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Our model combines electrostatics and chemical reactions within well-defined energetic and dissipative functionals. Extending the energy variation method to open systems consisting of charge, mass, and energy inputs, this model explores energy transformation from one form to another. Electronic devices and biological channels and transporters are open systems. By applying this generalized approach, we investigate the conversion of an electrical current to a proton flow by cytochrome c oxidase, a vital mitochondrial enzyme contributing to ATP production, the \u2018energetic currency of life\u2019. This model shows how the enzyme\u2019s structure directs currents and mass flows governed by energetic and dissipative functionals. The interplay between electron and proton flows, guided by Kirchhoff\u2019s current law within the mitochondrial membrane and the mitochondria itself, determines the function of the systems, where electron flows are converted into proton flows and gradients. This important biological system serves as a practical example of the use of energy variation methods to deal with electrochemical reactions in open systems. We combine chemical reactions and Kirchhoff\u2019s law in a model that is much simpler to implement than a full accounting of all the charges in a chemical system.<\/jats:p>","DOI":"10.3390\/computation11120253","type":"journal-article","created":{"date-parts":[[2023,12,11]],"date-time":"2023-12-11T11:24:43Z","timestamp":1702293883000},"page":"253","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Mathematical Model for Chemical Reactions in Electrolytes Applied to Cytochrome c Oxidase: An Electro-Osmotic Approach"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8207-7313","authenticated-orcid":false,"given":"Shixin","family":"Xu","sequence":"first","affiliation":[{"name":"Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, 8 Duke Ave., Kunshan 215316, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4860-5434","authenticated-orcid":false,"given":"Robert","family":"Eisenberg","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA"},{"name":"Department of Physiology and Biophysics, Rush University, Chicago, IL 60612, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zilong","family":"Song","sequence":"additional","affiliation":[{"name":"Math and Statistics Department, Utah State University, Old Main Hill, Logan, UT 84322, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huaxiong","family":"Huang","sequence":"additional","affiliation":[{"name":"Research Center for Mathematics, Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai 519088, China"},{"name":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai 519088, China"},{"name":"Laboratory of Mathematics and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China"},{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,11]]},"reference":[{"key":"ref_1","unstructured":"Brush, S.G. 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