New insights into the effects of small permanent charge on ionic flows: A higher order analysis

This study investigated how permanent charges influence the dynamics of ionic channels. Using a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP) model, we investigated the behavior of two distinct ion species-one positively charged and the other negatively charged. The spatial distribution of permanent charges was characterized by zero values at the channel ends and a constant charge $ Q_0 $ within the central region. By treating the classical PNP model as a boundary value problem (BVP) for a singularly perturbed system, the singular orbit of the BVP depended on $ Q_0 $ in a regular way. We therefore explored the solution space in the presence of a small permanent charge, uncovering a systematic dependence on this parameter. Our analysis employed a rigorous perturbation approach to reveal higher-order effects originating from the permanent charges. Through this investigation, we shed light on the intricate interplay among boundary conditions and permanent charges, providing insights into their impact on the behavior of ionic current, fluxes, and flux ratios. We derived the quadratic solutions in terms of permanent charge, which were notably more intricate compared to the linear solutions. Through computational tools, we investigated the impact of these quadratic solutions on fluxes, current-voltage relations, and flux ratios, conducting a thorough analysis of the results. These novel findings contributed to a deeper comprehension of ionic flow dynamics and hold potential implications for enhancing the design and optimization of ion channel-based technologies.