Mathematical Model for Chemical Reactions in Electrolytes Applied to Cytochrome c Oxidase: An Electro-Osmotic Approach

Abstract

This study introduces a mathematical model for electrolytic chemical reactions, employing an energy variation approach grounded in classical thermodynamics. 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 ‘energetic currency of life’. This model shows how the enzyme’s structure directs currents and mass flows governed by energetic and dissipative functionals. The interplay between electron and proton flows, guided by Kirchhoff’s 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’s law in a model that is much simpler to implement than a full accounting of all the charges in a chemical system.

Maxwell’s True Current

“…The complexity is hard to deal with in mitochondria where electrons, protons, and ions flow, each carrying significant current important in the generation of ATP, the main function of mitochondria. Despite this complexity, it is in fact possible to use conservation laws and descriptions of structure and structural components to compute currents directly comparable to experiments [9,56,[97][98][99][100][101][102].…”

“…The arrows are not identified as currents. The flows of electrons exist in this classical work, as in real mitochondria, but the laws of current flow or electrodynamics have not been used to describe those currents, as far as I know, except in the papers of our research group [56,97].…”

“…The Maxwell current law (3) applies to currents driven via chemical reactions, protonmotive forces, electrochemical potentials, convection, heat flow, and so on. The coupling can be embedded in models with the structural complexity of biological systems, albeit oversimplified [9,56,[97][98][99][100][101][102].…”

“…Methods advocated here are practical. The research group of Huaxiong Huang, Xu et al 2022, have performed actual calculations of coupled currents, without particular difficulties, for one of the complexes (Respiratory Complex 4, the cytochrome c oxidase system) using standard numerical methods of partial differential equations applied to 'multiphysics' systems, where migration and diffusion occur in specialized structures [9,56,[97][98][99][100][101][102].…”

“…Note that the units of protonmotive force are not the units of flux or current or ATP production. This difficulty of units in the original chemiosmotic hypothesis is not present when the electro-osmotic approach is used, because the electro-osmotic approach, like all circuit analysis, is automatically nonequilibrium, with velocity, flows, friction, and power transfer as automatic consequences of the analysis and mathematics [9,56,[97][98][99][100][101][102].…”