{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T12:07:15Z","timestamp":1770984435268,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T00:00:00Z","timestamp":1770940800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"start-up fund from Linyi University","award":["Z6124034"],"award-info":[{"award-number":["Z6124034"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We examine the effects from small, spatially localized permanent charges on ionic transport in narrow membrane channels. Our analysis is based on a one-dimensional steady-state Poisson\u2013Nernst\u2013Planck (PNP) model involving two oppositely charged ion species with constant diffusion coefficients under electroneutral boundary conditions. In the framework of geometric singular perturbation theory, the steady PNP system is reformulated as a fast\u2013slow dynamical system amenable to boundary-layer analysis. In the limit of vanishing permanent charge, the solution exhibits a singular structure with sharp boundary-layer segments and smooth bulk segments across regions of piecewise constant charge. Assuming the permanent charge strength Q is small, we carry out a regular perturbation expansion about Q=0 and derive explicit first-order corrections to each ion\u2019s flux. Closed-form expressions are obtained for both the leading-order (zero-charge) fluxes and the O(Q) flux corrections, revealing how even a small fixed charge can modulate the magnitude of individual ionic fluxes as a function of the applied transmembrane voltage and boundary concentration asymmetry. These results elucidate how permanent charge enhances or inhibits specific ionic flows, thereby influencing channel selectivity. Overall, our analysis provides clear asymptotic formulas and highlights the broader relevance of this perturbative approach to electro-diffusive transport modeling in biophysical systems.<\/jats:p>","DOI":"10.3390\/axioms15020135","type":"journal-article","created":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T10:52:21Z","timestamp":1770979941000},"page":"135","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Additive Effects of Small Permanent Charges on Ionic Flow Using Poisson\u2013Nernst\u2013Planck Systems"],"prefix":"10.3390","volume":"15","author":[{"given":"Jia","family":"Guo","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhantao","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-6869-2723","authenticated-orcid":false,"given":"Jie","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Linyi University, Linyi 276000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8249-4758","authenticated-orcid":false,"given":"Mingji","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1960","DOI":"10.1529\/biophysj.107.105478","article-title":"Steric selectivity in Na channels arising from protein polarization and mobile side chains","volume":"93","author":"Boda","year":"2007","journal-title":"Biophys. 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