{"paper_id":"06de2927-3667-4ef3-a696-9b37ad77d4b2","body_text":"1 \n \n28th Jan 2026 \nThe biophysical basis of enterocyte homeostasis \nPeter Hunter†, Jarrah Dowrick, Weiwei Ai, David Nickerson,  \nMohammad Hossein Shafieizadegan and Finbar Argus \nAuckland Bioengineering Ins�tute, University of Auckland, New Zealand \n†Email p.hunter@auckland.ac.nz for correspondence. \nABSTRACT \nWe present an approach to analysing cell homeostasis using a ‘bond graph’ modelling approach that \nensures that the conserva�on laws of physics (conserva�on of mass, charge, and energy, respec�vely) \nare sa�sfied for the interdependent biochemical, electrical, mechanical, and thermal energy storage \nmechanisms opera�ng within the cell. We apply the bond graph approach to several cell membrane \ntransport mechanisms and then consider how physics constrains intracellular electrolyte homeostasis \nfor enterocytes (the epithelial absorp�ve cells of the gut). The model includes the electrogenic sodium-\npotassium ATPase pump (NKA), the glucose transporter (GLUT2), and an inwardly rec�fying potassium \nchannel, all in the basolateral membrane, and the electrogenic sodium-driven glucose transporter \n(SGLT1) in the apical membrane . Glycolysis converts the imported glucose to ATP to drive NK A. For \nspecified levels of sodium, potassium , and glucose  in the blood, the model  demonstrates how \nenterocytes absorb sodium and glucose from the gut and transfer glucose to the blood  while \nmaintaining the membrane poten�al and homeostasis of intracellular sodium and potassium. The \nGibbs free energy available from the ATP hydrolysis ensures that the cell operates as a ‘sodium batery’ \nwith a high external to internal ra�o of sodium concentra�on in order to provide the energy for many \nother cellular transport processes.  We show that the 3:2 stoichiometry of Na+/K+ exchange in NKA, \ncoupled with 2:1 Na+/glucose cotransport in SGLT1, a 1:2:2 ra�o between glucose consump�on and \nATP and water produc�on in glycolysis, and K+ and glucose efflux through Kir and GLUT2, respec�vely, \nprovides a balanced system that maintains homeostasis of intracellular Na+, K+, glucose, ATP and water, \nand homeostasis of the membrane poten�al, under varying levels of transport of glucose from the gut \nto the blood . We also show how the flux expressions for SLC transporters, ATPase pumps and ion \nchannels can all be expressed in a consistent and thermodynamically valid way.    \nINTRODUCTION \nCellular physiology operates at the theore�cal limits set by physical laws – for example, ion channels \nare sensi�ve to the passage of a single elementary charge, and re�nae can detect one photon  [1]. \nEukaryo�c cells also exploit every form of energy storage mechanism available – biochemical (e.g., \nsolute concentra�ons and chemical bonds ), electrical (e.g., capaci�ve charge storage in the cell \nmembrane), mechanical (e.g., the elas�c compliance of cellular membranes), and thermal (the heat \nstorage essen�al for maintaining body temperature). The physical processes that maintain intracellular \nhomeostasis are conserva�on of mass, conserva�on of charge, and conserva�on of energy. In this \npaper we show how the ‘bond graph’ concept, invented over 50 years ago by Henry Paynter at MIT [2] \ncan be used to explain how cellular homeostasis depends on all forms of energy transmission, storage, \nand dissipa�on (chemical, electrical, mechanical and thermal)  via the applica�on of these basic \nbiophysical laws. The applica�on of bond graphs to a variety of biological mechanisms was pioneered \nby Oster, Perelson and Katchlsky in the 1970s [3] and then later expanded upon by Gawthrop, Crampin, \nand Pan at the University of Melbourne [4,5,6]. The formula�on presented here, including a focus on \nappropriate units, model reduc�on strategies, and a new graphical formula�on to simplify the analysis, \nwas pioneered by the present authors [7,8].  \nThe applica�on of bond graphs  to analysing cellular homeostasis has not, to our knowledge, been \nundertaken previously and relies on a new algebraic formula�on of the steady membrane fluxes [9]. \nWe present a new way of expressing the flux for a reac�on that is thermodynamically consistent and \napplicable across all transmembrane transport mechanisms (exchangers, cotransporters, ATPase \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n2 \n \npumps and ion channels). We demonstrate the importance of inward rec�fica�on in the Kir channel \nfor achieving homeostasis. Other novel contribu�ons in the paper include the demonstra�on of using \nbond graphs in whole cell modelling to ensure energy consistency, and an energy-conserving model of \nan enterocyte for analysis of cell homeostasis.  \nUNITS AND THE CONSERVATION LAWS OF PHYSICS \nBefore we introduce bond graph concepts, it is useful to first discuss units, since adop�ng a small \nnumber of appropriate units  and understanding their role in linking atomic-level quan��es with \nprocesses at the macroscale clarifies the applica�on of bond graphs to mul�ple physical domains. The \nSI base units (m, s, kg, mol, A, K, cd) [10] were largely chosen to reflect the technologies available at \nthe �me (1960) to measure them with sufficient accuracy (such as the standard kg mass held in a vault \nin Paris). However, since 2019 the magnitudes of all SI units have been defined by declaring that seven \ndefining constants (the speed of light in vacuum 𝑐𝑐 = 2.99792458×108 m.s-1, the Planck constant ℎ  = \n6.62607015×10−34 J.s, the elementary charge qe = 1.602176634×10−19 C, the hyperfine transi�on \nfrequency of caesium 𝛥𝛥𝜈𝜈𝐶𝐶𝐶𝐶 = 9.192631770×109 s-1, the Boltzmann constant kB = 1.380649x10-23 J.K-1, \nAvogadro’s constant NA = 6.02214076 x1023 mol-1, and the luminous efficacy 𝐾𝐾𝑐𝑐𝑐𝑐) have certain exact \nnumerical values when expressed in terms of their SI units [10]. The underlying units used to define \nthese constants are m , s, J, mol, C, K, and cd (the seven SI base units are then defined in terms of \nthese). The first three (m, s, J) establish the energized 4D space-�me world in which we reside. The \nnext two (mol, C) reflect atomic or molecular level physical en��es: a mole ( mol) of substance \nmeasures the number of atoms or molecules it contains (there are NA atoms in 12 g of carbon-12); a \nCoulomb (C) measures charge at the macroscale since  the charge on an electron q e scaled up by \nAvogadro’s number to give Faraday’s constant F= NA.qe is the charge in Coulombs per mole (C.mol-1). \nThe Kelvin (K) measures the temperature or thermal energy associated with Brownian mo�on by \nscaling up the energy per atom or molecule, given by Boltzmann’s constant kB (J.K-1), to the macroscale \nwith NA.kB.T = RT (J.mol-1) where R = NA.kB is the gas constant. The last unit (Candela or cd) is not needed \nunless photons need to be counted  since a photon of frequency 𝜈𝜈 (s-1) has energy ℎ𝜈𝜈 (J). Finally, the \nmagnitude of a rota�on is expressed in terms of the dimensionless unit radians (rad). The SI base unit \nkg is J.s2.m-2, and the SI base unit A is C.s-1. From here on, we use these six units: m, s, J, mol, C, and K.  \nIn the following sec�on, we dis�nguish units that measure the amount of a quan�ty 𝑞𝑞, such as m, mol, \nand C, from units of flux 𝑣𝑣=\n𝑐𝑐𝑑𝑑\n𝑐𝑐\n𝑑𝑑 (the amount per second) and poten�al 𝑢𝑢 (expressed as J per unit \nquan�ty) driving that flux. We also show that the three conserva�on laws of physics (conserva�on of \nmass, charge, and energy) can be consolidated into a single conservation of power law, where power \nis the product 𝑢𝑢. 𝑣𝑣.  \nB\nond graphs provide a useful  means of formula�ng and  visualising a thermodynamically valid \nbiophysical model because they ensure that mass, charge, and energy are each conserved, and they \nclearly dis�nguish the mechanisms for (i) transmission of power (the product of poten�al 𝑢𝑢 and flux \n𝑣𝑣), (ii) energy storage (mechanically in a spring, electrically in a capacitor, or chemically in a solute \ndissolved in a solvent), (iii) energy dissipation to heat (a mechanical damper, an electrical resistance or \na chemical reac�on), and (iv) energy transfer between mechanical, electrical or chemical domains.  \nMost importantly, they dis�nguish between the conserva�on laws of physics and the empirically \nmeasured cons�tu�ve rela�ons that characterise par�cular materials. For example, a chemical species \n𝑖𝑖 is stored as a solute 𝑞𝑞𝑗𝑗\n𝑖𝑖 (mol) in a solu�on at loca�on 𝑗𝑗 with a par�cular solubility that generates a \nchemical poten�al 𝑢𝑢𝑗𝑗\n𝑖𝑖 (J.mol-1). The diffusion of this solute through a dissipa�ve medium from one \nloca�on to another is quan�fied as a molar flux 𝑣𝑣𝑗𝑗\n𝑖𝑖=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑖𝑖\n𝑐𝑐𝑑𝑑  (mol.s-1) that depends both on the difference \nin chemical poten�al between the two loca�ons and on the diffusivity of that medium. The measured \nvalues for solubility and diffusivity are two dis�nct material constants, and both are quite separate \nfrom the equa�ons represen�ng mass and energy conserva�on. These different material parameters \nand conserva�on laws are lumped together in Fick’s law of diffusion. \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n3 \n \nWe use the  symbol 𝑞𝑞𝑗𝑗\n𝑖𝑖 to denote t he quan�ty in moles (mol) of a chemical species  𝑖𝑖 at loca�on 𝑗𝑗. \nSimilarly, 𝑞𝑞𝑗𝑗\n𝑒𝑒 represents electrical charge in Coulombs (C), and 𝑞𝑞𝑗𝑗\n𝑚𝑚 represents distance (m), volume (m3) \nor rota�on (rad). M olar flux  is denoted 𝑣𝑣𝑗𝑗\n𝑖𝑖=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑖𝑖\n𝑐𝑐𝑑𝑑 (mol.s-1) and 𝑢𝑢𝑗𝑗\n𝑖𝑖 (J.mol-1) is the chemical poten�al \ngenerated by storage of solute 𝑞𝑞𝑗𝑗\n𝑖𝑖 (mol). Electrical current is 𝑣𝑣𝑗𝑗\n𝑒𝑒=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑒𝑒\n𝑐𝑐𝑑𝑑 (C.s-1) and 𝑢𝑢𝑗𝑗\n𝑒𝑒 (J.C-1) is the electrical \npoten�al generated by capaci�ve storage of charge 𝑞𝑞𝑗𝑗\n𝑒𝑒 (C). In solid mechanics 𝑣𝑣𝑗𝑗\n𝑚𝑚=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑚𝑚\n𝑐𝑐𝑑𝑑 (m.s-1) is the \nvelocity and 𝑢𝑢𝑗𝑗\n𝑚𝑚 (J.m-1) is the force generated by displacement 𝑞𝑞𝑗𝑗\n𝑚𝑚 (m), or 𝑣𝑣𝑗𝑗\n𝑚𝑚=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑚𝑚\n𝑐𝑐𝑑𝑑 (rad.s-1) is the \nangular velocity and 𝑢𝑢𝑗𝑗\n𝑚𝑚 (J.rad-1) is the torque generated by rota�on 𝑞𝑞𝑗𝑗\n𝑚𝑚 (rad). Finally, in fluid mechanics \n𝑣𝑣𝑗𝑗\n𝑚𝑚=\n𝑐𝑐𝑑𝑑𝑗𝑗\n𝑚𝑚\n𝑐𝑐𝑑𝑑 (m3.s-1) is the fluid flow and 𝑢𝑢𝑗𝑗\n𝑚𝑚 (J.m-3) is the pressure (energy density) generated by fluid \nvolume 𝑞𝑞𝑗𝑗\n𝑚𝑚 (m3).  \nThese units therefore facilitate the expression of energy flux (J.s-1) as the product of a poten�al in \nJoules per unit quan�ty (mol, C, m, m3 or rad) that is driving the flow of that quan�ty in a way that is \ncommon to all physical systems. The product of chemical poten�al 𝑢𝑢𝑗𝑗\n𝑖𝑖 (J.mol-1) and flux 𝑣𝑣𝑗𝑗\n𝑖𝑖 (mol.s-1), or \nelectrical poten�al 𝑢𝑢𝑗𝑗\n𝑒𝑒 (J.C-1) and flux 𝑣𝑣𝑗𝑗\n𝑒𝑒 (C.s-1), is always power ( J.s-1). Similarly, the product of \nmechanical poten�al (force, pressure, or torque) and mechanical flux (velocity, fluid flow, or angular \nv e l o c i t y )  i s  p o w e r .  T h e  p r o d u c t  o f  h e a t  fl o w ,  w h i c h  i s  a n  e n t r o p y  fl u x  (entropy.s-1), and thermal \npoten�al (J.entropy-1 or temperature in Kelvin K) is also power.    \nTHE BOND GRAPH MODELLING FRAMEWORK  \nWe now introduce the basic ideas behind the bond graph framework introduced by Paynter [2]. Lines \nof power transmission called ‘bonds’ always have an associated flux 𝑣𝑣 and poten�al 𝑢𝑢 (see Figure 1). \nIf these bonds meet, the sum of powers must be zero: ∑ 𝑢𝑢. 𝑣𝑣= 0, to ensure power conserva�on. If \nthey share a common poten�al 𝑢𝑢 (called a ‘ 0:node’), power conserva�on 𝑢𝑢∑ 𝑣𝑣= 0 becomes just \n∑ 𝑣𝑣= 0 (for non-zero 𝑢𝑢), which is mass conservation if 𝑣𝑣 is a molar flux or mechanical flow and charge \nconservation if 𝑣𝑣 is an electrical flux. Alterna�vely, if they share a common flux 𝑣𝑣 (called a ‘1:node’), \npower conserva�on 𝑣𝑣∑ 𝑢𝑢= 0 becomes just ∑ 𝑢𝑢= 0 (for non-zero 𝑣𝑣), which is energy conservation. \nFor chemical reac�ons these correspond to mass conserva�on and chemical stoichiometric rela�ons, \nrespec�vely. For electrical circuits the y correspond to Kirch hoff’s current law and voltage law, \nrespec�vely. For solid mechanics systems they correspond to kinema�c consistency and force or \ntorque balance, respec�vely. For fluid systems they correspond to conserva�on of volume \n(conserva�on of mass which holds when density is assumed constant) and pressure balance.   \n \n (a)   (b)   (c)  (d)  \nFigure 1. Key bond graph concepts: (a) a bond, which transmits energy, carries a flow 𝑣𝑣𝑗𝑗\n𝑖𝑖 and a poten�al 𝑢𝑢𝑗𝑗\n𝑖𝑖; (b) a \n0:node is a bond junc�on where the poten�al is the same for all bonds and therefore the sum of flows is zero \n(conserva�on of mass or charge); (c) a 1:node is a bond junc�on where the flow is the same for all bonds and \ntherefore the sum of poten�als is zero (conserva�on of energy); (d) a 0:node is usually associated with capaci�ve \nenergy storage as well as flux balance (top) and can be more succinctly expressed by the red-bordered box where \nthe poten�al 𝑢𝑢𝑐𝑐\n1 is given by an empirically defined capaci�ve storage rela�onship 𝑢𝑢𝑐𝑐\n1\n= 𝑓𝑓(𝑞𝑞𝑐𝑐\n1). Note that in this \nfigure, the poten�als are coloured red and the kinema�c quan��es in green, just to empathise the difference. \nBy applying power conserva�on to 0:nodes and 1:nodes, bond graph models ensure that the \nconserva�on laws of physics are always obeyed. By using a common nota�on (quan�ty 𝑞𝑞, flux 𝑣𝑣, and \npoten�al 𝑢𝑢) with appropriate units (m, s, J, mol, C, K, rad), we can use bond graphs to examine \n𝒗𝒗𝒋𝒋\n𝒊𝒊  �=\n𝑐𝑐𝒒𝒒𝒋𝒋\n𝒊𝒊\n𝑐𝑐𝑑𝑑�  \n𝒖𝒖𝒋𝒋\n𝒊𝒊 \n 𝒗𝒗𝟏𝟏\n𝒊𝒊 \n𝒗𝒗𝟐𝟐\n𝒊𝒊 \n𝒗𝒗𝟑𝟑\n𝒊𝒊 \n𝒗𝒗𝟒𝟒\n𝒊𝒊 \n𝒗𝒗𝟓𝟓\n𝒊𝒊 \n0: 𝒖𝒖𝒄𝒄\n𝒊𝒊 \n 𝒖𝒖𝟏𝟏\n𝒊𝒊 \n𝒖𝒖𝟐𝟐\n𝒊𝒊 \n𝒖𝒖𝟑𝟑\n𝒊𝒊 \n𝒖𝒖𝟒𝟒\n𝒊𝒊 \n𝒖𝒖𝟓𝟓\n𝒊𝒊 \n1: 𝒗𝒗𝒄𝒄\n𝒊𝒊 \n \n𝑞𝑞𝑐𝑐\n𝑖𝑖 \n0: 𝒖𝒖𝒄𝒄\n𝒊𝒊 \n𝐶𝐶: 𝒒𝒒𝒄𝒄\n𝒊𝒊 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n4 \n \nphysiological processes that involve the exchange of energy between chemical, electrical, mechanical, \nand thermal forms. \nNote, however, that the mass or charge balance equa�on ∑ 𝑣𝑣= 0 involves only flux es 𝑣𝑣 (and not \npoten�als 𝑢𝑢), and that the energy balance equa�on ∑ 𝑢𝑢= 0 involves only poten�als 𝑢𝑢 (and not flux 𝑣𝑣 \nor quan�ty 𝑞𝑞). To apply these equa�ons to a specific mechanism, they must therefore be \nsupplemented with constitutive relations that couple poten�al 𝑢𝑢 with quan�ty 𝑞𝑞 or flux 𝑣𝑣. Cons�tu�ve \nrela�ons contain the ‘material’ parameters that characterise par�cular material proper�es, in contrast \nwith the conserva�on laws that express the generic constraints imposed by physics. Cons�tu�ve laws \n(and their corresponding material constants) are, for example, needed to describe the  sta�c storage \nmechanisms (solute  solubility, electrical charge capacitance, mechanical elas�city, thermal \ncapacitance) and the dissipa�ve mechanisms ( chemical reac�on kine�cs, electrical resistance, \nmechanical viscosity, and thermal conduc�vity). Both types of equa�on are needed but it is extremely \nimportant to dis�nguish the generally applicable conserva�on laws , captured with 0:nodes and \n1:nodes, from the materially specific cons�tu�ve equa�ons.    \nBOND GRAPH MODELS OF BASIC CELLULAR MECHANISMS  \nBefore considering intracellular homeostasis, we illustrate the applica�on of bond graphs to (i) a simple \nchemical reac�on, (ii) a reac�on with mul�ple reactants and products, (iii) an enzyme -catalysed \nreac�on, (iv) ATP hydrolysis, (v) the glucose transporter (GLUT2), (vi) the electrogenic sodium-glucose \ncotransporter (SGLT1), (vii) the sodium-potassium ATPase (NKA) pump, and (viii) an inwardly rec�fying \npotassium ion channel (Kir). Wherever experimentally valid, we make assump�ons that simplify the \nmodels, but always in a way that maintains conserva�on of mass, charge, and energy. These reduced \nexamples provide the mechanis�c bond graph models we subsequently use for analysis of intracellular \nhomeostasis. We also present a simplified graphical representa�on in order to facilitate clarity as we \nprogress to more advanced models.    \n(i) Simple chemical reaction \nWe use the simplest possible chemical reac�on, consis�ng of the reversible interconversion of molar \nquan��es 𝑞𝑞𝑐𝑐\n1 and 𝑞𝑞𝑐𝑐\n2 (Figure 2a), to derive the bond graph equa�ons and to discuss the use of bond \ngraph symbols that minimise the complexity of the diagrams.  \n  \n   (a)   (b)   (c)  \nFigure 2. (a) The chemical reac�on between 𝑞𝑞𝑐𝑐\n1 and 𝑞𝑞𝑐𝑐\n2; (b) the full bond graph representa�on; and (c) the \nsimplified bond graph representa�on in which the red-bordered symbol represents capaci�ve storage combined \nwith 0:node mass balance, and the black -bordered reac�on flux 𝑣𝑣𝑅𝑅𝑅𝑅 symbol includes the reac�on and 1:node \nenergy balance. It is some�mes more useful to replace 𝑞𝑞𝑐𝑐\n𝑖𝑖 in the red -bordered symbol with the poten�al 𝑢𝑢𝑐𝑐\n𝑖𝑖 \ncalculated from the 𝑞𝑞𝑐𝑐\n𝑖𝑖 by a Boltzmann equa�on (see text). \nIn order to formulate the bond graph describing the simple reac�on (Figure 2b), we first iden�fy the \nmolar quan��es 𝑞𝑞𝑐𝑐1 and 𝑞𝑞𝑐𝑐2 stored in solu�on as capaci�ve energy storage elements 𝐶𝐶: 𝑞𝑞𝑐𝑐1 and 𝐶𝐶: 𝑞𝑞𝑐𝑐2, \neach genera�ng, via a cons�tu�ve law (the Boltzmann equa�on), a chemical poten�al at a 0:node \n(0: 𝑢𝑢𝑐𝑐1 and 0: 𝑢𝑢𝑐𝑐2) where mass balance is imposed. In Figure 2c the 0:node is combined with the \ncapaci�ve storage element and writen as a single component with a red border (since every molar \nquan�ty in solu�on generates a poten�al through the Boltzmann equa�on). In Figure 2b the chemical \ninterconversion between species is a dissipa�ve reac�on element interposed between two 1:nodes, \nenforcing a common reac�on flux with associated forward (𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓)  a n d  r e v e r s e  (𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟) poten�als (the \narrowhead iden�fies the posi�ve flux direc�on). As above, we combine the dissipa�ve element and its \nneighbouring 1:nodes to form a single simplified element in Figure 2c. In so doing, the seven nodes of \n𝑞𝑞𝑐𝑐\n1 \n 𝑞𝑞𝑐𝑐\n2 \n𝑘𝑘𝑓𝑓 \n𝑘𝑘𝑟𝑟 \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 \n 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 \n𝐶𝐶: 𝒒𝒒𝒄𝒄𝟏𝟏 \n 𝐶𝐶: 𝒒𝒒𝒄𝒄𝟐𝟐 \n1: 𝒗𝒗𝑹𝑹𝑹𝑹 \n \n0: 𝒖𝒖𝒄𝒄𝟏𝟏 \n Reac�on Rx \n 0: 𝒖𝒖𝒄𝒄𝟐𝟐 \n1: 𝒗𝒗𝑹𝑹𝑹𝑹 \n \n𝒒𝒒𝒄𝒄𝟏𝟏 \n 𝒗𝒗𝑹𝑹𝑹𝑹 \n 𝒒𝒒𝒄𝒄𝟐𝟐 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n5 \n \nthe full bond graph representa�on in Figure 2b are reduced to three  in the simplified bond graph \n(Figure 2c). From here on, we use this simplified representa�on of the bond graph. \nSince the sum of fluxes entering or leaving the first 0:node must be zero: \n   𝑣𝑣𝑅𝑅𝑅𝑅+\n𝑐𝑐𝑑𝑑𝑐𝑐1\n𝑐𝑐𝑑𝑑= 0  \nand hence the flux entering the reac�on (𝑣𝑣𝑅𝑅𝑅𝑅) equals the rate of reduc�on of stored solute 𝑞𝑞𝑐𝑐1: \n  𝑣𝑣𝑅𝑅𝑅𝑅= −\n𝑐𝑐𝑑𝑑𝑐𝑐1\n𝑐𝑐𝑑𝑑. \nSimilarly, on the downstream side (the second 0:node),  \n   𝑣𝑣𝑅𝑅𝑅𝑅=\n𝑐𝑐𝑑𝑑𝑐𝑐2\n𝑐𝑐𝑑𝑑, \ngiven that the same flux 𝑣𝑣𝑅𝑅𝑅𝑅 passes through the reac�on (as highlighted in Figure 2c).  \nWith the bond graph representa�on in place, we can derive the associated system of equa�ons. If the \nsystem is opera�ng at constant temperature and pressure , the  Gibbs free energy can be used to \ncharacterise the poten�als on either side of the reac�on, and for a dilute system the chemical poten�al \n(the Gibbs free energy per mole) is given by the Boltzmann thermodynamic rela�on [12], \n 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑢𝑢0\n𝑖𝑖+ 𝑅𝑅𝑅𝑅ln\n𝑑𝑑𝑐𝑐𝑖𝑖\n𝑑𝑑𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡   (J.mol-1)   \nw\nhere 𝑞𝑞𝑐𝑐𝑖𝑖 is the number of moles of chemical species 𝑖𝑖, 𝑞𝑞𝑐𝑐𝑑𝑑𝑡𝑡𝑑𝑑 is the total number of moles of all \nsubstances in the mixture in compartment 𝑐𝑐. 𝑢𝑢0\n𝑖𝑖 is the (reference) poten�al when 𝑞𝑞𝑐𝑐𝑖𝑖= 𝑞𝑞𝑐𝑐𝑑𝑑𝑡𝑡 𝑑𝑑.  \nM\nore compactly,   \n 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑅𝑅𝑅𝑅ln 𝐾𝐾𝑐𝑐𝑖𝑖𝑞𝑞𝑐𝑐𝑖𝑖  (J.mol-1),    where 𝐾𝐾𝑐𝑐𝑖𝑖=\n1\n𝑑𝑑𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑒𝑒𝑢𝑢0\n𝑖𝑖𝑅𝑅𝑅𝑅⁄    (mol-1),  \nis the cons�tu�ve law for biochemical energy storage, and 𝐾𝐾𝑐𝑐𝑖𝑖 (mol-1) is a thermodynamic parameter.   \nTo simplify the equa�ons, we introduce the non-dimensional quan�ty \n  𝑞𝑞 �𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖𝑞𝑞𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖. 𝑉𝑉𝑐𝑐. 𝑐𝑐𝑐𝑐𝑖𝑖, (1) \nwhere 𝑐𝑐𝑐𝑐𝑖𝑖 is the concentra�on of chemical species 𝑖𝑖, 𝑉𝑉𝑐𝑐 is the volume of 𝑐𝑐. \nThe chemical poten�al is then  \n 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑅𝑅𝑅𝑅ln 𝑞𝑞 �𝑐𝑐𝑖𝑖  (J.mol-1). (2) \nEnergy balance at the 1:nodes is   \n  𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓= 𝑢𝑢𝑐𝑐1 and  𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟= 𝑢𝑢𝑐𝑐2, \nwhere 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 are the forward and reverse poten�als for reac�on 𝑅𝑅𝑅𝑅. In this case, with only one \nchemical species entering the reac�on, the energy balance is trivial, but i f there were mul�ple \nreactants and/or mul�ple products, the 1:node energy balance ensures the appropriate chemical \nstoichiometry for the reac�on.  \nThe molar flow for the reac�on 𝑅𝑅𝑅𝑅 can be given by the Marcelin-de Donder formula [4] (a cons�tu�ve \nrela�on but one that, like the Boltzmann rela�on, can be derived from assump�ons about the \ndistribu�on of par�cle veloci�es):  \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟𝑅𝑅𝑅𝑅⁄ �    (mol.s-1).  \nwhere 𝜅𝜅𝑅𝑅𝑅𝑅 (mol.s-1) is the e xperimentally determined reac�on rate constant (a cons�tu�ve \nparameter).  \nSubs�tu�ng for the poten�als gives \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n6 \n \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅� 𝑒𝑒𝑢𝑢𝑐𝑐1 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑐𝑐2 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅(𝑞𝑞 �𝑐𝑐1 − 𝑞𝑞 �𝑐𝑐2),   (mol.s-1) (3) \nwith equilibrium (zero flow) achieved when 𝑞𝑞 �𝑐𝑐1 = 𝑞𝑞 �𝑐𝑐2.  \nThe change in Gibbs free energy per mole for the reac�on is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓, which must be nega�ve \nfor the reac�on to proceed  (the second law of thermodynamics ), with heat output  rate \n𝑣𝑣𝑅𝑅𝑅𝑅. � 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓− 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟� . Note that heat output rate being the poten�al difference �mes the flow is the same \nin a biochemical reac�on as it is in an electrical resistance or a mechanical damper but the flux for a \nbiochemical reac�on depends on the difference in the exponentials of 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 (the Marcelin-de \nDonder formula above) , whereas  the flux through an electrical resistor or mechanical damper is \ndependent only on the poten�al difference across the resistor or damper (the voltage drop or net \nforce, respec�vely). At equilibrium, the change in Gibbs free energy is zero.    \nNote that, from now on, every reac�on is assumed to have associated poten�als 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 driving \nthe forward and reverse reac�ons, respec�vely, and these will not be explicitly labelled on the bond \ngraph diagram.  \nIn summary, the poten�als 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 that drive the reac�on in the forward and reverse direc�ons \nare obtained (via energy balance rela�ons) from the chemical solute poten�als 𝑢𝑢𝑐𝑐𝑖𝑖 that depend (via \nBoltzmann’s cons�tu�ve equa�on) on the nondimensional terms 𝑞𝑞 �𝑐𝑐𝑖𝑖 (which include the \nthermodynamic constants 𝐾𝐾𝑐𝑐𝑖𝑖). The nondimensional solute quan�ty 𝑞𝑞 �𝑐𝑐𝑖𝑖 is expressed in terms of the \nsolute concentra�on 𝑐𝑐𝑐𝑐𝑖𝑖 (in compartment c with volume 𝑉𝑉𝑐𝑐) by 𝑞𝑞 �𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖. 𝑉𝑉𝑐𝑐. 𝑐𝑐𝑐𝑐𝑖𝑖. The change in Gibbs free \nenergy occurring in the reac�on is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓≤ 0 (∆𝐺𝐺𝑅𝑅𝑅𝑅= 0 at equilibrium). The reac�on flux \n𝑣𝑣𝑅𝑅𝑅𝑅 is given (via the Marcelin-de Donder formula) by equa�on (3) and the heat output is −∆𝐺𝐺𝑅𝑅𝑅𝑅. 𝑣𝑣𝑅𝑅𝑅𝑅.  \n(ii) A reaction with multiple reactants and products \nMost reac�ons involve mul�ple reactants and products, o�en with variable stoichiometry. These can \nall be represented as illustrated in Figure 3 (including mul�ple instances of one chemical species, \nwhere, for example, 2 moles of one species combine with 1 mole of another species).  \n        (a)    (b)   \nFigure 3. (a) A reac�on with mul�ple reactants and products. ∑ . .𝑚𝑚\n𝑖𝑖=1 and ∏ . .𝑚𝑚\n𝑖𝑖=1\n imply a sum and product, \nrespec�vely, over the 𝑚𝑚 reactants, and ∑ . .𝑛𝑛\n𝑖𝑖=1\n and ∏ . .𝑛𝑛\n𝑖𝑖=1\n imply a sum and product over the 𝑛𝑛 reac�on products. \n(b) The corresponding bond graph diagram.  \nThe reac�on flux is \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅� ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1  −  ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n�  , (4) \nand the change in Gibbs free energy is \n ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n∏ 𝑞𝑞� 𝑐𝑐\n2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝑞𝑞� 𝑐𝑐\n1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� . \nIn terms of solute concentra�ons (using equa�on 1),  \n ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖.𝑛𝑛\n𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖.𝑚𝑚\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�(𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖.𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖.𝑚𝑚\n𝑖𝑖=1\n� + 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� . \nThis can be writen  \n𝒗𝒗𝑹𝑹𝑹𝑹 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟐𝟐 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 \n… \n𝒒𝒒𝒄𝒄𝟐𝟐𝟏𝟏 \n𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 \n𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 \n… \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 \n 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 \n𝑘𝑘𝑓𝑓 \n𝑘𝑘𝑟𝑟 \n� 𝒒𝒒 𝒄𝒄𝟏𝟏𝒊𝒊\n𝑚𝑚\n𝑖𝑖=1\n \n𝒒𝒒𝒄𝒄\n𝟏𝟏𝟏𝟏+ 𝒒𝒒𝒄𝒄\n𝟏𝟏𝟐𝟐+ . . +𝒒𝒒𝒄𝒄\n𝟏𝟏𝟏𝟏 \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓= � 𝒖𝒖 𝒄𝒄\n𝟏𝟏\n𝒊𝒊\n𝑚𝑚\n𝑖𝑖=1\n= 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�� 𝑞𝑞 � 𝑐𝑐\n1𝑖𝑖\n𝑚𝑚\n𝑖𝑖=1\n�  \n \n� 𝒒𝒒 𝒄𝒄𝟐𝟐𝒊𝒊\n𝑛𝑛\n𝑖𝑖=1\n \n𝒒𝒒𝒄𝒄\n𝟐𝟐𝟏𝟏+ 𝒒𝒒𝒄𝒄\n𝟐𝟐𝟐𝟐+ . . +𝒒𝒒𝒄𝒄\n𝟐𝟐𝟐𝟐 \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟= � 𝒖𝒖 𝒄𝒄\n𝟏𝟏\n𝒊𝒊\n𝑛𝑛\n𝑖𝑖=1\n= 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�� 𝑞𝑞 � 𝑐𝑐\n1𝑖𝑖\n𝑛𝑛\n𝑖𝑖=1\n�  \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n7 \n \n ∆𝐺𝐺𝑅𝑅𝑅𝑅= ∆𝐺𝐺𝑅𝑅𝑅𝑅\n𝑜𝑜− 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅 or  ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛� 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑞𝑞� − 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑞𝑞� , (5) \nwhere \n 𝑄𝑄𝑅𝑅𝑅𝑅=\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n    (6) \nis the ra�o of reactant concentra�ons to product concentra�ons, \n 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑= (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n , (7) \nis\n the equilibrium constant (i.e., when 𝑄𝑄𝑅𝑅𝑅𝑅= 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑, then \n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒= 1 and ∆𝐺𝐺𝑅𝑅𝑅𝑅= 0), and  \n ∆𝐺𝐺𝑅𝑅𝑅𝑅\n𝑡𝑡= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑�   \ni\ns the free energy needed to move the system from the reference state (all quan��es at the reference \nconcentra�on of 1M and hence 𝑄𝑄𝑅𝑅𝑅𝑅= 1 or 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅= 0) to the equilibrium state where 𝑄𝑄𝑅𝑅𝑅𝑅= 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑 \nand ∆𝐺𝐺𝑅𝑅𝑅𝑅= 0.  \nN\note that the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑, obtained from the measured ra�o of reactant concentra�ons \nto product concentra�ons at equilibrium (when 𝑣𝑣𝑅𝑅𝑅𝑅 and  ∆𝐺𝐺𝑅𝑅𝑅𝑅 are zero), along with the reac�on rate \n(see below), is an important parameter characterising a reac�on. It has a value of 1 for a simple \ndiffusive process, since the flow is zero when the reactant and product concentra�ons are equal. When \nthe flow 𝑣𝑣𝑅𝑅𝑅𝑅 is non-zero, the ra�o \n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄�𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒 (with a corresponding ∆𝐺𝐺𝑅𝑅𝑅𝑅= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄�𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒) characterizes how \nfar the reac�on is from equilibrium under those condi�ons.     \nSummarising, the change in Gibbs free energy (the free energy of the reaction) is   \n ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒.∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� = −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑞𝑞 , (8) \na r\nela�onship we will make much use of below. The thermodynamic constants for each chemical \nspecies have been combined into just one constant  𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑, whose value can be established simply by \nmeasuring the ra�o 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑= 𝑄𝑄𝑅𝑅𝑅𝑅�𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚= �\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n��\n𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚\n. For a reac�on t o proceed in \nthe forward direc�on requires ∆𝐺𝐺𝑅𝑅𝑅𝑅< 0 or  \n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒> 1 (the second law of thermodynamics). \nNote that the reac�on flux given by (4) can also be expressed in terms of concentra�ons: \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�(𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1 . ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n − (𝑉𝑉𝑐𝑐)𝑛𝑛. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n. ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n� , \nor \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝑘𝑘𝑓𝑓. � ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n�  − 𝑘𝑘𝑟𝑟. � ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n�  ,  \nwhere 𝑘𝑘𝑓𝑓= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n  and 𝑘𝑘𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑛𝑛. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n.  \nThis is the “mass ac�on ” equa�on widely used in the biochemical literature but note that the \nparameters 𝑘𝑘𝑓𝑓 and 𝑘𝑘𝑟𝑟 combine two completely different material quan��es: the thermodynamic \nBoltzmann coefficient 𝐾𝐾𝑐𝑐𝑖𝑖 associated with capaci�ve storage of solute 𝑖𝑖 (which generates the chemical \npoten�al 𝑢𝑢𝑐𝑐𝑖𝑖 via the empirical Boltzmann equa�on), and the reaction rate  constant 𝜅𝜅𝑅𝑅𝑅𝑅, which is a \nmaterial property of the reac�on.     \nA preferable way of represen�ng the reac�on is to define  \n 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n   \n(which depends on both the actual reac�on rate and the thermodynamic constants of the reactants) \nand then     \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅. �∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1 − 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑. ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n�,    (9) \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n8 \n \nhighligh�ng the fact that only two parameters are needed to describe the reac�on, one that can be \nobtained from steady state concentra�ons, and one that specifies how quickly the reac�on occurs. \nWe illustrate these concepts with a simple sodium-chloride reac�on in Figure 4.   \n   \n (a)    (b)    \nFigure 4. (a) A sodium-chloride reac�on. (b) The bond graph model in simplified form. \nThe 0:node mass balance equa�ons are: \n \n𝑐𝑐𝑑𝑑𝑐𝑐𝑁𝑁𝑁𝑁+\n𝑐𝑐𝑑𝑑= −𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅,     \n𝑐𝑐𝑑𝑑𝑐𝑐𝐶𝐶𝐶𝐶−\n𝑐𝑐𝑑𝑑= −𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅,     \n𝑐𝑐𝑑𝑑𝑐𝑐𝑁𝑁𝑁𝑁𝐶𝐶 𝐶𝐶\n𝑐𝑐𝑑𝑑= 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 \nThe reac�on flux (inser�ng the Boltzmann equa�ons for each chemical species) is  \n 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓/𝑅𝑅𝑅𝑅− 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟/𝑅𝑅𝑅𝑅� = 𝜅𝜅𝑅𝑅𝑅𝑅� 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+\n. 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎+\n. 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒−\n. 𝑞𝑞𝑐𝑐𝐶𝐶 𝑒𝑒−\n− 𝐾𝐾𝑐𝑐𝑁𝑁𝑎𝑎𝐶𝐶 𝑒𝑒. 𝑞𝑞𝑐𝑐𝑁𝑁𝑎𝑎 𝐶𝐶𝑒𝑒� . (10) \nW\nith 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎+\n= 𝑉𝑉𝑐𝑐. [𝑁𝑁𝑁𝑁+]𝑐𝑐, 𝑞𝑞𝑐𝑐𝐶𝐶 𝑒𝑒−\n= 𝑉𝑉𝑐𝑐. [𝐶𝐶 𝑅𝑅−]𝑐𝑐, and 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎𝐶𝐶𝑒𝑒= 𝑉𝑉𝑐𝑐. [𝑁𝑁𝑁𝑁𝐶𝐶 𝑅𝑅 ]𝑐𝑐 for compartment c of volume 𝑉𝑉𝑐𝑐, \nequa�on (10) becomes  \n 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅�[𝑁𝑁𝑁𝑁+]𝑐𝑐. [𝐶𝐶 𝑅𝑅−]𝑐𝑐− 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑. [𝑁𝑁𝑁𝑁𝐶𝐶 𝑅𝑅 ]𝑐𝑐� (11) \nwh\nere 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+\n. 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒−\n. 𝑉𝑉𝑐𝑐2 and 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑=\n𝐾𝐾𝑐𝑐𝑁𝑁𝑁𝑁𝐶𝐶 𝐶𝐶\n𝐾𝐾𝑐𝑐𝑁𝑁𝑁𝑁+.𝐾𝐾𝑐𝑐𝐶𝐶𝐶𝐶−.𝑉𝑉𝑐𝑐\n= �\n[𝑁𝑁 𝑎𝑎+]𝑐𝑐.[𝐶𝐶 𝑒𝑒−]𝑐𝑐\n[𝑁𝑁𝑎𝑎𝐶𝐶 𝑒𝑒]𝑐𝑐\n��\n𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚\n. \nT\nhis equa�on, now using concentra�ons, is the most useful form of the reac�on flux, but it is important \nto remember that the ‘reac�on rate’ constant 𝜅𝜅̂𝑅𝑅𝑅𝑅 is now a combina�on of the kine�c parameter 𝜅𝜅𝑅𝑅𝑅𝑅 \nand the thermodynamic parameters 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+\n, 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒−\n. In prac�ce 𝜅𝜅̂𝑅𝑅𝑅𝑅 is determined experimentally by \nmeasuring the non-equilibrium flux, and the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑 is determined by measuring the \nconcentra�ons of Na+, Cl- and NaCl at equilibrium.  \n(\niii) Enzyme-catalysed reactions \nAn enzyme-catalysed reac�on is illustrated in Figure 5.     \n \n   (a)    (b)  (c)    \nFigure 5. (a) An enzyme -catalysed reac�on in which a solute 𝑞𝑞𝑐𝑐\n1 combines (reac�on 1) with the enzyme 𝑞𝑞𝑐𝑐\n3 to \nform an intermediate 𝑞𝑞𝑐𝑐\n4 which then dissociates (reac�on 2) to form the solute product 𝑞𝑞𝑐𝑐\n2 and recycled enzyme \n𝑞𝑞𝑐𝑐\n3. (b) The enzyme-catalysed reac�on shown as a bond graph with the four red-bordered symbols represen�ng \nboth mass balance and storage, and the two reac�ons (with 1:nodes incorporated as in Figure 2) represen�ng \nthe points of energy balance (chemical stoichiometry). (b) A further simplifica�on of the bond graph model in \nwhich the steady-state flux 𝑣𝑣𝑐𝑐\n𝑅𝑅𝑅𝑅 is the final net flux, computed from (19), for the whole reac�on. \nThe flux balance equa�ons, defined at the four 0:nodes, are  \n(i)  \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐1 = −𝑣𝑣𝑅𝑅𝑅𝑅\n1 ; (ii)  \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐2 = 𝑣𝑣𝑅𝑅𝑅𝑅\n2 ; (iii)  \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐3 = −𝑣𝑣𝑅𝑅𝑅𝑅\n1 + 𝑣𝑣𝑅𝑅𝑅𝑅\n2 ; (iv)  \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐4 = 𝑣𝑣𝑅𝑅𝑅𝑅\n1 − 𝑣𝑣𝑅𝑅𝑅𝑅\n2 . \nN\note that, since \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑(𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4) = 0, the total amount of enzyme is constant. i.e., \n 𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4 = 𝐸𝐸0,  (12) \n𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵 \n𝒗𝒗𝒄𝒄𝑹𝑹𝑹𝑹 \n𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵+\n \n𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵−\n \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓\n \n 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 \n𝑁𝑁𝑁𝑁+ + 𝐶𝐶𝑅𝑅− \n 𝑁𝑁𝑁𝑁𝐶𝐶𝑅𝑅 \n𝐴𝐴𝑓𝑓 \n𝐴𝐴𝑟𝑟 \n𝑞𝑞𝑐𝑐\n1 + 𝑞𝑞𝑐𝑐\n3 \n 𝑞𝑞𝑐𝑐\n4 \n 𝑞𝑞𝑐𝑐\n2 + 𝑞𝑞𝑐𝑐\n3 \n𝑘𝑘1\n𝑓𝑓 \n𝑘𝑘1\n𝑟𝑟 \n𝑘𝑘2\n𝑓𝑓 \n𝑘𝑘2\n𝑟𝑟 \n 𝒒𝒒𝒄𝒄𝟑𝟑 \n𝒒𝒒𝒄𝒄𝟏𝟏 \n 𝒒𝒒𝒄𝒄𝟒𝟒 \n 𝒒𝒒𝒄𝒄𝟐𝟐 \n𝒗𝒗𝑹𝑹𝑹𝑹\n𝟏𝟏 \n 𝒗𝒗𝑹𝑹𝑹𝑹\n𝟐𝟐 \n𝒒𝒒𝒄𝒄𝟏𝟏 \n 𝒒𝒒𝒄𝒄𝟐𝟐 \n𝒗𝒗𝒄𝒄𝑹𝑹𝑹𝑹 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n9 \n \nwhere 𝐸𝐸0 is the ini�al quan�ty of enzyme.  \nThe energy balance equa�ons, defined at the four 1:nodes, are \n(i) 𝑢𝑢𝑅𝑅𝑅𝑅1\n𝑓𝑓 = 𝑢𝑢𝑐𝑐1 + 𝑢𝑢𝑐𝑐3;   (ii)  𝑢𝑢𝑅𝑅𝑅𝑅1\n𝑟𝑟 = 𝑢𝑢𝑐𝑐4; (iii)  𝑢𝑢𝑅𝑅𝑅𝑅2\n𝑓𝑓 = 𝑢𝑢𝑐𝑐4; (iv) 𝑢𝑢𝑅𝑅𝑅𝑅2\n𝑟𝑟 = 𝑢𝑢𝑐𝑐2 + 𝑢𝑢𝑐𝑐3. \nUsing these poten�als, the two reac�ons are  \n 𝑣𝑣𝑅𝑅𝑅𝑅\n1  = 𝜅𝜅𝑅𝑅𝑅𝑅\n1 �𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅1\n𝑓𝑓 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅1\n𝑟𝑟 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅\n1 (𝑞𝑞 �𝑐𝑐1𝑞𝑞 �𝑐𝑐3 − 𝑞𝑞 �𝑐𝑐4),   (13) \n 𝑣𝑣𝑅𝑅𝑅𝑅\n2   = 𝜅𝜅𝑅𝑅𝑅𝑅\n2 �𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅2\n𝑓𝑓 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅2\n𝑟𝑟 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅\n2 (𝑞𝑞 �𝑐𝑐4 − 𝑞𝑞 �𝑐𝑐2𝑞𝑞 �𝑐𝑐3).   (14) \nWe can also express these fluxes in “mass ac�on” form as  \n 𝑣𝑣𝑅𝑅𝑅𝑅\n1  =  𝑘𝑘1\n𝑓𝑓𝑐𝑐𝑐𝑐1𝑐𝑐𝑐𝑐3 − 𝑘𝑘1\n𝑟𝑟𝑐𝑐𝑐𝑐4 and  𝑣𝑣𝑅𝑅𝑅𝑅\n2 = 𝑘𝑘2\n𝑓𝑓𝑐𝑐𝑐𝑐4 − 𝑘𝑘2\n𝑟𝑟𝑐𝑐𝑐𝑐2𝑐𝑐𝑐𝑐3, \nwhere 𝑘𝑘1\n𝑓𝑓 = 𝜅𝜅𝑅𝑅𝑅𝑅\n1 . 𝐾𝐾𝑐𝑐1𝐾𝐾𝑐𝑐3. (𝑉𝑉𝑐𝑐)2 (m6.mol-1.s-1),   𝑘𝑘1\n𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅\n2 . 𝐾𝐾𝑐𝑐4. 𝑉𝑉𝑐𝑐 (m3.s-1), \nand  𝑘𝑘2\n𝑓𝑓 = 𝜅𝜅𝑅𝑅𝑅𝑅\n2 . 𝐾𝐾𝑐𝑐4. 𝑉𝑉𝑐𝑐 (m3.s-1),   𝑘𝑘2\n𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅\n2 . 𝐾𝐾𝑐𝑐2𝐾𝐾𝑐𝑐3. (𝑉𝑉𝑐𝑐)2 (m6.mol-1.s-1).  \nNote the inconsistent  units that result from combining reac�on rate constants (mol.s-1) with \nthermodynamic constants (mol-1).  \nThe Briggs-Haldane assumption [14] is that the amount of enzyme (𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4) is much less than the \namounts of substrate (𝑞𝑞𝑐𝑐1 and 𝑞𝑞𝑐𝑐2), so that the unbound enzyme 𝑞𝑞𝑐𝑐3 and the complex 𝑞𝑞𝑐𝑐4 quickly reach \na steady state (\n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐3 = 0 and \n𝑐𝑐\n𝑐𝑐\n𝑑𝑑𝑞𝑞𝑐𝑐4 = 0 ). The second and third equa�ons in (4) then give \n  𝑣𝑣𝑅𝑅𝑅𝑅\n1 = 𝑣𝑣𝑅𝑅𝑅𝑅\n2 = 𝑣𝑣𝑅𝑅𝑅𝑅  \nand hence, from (13) and (14),  \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅\n1 (𝑞𝑞 �𝑐𝑐1𝑞𝑞 �𝑐𝑐3 − 𝑞𝑞 �𝑐𝑐4) = 𝜅𝜅𝑅𝑅𝑅𝑅\n2 (𝑞𝑞 �𝑐𝑐4 − 𝑞𝑞 �𝑐𝑐2𝑞𝑞 �𝑐𝑐3)  (15) \nor \n   𝑞𝑞 �𝑐𝑐4 =\n� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2� 𝑑𝑑� 𝑐𝑐3\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 .  (16) \nAlso, from (12), \n 𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4 = 𝐸𝐸0 or    \n𝑑𝑑� 𝑐𝑐3\n𝐾𝐾𝑐𝑐3 +\n𝑑𝑑� 𝑐𝑐4\n𝐾𝐾𝑐𝑐4 = 𝐸𝐸0 or   𝑞𝑞 �𝑐𝑐3 = 𝐾𝐾𝑐𝑐3𝐸𝐸0 −\n𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4 𝑞𝑞 �𝑐𝑐4. (17) \nSubs�tu�ng (17) into (16),  \n 𝑞𝑞 �𝑐𝑐4 =\n� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 �𝐾𝐾𝑐𝑐3. 𝐸𝐸0 −\n𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4 𝑞𝑞 �𝑐𝑐4� , \nor \n 𝑞𝑞 �𝑐𝑐4 = 𝐾𝐾𝑐𝑐3. 𝐸𝐸0.\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n (18) \nand hence, from (17),  \n 𝑞𝑞 �𝑐𝑐3 = 𝐾𝐾𝑐𝑐3𝐸𝐸0 � 1 −\n𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4 .\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n� = 𝐾𝐾𝑐𝑐3𝐸𝐸0\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n. (19) \nSubs�tu�ng (18) and (19) back into the first equa�on in (15), \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝐾𝐾𝑐𝑐3𝐸𝐸0 �\n𝑑𝑑� 𝑐𝑐1� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n−\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n� , \nwhich simplifies to \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n10 \n \n 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝐾𝐾𝑐𝑐3𝐸𝐸0.\n𝑑𝑑� 𝑐𝑐1−𝑑𝑑� 𝑐𝑐2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2 +𝐾𝐾𝑐𝑐3\n𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅\n2 𝑑𝑑� 𝑐𝑐2�\n. \nA convenient way of expressing this enzyme-catalysed reac�on flux is \n 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅.\n𝑑𝑑� 𝑐𝑐1−𝑑𝑑� 𝑐𝑐2\n1+ 𝑒𝑒� 𝑐𝑐\n1\n𝑘𝑘𝑅𝑅𝑅𝑅\n1\n+ 𝑒𝑒� 𝑐𝑐2\n𝑘𝑘𝑅𝑅𝑅𝑅\n2\n, (20) \nwhere the three parameters \n 𝜅𝜅𝑅𝑅𝑅𝑅= 𝐾𝐾𝑐𝑐3𝐸𝐸0.\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 𝜅𝜅𝑅𝑅𝑅𝑅\n2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2  (mol.s-1);  𝑘𝑘𝑅𝑅𝑅𝑅\n1 =\n𝐾𝐾𝑐𝑐4\n𝐾𝐾𝑐𝑐3 .\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 ;  𝑘𝑘𝑅𝑅𝑅𝑅\n2 =\n𝐾𝐾𝑐𝑐4\n𝐾𝐾𝑐𝑐3 .\n𝜅𝜅𝑅𝑅𝑅𝑅\n1 +𝜅𝜅𝑅𝑅𝑅𝑅\n2\n𝜅𝜅𝑅𝑅𝑅𝑅\n2   (21) \ncharacterise the maximum flux, the value of 𝑞𝑞 �𝑐𝑐1 that corresponds to half that maximum when 𝑞𝑞 �𝑐𝑐2 = 0, \nand the value of 𝑞𝑞 �𝑐𝑐2 that corresponds to half that maximum when 𝑞𝑞 �𝑐𝑐1 = 0. The flux term (20) shows \nsatura�on at high values of 𝑞𝑞 �𝑐𝑐1 or 𝑞𝑞 �𝑐𝑐2. Note that we use 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 in (20) and in the bond graph diagrams in \nFigure 6, with 𝑅𝑅𝑅𝑅 as a superscript to indicate that this is the steady-state flux for the whole enzyme-\ncatalysed reac�on. This conven�on greatly simplifies subsequent mul�-enzyme reac�ons (e.g., for \nmembrane transporters).      \nIf  𝑞𝑞 �𝑐𝑐2 is assumed to be much smaller than 𝑞𝑞 �𝑐𝑐1 and is set to zero, equa�on (20) becomes the familiar \nMichaelis-Menten equa�on [14], but the reac�on is then irreversible. The validity of the Briggs-\nHaldane assump�on can be tested by comparing the solu�on of the full bond graph model with the \nreduced model given by (20).  \nFigure 6b shows the bond graph for an enzyme-catalysed reac�on which has 𝑚𝑚 reactants (assumed to \nbind simultaneously) and 𝑛𝑛 products (assumed to unbind simultaneously). \n   \n      (a)    (b)   \nFigure 6. (a) The bond graph diagram for an enzyme-catalysed reac�on under the Briggs-Haldane assump�on \nwhere the flux term 𝑣𝑣𝑐𝑐\n𝑅𝑅𝑅𝑅 (given by equa�on (19)) now incorporates the ac�on of the enzyme. (b) An enzyme-\ncatalysed reac�on with 𝑚𝑚 reactants and 𝑛𝑛 products.  \nFollowing the same process described above for the deriva�on of the full bond graph model, and then \nthe corresponding reduced Briggs-Haldane model,   \n 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅.\n∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1  − ∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n1+\n∏ 𝑒𝑒� 𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n𝑘𝑘𝑅𝑅𝑅𝑅\n1 +\n∏ 𝑒𝑒� 𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n𝑘𝑘𝑅𝑅𝑅𝑅\n2\n , (22) \nwhere the parameters 𝜅𝜅𝑅𝑅𝑅𝑅, 𝑘𝑘𝑅𝑅𝑅𝑅\n1  and 𝑘𝑘𝑅𝑅𝑅𝑅\n2 , which characterise the catalysed reac�on are given by (22).  \nThe zero-flux state (𝑣𝑣𝑅𝑅𝑅𝑅= 0) is reached when ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1 =  ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n.  \nThe free energy of the reac�on is \n  𝛥𝛥𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑐𝑐𝑟𝑟− 𝑢𝑢𝑐𝑐\n𝑓𝑓= 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛� ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n� − 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛� ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n� = 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛�\n∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n� = −𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒, \nwhere \n  𝑄𝑄𝑅𝑅𝑅𝑅=\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n,   and 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑= (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n= 𝑄𝑄𝑅𝑅𝑅𝑅|𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚.  (23) \n𝒗𝒗𝒄𝒄\n𝑹𝑹𝑹𝑹 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟐𝟐 \n𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 \n… \n𝒒𝒒𝒄𝒄𝟐𝟐𝟏𝟏 \n𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 \n𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 \n… \n𝒒𝒒𝒄𝒄𝟏𝟏 \n 𝒗𝒗𝒄𝒄\n𝑹𝑹𝑹𝑹 \n 𝒒𝒒𝒄𝒄𝟐𝟐 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n11 \n \n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑 is the equilibrium constant for the reac�on, which replaces the 𝑚𝑚+ 𝑛𝑛 thermodynamic \ncoefficients. \nTh\ne zero-flux state for the reac�on corresponds to \n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒= 1 and 𝛥𝛥𝐺𝐺𝑅𝑅𝑅𝑅= 0. \nUsing (1) and (21), the flux expression (20) can be rewriten in terms of concentra�ons: \n 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅.\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1  −𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒.∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n1+\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n𝑘𝑘� 𝑅𝑅𝑅𝑅\n1 +\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n𝑘𝑘� 𝑅𝑅𝑅𝑅\n2\n ,  (24) \nwhere  \n 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1 ,  𝑘𝑘� 𝑅𝑅𝑅𝑅\n1 =\n𝑘𝑘𝑅𝑅𝑅𝑅\n1\n(𝑉𝑉𝑐𝑐)𝑚𝑚.∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n,  𝑘𝑘� 𝑅𝑅𝑅𝑅\n2 =\n𝑘𝑘𝑅𝑅𝑅𝑅\n2\n(𝑉𝑉𝑐𝑐)𝑛𝑛.∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n. (25) \nTh\nere are now four  constants available ( 𝜅𝜅̂𝑅𝑅𝑅𝑅, 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑, 𝑘𝑘� 𝑅𝑅𝑅𝑅\n1 , 𝑘𝑘� 𝑅𝑅𝑅𝑅\n2 ) to fit the general enzyme catalysed \nreac�on (24) to experimental data on the rela�onship between flux and the reactant and product \nconcentra�ons.  \nFo\nr a very fast reac�on (when the reac�on rate constant 𝜅𝜅𝑅𝑅𝑅𝑅→ ∞), a finite (non-zero) value for the \nsteady enzyme turn-over flux 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 implies that the reac�on is close to equilibrium (\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒→ 1).  \n(iv) ATP hydrolysis \nATP hydrolysis supplies the energy (usually via the ‘sodium batery’) to drive nearly all physiological \nprocesses [13]. The hydrolysis of ATP to ADP, Pi (inorganic phosphate), and H+ is represented chemically \nin Figure 7a and as an enzyme-catalysed bond graph process in Figure 7b.   \n  \n    (a)    (b)   \nF\nigure 7. The bond graph representa�on of ATP hydrolysis. At pH 7, inorganic phosphate (𝑃𝑃𝑖𝑖) is a mixture of the \nprotonated forms of phosphate: 𝐻𝐻2𝑃𝑃𝑃𝑃4\n− and 𝐻𝐻𝑃𝑃𝑃𝑃4\n2−. Here we assume 𝐻𝐻2𝑃𝑃𝑃𝑃4\n− only. \nThis is an enzyme-catalysed reac�on with an equilibrium constant, from (25), defined as \n 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑= (𝑉𝑉𝑐𝑐)3−2.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖3\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖2\n𝑖𝑖=1\n= 𝑉𝑉𝑐𝑐.\n𝐾𝐾𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴.𝐾𝐾𝑐𝑐\n𝑃𝑃𝑖𝑖.𝐾𝐾𝑐𝑐𝐻𝐻+\n𝐾𝐾𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴.𝐾𝐾𝑐𝑐\n𝐻𝐻2𝑃𝑃, \nbut determined from the experimentally measured concentra�ons at equilibrium: \n 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑= 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴|𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒= �\n𝑅𝑅𝑒𝑒 𝑎𝑎𝑐𝑐𝑑𝑑𝑎𝑎𝑛𝑛𝑑𝑑 𝑐𝑐𝑡𝑡𝑛𝑛𝑐𝑐𝑒𝑒 𝑛𝑛𝑑𝑑𝑟𝑟𝑎𝑎𝑑𝑑𝑖𝑖𝑡𝑡𝑛𝑛𝐶𝐶\n𝐴𝐴𝑟𝑟𝑡𝑡𝑐𝑐𝑢𝑢𝑐𝑐𝑑𝑑 𝑐𝑐𝑡𝑡𝑛𝑛𝑐𝑐𝑒𝑒\n𝑛𝑛𝑑𝑑𝑟𝑟𝑎𝑎𝑑𝑑𝑖𝑖𝑡𝑡𝑛𝑛𝐶𝐶��\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒\n= �\n∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n��\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒\n= �\n[𝐴𝐴𝑅𝑅𝑃𝑃].[𝐻𝐻2𝑂𝑂]\n[𝐴𝐴𝐴𝐴𝑃𝑃].[𝑃𝑃𝑖𝑖].[𝐻𝐻+]��\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒\n, \nwh\nere the concentra�ons [. . ] are defined in units of mol.m-3, which is the same as millimol.L-1 or mM. \nThe concentra�on of water is 55.5 mM (the density 1 kg.L-1 divided by 0.018 kg.mol-1), but since the \namount of water absorbed by the reac�on is small compared to the quan�ty of water in which the \nreac�on is taking place, a modified equilibrium constant is measured using  \n   𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑= �\n[𝐴𝐴𝑅𝑅𝑃𝑃]\n[𝐴𝐴𝐴𝐴𝑃𝑃].[𝑃𝑃𝑖𝑖].[𝐻𝐻+]��\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒\n. \nWh\nen measured at 298.15 K (25 oC) at pH 7 ([𝐻𝐻+] = 10-4 mM and 1 mM [𝑀𝑀𝑀𝑀2+] [18]), the equilibrium \nconstant has the value 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑≈ 0.5x10-5 (assuming mM units) or, with 𝑅𝑅𝑅𝑅 = 2.5 kJ.mol-1, \n𝐴𝐴𝑅𝑅𝑃𝑃4− + 𝐻𝐻2𝑃𝑃 ⇄  𝐴𝐴𝐴𝐴𝑃𝑃4− + 𝐻𝐻2𝑃𝑃𝑃𝑃4\n− + 𝐻𝐻+  \nwhere ATP is C₁₀H₁₆N₅O₁₃P₃ and ADP is C₁₀H₁₅N₅O₁₀P₂ \n𝑘𝑘𝑓𝑓 \n𝑘𝑘𝑟𝑟 \n 𝒗𝒗𝒄𝒄\n𝑹𝑹𝑹𝑹 \n𝑢𝑢𝑅𝑅𝑅𝑅\n𝑓𝑓 \n 𝑢𝑢𝑅𝑅𝑅𝑅\n𝑟𝑟 \n𝑞𝑞𝑐𝑐𝐴𝐴𝑅𝑅 𝐴𝐴 \n𝑞𝑞𝑐𝑐\n𝐻𝐻2𝑂𝑂 \n𝑞𝑞𝑐𝑐\n𝐴𝐴𝑖𝑖 \n𝑞𝑞𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 \n𝑞𝑞𝑐𝑐𝐻𝐻+\n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n12 \n \n  ∆𝐺𝐺𝐴𝐴𝑅𝑅𝐴𝐴\n𝑡𝑡 = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑� = −2.5 𝑅𝑅𝑛𝑛(2 . 105)= -30.5 kJ.mol-1.  \nAs we show below for the ATPase-driven NKA pump, in many situa�ons ATP hydrolysis operates at ∆𝐺𝐺s \nthat are well in excess of this equilibrium. \n(v) Glucose transporter (GLUT2) \nUs\ning a similar analysis (see [7] for details), the bond graph model of facilitated transport of glucose \n(‘Glc’) through the SLC2A2-encoded membrane transporter GLUT2 (see Figure 8), under the Briggs-\nHaldane assump�on of steady state enzyme cycling and fast binding and unbinding, is given in terms \nof the non-dimensional intracellular and extracellular glucose amounts (𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒𝑐𝑐, 𝑞𝑞 �𝑡𝑡\n𝐺𝐺𝑒𝑒𝑐𝑐) by   \n 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 = 𝜅𝜅𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2.\n𝑞𝑞� 𝑖𝑖\n𝐺𝐺\n𝑅𝑅𝑐𝑐−𝑞𝑞� 𝑜𝑜\n𝐺𝐺\n𝑅𝑅𝑐𝑐\n1+ \n𝑞𝑞� 𝑖𝑖\n𝐺𝐺\n𝑅𝑅𝑐𝑐 \n𝜅𝜅𝑚𝑚1  + \n𝑞𝑞� 𝑜𝑜\n𝐺𝐺\n𝑅𝑅𝑐𝑐 \n𝜅𝜅𝑚𝑚2  + \n𝑞𝑞� 𝑖𝑖\n𝐺𝐺\n𝑅𝑅𝑐𝑐.𝑞𝑞� 𝑜𝑜\n𝐺𝐺\n𝑅𝑅𝑐𝑐\n𝜅𝜅𝑚𝑚3\n    (26) \na\nnd, in terms of the intracellular and extracellular glucose concentra�ons ([𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡),  \n 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 = 𝜅𝜅̂𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝐴𝐴2\n𝑒𝑒𝑒𝑒.[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑡𝑡\n1+ \n[𝐺𝐺 𝐶𝐶𝑐𝑐]𝑖𝑖\n𝑘𝑘𝑖𝑖\n𝐺𝐺𝐺𝐺\n𝐺𝐺𝐴𝐴2 + [𝐺𝐺 𝐶𝐶𝑐𝑐]𝑡𝑡\n𝑘𝑘𝑡𝑡𝐺𝐺𝐺𝐺\n𝐺𝐺𝐴𝐴2 + \n[𝐺𝐺 𝐶𝐶𝑐𝑐]𝑖𝑖.[𝐺𝐺 𝐶𝐶𝑐𝑐]𝑡𝑡\n𝑘𝑘𝑖𝑖\n𝑡𝑡\n𝐺𝐺𝐺𝐺\n𝐺𝐺𝐴𝐴2\n.    fmol.mm-2.s-1 (27)     \nF\nor the analysis of cellular processes, we use units fmol (10-15 mol) for molar quan�ty, mm2 (10-6 m2) \nfor area, and pL (10-12 L = 10-15 m3) for volume (1 fmol.pL-1 = 1 mol.m-3 = 1 mM). Since 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 is a \ntransmembrane flux, it is convenient to express as a molar  flux per unit area of membrane , and \ntherefore to use units of fmol.mm-2.s-1. S ince the solute concentra�ons are in mM (mol.m-3), the \nreac�on rate 𝜅𝜅̂𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 has units of fmol.mm-2.s-1/mol.m-3 or pL.mm-2.s-1 This parameter, fited to whole \ncell data, is therefore a combina�on of reac�on rate for the individual membrane protein transporter \nand a scaling constant that takes into account the area l protein density. When we later place this \ntransporter into the whole cell model, we introduce an area -to-volume ra�o for the cell that links \nconcentra�ons and per-unit-area fluxes to changes in molar amounts.   \n \n  (a)   (b) (c)   (d) \nF\nigure 8. The GLUT2 transporter. (a) Facilitated transport of glucose across the cell membrane, shown by the \ndouble line ( green background is inside the cell, blue outside); (b) Full bond graph model with reac�ons for 1. \nglucose binding, 2. transloca�on of the ligand-bound enzyme from inward-facing to outward-facing, 3. release \nof glucose, and 4. return of the enzyme to inward facing; (c) Reduced model where the flux 𝑣𝑣𝑚𝑚\n𝐺𝐺𝐺𝐺𝐺𝐺\n𝑅𝑅2 through the \nmembrane is an algebraic func�on (equa�on 24) of the quan��es of external and internal glucose; (d) the \nsimpler representa�on in which the 1:nodes are included in the reac�on.    \nNote that, from (7), the equilibrium constant is \n 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2\n𝑒𝑒\n𝑑𝑑 = (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n=\n𝐾𝐾𝑐𝑐\n𝐺𝐺\n𝐶𝐶𝑐𝑐𝑡𝑡\n𝐾𝐾𝑐𝑐\n𝐺𝐺\n𝐶𝐶𝑐𝑐𝑖𝑖= 1, \nsince the Boltzmann thermodynamic constant 𝐾𝐾𝑐𝑐 is the same for internal and external glucose.    \nT\nhe parameters 𝑘𝑘� 𝑚𝑚\n1 , 𝑘𝑘� 𝑚𝑚\n2  and 𝑘𝑘� 𝑚𝑚\n3  scale the rela�ve contribu�ons of extracellular and intracellular \nglucose and their product, respec�vely (see Table 1 for the value of these parameters fited to data \nfrom Lowe and Walmsley [15]). 𝑘𝑘� 𝑚𝑚1  is the value of [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 that reduces the flux to 50% of its maximum \nvalue when [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡 is zero, and 𝑘𝑘� 𝑚𝑚2  is the value of [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡 that reduces the flux to 50% of its maximum \nvalue when [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 is zero. Note that 𝑘𝑘� 𝑚𝑚\n2  is an order of magnitude larger than 𝑘𝑘� 𝑚𝑚\n1  and that 𝑘𝑘� 𝑚𝑚\n3  is a further \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n13 \n \norder of magnitude larger than 𝑘𝑘� 𝑚𝑚\n2 . These parameters are expressed in terms of the thermodynamic \nand kine�c constants for the transporter as   \nParameters Value Unit Parameters Value Unit \n𝐾𝐾𝐺𝐺𝑒𝑒𝑐𝑐 ? fmol-1    \n𝑘𝑘𝑚𝑚1  1.474 dimensionless 𝑘𝑘𝑖𝑖\n𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.1094 mM \n𝑘𝑘𝑚𝑚2  21.67 dimensionless 𝑘𝑘𝑡𝑡𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 1.609 mM \n𝑘𝑘𝑚𝑚3  235.1 dimensionless 𝑘𝑘𝑖𝑖𝑡𝑡\n𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 1.296 (mM)2 \n𝜅𝜅𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.003284 fmol.s-1 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2\n𝑒𝑒𝑑𝑑 1 dimensionless \n   𝜅𝜅̂𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.0442 pL.s-1 \nTable 1. Parameter values used in the  expression for flux 𝑣𝑣𝑚𝑚\n𝐺𝐺𝐺𝐺𝐺𝐺\n𝑅𝑅2 through the GLUT2 transporter. Values on the \nle� are obtained by fi�ng (24) to data from Lowe and Walmsley [19]. Values on the right have been adjusted to \nreflect use of concentra�ons in (25). \nFigure 9 illustrates the flux through the steady-state (SS) bond graph model as a func�on of intracellular \nconcentra�on, using equa�on (27) with the parameters given in Table 1.   \n \n \nFigure 9. The GLUT2 bond graph model fited to flux data from [19]. Reproduced from [7] with permission). \n(vi) Electrogenic sodium-glucose cotransporter (SGLT1) \nGlucose transport through the SLC5A1-encoded membrane co -transporter SGLT1 (see Figure 10), \nunder the same assump�ons used above (Briggs-Haldane and fast binding/unbinding) together with \nthe assump�on of no slippage (see [7]), is given in terms of non-dimensional quan��es by \n 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝜅𝜅6𝐾𝐾1𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑�� 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐− �𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒\n𝑐𝑐. 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� /𝐵𝐵 , (28) \nwh\nere \n 𝐵𝐵= �𝑒𝑒2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ � 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+\n�\n2\n�\n𝐾𝐾1\n𝐾𝐾2\n+\n𝐾𝐾1\n𝐾𝐾3\n. 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐� � ��𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒\n𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+\n𝜅𝜅6\n𝜅𝜅3\n�  \n  + �\n𝐾𝐾1\n𝐾𝐾6\n+ �𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n�\n𝐾𝐾1\n𝐾𝐾5\n+\n𝐾𝐾1\n𝐾𝐾4\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒\n𝑐𝑐� � �� 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+\n𝜅𝜅6\n𝜅𝜅3\n𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� \nand 𝑧𝑧1 = 𝑧𝑧2 = 0.5.  \nD\nividing numerator and denominator in (2 8) by  �𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒𝑐𝑐, using concentra�ons rather than \ndimensionless molar quan��es, and iden�fying the external compartment as ‘gut lumen’, (28) \nbecomes  \n 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 =\n𝜅𝜅𝑚𝑚𝑆𝑆 𝐺𝐺𝐺𝐺𝐴𝐴1\nΦ𝑆𝑆𝐺𝐺𝐺𝐺𝐴𝐴1 � �\n[𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n2\n.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n− 𝑄𝑄𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1\n𝑒𝑒\n𝑑𝑑 .  𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�    fmol.mm-2.s-1   (29) \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n14 \n \nwhere \n 𝑄𝑄𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1\n𝑒𝑒\n𝑑𝑑 = (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚.\n∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛\n𝑖𝑖=1\n∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚\n𝑖𝑖=1\n= �\n𝐾𝐾𝑐𝑐\n𝑁𝑁𝑁𝑁𝑡𝑡+\n𝐾𝐾𝑐𝑐\n𝑁𝑁𝑁𝑁𝑖𝑖\n+ �\n2\n.\n𝐾𝐾𝑐𝑐\n𝐺𝐺\n𝐶𝐶𝑐𝑐𝑡𝑡\n𝐾𝐾𝑐𝑐\n𝐺𝐺\n𝐶𝐶𝑐𝑐𝑖𝑖= 1. (30) \nand the denominator term is  \n Φ𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = �� 𝑒𝑒2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ � 𝑞𝑞� 𝑀𝑀𝑢𝑢𝑔𝑔\n𝑁𝑁𝑁𝑁+\n�\n2\n�\n𝐾𝐾1\n𝐾𝐾2\n+\n𝐾𝐾1\n𝐾𝐾3\n. 𝑞𝑞� 𝑀𝑀𝑢𝑢𝑔𝑔\n𝐺𝐺𝑅𝑅𝑐𝑐� � �� 𝑞𝑞� 𝑖𝑖\n𝑁𝑁𝑁𝑁+\n�\n2\n. 𝑞𝑞� 𝑖𝑖\n𝐺𝐺\n𝑅𝑅𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+\n𝜅𝜅6\n𝜅𝜅3\n�  \n + �\n𝐾𝐾1\n𝐾𝐾6\n+ �𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n�\n𝐾𝐾1\n𝐾𝐾5\n+\n𝐾𝐾1\n𝐾𝐾4\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒𝑐𝑐�� ��𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝐺𝐺𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+\n𝜅𝜅6\n𝜅𝜅3\n𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�� /�𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺𝑒𝑒𝑐𝑐 (31) \nNote that 𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝜅𝜅6𝐾𝐾1𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑 (fmol.mm-2.s-1) is a rate parameter also reflec�ng the overall expression \nlevel for the transporter and its reac�on rate. \n \n  (a)  (b)     (c)  (d) \nF\nigure 10. The sodium-glucose transporter SGLT1. (a) Symbolic representa�on for the transport of two moles of \nsodium for every one mole of glucose; (b) Full bond graph model; and (c) Reduced model where the flux 𝑣𝑣𝑚𝑚\n𝑆𝑆\n𝐺𝐺𝐺𝐺𝑅𝑅1  \nis an algebraic func�on (27) of the quan��es of sodium and glucose either side of the membrane (see [7]); (d) \nthe alterna�ve representa�on in which the 1:nodes are included in the reac�on.   \nParameters Value Unit Solute & enzyme states \n𝐾𝐾𝑁𝑁𝑎𝑎+\n 0. 0159 fmol-1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑\n𝑁𝑁\n𝑎𝑎+\n, 𝑞𝑞𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n \n𝐾𝐾𝐺𝐺𝑒𝑒𝑐𝑐 ? f mol-1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑\n𝐺𝐺\n𝑒𝑒𝑐𝑐, 𝑞𝑞𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐 \n𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 3. 5e6 fmol.mm-2. s-1  \n𝐾𝐾1 3.66 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡  \n𝐾𝐾2 330 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+\n  \n𝐾𝐾3 321 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐  \n 𝐾𝐾4 321 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐  \n𝐾𝐾5 905 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+\n  \n𝐾𝐾6 0.314 fmol-1  𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖  \n𝜅𝜅3 0.156 fmol.s-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐→ 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐 \n𝜅𝜅6 9.562 fmol.s-1    𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖→ 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡 \nTable 2. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚\n𝑆𝑆\n𝐺𝐺𝐺𝐺𝑅𝑅1 through the SGLT1 transporter. Values obtained \nby fi�ng (29) and (31) to data from Lowe and Walmsley [19] (see [7]}. The enzyme states corresponding to the \nthermodynamic parameters labelled 𝐾𝐾1.. 𝐾𝐾6, and the reac�ons corresponding to the kine�c parameters 𝜅𝜅3 and \n𝜅𝜅6, are shown in the right-hand column.  \nFigure 11 shows the flux through the steady-state (SS) bond graph model using the parameters given \nin Table 2.   \n𝒗𝒗𝟏𝟏𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝟏𝟏 \n𝒒𝒒𝒐𝒐\n𝑺𝑺𝑵𝑵𝒄𝒄 \n𝒒𝒒𝒊𝒊\n𝑺𝑺𝑵𝑵𝒄𝒄 \n𝒒𝒒𝒐𝒐\n𝑵𝑵𝑵𝑵+\n \n𝒒𝒒𝒊𝒊\n𝑵𝑵𝑵𝑵+\n \n2 \n2 \n 𝑢𝑢𝑚𝑚\n𝑒𝑒 \n2F \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n15 \n \n  \nFigure 11. Shows SGLT1 model fited to data (normalised flux). Reproduced from [7] with permission. \nFrom (29) the reversal poten�al for SGLT1 (i.e. when 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1= 0) is given by \n 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 =\n𝑅𝑅𝑅𝑅\n2𝐹𝐹𝑅𝑅𝑛𝑛��\n[𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n2\n.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n�     (32) \nS\nubs�tu�ng typical values (following a carbohydrate-rich meal) of [𝑁𝑁𝑁𝑁+]𝑔𝑔𝑢𝑢𝑑𝑑= 140 mM, [𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 \nmM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑔𝑔𝑢𝑢𝑑𝑑= 40 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖= 1 mM (with 𝑅𝑅𝑅𝑅/𝐹𝐹= 25 mV) gives 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1= 102 mV. Given a typical \nepithelial cell membrane poten�al of about –50 mV, the SGLT1 cotransporter operates well away from \nthe reversal poten�al. Note that at 𝑢𝑢𝑚𝑚𝑒𝑒= -50 mV, and with these sodium and glucose concentra�ons, \nthe ra�o of forward to reverse flux is  �\n[𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n2\n.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n/ 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅= 1.9x105, which means that  \n𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 is not impeded by the rising [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 (which is also rapidly transferred to the blood by GLUT2). \n(\nvii) Sodium-potassium ATPase pump (NKA) \nThe Na+-K+ ATPase NKA (a P-type ATPase) pumps 3 sodium ions out of the cell and 2 potassium ions in \nagainst their concentra�on gradients, using the energy supplied by the enzyme-catalysed hydrolysis of \nATP [16]. The pump is electrogenic since there is a net movement of 1 charge for every ATP hydrolysed. \nBy maintaining the sodium gradient across cell membranes, the NKA pump maintains the energy \ngradient used by many SLC transporters and is responsible for nearly 30% of the body’s res�ng \nmetabolism. We have developed a 6-state model of NKA following the Post-Albers scheme [ 17], as \nillustrated in Figure 12 (see [9]). \nThe 6 states of the NKA protein, each followed by a reac�on (defining the state transi�on), are (in \nclockwise order star�ng at the top right):  \n(1) The first is the unbound inward-facing state 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖 is followed by the membrane reac�on 𝑅𝑅𝑅𝑅𝑚𝑚1  in \nwhich three intracellular sodium ions and one ATP molecule bind. Note that sodium binding is \ncoopera�ve and that A TP binds to a cataly�c site on the enzyme’s cytoplasmic domain with high \naffinity in the presence of bound sodium. The voltage sensi�vity is minimal at this stage, as the \nbinding of the three charged 𝑁𝑁𝑁𝑁+s occurs on the cytoplasmic side without crossing the \nmembrane’s electric field. \n(2\n) The second (inward facing) state 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.3𝑁𝑁 𝑎𝑎+.𝐴𝐴 𝑅𝑅𝐴𝐴 is followed by the 𝑅𝑅𝑅𝑅𝑚𝑚2  reac�on in which bound ATP \nis hydrolysed and ADP is ejected. The other products of ATP hydrolysis, inorganic phosphate 𝑃𝑃𝑖𝑖 and \na proton 𝐻𝐻+, remain bound.  The phosphoryla�on associated with ATP hydrolysis is thought to \nocclude the sodium ions within the membrane.  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n16 \n \n(3) The third (inward facing) state is therefore 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.3𝑁𝑁 𝑎𝑎+.𝐴𝐴𝑖𝑖.𝐻𝐻+\n and is followed by reac�on 𝑅𝑅𝑅𝑅𝑚𝑚3  in which \nthe protein transi�ons to outward facing and the three sodium ions are ejected (since the affinity \nfor 𝑁𝑁𝑁𝑁+ is greatly reduced by the inward-facing to outward-facing transi�on). This transi�on is \nhighly voltage dependent as the three charged sodium ions are being transported outward against \nthe membrane’s electric field gradient (and against a large 𝑁𝑁𝑁𝑁+ concentra�on gradient). A more \nnega�ve highly charged membrane (i.e., more nega�ve membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒) facilitates the \nreac�on.  \n(4\n) The fourth (now outward facing) state 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+\n is followed by reac�on 𝑅𝑅𝑅𝑅𝑚𝑚4 , which binds two \nextracellular potassium ions. Note that cardiac glycosides, such as ouabain, bind to the 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+\n \nstate, stabilizing it and blocking 𝐾𝐾+ binding. \n(5\n) The fi�h (outward facing) state is  therefore 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+\n. Potassium binding triggers \ndephosphoryla�on (𝑅𝑅𝑅𝑅𝑚𝑚5 ) which occludes the bound 𝐾𝐾+ ions and ejects the remaining hydrolysis \nproducts (𝑃𝑃𝑖𝑖, 𝐻𝐻+). There is minimal voltage sensi�vity.  \n(6\n) The sixth and final (outward facing) state is 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.2𝐾𝐾+\n. In the final reac�on 𝑅𝑅𝑅𝑅𝑚𝑚6 , the two potassium \nions are ejected into the intracellular space leaving the unbound protein ready to begin the next \ncycle. This 𝐾𝐾\n+ release involves moving 2 posi�ve charges inward, facilitated by the membrane’s \nelectric field gradient.    \n \n (a)  (b)  (c) \nF\nigure 12. The sodium-potassium ATPase pump. (a) Symbolic representa�on for the transport of three moles of \nsodium outwards and two moles of potassium inwards for every mole of ATP hydrolysed; (b) The 6-state bond \ngraph model; and (c) the reduced model for the flux 𝑣𝑣𝑚𝑚\n𝑁𝑁𝐾𝐾𝐴𝐴.  \nThe expression for NKA flux [9] is  \n 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴=\n𝜅𝜅6𝐾𝐾6𝑑𝑑𝑡𝑡𝑡𝑡𝑡𝑡\n𝐶𝐶−𝐴𝐴𝐴𝐴��\n𝑑𝑑� 𝑖𝑖\n𝑁𝑁𝑁𝑁+\n𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+ �\n3\n.\n𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴\n𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴.𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖\n𝐻𝐻+ − �\n𝑑𝑑� 𝑖𝑖\n𝐾𝐾+\n𝑑𝑑� 𝑡𝑡𝐾𝐾+ �\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴�     (33) \nw\nhere the flux rate depends on the non-dimensional quan��es 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐴𝐴 g\niven in Table 4. See [9] for \nthe meaning of each variable and parameter, and the deriva�on of these expressions. Model \nparameters are given in Table 3.  \nS\nubs�tu�ng for the concentra�ons using (1), equa�on (33) becomes \n 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴=\n𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴\nΦ𝑁𝑁𝐾𝐾𝐴𝐴. ��\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n.\n[𝐴𝐴 𝑅𝑅𝐴𝐴]\n[𝐴𝐴𝐴𝐴\n𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+] − 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑. �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴�     fmol.mm-2.s-1  \nor \n 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴=\n𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴\nΦ𝑁𝑁𝐾𝐾𝐴𝐴. ��\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n. 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴− 𝑄𝑄�𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑. �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴�     fmol.mm-2.s-1 (34) \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n17 \n \nwhere 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴=\n[𝐴𝐴 𝑅𝑅𝐴𝐴]\n[𝐴𝐴𝐴𝐴\n𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+] is the ra�o of reactants to products for the intracellular components of \nATP hydrolysis (assuming [𝐻𝐻2𝑃𝑃] c onstant)  and 𝑄𝑄�𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑= 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴\n𝑒𝑒\n𝑑𝑑 is the corresponding ra�o of these \ncomponents that atains equilibrium for ATP hydrolysis (i.e., the equilibrium constant  for ATP \nh y dr oly sis). N ot e  tha t  othe r solut e t erms d o n ot c on t ribut e t o 𝑄𝑄�𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑 because the Boltzmann terms \n𝐾𝐾𝑁𝑁𝑎𝑎+\n and 𝐾𝐾𝐾𝐾+\n cancel out in the concentra�on ra�o terms. The dimensionless ra�o \n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒 provides the \nac�ve driving force for the reac�on. 𝜅𝜅𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 𝜅𝜅6𝐾𝐾6𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑 (fmol.mm-2.s-1) is a reac�on rate constant, and \nΦ𝑁𝑁𝐾𝐾𝐴𝐴= 𝐶𝐶 −𝐴𝐴𝐴𝐴 is a nondimensional scaling factor for flux that is dependent on the concentra�ons \nof all solutes involved in the reac�on (and is given by the expressions for 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, & 𝐴𝐴 in\n Table 4).   \nParameters Value Unit Solute & enzyme states \n𝐾𝐾𝐾𝐾+\n 0.0239 fmol-1 𝑞𝑞𝑖𝑖\n𝐾𝐾+\n, 𝑞𝑞𝑡𝑡𝐾𝐾+\n \n𝐾𝐾𝑁𝑁𝑎𝑎+\n 0. 0159 fmol-1 𝑞𝑞𝑖𝑖\n𝑁𝑁𝑎𝑎+\n, 𝑞𝑞𝑡𝑡𝑁𝑁𝑎𝑎+\n \n𝐾𝐾𝐴𝐴𝑅𝑅 𝐴𝐴 45. 06 fmol-1 𝑞𝑞𝑖𝑖\n𝐴𝐴𝑅𝑅 𝐴𝐴  \n𝐾𝐾𝐴𝐴𝐴𝐴𝐴𝐴 0.0015 fmol-1 𝑞𝑞𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴  \n𝐾𝐾𝐴𝐴𝑖𝑖 0. 8673 fmol-1 𝑞𝑞𝑖𝑖\n𝐴𝐴𝑖𝑖  \n𝐾𝐾𝐻𝐻+\n 0.8673 fmol-1 𝑞𝑞𝑖𝑖\n𝐻𝐻  \n𝐾𝐾1 0.01673 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖   \n𝐾𝐾2 4.477e3 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+\n  \n𝐾𝐾3 46.64e3 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁𝑎𝑎+)  \n𝐾𝐾4 720.6 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+\n  \n𝐾𝐾5 1.607 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+\n  \n𝐾𝐾6 30.25e3 fmol-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡. (2𝐾𝐾+)  \n𝜅𝜅1 2.839e3 fmol.s-1                     𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖 → 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+\n \n𝜅𝜅2 51.10 fmol.s-1      𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+\n→ 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁 𝑎𝑎+)  \n𝜅𝜅3 469.3 fmol.s-1 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁𝑎𝑎+) → 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+\n  \n𝜅𝜅4 619.1e3 fmol.s-1            𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+\n→ 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+\n  \n𝜅𝜅5 35.87 fmol.s-1     𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+\n→ 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡. (2𝐾𝐾+)  \n𝜅𝜅6 868.4 fmol.s-1           𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡. (2𝐾𝐾+) → 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖   \n𝑧𝑧1 0.9450 dim  \n𝑧𝑧2 𝑧𝑧1 − 1 dim  \nTable 3. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴 through the NKA transporter. Values obtained by \nfi�ng (34) to data from [18]. The enzyme states corresponding to the thermodynamic parameters labelled 𝐾𝐾1.. \n𝐾𝐾6, and the reac�ons corresponding to the kine�c parameters 𝜅𝜅1.. 𝜅𝜅6, are shown in the right-hand column. \nFigure 13 illustrates the dependence of flux (34) on 𝑢𝑢𝑚𝑚𝑒𝑒 for varying values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 between 10 mM \nand 20 mM, with fixed values of [𝑁𝑁𝑁𝑁+]𝑡𝑡=  140 mM, [𝐾𝐾+]𝑖𝑖= 145 mM, [𝐾𝐾+]𝑡𝑡 = 4.5 mM and a Gibbs free \nenergy change of ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�  \n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒�  = -61 kJ.mol-1.  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n18 \n \n \nFigure 13. NKA transporter flux 𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴 ploted against 𝑢𝑢𝑚𝑚\n𝑒𝑒 for three values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 (10 mM, 15 mM and 20 mM). \nOther concentra�ons are [𝑁𝑁𝑁𝑁+]𝑡𝑡 = 140 mM, [𝐾𝐾+]𝑖𝑖 = 140 mM, [𝐾𝐾+]𝑡𝑡= 5 mM . T he Gibbs free energy  made \navailable by ATP hydrolysis to drive the pump is ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�  \n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒� = -61 kJ.mol-1. The reversal poten�als \ncorresponding to the intracellular sodium concentra�ons are (-238 mV, -271 mV, and -293 mV, respec�vely).  \nAt equilibrium (𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 0), the ra�o �\n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒� �\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒\n needed to maintain transmembrane ion gradients is  \n �\n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒� �\n𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒\n= �\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n3\n. �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴 . (35) \nApplying −\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛 .. t o both sides gives   \n \n∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒𝑒𝑒\n𝐹𝐹 = −3. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎+ 2. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾+ 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸 \nwh\nere 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎=\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛\n[𝑁𝑁 𝑎𝑎+]0\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n= 58 mV is the reversal (Nernst) poten�al for sodium (using \n𝑅𝑅𝑅𝑅\n𝐹𝐹= 26 mV), \n𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾=\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛\n[𝐾𝐾+]0\n[𝐾𝐾+]𝑖𝑖\n= -87 mV is the reversal (Nernst) poten�al for potassium, ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒𝑑𝑑\n𝑢𝑢𝑖𝑖𝑒𝑒= -55 kJ.mol-1 is \nthe Gibbs free energy driving the equilibrium reac�on, and 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴 is the reversal poten�al for the NKA \npump obtained from \n �\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n.\n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒= �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴  \nor \n 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= −\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛�\n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒. �\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n. �\n[𝐾𝐾+]𝑡𝑡\n[𝐾𝐾+]𝑖𝑖\n�\n2\n� (36) \nSubs�tu�ng the standard concentra�ons into (36),  \n 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= −\n55𝑒𝑒3\n0.96𝑒𝑒5+ 3x0.058 + 2x0.087 = −0.225 J.C-1 or 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= -225 mV. \nN\note that a Gibbs free energy of –55 kJ.mol-1 corresponds to  \n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒= 𝑒𝑒55/𝑅𝑅𝑅𝑅= 𝑒𝑒55/2.5 ≈ 3.6𝑒𝑒9, \nshowing just how enormous the driving force is for maintaining intracellular sodium.  \nEqua�on (34) also illustrates the par��oning of Gibbs free energy: \n ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑= −𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛��\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n3\n. �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴� = −3𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n� − 2𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n� + 𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 \nor \n ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑= ∆𝐺𝐺𝑁𝑁𝑎𝑎+ ∆𝐺𝐺𝐾𝐾+ ∆𝐺𝐺𝑒𝑒, (37) \nw\nhere ∆𝐺𝐺𝑁𝑁𝑎𝑎= −3𝐹𝐹. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎 = -16.7 kJ.mol-1, ∆𝐺𝐺𝐾𝐾= −2𝐹𝐹. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾 = -16.7 kJ.mol-1 and ∆𝐺𝐺𝑒𝑒 = -21.6 kJ.mol-1 \n(which add to give ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑 = -55 kJ.mol-1 as above). Note that 39% (21.6/55) of the Gibbs free energy \n[𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 𝑚𝑚𝑀𝑀 \n[𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 𝑚𝑚𝑀𝑀 \n[𝑁𝑁𝑁𝑁+]𝑖𝑖= 20 𝑚𝑚𝑀𝑀 \n𝑣𝑣𝑚𝑚\n𝑁𝑁𝐾𝐾𝐴𝐴   \n(fmol.mm-2.s-1) \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n19 \n \nsupplied by ATP hydrolysis at equilibrium goes into storing electrical energy in the cell membrane (by \nhyperpolarising it).  \nF\nor this calcula�on of ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 under equilibrium condi�ons, there is no heat loss because the NKA flux \nis zero; however, when the membrane voltage has the typically observed value of –50 mV, rather than \nbeing at the reversal poten�al of –225 mV, the free energy stored in the membrane capacitor drops to \n∆𝐺𝐺𝑒𝑒= 𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 = -4.8 kJ.mol-1, with the remaining 16.8 kJ.mol-1 dissipated as heat. The ‘efficiency’ of the \npump is now \n16.7+16.7+4.8\n55   ≈70%, but of course the heat is helping to maintain body temperature, so \nit is more accurate to say that of the 55 kJ.mol-1 available from ATP hydrolysis, 61% goes to maintaining \nthe electrolyte gradients, 9% goes to electrical energy storage in the membrane capacitance and 30% \ngoes to thermal energy storage in the heat capacity of the cell and its surroundings.   \n(viii) The inwardly rectifying potassium ion channel (Kir) \nW\ne next consider the inwardly rec�fying ion channel Kir (e.g., the Kir7.1 subtype). Highly selective, \nlow-conductance single -ion channels can be considered as three - or four-state transporters \n(depending on the number of necessary intermediate steps). As illustrated in Figure 14, we use a \nthree-state model with a bound state 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑖𝑖𝐾𝐾+\n (𝑞𝑞1), a meta-stable intermediate state 𝑞𝑞𝑚𝑚\n𝐸𝐸𝑡𝑡𝐾𝐾+\n (𝑞𝑞1), and \nan empty state 𝑞𝑞𝑚𝑚𝐸𝐸  (𝑞𝑞3). It is tempting to assume that the electrogenic step is associated with the \ntransition between states 1 and 2 since this is where the ion transport occurs. But the bond graph \nmodel, while ensuring that physical conservation laws are satisfied for the protein as a whole, is \nlimited in its ability to capture the details of the electrostatic field operating over the membrane \nand its interaction with the potassium ion moving through that field. The best we can do is to allow \ncharge movement to be associated with all three reactions by including electrogenic effects via \nthe unknown 𝑧𝑧𝑖𝑖 (for 𝑖𝑖=1..6) terms as shown in Figure 14. The parameter estimation process  \ndescribed in Supplementary Material  is used to determine these parameters (subject to the \nrequirement that for each ion that crosses the membrane, the overall charge  displacement for \nthe membrane is -1).    \n  \nFigure 14. A 3-state bond graph model of the Kir inward rec�fier with provision for charge movement by all \nreac�ons (represented by the purple transforming factors 𝑧𝑧𝑖𝑖𝐹𝐹). The binding and unbinding reac�ons (𝑅𝑅𝑅𝑅𝑚𝑚\n1  and \n𝑅𝑅𝑅𝑅𝑚𝑚\n3 ) are assumed to be much faster than the protein transi�on from inward facing to outward facing (𝑅𝑅𝑅𝑅𝑚𝑚\n2 ). \nFrom Figure 14, the flux equa�ons associated with the three reac�ons are  \n 𝑣𝑣1 =  𝜅𝜅1 �𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾1𝑞𝑞1. 𝑒𝑒\n𝑧𝑧2𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  (38) \n 𝑣𝑣2 = 𝜅𝜅2 �𝐾𝐾1𝑞𝑞1. 𝑒𝑒\n𝑧𝑧3𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾2𝑞𝑞2. 𝑒𝑒\n𝑧𝑧4𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  (39) \n 𝑣𝑣3 = 𝜅𝜅3 �𝐾𝐾2𝑞𝑞2. 𝑒𝑒\n𝑧𝑧5𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+\n. 𝑒𝑒\n𝑧𝑧6𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  (40) \nAssuming rapid binding and unbinding of K+ (𝜅𝜅1, 𝜅𝜅3 → ∞), for finite values of 𝑣𝑣1 and 𝑣𝑣3, the bracketed \nterms on the right in (38) and (39) must be zero, and hence  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n20 \n \n 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾1𝑞𝑞1. 𝑒𝑒\n𝑧𝑧2𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 = 0 (41) \n 𝐾𝐾2𝑞𝑞2. 𝑒𝑒\n𝑧𝑧5𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+\n. 𝑒𝑒\n𝑧𝑧6𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 = 0 (42) \nIf we also assume (Briggs-Haldane) that the enzyme states are at steady-state, 𝑣𝑣1 = 𝑣𝑣2 = 𝑣𝑣3 = 𝑣𝑣 and \nwith subs�tu�ons for 𝐾𝐾1𝑞𝑞1 and 𝐾𝐾2𝑞𝑞2 from (41) and (42), respec�vely, (39) gives \n 𝑣𝑣= 𝑣𝑣2 = 𝜅𝜅2 �𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+\n. 𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5+𝑧𝑧4)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 �   \nor \n 𝑞𝑞3 =\n𝑟𝑟\n𝐾𝐾3𝜅𝜅2\n/ �𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝑞𝑞 �𝑡𝑡K+\n. 𝑒𝑒\n(𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 �  (43) \nF\nor a fixed (and specified) number of ion channels, 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑, and using (41) and (42),  \n 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑= 𝑞𝑞1 + 𝑞𝑞2 + 𝑞𝑞3 = 𝑞𝑞3. �\n𝐾𝐾3\n𝐾𝐾1\n𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 +\n𝐾𝐾3\n𝐾𝐾2\n𝑞𝑞 �𝑡𝑡K+\n. 𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 + 1�  \nor, with (43), \n 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑=\n𝑟𝑟\n𝐾𝐾3𝜅𝜅2\n⋅ �\n𝐾𝐾3\n𝐾𝐾1\n𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴 +\n𝐾𝐾3\n𝐾𝐾2\n𝑞𝑞 �𝑡𝑡\nK+\n. 𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 + 1� / �𝑞𝑞 �𝑖𝑖\nK+\n. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝑞𝑞 �𝑡𝑡\nK+\n. 𝑒𝑒\n(𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 �  \nRearranging for 𝑣𝑣,  \n 𝑣𝑣= 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑𝐾𝐾3𝜅𝜅2.\n𝑑𝑑� 𝑖𝑖\nK+\n.𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 − 𝑑𝑑� 𝑡𝑡K+\n.𝑒𝑒\n(𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴\n1 + 𝐾𝐾3\n𝐾𝐾1\n.𝑑𝑑� 𝑖𝑖\nK+.𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴 + 𝐾𝐾3\n𝐾𝐾2\n.𝑑𝑑� 𝑡𝑡K+.𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴\n. (44) \nF\ninally, with 𝑞𝑞 �𝑖𝑖\nK+\n= 𝐾𝐾𝑖𝑖\n𝐾𝐾+\n. 𝑉𝑉𝑖𝑖. [𝐾𝐾+]𝑖𝑖, 𝑞𝑞 �𝑡𝑡K+\n= 𝐾𝐾𝑡𝑡𝐾𝐾+\n. 𝑉𝑉𝑡𝑡. [𝐾𝐾+]𝑡𝑡, and defining 𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖\n𝑟𝑟=\n𝑘𝑘𝑚𝑚1\n𝐾𝐾𝑖𝑖\n𝐾𝐾+\n.𝑉𝑉𝑖𝑖\n,  𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖 𝑟𝑟=\n𝑘𝑘𝑚𝑚2\n𝐾𝐾𝑡𝑡𝐾𝐾+.𝑉𝑉𝑡𝑡\n, \n𝑄𝑄𝐾𝐾𝑖𝑖 𝑟𝑟\n𝑒𝑒\n𝑑𝑑=\n𝐾𝐾𝑡𝑡𝐾𝐾+\n.𝑉𝑉𝑡𝑡\n𝐾𝐾𝑖𝑖\n𝐾𝐾+\n.𝑉𝑉𝑖𝑖\n  and 𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟= 𝐾𝐾𝐾𝐾+\n𝑉𝑉𝑖𝑖. 𝜅𝜅𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟, (44) becomes \n 𝑣𝑣= 𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟.\n[𝐾𝐾+]𝑖𝑖.𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴  − 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟\n𝑒𝑒𝑒𝑒.[𝐾𝐾+]𝑡𝑡.𝑒𝑒\n(𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴\n1 + \n�𝐾𝐾+�𝑖𝑖\n𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖𝑟𝑟.𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴  +  \n�𝐾𝐾+�𝑡𝑡\n𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟.𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴\n    fmol.mm-2.s-1  \nor, with −𝑧𝑧1 + 𝑧𝑧2 − 𝑧𝑧3 + 𝑧𝑧4 − 𝑧𝑧5 + 𝑧𝑧6 = −1 \n 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟=\n𝜅𝜅� 𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟\nΦ . �[𝐾𝐾+]𝑖𝑖− 𝑄𝑄𝐾𝐾𝑖𝑖 𝑟𝑟\n𝑒𝑒\n𝑑𝑑. [𝐾𝐾+]𝑡𝑡. 𝑒𝑒\n−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴�,       fmol.mm-2.s-1 (45) \nwhere  \n Φ = 𝑒𝑒\n(−𝑧𝑧1+𝑧𝑧2−𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 . �1 +  \n[𝐾𝐾+]𝑖𝑖\n𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴  +  \n[𝐾𝐾+]𝑡𝑡\n𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚\n𝑒𝑒\n𝑅𝑅\n𝐴𝐴 �. (46) \nTh\ne electrical current is  𝐼𝐼𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟= 𝐹𝐹𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟 (fC.mm-2.s-1) and the reversal or ‘Nernst’ poten�al is   \n 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟= − 𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛�\n�𝐾𝐾+�𝑖𝑖\n�𝐾𝐾+�𝑜𝑜\n� = −26.5𝑅𝑅𝑛𝑛� 140\n4.5 � = -91 mV (47) \nFigure 15 shows the current-voltage rela�on for the Kir channel.  \nThe fluxes out of the intracellular solu�on and into the extracellular solu�on are   \n \n𝑐𝑐𝑑𝑑𝑖𝑖\nK+\n𝑐𝑐𝑑𝑑= −𝑣𝑣1 = −𝑣𝑣,   and  \n𝑐𝑐𝑑𝑑𝑖𝑖\nK+\n𝑐𝑐𝑑𝑑= 𝑣𝑣3 = 𝑣𝑣. \n \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n21 \n \nParameters Value Unit \n𝐾𝐾𝐾𝐾+\n 0.0239 fmol-1 \n𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 0.0442 pL.s-1 \n𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟\n𝑒𝑒𝑑𝑑 1 dimensionless \n𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖𝑟𝑟 1 mM \n𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟 10 mM \nTable 1. Parameter values used in the  expression for flux 𝑣𝑣𝑚𝑚\n𝐺𝐺𝐺𝐺𝐺𝐺\n𝑅𝑅2 through the GLUT2 transporter. Values on the \nle� are obtained by fi�ng (24) to data from Lowe and Walmsley [19]. Values on the right have been adjusted to \nreflect use of concentra�ons in (25). \n \n  \nFigure 15. The current-voltage rela�on predicted for Kir by (45) and (46). Note the reversal poten �al at -91 mV \nand the inward rec�fica�on above this poten�al. Current-voltage curves are shown for 3 values of the parameter \n𝑘𝑘𝑡𝑡\n𝐾𝐾𝑖𝑖𝑟𝑟. The curve steepens as this parameter is increased from 1 mM to 10 mM and then hardly changes for higher \nvalues. Changing the parameter 𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖𝑟𝑟 is equivalent to changes in 𝜅𝜅̂𝑚𝑚\n𝐾𝐾𝑖𝑖𝑟𝑟 so this parameter is set to 1 mM.   \nA COMMON FRAMEWORK FOR ALL TRANSPORTERS AND ATPase PUMPS \nTo provide a single unifying framework for all enzyme -catalysed reac�ons (including all membrane \ntransporters, pumps and ion channels), we will express all steady-state reac�on fluxes in the form: \n  𝑣𝑣𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚\n𝑒𝑒) =\n𝜅𝜅𝑅𝑅𝑅𝑅\nΦ𝑅𝑅𝑅𝑅(𝒒𝒒,𝑢𝑢𝑚𝑚\n𝑒𝑒) �𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) − 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑. 𝑃𝑃𝑅𝑅𝑅𝑅([𝒒𝒒]). 𝑒𝑒𝑧𝑧𝐹𝐹𝑢𝑢𝑚𝑚\n𝑒𝑒/𝑅𝑅𝑅𝑅� (49) \nwhere 𝒒𝒒 is a state vector that includes all solutes (with concentra�ons [𝒒𝒒]) par�cipa�ng in the reac�on, \n𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) and 𝑃𝑃𝑅𝑅𝑅𝑅([𝒒𝒒]) are products of reactant and product concentra�ons, 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑 is the equilibrium \nconstant given by 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑=\n𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒])\n𝐴𝐴𝑅𝑅𝑅𝑅([𝒒𝒒])�\n𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚\n, and Φ𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) is an algebraic expression that affects \nthe speed of 𝑅𝑅𝑅𝑅 when the reac�on is not at equilibrium. The constant 𝜅𝜅𝑅𝑅𝑅𝑅 scales the reac�on flux and \nreflects the level of protein expression in a cell. Expressing all fluxes in this form clearly iden�fies:  \n1) th\ne {}-bracketed component that governs equilibrium, including the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒\n𝑑𝑑, \n2) th\ne explicit dependence on membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒 (for electrogenic reac�ons) that provides the \nreversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑅𝑅𝑅𝑅=\n𝑅𝑅𝑅𝑅\n𝑧𝑧\n𝐹𝐹𝑅𝑅𝑛𝑛�\n𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒])\n𝑄𝑄𝑅𝑅𝑅𝑅\n𝑒𝑒𝑒𝑒.𝐴𝐴𝑅𝑅𝑅𝑅([𝒒𝒒])� ,  \n3) the state-dependent term Φ𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) that governs the flux magnitude when not at equilibrium,  \n4) the scaling constant 𝜅𝜅𝑅𝑅𝑅𝑅 that reflects the expression levels for the protein.  \nTable 4 summarises the transmembrane fluxes, all expressed in this form.  \n  \n𝐼𝐼𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 (fC.mm-2.s-1) \n𝑢𝑢𝑚𝑚\n𝑒𝑒 (mV) \n𝑘𝑘𝑡𝑡\n𝐾𝐾𝑖𝑖𝑟𝑟= 1 mM \n𝑘𝑘𝑡𝑡\n𝐾𝐾𝑖𝑖𝑟𝑟= 10 mM \n𝑘𝑘𝑡𝑡\n𝐾𝐾𝑖𝑖𝑟𝑟= 100 mM \n \nReversal poten�al for Kir \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n22 \n \nProtein 𝒒𝒒 Flux Eq # \nGLUT2 𝑞𝑞𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐 \n𝑞𝑞𝑡𝑡𝐺𝐺 𝑒𝑒𝑐𝑐 \n𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2(𝒒𝒒) =\n𝜅𝜅� 𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝐴𝐴2\nΦ𝑅𝑅𝑅𝑅(𝒒𝒒) . �[𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2\n𝑒𝑒\n𝑑𝑑 . [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡�  fmol.mm-2.s-1     \n𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2\n𝑒𝑒\n𝑑𝑑 = 1;  \nΦ𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2(𝒒𝒒) = 1 +  \n[𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖\n𝑘𝑘𝑖𝑖\n𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2  +  \n[𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡\n𝑘𝑘𝑡𝑡\n𝐺𝐺\n𝐺𝐺𝐺𝐺\n𝑅𝑅2  +  \n[𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖. [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡\n𝑘𝑘𝑖𝑖𝑡𝑡\n𝐺𝐺𝐺𝐺\n𝐺𝐺𝑅𝑅2  \n27 \nSGLT1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑\n𝑁𝑁\n𝑎𝑎+\n \n𝑞𝑞𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n \n𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑\n𝐺𝐺\n𝑒𝑒𝑐𝑐 \n𝑞𝑞𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐 \n \n𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) =\n𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝐴𝐴1\nΦ𝑆𝑆𝐺𝐺𝐺𝐺 𝐴𝐴 1(𝒒𝒒) ��\n[𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n2\n.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n− 𝑄𝑄𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1\n𝑒𝑒\n𝑑𝑑 .  𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�  fmol.mm-2.s-1       \n𝑄𝑄𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1\n𝑒𝑒\n𝑑𝑑 = 1  \nΦ𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = {�𝑒𝑒\n2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅+ �𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝑁𝑁\n𝑎𝑎+\n�\n2\n� 𝐾𝐾1\n𝐾𝐾2\n+ 𝐾𝐾1\n𝐾𝐾3\n. 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝐺𝐺\n𝑒𝑒𝑐𝑐�� . ��𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6\n𝜅𝜅3\n� \n + �\n𝐾𝐾1\n𝐾𝐾6\n+ �𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n� 𝐾𝐾1\n𝐾𝐾5\n+\n𝐾𝐾1\n𝐾𝐾4\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐� �.��𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑\n𝐺𝐺\n𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+\n𝜅𝜅6\n𝜅𝜅3\n𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�}/�𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n2\n. 𝑞𝑞 �𝑖𝑖\n𝐺𝐺\n𝑒𝑒𝑐𝑐 \nR\neversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 =\n𝑅𝑅𝑅𝑅\n2𝐹𝐹𝑅𝑅𝑛𝑛��\n[𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n�\n2\n.\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡\n[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n� = 102 mV \n29 \n \n \n31 \n \n \n32 \nNKA 𝑞𝑞𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n \n𝑞𝑞𝑡𝑡𝑁𝑁 𝑎𝑎+\n \n𝑞𝑞𝑖𝑖\n𝐾𝐾+\n \n𝑞𝑞𝑡𝑡𝐾𝐾+\n \n \n𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) =\n𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝑁𝑁\nΦ𝑁𝑁𝐾𝐾𝑁𝑁(𝒒𝒒,𝑢𝑢𝑚𝑚𝑒𝑒) . ��\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n. 𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴− 𝑄𝑄𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑. �\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n�\n2\n. 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  fmol.mm-2.s-1        \n𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴=\n[𝐴𝐴𝑅𝑅 𝐴𝐴]\n[𝐴𝐴𝐴𝐴𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+]      and   𝑄𝑄𝑁𝑁𝐾𝐾𝐴𝐴\n𝑒𝑒\n𝑑𝑑= 𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴\n𝑒𝑒\n𝑑𝑑=\n[𝐴𝐴𝑅𝑅 𝐴𝐴]\n[𝐴𝐴𝐴𝐴𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+]�\n𝑒𝑒𝑑𝑑𝑢𝑢 𝑖𝑖𝑒𝑒\n \n∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�  \n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒�  = -55 kJ/mol-1 \nΦ𝑁𝑁𝐾𝐾𝐸𝐸(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = C − AD, where \n𝐴𝐴= � 𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ �\n3\n. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴\n𝑅𝑅𝐴𝐴\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+ − � 𝑞𝑞 �𝑖𝑖\n𝐾𝐾+\n𝑞𝑞 �𝑡𝑡𝐾𝐾+ �\n2\n. 𝑒𝑒−𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅 \n𝐵𝐵= � 𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ �\n3\n. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴\n𝑅𝑅𝐴𝐴\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+ . 𝑒𝑒−𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅�Λ4 + � Λ5 + 1\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+ � . � 𝑞𝑞 �𝑡𝑡\n𝐾𝐾+\n�\n2\n� \n𝐶𝐶= �\nΛ1\n� 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+�\n3\n.𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖\n𝐻𝐻+ + (Λ2. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴+ Λ3). �\n𝑑𝑑� 𝑖𝑖\n𝑁𝑁𝑁𝑁+\n𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+�\n3\n.\n𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴\n𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴.𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖\n𝐻𝐻+ +\n𝐵𝐵\n� 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+�\n3�   \n       \nx �\n𝜆𝜆1+𝜆𝜆2\n𝑑𝑑� 𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴+ 𝜆𝜆3 + � 𝜆𝜆4 + 𝜆𝜆5 + 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+\n� .\n� 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+�\n3\n� 𝑑𝑑� 𝑡𝑡𝐾𝐾+�\n2 . 𝑒𝑒\n𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  \n𝐴𝐴= Λ2. 𝜆𝜆1 + Λ5. 𝜆𝜆4 + 𝜆𝜆4 + 𝜆𝜆5\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+ + Λ3. 𝜆𝜆1 + 𝜆𝜆2\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴+ �(𝜆𝜆1 + 𝜆𝜆2). 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴\n𝑅𝑅𝐴𝐴+ 𝜆𝜆3\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖\n𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖\n𝐻𝐻+\n𝑞𝑞 �𝑖𝑖\n𝐴𝐴\n𝑅𝑅𝐴𝐴 � . 𝐵𝐵\n� 𝑞𝑞 �𝑖𝑖\n𝑁𝑁\n𝑎𝑎+\n�\n3 \nand 𝜆𝜆𝑖𝑖=\n𝜅𝜅6\n𝜅𝜅𝑖𝑖\n, Λ𝑖𝑖=\n𝐾𝐾6\n𝐾𝐾𝑖𝑖\n, for 𝑖𝑖= 1. .5. \nReversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= −\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛�\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒. �\n[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n[𝑁𝑁 𝑎𝑎+]𝑡𝑡\n�\n3\n. �\n[𝐾𝐾+]𝑡𝑡\n[𝐾𝐾+]𝑖𝑖\n�\n2\n� = -225 mV \n \n34 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n36 \n \nKir 𝑞𝑞𝑖𝑖\n𝐾𝐾+\n \n𝑞𝑞𝑡𝑡𝐾𝐾+\n \n \n𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) =\n𝜅𝜅� 𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟\nΦ𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒) . �[𝐾𝐾+]𝑖𝑖− 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟\n𝑒𝑒\n𝑑𝑑. [𝐾𝐾+]𝑡𝑡. 𝑒𝑒\n−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒\n𝑅𝑅\n𝐴𝐴�  fmol.mm-2.s-1        \n𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟\n𝑒𝑒\n𝑑𝑑= 1 \nΦ𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = 𝑒𝑒\n(−𝑧𝑧1+𝑧𝑧2−𝑧𝑧3)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅 � 1 +  \n[𝐾𝐾+]𝑖𝑖\n𝑘𝑘𝑖𝑖\n𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒\n(𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅 +  \n[𝐾𝐾+]𝑡𝑡\n𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒\n(𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒\n𝑅𝑅𝑅𝑅�  \nR\neversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟= −\n𝑅𝑅𝑅𝑅\n𝐹𝐹𝑅𝑅𝑛𝑛�\n[𝐾𝐾+]𝑖𝑖\n[𝐾𝐾+]𝑡𝑡\n� = -91 mV \n45 \n \n \n46 \n \n47 \nTable 4. A list of the flux expressions in standardised form for the SLC transporters GLUT2 and SGL T1, the NKA \npump, and the ion channel/transporter Kir. The reversal poten�als shown for each membrane protein are \ncalculated using values of [𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 mM, [𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖= 1 mM, \n[𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡= 40 mM, and \n𝑅𝑅𝑅𝑅\n𝐹𝐹=26.5 mV.   \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n23 \n \nA MODEL OF THE ENTEROCYTE \nWe now combine the GLUT2, SGLT1, NKA, and Kir models discussed above to create  a model of \nhomeostasis in the enterocyte, the absorp�ve cell of the epithelium in the small intes�nes. The NKA \npump maintains a low intracellular [𝑁𝑁𝑁𝑁+]𝑖𝑖 such that the  SGLT1 cotransporter can use  the sodium \ngradient between the gut lumen and the intracellular space to bring glucose into the cell. Glucose is \nmetabolised into the ATP used to drive NKA  and also flows out of the cell via GLUT2 facilitated \ndiffusion. The NKA-driven potassium flux into the cell is balanced by potassium efflux through the Kir \nion channel, as illustrated in Figure 16.     \n \nFigure 16. Equilibrium between the outward flow of 𝐾𝐾+ ions through Kir (when the membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 is \nabove the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟\n𝐾𝐾𝑖𝑖𝑟𝑟 = -87 mV) and the inward flow of 𝐾𝐾+ ions through NKA (when the membrane \npoten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 is above the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟\n𝑁𝑁\n𝐾𝐾𝐸𝐸 = -225 mV for NKA). Equilibrium is achieved when these two \nopposing currents are equal. The purple line shows the 𝐾𝐾+ ion efflux (𝑣𝑣𝑚𝑚\n𝐾𝐾+(𝐾𝐾𝑖𝑖𝑟𝑟)) from Kir (equa�on [33]), and the \ndark brown line shows the influx 𝑣𝑣𝑚𝑚\n𝐾𝐾+(𝑁𝑁 𝐾𝐾𝐴𝐴) associated with NKA (equa�on (33) with two 𝐾𝐾+ ions transported for \neach NKA cycle). 𝑢𝑢𝑒𝑒𝑑𝑑𝑢𝑢𝑖𝑖𝑒𝑒\n𝐾𝐾  is membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 at this equilibrium.  \nFigure 16 illustrates the balance of inward sodium flux from SGLT1 with outward flux by NKA.  \n \nFigure 17. Equilibrium between the inward flow of 𝑁𝑁𝑁𝑁+ ions through SGLT1 (when the membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 \nis well below the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟\n𝑆𝑆\n𝐺𝐺𝐺𝐺𝑅𝑅1 = 102 mV) and the outward flow of 𝑁𝑁𝑁𝑁+ ions through NKA (when the \nmembrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 is above the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟\n𝑁𝑁\n𝐾𝐾𝐸𝐸 = -225 mV for NKA). Equilibrium is achieved when \nthese two opposing currents are equal. The green line shows the 𝑁𝑁𝑁𝑁+ ion influx (𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝑎𝑎+(𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1)) from SGLT1 \n(equa�on 29 with two 𝑁𝑁𝑁𝑁+ ions transported for each SGLT1 cycle) and the dark brown line shows the effl ux \n𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝑎𝑎+(𝑁𝑁 𝐾𝐾𝐸𝐸) associated with NKA pump (equa�on (33) with three 𝑁𝑁𝑁𝑁+ ions transported for each NKA cycle). 𝑢𝑢𝑒𝑒𝑑𝑑𝑢𝑢𝑖𝑖𝑒𝑒\n𝑁𝑁\n𝑎𝑎 \nis the value of membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 at this equilibrium.  \nFigure 18 shows the various membrane processes and the glycolysis pathway that regenerates the ATP \nused by ATP hydrolysis in the NKA pump.   \n𝒗𝒗𝟏𝟏\n𝑲𝑲+(𝑲𝑲 𝒊𝒊𝑲𝑲) \nOpera�ng range for cell \nOutward 𝐾𝐾+current \nInward 𝐾𝐾+current \n𝒖𝒖𝟏𝟏\n𝒆𝒆 \n𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟=-87mV \n 𝒖𝒖𝒆𝒆𝒒𝒒 𝒖𝒖𝒊𝒊𝑵𝑵\n𝑲𝑲  \n𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸=-225mV \n𝒗𝒗𝟏𝟏\n𝑲𝑲+(𝑵𝑵𝑲𝑲 𝑵𝑵) = −𝟐𝟐𝒗𝒗𝟏𝟏\n𝑵𝑵𝑲𝑲\n𝑵𝑵 \n𝒖𝒖𝒆𝒆𝒒𝒒 𝒖𝒖𝒊𝒊𝑵𝑵\n𝑵𝑵𝑵𝑵 \n𝒗𝒗𝟏𝟏\n𝑵𝑵𝑵𝑵+(𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝟏𝟏) = 𝟐𝟐𝒗𝒗𝟏𝟏\n𝑺𝑺\n𝑺𝑺𝑺𝑺𝑺𝑺\n𝟏𝟏 \nOpera�ng range for cell \nOutward 𝑁𝑁𝑁𝑁+current \nInward 𝑁𝑁𝑁𝑁+current \n𝒖𝒖𝟏𝟏\n𝒆𝒆 \n𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑆𝑆𝐺𝐺𝐺𝐺𝑅𝑅1=102mV \n𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸=-225mV \n𝒗𝒗𝟏𝟏\n𝑵𝑵𝑵𝑵+(𝑵𝑵𝑲𝑲 𝑵𝑵) = 𝟑𝟑𝒗𝒗𝟏𝟏\n𝑵𝑵𝑲𝑲\n𝑵𝑵 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n24 \n \n \nFigure 18. The enterocyte model includes NKA, Kir, and GLUT2 in the basolateral membrane and SGLT1 in the \napical membrane. Energy is transferred between biochemical storage  in ATP and solute concentra�ons, \nelectrical storage in membrane capacitance, mechanical storage in the cell wall compliance (which determines \nthe pressure-volume behaviour associated with water movement), and heat storage in the thermal capacity of \nwater in the cell and its surroundings.  One mole of glucose is assumed to instantaneously convert 2 moles of \nADP to ATP via glycolysis with the genera�on of 2 moles of water.      \nThe membrane voltage is 𝑢𝑢𝑚𝑚𝑒𝑒=\n𝑑𝑑𝑚𝑚𝑒𝑒\n𝐶𝐶𝑚𝑚\n  is calculated from net influx of membrane charge:   \n \n𝑐𝑐𝑢𝑢𝑚𝑚𝑒𝑒\n𝑐𝑐𝑑𝑑=\n𝐹𝐹\n𝐶𝐶𝑚𝑚\n� 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴− 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟� ,    J.C-1.s-1 (50) \nwh\nere 𝐶𝐶𝑚𝑚 is the membrane capacitance (assumed to be 1 µF.cm-2 or 10-8 C2.J-1.mm-2). Note that \n  \n𝐹𝐹\n𝐶𝐶𝑚𝑚\n = 105 C.mol-1 x10-8 C-2.J.mm2 x 10-15 mol.fmol-1 = 10-18 J.C-1.mm2.fmol-1, so the RHS is J.C-1.s-1. \nIncreases in intracellular potassium and sodium ion concentra�ons are given by  \n \n𝑐𝑐[𝐾𝐾+]𝑖𝑖\n𝑐𝑐𝑑𝑑= Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. � 2𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴− 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟�     mM.s-1       (51) \na\nnd \n \n𝑐𝑐[𝑁𝑁 𝑎𝑎+]𝑖𝑖\n𝑐𝑐𝑑𝑑 = Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. (2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 3𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴),  mM.s-1 (52) \nwh\nere Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒 (m-1) is the ra�o of membrane area A𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒 to cell volume V𝑐𝑐𝑒𝑒 𝑒𝑒𝑒𝑒. Note that we are assuming \nthe same areal density of the NKA pump, Kir channel, and GLUT2 transporter (all in the basolateral \nmembrane in contact with inters��al fluid and effec�vely with blood) and the SGLT2 cotransporter (in \nthe apical membrane in hence in contact with the intes�nal lumen).    \nS\nince 1 mol of glucose is assumed to replace 2 mol o f  A D P  w i t h  2  mol ATP instantaneously, the \nintracellular glucose is given by  \n  \n𝑐𝑐[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖\n𝑐𝑐𝑑𝑑= Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. (𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2) −\nV𝑐𝑐𝑒𝑒𝐶𝐶𝐶𝐶\n32 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴.   mM.s-1  (53) \n \n  \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n25 \n \nTo ex amine how these four membrane proteins achieve homeostasis of sodium, potassium and \nglucose in the enterocyte, we first consider the two transporters Kir and NKA. Figure 19(a) shows the \nreac�on fluxes 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟 and 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴 as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒 from -120 mV to 0 mV under \nphysiologically normal condi�ons ([𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 mM, [𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 \nmM and ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�  \n𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴\n𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴\n𝑒𝑒𝑒𝑒�  = -61 kJ.mol-1). The reversal poten�als for the two membrane \nproteins are 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾𝑖𝑖 𝑟𝑟= -91 mV and 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= -225 mV, respec�vely. The effect on 𝐾𝐾+ flux of these two \ntransporters is shown in Figure 19(b) where the net 𝐾𝐾+ flux (𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟− 2𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴) is ploted against 𝑢𝑢𝑚𝑚𝑒𝑒 for \ndifferent values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 from 0 mM to 12 mM. For [𝑁𝑁𝑁𝑁+]𝑖𝑖= 0 mM the net flux is zero at -91 mV \nand then occurs at progressively less nega�ve membrane poten�als as [𝑁𝑁𝑁𝑁+]𝑖𝑖 increases, up to a limit \nof litle over 10 mM. Beyond this there is no balance  possible because the influx of 𝐾𝐾+ by NKA \n(associated with the high efflux of 𝑁𝑁𝑁𝑁+) exceeds the efflux of 𝐾𝐾+ achieved by Kir at all poten�als.  \n \n (a)  (b)  \nF\nigure 19. (a) Fluxes 𝑣𝑣𝑚𝑚\n𝐾𝐾\n𝑖𝑖𝑟𝑟 and 𝑣𝑣𝑚𝑚\n𝑁𝑁𝐾𝐾𝐴𝐴 through Kir and NKA as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒. The reversal \npoten�als for Kir and NKA under standard physiological condi�ons are -91 mV and -225 mV, respec�vely. (b) The \nnet outward potassium flux 𝑣𝑣𝑚𝑚\n𝐾𝐾𝑖𝑖𝑟𝑟− 2𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴 shown as a func�on of 𝑢𝑢𝑚𝑚\n𝑒𝑒 for values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 from 0 mM to 12 mM. \nFor the specified electrolyte concentra�ons ([𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 mM) there is a well \ndefined zero net 𝐾𝐾+ flux for values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 from 0 mM to 10 mM, but not for values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 above 10 mM. \nThe flux balance for 𝑁𝑁𝑁𝑁+ depends on both NKA and SGLT1. Figure 20 shows how the balance of sodium \nefflux 3𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴 and sodium influx 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 (i.e. a net efflux of 3𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1) depends on membrane \npoten�al 𝑢𝑢𝑚𝑚𝑒𝑒.   \n \n (a)  (b)  \nF\nigure 20. (a) Fluxes 𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴 and 𝑣𝑣𝑚𝑚\n𝑆𝑆𝐺𝐺𝐺𝐺\n𝑅𝑅1 through NKA and SGLT1 as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒. Both \nfluxes are computed for 3 values of [𝑁𝑁𝑁𝑁+]𝑖𝑖, but this has litle effect on 𝑣𝑣𝑚𝑚\n𝑆𝑆\n𝐺𝐺𝐺𝐺𝑅𝑅1.(b) The corresponding net outward \n𝑁𝑁𝑁𝑁+ flux (calculated from 3𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚\n𝑆𝑆𝐺𝐺𝐺𝐺\n𝑅𝑅1). \nAchieving zero net flux for both [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖 also ensures zero net flow of charge across the \nmembrane since all charge transfer is associated with those ions.   \n \n[𝑵𝑵𝑵𝑵+]𝒊𝒊  \n 2 mM \n 10 mM \n 12 mM \n 0 mM \nReversal poten�al for Kir \n Net outward 𝑲𝑲+ flux (fmol.mm-2.s-1) \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \nPhysiological poten�al \n Reac�on fluxes for Kir & NKA at [𝑵𝑵𝑵𝑵+]𝒊𝒊= 10 mM  \n        (fmol.mm-2.s-1) \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \n 𝒗𝒗𝟏𝟏𝑵𝑵𝑲𝑲 𝑵𝑵 \n 𝒗𝒗𝟏𝟏𝑲𝑲𝒊𝒊𝑲𝑲 \nReversal poten�al for Kir \n[𝑵𝑵𝑵𝑵+]𝒊𝒊  \n 5 mM \n 10 mM \n 12 mM \n Net outward 𝑵𝑵𝑵𝑵+ flux (fmol.mm-2.s-1) \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \n Reac�on fluxes for NKA & SGLT1 (fmol.mm-2.s-1) \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \n[𝑵𝑵𝑵𝑵+]𝒊𝒊  \n 5 mM \n 10 mM \n 12 mM \nPhysiological poten�al \n 𝒗𝒗𝟏𝟏𝑵𝑵𝑲𝑲 𝑵𝑵  𝒗𝒗𝟏𝟏𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝟏𝟏 \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n26 \n \nAt [𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 mM, both the net 𝐾𝐾+ flux and the net 𝑁𝑁𝑁𝑁+ flux are zero at the same poten�al (𝑢𝑢𝑚𝑚\n𝑒𝑒= -60 mV). \nThese values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and 𝑢𝑢𝑚𝑚\n𝑒𝑒 were calculated from (50) and (51) for a fixed values of 𝜅𝜅𝑚𝑚\n𝑅𝑅𝑅𝑅 for each of the three \ntransporters. An alterna�ve is to set the le� hand sides of (50) and (51) to zero for all values of 𝑢𝑢𝑚𝑚\n𝑒𝑒 and compute \nthe corresponding scaling on the 𝜅𝜅𝑚𝑚\n𝑅𝑅𝑅𝑅 values (as a func�on of 𝑢𝑢𝑚𝑚\n𝑒𝑒) in order to achieve zero net flux. The computed \nscale factors are shown as a func�on of 𝑢𝑢𝑚𝑚\n𝑒𝑒 in the botom panel of Figure 21. The unscaled fluxes (𝑣𝑣𝑚𝑚\n𝑅𝑅𝑅𝑅 for each \nof the transporters) are shown in the top panel.     \n  \nF\nigure 21. The transporter fluxes (top) and scaling on the reac�on rate constants (botom) needed for Kir and \nSGLT1 (rela�ve to that on NKA) to achieve homeostasis of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖. These rela�ve scaling values depend \non the membrane poten�al 𝑢𝑢𝑚𝑚\n𝑒𝑒 at which the net fluxes 𝑣𝑣𝑚𝑚\n𝐾𝐾𝑖𝑖𝑟𝑟− 2𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴 and 3𝑣𝑣𝑚𝑚\n𝑁𝑁\n𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚\n𝑆𝑆\n𝐺𝐺𝐺𝐺𝑅𝑅1 are both zero (i.e. \nthese net fluxes are zero for all values of 𝑢𝑢𝑚𝑚\n𝑒𝑒 shown. Note the rapid increase in scaling on 𝜅𝜅𝑚𝑚\n𝐾𝐾𝑖𝑖𝑟𝑟 in the botom \ngraph as 𝑢𝑢𝑚𝑚\n𝑒𝑒 approaches the reversal poten�al for Kir (i.e. 𝑣𝑣𝑚𝑚\n𝐾𝐾𝑖𝑖𝑟𝑟 approaches zero on top graph).    \nThe rela�ve scale factors for SGLT1 and Kir shown in the botom panel of Figure 21 would correspond \nto either regula�on of the protein kine�cs (e.g., by phosphoryla�on) or to the protein expression levels \n(under transcrip�onal control).   \nDISCUSSION \nEpithelial cells lining the small intes�nes take up glucose and sodium (delivered via ingested food) via \nthe SGLT1 transporter, using solute gradients to drive the transmembrane flux (two sodium ions for \neach glucose molecule). Intracellular glucose is immediately converted to ATP via glycolysis (rather \nthan oxida�ve metabolism), which then drives the NKA pump to maintain an intracellular sodium \nconcentra�on of about 10 mM in the face of 140 mM sodium in the capillaries. The 140 mM/10 mM \nsodium difference provides a ‘batery’ that drives many other transmembrane transport processes. \nNote that a ‘batery’ is a source of poten�al energy and only loses energy when current flows (sodium \nions in this case). The ‘sodium batery’ (i.e., the Gibbs free energy available from the sodium gradient) \nis therefore maintained by the Gibbs free energy of ATP hydrolysis – essen�ally the ’top-up batery’ \nneeded to maintain the sodium batery. \nIn this paper, we have explained how the physical principles of conserva�on of mass, conserva�on of \ncharge, and conserva�on of energy are conveniently implemented in models of biological processes \nusing bond graphs. The bond graph approach is well-known in the engineering literature for problems \nthat involve energy transfer between the four available forms of energy storage – chemical, electrical, \nmechanical, and thermal. Physiological processes almost always involve energy transmission, \nexchange, and conversion (including dissipa�on) between all four physical domains, and bond graphs \ntherefore provide an ideal framework for modelling these processes. We show how an appropriate \n Scale factors needed \nto achieve equilibrium  \n 𝒗𝒗𝟏𝟏𝑹𝑹𝑹𝑹 for transporters \n(fmol.mm-2.s-1) \n(no scaling) \nSGLT1 \nNKA \nKir \nSGLT1 \nNKA \nKir \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \n 𝒖𝒖𝟏𝟏𝒆𝒆 mV \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint \n\n27 \n \nchoice of a small number of units (joules, meters, seconds, moles, coulombs, and entropy or Kelvin) \nprovides a common founda�on for describing all physical processes, and we illustrate the applica�on \nof bond graphs to basic biochemical processes where Gibbs free energy is key to understanding and \nmodelling reac�ons.   \nWe then derived biophysically based bond graph models of GLUT2, SGLT1, Kir, and the NKA ATPase \npump and coupled NKA with the transport of glucose via SGLT1 and GLUT2. Glycolysis supplies the ATP \nneeded by NKA for maintaining homeostasis of [𝑁𝑁𝑁𝑁\n+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖. \nGiven the 3:2 ra�o of sodium efflux to potassium entry in NKA, the 2:1 ra�o of coupled sodium to \nglucose entry by SGLT1, and the single potassium efflux by Kir, the cycling rates of SGLT1 and Kir must \nbe 1.5x and 2x that of NKA, respec�vely, in order to maintain homeostasis of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖, and \nhence also of the membrane res�ng poten�al 𝑢𝑢𝑚𝑚𝑒𝑒. This balance of the transporter fluxes can either be \nachieved by adjus�ng the levels of [𝑁𝑁𝑁𝑁+]𝑖𝑖, [𝐾𝐾+]𝑖𝑖, and 𝑢𝑢𝑚𝑚𝑒𝑒, or by altering either the phosphoryla�on \nstatus or expression levels of the transporters, as illustrated in Figure 21. Future work will explore the \nregula�on of these membrane transport proteins.   \nREFERENCES \n1. Tinsley JN, et al. Direct detec�on of a single photon by humans. Nature Communications. 7, 12172, \n2016.  \n2. Paynter HM. Analysis and design of engineering systems, MIT press, 1961. \n3. Oster GF , Perelson AS and Katchalsky A. Network thermodynamics: dynamic modelling of \nbiophysical systems. Quarterly reviews of Biophysics, vol. 6, p. 1–134, 1973.  \n4. Gawthrop PJ and Crampin EJ. Energy-based analysis of biochemical cycles using bond graphs. Proc. \nRoy. Soc. A, vol. 470, p. 20140459, 2014. \n5. Gawthrop PJ, Cursons J and Crampin EJ. Hierarchical bond graph modelling of biochemical \nnetworks. Proc. Roy. Soc. A, vol. 471, p.20150642, 2015. htps://doi.org/10.1098/rspa.2015.0642      \n6. Gawthrop PJ. Bond graph modeling of chemiosmo�c biomolecular energy transduc�on. IEEE \nTrans. NanoBiosci.  16(3):177–188, 2017. htps://doi.org/10.1109/TNB.2017.2674683  PMID: \n28252411 \n7. Hunter PJ, Ai W, Nickerson D. Energy -based bond graph models of glucose transport with SLC \ntransporters. Biophysical J, Vol 124, 2, p316-335, 2025. https://doi.org/10.1016/j.bpj.2024.12.006 \n8. Hunter PJ, de Bono B, Brooks D, Chris�e R, Hussan J, Lin M. The physiome project and digital \ntwins. IEEE Reviews in Biomed Eng, Vol 18, pp300-315, 2024. \nhtps://doi.org/10.1109/RBME.2024.3490455    \n9. Weiwei BJ paper on NKA \n10. Interna�onal System of Units - Wikipedia \n11.  \n12. Atkins P , De Paula J, Keeler J. Physical Chemistry. Oxford University Press UK. 11th Edn, 2018. \n13. Keener JP and Sneyd J, Mathematical physiology, I: Cellular Physiology (2nd ed.). Springer New \nYork, NY , USA, 2009. \n14. Sauro HM. Enzyme kine�cs for systems biology, Future Skill So�ware, 2011.   \n15. Lowe AG and Walmsley AR. The kine�cs of glucose transport in human red blood cells. Biochimica \net Biophysica Acta (BBA)-Biomembranes, 857:146–154, 1986. \n16. W. F. Boron and E. L. Boulpaep, Medical physiology E-book, Elsevier Health Sciences, 2016.  \n17. Post RL, Merrit CR, Kinsolving CR and Albright CD. Membrane adenosine triphosphatase as a \npar�cipant in the ac�ve transport of sodium and potassium in the human erythrocyte. Journal of \nBiological Chemistry, 235(6), 1796–1802, 1960. \n18. Reference to data for NKA \n \n \n \n.CC-BY 4.0 International licenseperpetuity. It is made available under a \npreprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in \nThe copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}