{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T07:22:22Z","timestamp":1765610542962,"version":"build-2065373602"},"reference-count":89,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,19]],"date-time":"2025-01-19T00:00:00Z","timestamp":1737244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Simons Foundation USA","award":["628308","Z6124034","12172199"],"award-info":[{"award-number":["628308","Z6124034","12172199"]}]},{"name":"Linyi University","award":["628308","Z6124034","12172199"],"award-info":[{"award-number":["628308","Z6124034","12172199"]}]},{"name":"National Natural Science Foundation of China","award":["628308","Z6124034","12172199"],"award-info":[{"award-number":["628308","Z6124034","12172199"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We examine the qualitative properties of ionic flows through membrane channels via Poisson\u2013Nernst\u2013Planck (PNP) type models with steric effects under relaxed electroneutrality boundary conditions, and more realistic setups in the study of ion channel problems. Of particular interest are the vital roles played by some critical potentials identified for both individual fluxes and current\u2013voltage relations. These critical potentials split the whole electric potential interval into different subintervals, over which distinct dynamics of ionic flows are observed. The discussion provides an efficient way to control the boundary conditions to observe distinct dynamics of ionic flows through membrane channels. This is important for future analytical studies and critical for future numerical and even experimental studies of ion channel problems.<\/jats:p>","DOI":"10.3390\/axioms14010069","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T12:32:52Z","timestamp":1737376372000},"page":"69","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Roles Played by Critical Potentials in the Study of Poisson\u2013Nernst\u2013Planck Models with Steric Effects Under Relaxed Neutral Boundary Conditions"],"prefix":"10.3390","volume":"14","author":[{"given":"Xiangshuo","family":"Liu","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266510, China"}],"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"}]},{"given":"Lijun","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266510, 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":[[2025,1,19]]},"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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