{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:41:21Z","timestamp":1760143281232,"version":"build-2065373602"},"reference-count":157,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T00:00:00Z","timestamp":1706659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Maxwell defined a \u2018true\u2019 or \u2018total\u2019 current in a way not widely used today. He said that \u201c\u2026 true electric current \u2026 is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity\u201d. We show that the true or total current is a universal property of electrodynamics independent of the properties of matter. We use mathematics without the approximation of a dielectric constant. The resulting Maxwell current law is a generalization of the Kirchhoff law of current used in circuit analysis, that also includes the displacement current. The generalization is not a long-time low-frequency approximation in contrast to the traditional presentation of Kirchhoff\u2019s law.<\/jats:p>","DOI":"10.3390\/computation12020022","type":"journal-article","created":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T09:41:59Z","timestamp":1706694119000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Maxwell\u2019s True Current"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4860-5434","authenticated-orcid":false,"given":"Robert S.","family":"Eisenberg","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA"},{"name":"Department of Physiology & Biophysics, Rush University Medical Center, Chicago, IL 60612, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,31]]},"reference":[{"key":"ref_1","unstructured":"Maxwell, J.C. (1865). A Treatise on Electricity and Magnetism (Reprinted 1954), Dover Publications. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1137\/22M1477891","article-title":"A Maxwell\u2013Amp\u00e8re Nernst\u2013Planck Framework for Modeling Charge Dynamics","volume":"83","author":"Qiao","year":"2023","journal-title":"SIAM J. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111845","DOI":"10.1016\/j.jcp.2022.111845","article-title":"Structure-preserving numerical method for Maxwell-Amp\u00e8re Nernst-Planck model","volume":"475","author":"Qiao","year":"2023","journal-title":"J. Comput. Phys."},{"key":"ref_4","unstructured":"Qiao, Z., Xu, Z., Yin, Q., and Zhou, S. (2022). An Ampere-Nernst-Planck Framework for Modeling Charge Dynamics. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Qiao, Z., Xu, Z., Yin, Q., and Zhou, S. (2022). Structure-Preserving Numerical Method for Ampere-Nernst-Planck Model. SSRN Electron. J.","DOI":"10.2139\/ssrn.4103018"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"115784","DOI":"10.1016\/j.cam.2024.115784","article-title":"A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson\u2013Nernst\u2013Planck (PNP) system","volume":"444","author":"Dong","year":"2024","journal-title":"J. Comput. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2071","DOI":"10.1090\/mcom\/3642","article-title":"A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system","volume":"90","author":"Liu","year":"2021","journal-title":"Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"104104","DOI":"10.1063\/1.3476262","article-title":"Energy Variational Analysis EnVarA of Ions in Water and Channels: Field Theory for Primitive Models of Complex Ionic Fluids","volume":"133","author":"Eisenberg","year":"2010","journal-title":"J. Chem. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Eisenberg, R. (2022). Structural Analysis of Fluid Flow in Complex Biological Systems. Model. Artif. Intell. Ophthalmol.","DOI":"10.14293\/S2199-1006.1.SOR-.PPLLJDK.v1"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Eisenberg, B. (2022). Setting Boundaries for Statistical Mechanics, Version 2: MDPI Molecules. Molecules, 27.","DOI":"10.3390\/molecules27228017"},{"key":"ref_11","first-page":"203","article-title":"Structural analysis of electrical properties of cells and tissues","volume":"4","author":"Eisenberg","year":"1980","journal-title":"CRC Crit. Rev. Bioeng."},{"key":"ref_12","unstructured":"Eisenberg, R.S. (2019). Kirchhoff\u2019s Law can be Exact. arXiv."},{"key":"ref_13","unstructured":"Eisenberg, B., Gold, N., Song, Z., and Huang, H. (2018). What Current Flows Through a Resistor?. arXiv."},{"key":"ref_14","unstructured":"Horowitz, P., and Hill, W. (2015). The Art of Electronics, Cambridge University Press. [3rd ed.]."},{"key":"ref_15","unstructured":"Feynman, R.P., Leighton, R.B., and Sands, M. (1963). The Feynman: Lectures on Physics, Vol, 1, 2, 3, Addison-Wesley Publishing Co.. Available online: http:\/\/www.feynmanlectures.caltech.edu\/II_toc.html."},{"key":"ref_16","first-page":"444","article-title":"The electrical constants of a crustacean nerve fiber","volume":"133","author":"Hodgkin","year":"1946","journal-title":"Proc. R. Soc. (Lond.) Ser. B"},{"key":"ref_17","first-page":"442","article-title":"Contribution to the Mathematical Theory of the electrotonus","volume":"131","author":"Davis","year":"1947","journal-title":"Stud. Rockefeller Inst. Med. Res."},{"key":"ref_18","unstructured":"Huxley, A. (1972). Nobel Lectures, Physiology or Medicine 1963\u20131970, Elsevier."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Morelli, A.M., Ravera, S., Calzia, D., and Panfoli, I. (2019). An update of the chemiosmotic theory as suggested by possible proton currents inside the coupling membrane. Open Biol., 9.","DOI":"10.1098\/rsob.180221"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"zqac018","DOI":"10.1093\/function\/zqac018","article-title":"Setting the Record Straight: A New Twist on the Chemiosmotic Mechanism of Oxidative Phosphorylation","volume":"3","author":"Juhaszova","year":"2022","journal-title":"Function"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0968-0004(88)90005-9","article-title":"Bioenergetic coupling to protonmotive force: Should we be considering hydronium ion coordination and not group protonation?","volume":"13","author":"Boyer","year":"1988","journal-title":"Trends Biochem. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2196","DOI":"10.1021\/cr500448t","article-title":"New perspectives on proton pumping in cellular respiration","volume":"115","author":"Wikstrom","year":"2015","journal-title":"Chem. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1038\/213137a0","article-title":"Chemiosmotic hypothesis of oxidative phosphorylation","volume":"213","author":"Mitchell","year":"1967","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/0014-5793(75)80359-0","article-title":"The protonmotive Q cycle: A general formulation","volume":"59","author":"Mitchell","year":"1975","journal-title":"FEBS Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1146\/annurev.bi.46.070177.005024","article-title":"Vectorial chemiosmotic processes","volume":"46","author":"Mitchell","year":"1977","journal-title":"Annu. Rev. Biochem."},{"key":"ref_26","unstructured":"Stryer, L. (1995). Biochemistry, W.H. Freeman. [4th ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1038\/191144a0","article-title":"Coupling of phosphorylation to electron and hydrogen transfer by a chemi-osmotic type of mechanism","volume":"191","author":"Mitchell","year":"1961","journal-title":"Nature"},{"key":"ref_28","unstructured":"Berg, J.M., Tymoczko, J.L., and Stryer, L. (2010). Biochemistry, W. H. Freeman. [7th ed.]."},{"key":"ref_29","unstructured":"Nicholls, D.G. (2013). Bioenergetics, Academic Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"10004","DOI":"10.1016\/S0021-9258(18)90918-X","article-title":"The relation between the internal phosphorylation potential and the proton motive force in mitochondria during ATP synthesis and hydrolysis","volume":"259","author":"Ogawa","year":"1984","journal-title":"J. Biol. Chem."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2308","DOI":"10.1002\/(SICI)1521-3773(19980918)37:17<2308::AID-ANIE2308>3.0.CO;2-W","article-title":"ATP Synthesis by Rotary Catalysis (Nobel lecture)","volume":"37","author":"Walker","year":"1998","journal-title":"Angew. Chem. Int. Ed. Engl."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"DeCoursey, T.E. (2023). Transcendent Aspects of Proton Channels. Annu. Rev. Physiol., 86.","DOI":"10.1146\/annurev-physiol-042222-023242"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"20130799","DOI":"10.1098\/rsif.2013.0799","article-title":"Philosophy of voltage-gated proton channels","volume":"11","author":"DeCoursey","year":"2014","journal-title":"J. R. Soc. Interface"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Chaves, G., Jardin, C., Derst, C., and Musset, B. (2023). Voltage-Gated Proton Channels in the Tree of Life. Biomolecules, 13.","DOI":"10.3390\/biom13071035"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1021\/ja209176e","article-title":"Insights into the Mechanism of Proton Transport in Cytochrome c Oxidase","volume":"134","author":"Yamashita","year":"2012","journal-title":"J. Am. Chem. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1021\/ar200140h","article-title":"The Curious Case of the Hydrated Proton","volume":"45","author":"Knight","year":"2011","journal-title":"Acc. Chem. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3223","DOI":"10.1021\/ct1004438","article-title":"Defining Condensed Phase Reactive Force Fields from ab Initio Molecular Dynamics Simulations: The Case of the Hydrated Excess Proton","volume":"6","author":"Knight","year":"2010","journal-title":"J. Chem. Theory Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1529\/biophysj.106.091934","article-title":"Charge delocalization in proton channels, I: The aquaporin channels and proton blockage","volume":"92","author":"Chen","year":"2007","journal-title":"Biophys. J."},{"key":"ref_39","first-page":"459","article-title":"A Dynamical Theory of the Electromagnetic Field","volume":"155","author":"Maxwell","year":"1865","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Griffiths, D.J. (2017). Introduction to Electrodynamics, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/9781108333511"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Purcell, E.M., and Morin, D.J. (2013). Electricity and Magnetism, Cambridge University Press.","DOI":"10.1017\/CBO9781139012973"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Zangwill, A. (2013). Modern Electrodynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139034777"},{"key":"ref_43","unstructured":"Eisenberg, R.S. (2019). Dielectric Dilemma. arXiv."},{"key":"ref_44","unstructured":"Eisenberg, R.S. (2019). Updating Maxwell with Electrons, Charge, and More Realistic Polarization. arXiv."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Eisenberg, R.S. (2020). Maxwell Equations for Material Systems. arXiv.","DOI":"10.20944\/preprints202011.0201.v1"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Eisenberg, R.S. (2021). Maxwell Equations Without a Polarization Field, Using a Paradigm from Biophysics. Entropy, 23.","DOI":"10.3390\/e23020172"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.4310\/CMS.2022.v20.n6.a3","article-title":"On Variational Principles for Polarization Responses in Electromechanical Systems","volume":"20","author":"Eisenberg","year":"2022","journal-title":"Commun. Math. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1016\/j.bpj.2013.03.049","article-title":"Interacting ions in Biophysics: Real is not ideal","volume":"104","author":"Eisenberg","year":"2013","journal-title":"Biophys. J."},{"key":"ref_49","unstructured":"Barthel, J., Buchner, R., and M\u00fcnsterer, M. (1995). Electrolyte Data Collection, DECHEMA. Volume 12, Part 2: Dielectric Properties of Water and Aqueous Electrolyte Solutions."},{"key":"ref_50","unstructured":"Barthel, J., Krienke, H., and Kunz, W. (1998). Physical Chemistry of Electrolyte Solutions: Modern Aspects, Springer."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Kremer, F., and Sch\u00f6nhals, A. (2003). Broadband Dielectric Spectroscopy, Springer.","DOI":"10.1007\/978-3-642-56120-7"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Barsoukov, E., and Macdonald, J.R. (2018). Impedance Spectroscopy: Theory, Experiment, and Applications, John Wiley & Sons.","DOI":"10.1002\/9781119381860"},{"key":"ref_53","first-page":"78","article-title":"Dynamics of Current, Charge, and Mass","volume":"5","author":"Eisenberg","year":"2017","journal-title":"Mol. Based Math. Biol."},{"key":"ref_54","unstructured":"Eisenberg, R., Oriols, X., and Ferry, D.K. (2022). Kirchhoff\u2019s Current Law with Displacement Current. arXiv."},{"key":"ref_55","unstructured":"Whittaker, E. (1951). A History of the Theories of Aether & Electricity, Harper."},{"key":"ref_56","unstructured":"Xu, S., Eisenberg, R., Song, Z., and Huang, H. (2023). Mathematical models for electrochemistry: Law of mass action revisited. arXiv."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Xu, S., Eisenberg, R., Song, Z., and Huang, H. (2022). Mathematical Model for Chemical Reactions in Electrolyte Applied to Cytochrome c Oxidase: An Electro-osmotic Approach. arXiv.","DOI":"10.3390\/computation11120253"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00249-012-0798-4","article-title":"Ionizable side chains at catalytic active sites of enzymes","volume":"41","author":"Liang","year":"2012","journal-title":"Eur. Biophys. J."},{"key":"ref_59","unstructured":"Banwell, C.N., and McCash, E.M. (1994). Fundamentals of Molecular Spectroscopy, McGraw-Hill."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Stuart, B. (2005). Infrared Spectroscopy, Wiley Online Library.","DOI":"10.1002\/0471238961.0914061810151405.a01.pub2"},{"key":"ref_61","unstructured":"Jaff\u00e9, H.H., and Orchin, M. (1962). Theory and Applications of Ultraviolet Spectroscopy, Wiley."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Parsegian, V.A. (2006). Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge University Press.","DOI":"10.1017\/CBO9780511614606"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s10825-013-0461-z","article-title":"Quantum transport beyond DC","volume":"12","author":"Oriols","year":"2013","journal-title":"J. Comput. Electron."},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Ayers, J.E. (2018). Digital Integrated Circuits: Analysis and Design, CRC Press. [2nd ed.].","DOI":"10.1201\/b12491"},{"key":"ref_65","unstructured":"Gielen, G., and Sansen, W.M. (2012). Symbolic Analysis for Automated Design of Analog Integrated Circuits, Springer Science & Business Media."},{"key":"ref_66","unstructured":"Gray, P.R., Hurst, P.J., Lewis, S.H., and Meyer, R.G. (2009). Analysis and Design of Analog Integrated Circuits, Wiley."},{"key":"ref_67","unstructured":"Hall, S.H., and Heck, H.L. (2011). Advanced Signal Integrity for High-Speed Digital Designs, John Wiley & Sons."},{"key":"ref_68","unstructured":"Scherz, P., and Monk, S. (2006). Practical Electronics for Inventors, McGraw-Hill, Inc."},{"key":"ref_69","unstructured":"Sedra, A.S., Smith, K.C., Chan, T., Carusone, T.C., and Gaudet, V. (2020). Microelectronic Circuits, Oxford University Press, Incorporated."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its application to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1113\/jphysiol.1937.sp003508","article-title":"Evidence for electrical transmission in nerve: Part II","volume":"90","author":"Hodgkin","year":"1937","journal-title":"J. Physiol."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1113\/jphysiol.1937.sp003507","article-title":"Evidence for electrical transmission in nerve: Part I","volume":"90","author":"Hodgkin","year":"1937","journal-title":"J. Physiol."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01869101","article-title":"Channels as enzymes: Oxymoron and Tautology","volume":"115","author":"Eisenberg","year":"1990","journal-title":"J. Membr. Biol."},{"key":"ref_74","doi-asserted-by":"crossref","unstructured":"Kuhn, T.S. (2012). The Structure of Scientific Revolutions, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226458144.001.0001"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1063\/1.857865","article-title":"The equations of nearly incompressible fluids. I. Hydrodynamics, turbulence, and waves","volume":"3","author":"Zank","year":"1991","journal-title":"Phys. Fluids A Fluid Dyn."},{"key":"ref_76","unstructured":"Christodoulou, D., and Miao, S. (2014). Compressible Flow and Euler\u2019s Equations, International Press."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1088\/0031-8949\/1992\/T42\/020","article-title":"Conductance from transmission: Common sense points","volume":"1992","author":"Landauer","year":"1992","journal-title":"Phys. Scr."},{"key":"ref_78","unstructured":"Bush, V., and Wiener, N. (1929). Operational Circuit Analysis: With an Appendix by Norbert Wiener, Chapman & Hall."},{"key":"ref_79","unstructured":"Tuttle, D.F. (1958). Network Synthesis, Wiley."},{"key":"ref_80","unstructured":"Ghausi, M.S., and Kelly, J.J. (1968). Introduction to Distributed-Parameter Networks: With Application to Integrated Circuits, Holt, Rinehart and Winston."},{"key":"ref_81","unstructured":"Guillemin, E.A. (1931). Communications Networks Vol. 1 The Classical Theory of Lumped Constant Networks, John Wiley."},{"key":"ref_82","unstructured":"Guillemin, E.A. (2013). Theory of Linear Physical Systems: Theory of Physical Systems from the Viewpoint of Classical Dynamics, Including Fourier Methods, Dover Publications."},{"key":"ref_83","unstructured":"Balabanian, N., and Bickart, T.A. (1969). Electrical Network Theory, Wiley."},{"key":"ref_84","unstructured":"Weinberg, L. (1975). Network Analysis and Synthesis, Krieger Pub. Co."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1109\/MPUL.2013.2240078","article-title":"James Clerk Maxwell, Kirchhoff\u2019s Laws, and their implications on modeling physiology","volume":"4","author":"Valentinuzzi","year":"2013","journal-title":"IEEE Pulse"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"3772","DOI":"10.1109\/TAP.2011.2163787","article-title":"Kirchhoff\u2019s Laws as a Finite Volume Method for the Planar Maxwell Equations","volume":"59","author":"Bhat","year":"2011","journal-title":"Antennas Propag. IEEE Trans."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Eisenberg, R. (2022). A Necessary Addition to Kirchhoff\u2019s Current Law of Circuits, Version 2. Eng. Arch. EngArXiv.","DOI":"10.31224\/2234"},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Eisenberg, R. (2023). Circuits, Currents, Kirchhoff, and Maxwell. Qeios.","DOI":"10.32388\/L9QQSH"},{"key":"ref_89","unstructured":"Eisenberg, R.S. (2020). Electrodynamics Correlates Knock-on and Knock-off: Current is Spatially Uniform in Ion Channels. arXiv."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1113\/jphysiol.1955.sp005291","article-title":"The potassium permeability of a giant nerve fibre","volume":"128","author":"Hodgkin","year":"1955","journal-title":"J. Physiol."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1085\/jgp.72.4.409","article-title":"Potassium channels as multi-ion single-file pores","volume":"72","author":"Hille","year":"1978","journal-title":"J. Gen. Physiol."},{"key":"ref_92","unstructured":"Hille, B. (2001). Ion Channels of Excitable Membranes, Sinauer Associates Inc.. [3rd ed.]."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1038\/s41557-018-0105-9","article-title":"Direct knock-on of desolvated ions governs strict ion selectivity in K+ channels","volume":"10","author":"Kopec","year":"2018","journal-title":"Nat. Chem."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Luchinsky, D.G., Tindjong, R., Kaufman, I., McClintock, P.V.E., Peter, V.E., Khovanov, I.A., and Eisenberg, B. (2014). Observation Of \u201cRemote Knock-On\u201d, A New Permeation-Enhancement Mechanism In Ion Channels. Biophys. J., 106.","DOI":"10.1016\/j.bpj.2013.11.783"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1126\/science.1254840","article-title":"Ion permeation in K+ channels occurs by direct Coulomb knock-on","volume":"346","author":"Song","year":"2014","journal-title":"Science"},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1039\/b613668k","article-title":"Insight into the origins of the barrier-less knock-on conduction in the KcsA channel: Molecular dynamics simulations and ab initio calculations","volume":"9","author":"Kraszewski","year":"2007","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Xu, S., Eisenberg, R., Song, Z., and Huang, H. (2023). Mathematical Model for Chemical Reactions in Electrolytes Applied to Cytochrome c Oxidase: An Electro-Osmotic Approach. Computation, 11.","DOI":"10.3390\/computation11120253"},{"key":"ref_98","unstructured":"Song, Z., Eisenberg, R., Xu, S., and Huang, H. (2022). A Bubble Model for the Gating of K Channels. arXiv."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"3008","DOI":"10.1016\/j.bpj.2021.06.020","article-title":"A tridomain model for potassium clearance in optic nerve of Necturus","volume":"120","author":"Zhu","year":"2021","journal-title":"Biophys. J."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"041906","DOI":"10.1063\/5.0046323","article-title":"Optic nerve microcirculation: Fluid flow and electrodiffusion","volume":"33","author":"Zhu","year":"2021","journal-title":"Phys. Fluids"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1016\/j.bpj.2019.02.007","article-title":"A Bidomain Model for Lens Microcirculation","volume":"116","author":"Zhu","year":"2019","journal-title":"Biophys. J."},{"key":"ref_102","unstructured":"Xu, S., Eisenberg, B., Song, Z., and Huang, H. (2018). Osmosis through a Semi-permeable Membrane: A Consistent Approach to Interactions. arXiv."},{"key":"ref_103","unstructured":"Alberts, B., Bray, D., Johnson, A., Lewis, J., Raff, M., and Roberts, K. (1998). Essential Cell Biology, Garland. [3rd ed.]."},{"key":"ref_104","first-page":"382","article-title":"On the theory of the electric telegraph","volume":"7","author":"Kelvin","year":"1855","journal-title":"Proc. R. Soc."},{"key":"ref_105","first-page":"146","article-title":"On the theory of the electric telegraph","volume":"11","author":"Kelvin","year":"1856","journal-title":"Philos. Mag."},{"key":"ref_106","unstructured":"Gordon, J.S. (2008). A Thread Across the Ocean: The Heroic Story of the Transatlantic Cable, Paw Prints."},{"key":"ref_107","unstructured":"Okoshi, T. (2012). Planar Circuits for Microwaves and Lightwaves, Springer Science & Business Media."},{"key":"ref_108","unstructured":"Schwierz, F., and Liou, J.J. (2003). Modern Microwave Transistors: Theory, Design, and Performance, Wiley-Interscience."},{"key":"ref_109","doi-asserted-by":"crossref","unstructured":"Fukunaga, K., and Kurahashi, S. (2007, January 17\u201321). Dielectric properties of printed circuit board insulations at microwaves and millimetre waves. Proceedings of the 2007 International Conference on Electromagnetics in Advanced Applications (ICEAA 2007), Turin, Italy.","DOI":"10.1109\/ICEAA.2007.4387305"},{"key":"ref_110","unstructured":"Mei, K.K. (2000, January 12\u201315). From Kirchoff to Lorentz modifying-circuit theory for microwave and mm-wave structures. Proceedings of the 25th International Conference on Infrared and Millimeter Waves, Beijing, China."},{"key":"ref_111","doi-asserted-by":"crossref","unstructured":"Kevorkian, J., and Cole, J.D. (1996). Multiple Scale and Singular Perturbation Methods, Springer.","DOI":"10.1007\/978-1-4612-3968-0"},{"key":"ref_112","unstructured":"Desoer, C.A., and Kuh, E.S. (1969). Basic Circuit Theory, McGraw Hill."},{"key":"ref_113","unstructured":"Dobkin, B., and Williams, J. (2012). Analog Circuit Design Volume 2: Immersion in the Black Art of Analog Design, Newnes."},{"key":"ref_114","doi-asserted-by":"crossref","unstructured":"Lienig, J., and Scheible, J. (2020). Fundamentals of Layout Design for Electronic Circuits, Springer Nature.","DOI":"10.1007\/978-3-030-39284-0"},{"key":"ref_115","unstructured":"Muller, R.S., Chan, M., and Kamins, T.I. (2003). Device Electronics For Integrated Circuits, Wiley India Pvt. Limited. [3rd ed.]."},{"key":"ref_116","doi-asserted-by":"crossref","unstructured":"Eisenberg, R.S. (2020). Thermostatics vs. Electrodynamics. arXiv.","DOI":"10.20944\/preprints202009.0349.v1"},{"key":"ref_117","unstructured":"Howe, R.T., and Sodini, C.G. (1997). Microelectronics: An Integrated Approach, Prentice Hall."},{"key":"ref_118","unstructured":"Hastings, A. (2001). The Art of Analog Layout, Prentice-Hall. Inc."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1137\/0121036","article-title":"A singular perturbation analysis of induced electric fields in nerve cells","volume":"21","author":"Barcilon","year":"1971","journal-title":"SIAM J. Appl. Math."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1085\/jgp.63.5.625","article-title":"Longitudinal impedance of skinned frog muscle fibers","volume":"63","author":"Mobley","year":"1974","journal-title":"J. Gen. Physiol."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1085\/jgp.65.1.97","article-title":"Longitudinal impedance of single frog muscle fibers","volume":"65","author":"Mobley","year":"1975","journal-title":"J. Gen. Physiol."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1113\/jphysiol.1965.sp007733","article-title":"The effect of changing the internal solution on sodium inactivation and related phenomena in giant axons","volume":"180","author":"Chandler","year":"1965","journal-title":"J. Physiol."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1113\/jphysiol.1962.sp007025","article-title":"Replacement of the axoplasm of giant nerve fibres with artificial solutions","volume":"164","author":"Baker","year":"1962","journal-title":"J. Physiol."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1113\/jphysiol.1962.sp007026","article-title":"The effects of changes in internal ionic concentrations on the electrical properties of perfused giant axons","volume":"164","author":"Baker","year":"1962","journal-title":"J. Physiol."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1113\/jphysiol.1952.sp004716","article-title":"Measurement of current- voltage relations in the membrane of the giant axon of Loligo","volume":"116","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"736","DOI":"10.1085\/jgp.55.6.736","article-title":"The spatial variation of membrane potential near a small source of current in a spherical cell","volume":"55","author":"Eisenberg","year":"1970","journal-title":"J. Gen. Physiol."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/S0006-3495(72)86091-0","article-title":"The interpretation of current-voltage relations recorded from a spherical cell with a single microelectrode","volume":"12","author":"Engel","year":"1972","journal-title":"Biophys. J."},{"key":"ref_128","unstructured":"Hellmann, H. (1937). Introduction to Quantum Chemistry, Deuticke."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1103\/PhysRev.56.340","article-title":"Forces in molecules","volume":"56","author":"Feynman","year":"1939","journal-title":"Phys. Rev."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"16207","DOI":"10.1103\/PhysRevB.61.16207","article-title":"Hellmann-Feynman theorem and the definition of forces in quantum time-dependent and transport problems","volume":"61","author":"Pantelides","year":"2000","journal-title":"Phys. Rev. B"},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1007\/s00894-018-3784-7","article-title":"The Hellmann-Feynman theorem: A perspective","volume":"24","author":"Politzer","year":"2018","journal-title":"J. Mol. Model."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"014104","DOI":"10.1063\/5.0130668","article-title":"Accurate Hellmann\u2013Feynman forces from density functional calculations with augmented Gaussian basis sets","volume":"158","author":"Pathak","year":"2023","journal-title":"J. Chem. Phys."},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1113\/jphysiol.1969.sp008778","article-title":"Sodium influxes in internally perfused squid giant axon during voltage clamp","volume":"201","author":"Atwater","year":"1969","journal-title":"J. Physiol."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1113\/jphysiol.1970.sp009302","article-title":"Time course of the sodium permeability change during a single membrane action potential","volume":"211","author":"Atwater","year":"1970","journal-title":"J. Physiol."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1113\/jphysiol.1970.sp009301","article-title":"Sodium and potassium conductance changes during a membrane action potential","volume":"211","author":"Bezanilla","year":"1970","journal-title":"J. Physiol."},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1113\/jphysiol.1970.sp009054","article-title":"Time course of the sodium influx in squid giant axon during a single voltage clamp pulse","volume":"207","author":"Bezanilla","year":"1970","journal-title":"J. Physiol."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1038\/225747a0","article-title":"Demonstration of sodium and potassium conductance changes during a nerve action potential","volume":"225","author":"Rojas","year":"1970","journal-title":"Nature"},{"key":"ref_138","unstructured":"Eisenberg Robert, S. (2011). Comprehensive Physiology, Republished by the American Physiological Society, as Volume 1, Supplement 27 of Handbook of Physiology, Part of Comprehensive Physiology, American Physiological Society and Wiley OnLine. Handbook of Physiology."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0079-6107(70)90013-1","article-title":"Three dimensional electrical field problem in physiology","volume":"20","author":"Eisenberg","year":"1970","journal-title":"Prog. Biophys. Mol. Biol."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1085\/jgp.50.6.1785","article-title":"The equivalent circuit of single crab muscle fibers as determined by impedance measurements with intracellular electrodes","volume":"50","author":"Eisenberg","year":"1967","journal-title":"J. Gen. Physiol."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1146\/annurev.bb.02.060173.000433","article-title":"Interpretation of some microelectrode measurements of electrical properties of cells","volume":"2","author":"Peskoff","year":"1973","journal-title":"Annu. Rev. Biophys. Bioeng."},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0006-3495(79)85283-2","article-title":"Electrical properties of spherical syncytia","volume":"25","author":"Eisenberg","year":"1979","journal-title":"Biophys. J."},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1113\/jphysiol.1964.sp007348","article-title":"The Effect of Diluting the Internal Solution on the Electrical Properties of a Perfused Giant Axon","volume":"170","author":"Baker","year":"1964","journal-title":"J. Physiol."},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/0022-5193(65)90042-1","article-title":"The variation of capacitance and conductance of bimolecular lipid membranes with area","volume":"9","author":"Hanai","year":"1965","journal-title":"J. Theor. Biol."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/0022-5193(65)90041-X","article-title":"The influence of lipid composition and of some adsorbed proteins on the capacitance of black hydrocarbon membranes","volume":"9","author":"Hanai","year":"1965","journal-title":"J. Theor. Biol."},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/0022-5193(68)90084-2","article-title":"Electrical capacitance of a lipid membrane separating two aqueous phases","volume":"18","author":"Everitt","year":"1968","journal-title":"J. Theor. Biol."},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1085\/jgp.9.2.137","article-title":"The electric capacity of suspensions with special reference to blood","volume":"9","author":"Fricke","year":"1925","journal-title":"J. Gen. Physiol."},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1103\/PhysRev.26.682","article-title":"The electric capacity of suspensions of red corpuscles of a dog","volume":"26","author":"Fricke","year":"1925","journal-title":"Phys. Rev."},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1085\/jgp.9.2.153","article-title":"The electric resistance and capacity of blood for frequencies between 800 and 4\u00bd million cycles","volume":"9","author":"Fricke","year":"1925","journal-title":"J. Gen. Physiol."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1085\/jgp.22.5.649","article-title":"Electric impedance of the squid giant axon during activity","volume":"22","author":"Cole","year":"1939","journal-title":"J. Gen. Physiol."},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1085\/jgp.201812090","article-title":"Gating currents","volume":"150","author":"Bezanilla","year":"2018","journal-title":"J. Gen. Physiol."},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"3983","DOI":"10.1016\/j.bpj.2021.08.015","article-title":"Gating current noise produced by Brownian models of a voltage sensor","volume":"120","author":"Catacuzzeno","year":"2021","journal-title":"Biophys. J."},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"e202012706","DOI":"10.1085\/jgp.202012706","article-title":"Multiscale modeling shows that dielectric differences make NaV channels faster than KV channels","volume":"153","author":"Catacuzzeno","year":"2021","journal-title":"J. Gen. Physiol."},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"3227","DOI":"10.1113\/JP282780","article-title":"The 70-year search for the voltage sensing mechanism of ion channels","volume":"600","author":"Catacuzzeno","year":"2022","journal-title":"J. Physiol."},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1085\/jgp.20028660","article-title":"Voltage Sensor Movements","volume":"120","author":"Bezanilla","year":"2002","journal-title":"J. Gen. Physiol."},{"key":"ref_156","unstructured":"Hill, A.V. (1932). Chemical Wave Transmission in Nerve, Cambridge University Press."},{"key":"ref_157","doi-asserted-by":"crossref","unstructured":"Chazalviel, J.-N. (1999). Coulomb Screening by Mobile Charges, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1762-6"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/2\/22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:52:24Z","timestamp":1760104344000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/12\/2\/22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,31]]},"references-count":157,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["computation12020022"],"URL":"https:\/\/doi.org\/10.3390\/computation12020022","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2024,1,31]]}}}