that depends on the electric potential applied at the boundary. We further identify and compare several critical electric potentials, providing new insights into the transport mechanisms operative across different potential intervals. Numerical simulations on selected cases yield results in excellent agreement with the theory, confirming and supporting our analytical conclusions.