The biophysical basis of enterocyte homeostasis

We present an approach to analysing cell homeostasis using a ‘bond graph’ modelling approach that ensures that the conserva�on laws of physics (conserva�on of mass, charge, and energy, respec�vely) are sa�sfied for the interdependent biochemical, electrical, mechanical, and thermal energy storage mechanisms opera�ng within the cell. We apply the bond graph approach to several cell membrane transport mechanisms and then consider how physics constrains intracellular electrolyte homeostasis for enterocytes (the epithelial absorp�ve cells of the gut). The model includes the electrogenic sodium- potassium ATPase pump (NKA), the glucose transporter (GLUT2), and an inwardly rec�fying potassium channel, all in the basolateral membrane, and the electrogenic sodium-driven glucose transporter (SGLT1) in the apical membrane . Glycolysis converts the imported glucose to ATP to drive NK A. For specified levels of sodium, potassium , and glucose in the blood, the model demonstrates how enterocytes absorb sodium and glucose from the gut and transfer glucose to the blood while maintaining the membrane poten�al and homeostasis of intracellular sodium and potassium. The Gibbs free energy available from the ATP hydrolysis ensures that the cell operates as a ‘sodium batery’ with a high external to internal ra�o of sodium concentra�on in order to provide the energy for many other cellular transport processes. We show that the 3:2 stoichiometry of Na+/K+ exchange in NKA, coupled with 2:1 Na+/glucose cotransport in SGLT1, a 1:2:2 ra�o between glucose consump�on and ATP and water produc�on in glycolysis, and K+ and glucose efflux through Kir and GLUT2, respec�vely, provides a balanced system that maintains homeostasis of intracellular Na+, K+, glucose, ATP and water, and homeostasis of the membrane poten�al, under varying levels of transport of glucose from the gut to the blood . We also show how the flux expressions for SLC transporters, ATPase pumps and ion channels can all be expressed in a consistent and thermodynamically valid way.

Cellular physiology operates at the theore�cal limits set by physical laws – for example, ion channels are sensi�ve to the passage of a single elementary charge, and re�nae can detect one photon [1]. Eukaryo�c cells also exploit every form of energy storage mechanism available – biochemical (e.g., solute concentra�ons and chemical bonds ), electrical (e.g., capaci�ve charge storage in the cell membrane), mechanical (e.g., the elas�c compliance of cellular membranes), and thermal (the heat storage essen�al for maintaining body temperature). The physical processes that maintain intracellular homeostasis are conserva�on of mass, conserva�on of charge, and conserva�on of energy. In this paper we show how the ‘bond graph’ concept, invented over 50 years ago by Henry Paynter at MIT [2] can be used to explain how cellular homeostasis depends on all forms of energy transmission, storage, and dissipa�on (chemical, electrical, mechanical and thermal) via the applica�on of these basic biophysical laws. The applica�on of bond graphs to a variety of biological mechanisms was pioneered by Oster, Perelson and Katchlsky in the 1970s [3] and then later expanded upon by Gawthrop, Crampin, and Pan at the University of Melbourne [4,5,6]. The formula�on presented here, including a focus on appropriate units, model reduc�on strategies, and a new graphical formula�on to simplify the analysis, was pioneered by the present authors [7,8]. The applica�on of bond graphs to analysing cellular homeostasis has not, to our knowledge, been undertaken previously and relies on a new algebraic formula�on of the steady membrane fluxes [9]. We present a new way of expressing the flux for a reac�on that is thermodynamically consistent and applicable across all transmembrane transport mechanisms (exchangers, cotransporters, ATPase .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 2 pumps and ion channels). We demonstrate the importance of inward rec�fica�on in the Kir channel for achieving homeostasis. Other novel contribu�ons in the paper include the demonstra�on of using bond graphs in whole cell modelling to ensure energy consistency, and an energy-conserving model of an enterocyte for analysis of cell homeostasis. UNITS AND THE CONSERVATION LAWS OF PHYSICS Before we introduce bond graph concepts, it is useful to first discuss units, since adop�ng a small number of appropriate units and understanding their role in linking atomic-level quan��es with processes at the macroscale clarifies the applica�on of bond graphs to mul�ple physical domains. The SI base units (m, s, kg, mol, A, K, cd) [10] were largely chosen to reflect the technologies available at the �me (1960) to measure them with sufficient accuracy (such as the standard kg mass held in a vault in Paris). However, since 2019 the magnitudes of all SI units have been defined by declaring that seven defining constants (the speed of light in vacuum 𝑐𝑐 = 2.99792458×108 m.s-1, the Planck constant ℎ = 6.62607015×10−34 J.s, the elementary charge qe = 1.602176634×10−19 C, the hyperfine transi�on frequency of caesium 𝛥𝛥𝜈𝜈𝐶𝐶𝐶𝐶 = 9.192631770×109 s-1, the Boltzmann constant kB = 1.380649x10-23 J.K-1, Avogadro’s constant NA = 6.02214076 x1023 mol-1, and the luminous efficacy 𝐾𝐾𝑐𝑐𝑐𝑐) have certain exact numerical values when expressed in terms of their SI units [10]. The underlying units used to define these constants are m , s, J, mol, C, K, and cd (the seven SI base units are then defined in terms of these). The first three (m, s, J) establish the energized 4D space-�me world in which we reside. The next two (mol, C) reflect atomic or molecular level physical en��es: a mole ( mol) of substance measures the number of atoms or molecules it contains (there are NA atoms in 12 g of carbon-12); a Coulomb (C) measures charge at the macroscale since the charge on an electron q e scaled up by Avogadro’s number to give Faraday’s constant F= NA.qe is the charge in Coulombs per mole (C.mol-1). The Kelvin (K) measures the temperature or thermal energy associated with Brownian mo�on by scaling up the energy per atom or molecule, given by Boltzmann’s constant kB (J.K-1), to the macroscale with NA.kB.T = RT (J.mol-1) where R = NA.kB is the gas constant. The last unit (Candela or cd) is not needed unless photons need to be counted since a photon of frequency 𝜈𝜈 (s-1) has energy ℎ𝜈𝜈 (J). Finally, the magnitude of a rota�on is expressed in terms of the dimensionless unit radians (rad). The SI base unit kg 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. In the following sec�on, we dis�nguish units that measure the amount of a quan�ty 𝑞𝑞, such as m, mol, and C, from units of flux 𝑣𝑣= 𝑐𝑐𝑑𝑑 𝑐𝑐 𝑑𝑑 (the amount per second) and poten�al 𝑢𝑢 (expressed as J per unit quan�ty) driving that flux. We also show that the three conserva�on laws of physics (conserva�on of mass, charge, and energy) can be consolidated into a single conservation of power law, where power is the product 𝑢𝑢. 𝑣𝑣. B ond graphs provide a useful means of formula�ng and visualising a thermodynamically valid biophysical model because they ensure that mass, charge, and energy are each conserved, and they clearly dis�nguish the mechanisms for (i) transmission of power (the product of poten�al 𝑢𝑢 and flux 𝑣𝑣), (ii) energy storage (mechanically in a spring, electrically in a capacitor, or chemically in a solute dissolved in a solvent), (iii) energy dissipation to heat (a mechanical damper, an electrical resistance or a chemical reac�on), and (iv) energy transfer between mechanical, electrical or chemical domains. Most importantly, they dis�nguish between the conserva�on laws of physics and the empirically measured cons�tu�ve rela�ons that characterise par�cular materials. For example, a chemical species 𝑖𝑖 is stored as a solute 𝑞𝑞𝑗𝑗 𝑖𝑖 (mol) in a solu�on at loca�on 𝑗𝑗 with a par�cular solubility that generates a chemical poten�al 𝑢𝑢𝑗𝑗 𝑖𝑖 (J.mol-1). The diffusion of this solute through a dissipa�ve medium from one loca�on to another is quan�fied as a molar flux 𝑣𝑣𝑗𝑗 𝑖𝑖= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑖𝑖 𝑐𝑐𝑑𝑑 (mol.s-1) that depends both on the difference in chemical poten�al between the two loca�ons and on the diffusivity of that medium. The measured values for solubility and diffusivity are two dis�nct material constants, and both are quite separate from the equa�ons represen�ng mass and energy conserva�on. These different material parameters and conserva�on laws are lumped together in Fick’s law of diffusion. .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 3 We use the symbol 𝑞𝑞𝑗𝑗 𝑖𝑖 to denote t he quan�ty in moles (mol) of a chemical species 𝑖𝑖 at loca�on 𝑗𝑗. Similarly, 𝑞𝑞𝑗𝑗 𝑒𝑒 represents electrical charge in Coulombs (C), and 𝑞𝑞𝑗𝑗 𝑚𝑚 represents distance (m), volume (m3) or rota�on (rad). M olar flux is denoted 𝑣𝑣𝑗𝑗 𝑖𝑖= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑖𝑖 𝑐𝑐𝑑𝑑 (mol.s-1) and 𝑢𝑢𝑗𝑗 𝑖𝑖 (J.mol-1) is the chemical poten�al generated by storage of solute 𝑞𝑞𝑗𝑗 𝑖𝑖 (mol). Electrical current is 𝑣𝑣𝑗𝑗 𝑒𝑒= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑒𝑒 𝑐𝑐𝑑𝑑 (C.s-1) and 𝑢𝑢𝑗𝑗 𝑒𝑒 (J.C-1) is the electrical poten�al generated by capaci�ve storage of charge 𝑞𝑞𝑗𝑗 𝑒𝑒 (C). In solid mechanics 𝑣𝑣𝑗𝑗 𝑚𝑚= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑚𝑚 𝑐𝑐𝑑𝑑 (m.s-1) is the velocity and 𝑢𝑢𝑗𝑗 𝑚𝑚 (J.m-1) is the force generated by displacement 𝑞𝑞𝑗𝑗 𝑚𝑚 (m), or 𝑣𝑣𝑗𝑗 𝑚𝑚= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑚𝑚 𝑐𝑐𝑑𝑑 (rad.s-1) is the angular velocity and 𝑢𝑢𝑗𝑗 𝑚𝑚 (J.rad-1) is the torque generated by rota�on 𝑞𝑞𝑗𝑗 𝑚𝑚 (rad). Finally, in fluid mechanics 𝑣𝑣𝑗𝑗 𝑚𝑚= 𝑐𝑐𝑑𝑑𝑗𝑗 𝑚𝑚 𝑐𝑐𝑑𝑑 (m3.s-1) is the fluid flow and 𝑢𝑢𝑗𝑗 𝑚𝑚 (J.m-3) is the pressure (energy density) generated by fluid volume 𝑞𝑞𝑗𝑗 𝑚𝑚 (m3). These units therefore facilitate the expression of energy flux (J.s-1) as the product of a poten�al in Joules per unit quan�ty (mol, C, m, m3 or rad) that is driving the flow of that quan�ty in a way that is common to all physical systems. The product of chemical poten�al 𝑢𝑢𝑗𝑗 𝑖𝑖 (J.mol-1) and flux 𝑣𝑣𝑗𝑗 𝑖𝑖 (mol.s-1), or electrical poten�al 𝑢𝑢𝑗𝑗 𝑒𝑒 (J.C-1) and flux 𝑣𝑣𝑗𝑗 𝑒𝑒 (C.s-1), is always power ( J.s-1). Similarly, the product of mechanical poten�al (force, pressure, or torque) and mechanical flux (velocity, fluid flow, or angular v 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 poten�al (J.entropy-1 or temperature in Kelvin K) is also power. THE BOND GRAPH MODELLING FRAMEWORK We now introduce the basic ideas behind the bond graph framework introduced by Paynter [2]. Lines of power transmission called ‘bonds’ always have an associated flux 𝑣𝑣 and poten�al 𝑢𝑢 (see Figure 1). If these bonds meet, the sum of powers must be zero: ∑ 𝑢𝑢. 𝑣𝑣= 0, to ensure power conserva�on. If they share a common poten�al 𝑢𝑢 (called a ‘ 0:node’), power conserva�on 𝑢𝑢∑ 𝑣𝑣= 0 becomes just ∑ 𝑣𝑣= 0 (for non-zero 𝑢𝑢), which is mass conservation if 𝑣𝑣 is a molar flux or mechanical flow and charge conservation if 𝑣𝑣 is an electrical flux. Alterna�vely, if they share a common flux 𝑣𝑣 (called a ‘1:node’), power conserva�on 𝑣𝑣∑ 𝑢𝑢= 0 becomes just ∑ 𝑢𝑢= 0 (for non-zero 𝑣𝑣), which is energy conservation. For chemical reac�ons these correspond to mass conserva�on and chemical stoichiometric rela�ons, respec�vely. For electrical circuits the y correspond to Kirch hoff’s current law and voltage law, respec�vely. For solid mechanics systems they correspond to kinema�c consistency and force or torque balance, respec�vely. For fluid systems they correspond to conserva�on of volume (conserva�on of mass which holds when density is assumed constant) and pressure balance. (a) (b) (c) (d) Figure 1. Key bond graph concepts: (a) a bond, which transmits energy, carries a flow 𝑣𝑣𝑗𝑗 𝑖𝑖 and a poten�al 𝑢𝑢𝑗𝑗 𝑖𝑖; (b) a 0:node is a bond junc�on where the poten�al is the same for all bonds and therefore the sum of flows is zero (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 therefore the sum of poten�als is zero (conserva�on of energy); (d) a 0:node is usually associated with capaci�ve energy storage as well as flux balance (top) and can be more succinctly expressed by the red-bordered box where the poten�al 𝑢𝑢𝑐𝑐 1 is given by an empirically defined capaci�ve storage rela�onship 𝑢𝑢𝑐𝑐 1 = 𝑓𝑓(𝑞𝑞𝑐𝑐 1). Note that in this figure, the poten�als are coloured red and the kinema�c quan��es in green, just to empathise the difference. By applying power conserva�on to 0:nodes and 1:nodes, bond graph models ensure that the conserva�on laws of physics are always obeyed. By using a common nota�on (quan�ty 𝑞𝑞, flux 𝑣𝑣, and poten�al 𝑢𝑢) with appropriate units (m, s, J, mol, C, K, rad), we can use bond graphs to examine 𝒗𝒗𝒋𝒋 𝒊𝒊 �= 𝑐𝑐𝒒𝒒𝒋𝒋 𝒊𝒊 𝑐𝑐𝑑𝑑� 𝒖𝒖𝒋𝒋 𝒊𝒊 𝒗𝒗𝟏𝟏 𝒊𝒊 𝒗𝒗𝟐𝟐 𝒊𝒊 𝒗𝒗𝟑𝟑 𝒊𝒊 𝒗𝒗𝟒𝟒 𝒊𝒊 𝒗𝒗𝟓𝟓 𝒊𝒊 0: 𝒖𝒖𝒄𝒄 𝒊𝒊 𝒖𝒖𝟏𝟏 𝒊𝒊 𝒖𝒖𝟐𝟐 𝒊𝒊 𝒖𝒖𝟑𝟑 𝒊𝒊 𝒖𝒖𝟒𝟒 𝒊𝒊 𝒖𝒖𝟓𝟓 𝒊𝒊 1: 𝒗𝒗𝒄𝒄 𝒊𝒊 𝑞𝑞𝑐𝑐 𝑖𝑖 0: 𝒖𝒖𝒄𝒄 𝒊𝒊 𝐶𝐶: 𝒒𝒒𝒄𝒄 𝒊𝒊 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 4 physiological processes that involve the exchange of energy between chemical, electrical, mechanical, and thermal forms. Note, however, that the mass or charge balance equa�on ∑ 𝑣𝑣= 0 involves only flux es 𝑣𝑣 (and not poten�als 𝑢𝑢), and that the energy balance equa�on ∑ 𝑢𝑢= 0 involves only poten�als 𝑢𝑢 (and not flux 𝑣𝑣 or quan�ty 𝑞𝑞). To apply these equa�ons to a specific mechanism, they must therefore be supplemented with constitutive relations that couple poten�al 𝑢𝑢 with quan�ty 𝑞𝑞 or flux 𝑣𝑣. Cons�tu�ve rela�ons contain the ‘material’ parameters that characterise par�cular material proper�es, in contrast with the conserva�on laws that express the generic constraints imposed by physics. Cons�tu�ve laws (and their corresponding material constants) are, for example, needed to describe the sta�c storage mechanisms (solute solubility, electrical charge capacitance, mechanical elas�city, thermal capacitance) and the dissipa�ve mechanisms ( chemical reac�on kine�cs, electrical resistance, mechanical viscosity, and thermal conduc�vity). Both types of equa�on are needed but it is extremely important to dis�nguish the generally applicable conserva�on laws , captured with 0:nodes and 1:nodes, from the materially specific cons�tu�ve equa�ons. BOND GRAPH MODELS OF BASIC CELLULAR MECHANISMS Before considering intracellular homeostasis, we illustrate the applica�on of bond graphs to (i) a simple chemical reac�on, (ii) a reac�on with mul�ple reactants and products, (iii) an enzyme -catalysed reac�on, (iv) ATP hydrolysis, (v) the glucose transporter (GLUT2), (vi) the electrogenic sodium-glucose cotransporter (SGLT1), (vii) the sodium-potassium ATPase (NKA) pump, and (viii) an inwardly rec�fying potassium ion channel (Kir). Wherever experimentally valid, we make assump�ons that simplify the models, but always in a way that maintains conserva�on of mass, charge, and energy. These reduced examples provide the mechanis�c bond graph models we subsequently use for analysis of intracellular homeostasis. We also present a simplified graphical representa�on in order to facilitate clarity as we progress to more advanced models. (i) Simple chemical reaction We use the simplest possible chemical reac�on, consis�ng of the reversible interconversion of molar quan��es 𝑞𝑞𝑐𝑐 1 and 𝑞𝑞𝑐𝑐 2 (Figure 2a), to derive the bond graph equa�ons and to discuss the use of bond graph symbols that minimise the complexity of the diagrams. (a) (b) (c) Figure 2. (a) The chemical reac�on between 𝑞𝑞𝑐𝑐 1 and 𝑞𝑞𝑐𝑐 2; (b) the full bond graph representa�on; and (c) the simplified bond graph representa�on in which the red-bordered symbol represents capaci�ve storage combined with 0:node mass balance, and the black -bordered reac�on flux 𝑣𝑣𝑅𝑅𝑅𝑅 symbol includes the reac�on and 1:node energy balance. It is some�mes more useful to replace 𝑞𝑞𝑐𝑐 𝑖𝑖 in the red -bordered symbol with the poten�al 𝑢𝑢𝑐𝑐 𝑖𝑖 calculated from the 𝑞𝑞𝑐𝑐 𝑖𝑖 by a Boltzmann equa�on (see text). In order to formulate the bond graph describing the simple reac�on (Figure 2b), we first iden�fy the molar quan��es 𝑞𝑞𝑐𝑐1 and 𝑞𝑞𝑐𝑐2 stored in solu�on as capaci�ve energy storage elements 𝐶𝐶: 𝑞𝑞𝑐𝑐1 and 𝐶𝐶: 𝑞𝑞𝑐𝑐2, each genera�ng, via a cons�tu�ve law (the Boltzmann equa�on), a chemical poten�al at a 0:node (0: 𝑢𝑢𝑐𝑐1 and 0: 𝑢𝑢𝑐𝑐2) where mass balance is imposed. In Figure 2c the 0:node is combined with the capaci�ve storage element and writen as a single component with a red border (since every molar quan�ty in solu�on generates a poten�al through the Boltzmann equa�on). In Figure 2b the chemical interconversion between species is a dissipa�ve reac�on element interposed between two 1:nodes, enforcing a common reac�on flux with associated forward (𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓) a n d r e v e r s e (𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟) poten�als (the arrowhead iden�fies the posi�ve flux direc�on). As above, we combine the dissipa�ve element and its neighbouring 1:nodes to form a single simplified element in Figure 2c. In so doing, the seven nodes of 𝑞𝑞𝑐𝑐 1 𝑞𝑞𝑐𝑐 2 𝑘𝑘𝑓𝑓 𝑘𝑘𝑟𝑟 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 𝐶𝐶: 𝒒𝒒𝒄𝒄𝟏𝟏 𝐶𝐶: 𝒒𝒒𝒄𝒄𝟐𝟐 1: 𝒗𝒗𝑹𝑹𝑹𝑹 0: 𝒖𝒖𝒄𝒄𝟏𝟏 Reac�on Rx 0: 𝒖𝒖𝒄𝒄𝟐𝟐 1: 𝒗𝒗𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝟏𝟏 𝒗𝒗𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝟐𝟐 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 5 the full bond graph representa�on in Figure 2b are reduced to three in the simplified bond graph (Figure 2c). From here on, we use this simplified representa�on of the bond graph. Since the sum of fluxes entering or leaving the first 0:node must be zero: 𝑣𝑣𝑅𝑅𝑅𝑅+ 𝑐𝑐𝑑𝑑𝑐𝑐1 𝑐𝑐𝑑𝑑= 0 and hence the flux entering the reac�on (𝑣𝑣𝑅𝑅𝑅𝑅) equals the rate of reduc�on of stored solute 𝑞𝑞𝑐𝑐1: 𝑣𝑣𝑅𝑅𝑅𝑅= − 𝑐𝑐𝑑𝑑𝑐𝑐1 𝑐𝑐𝑑𝑑. Similarly, on the downstream side (the second 0:node), 𝑣𝑣𝑅𝑅𝑅𝑅= 𝑐𝑐𝑑𝑑𝑐𝑐2 𝑐𝑐𝑑𝑑, given that the same flux 𝑣𝑣𝑅𝑅𝑅𝑅 passes through the reac�on (as highlighted in Figure 2c). With the bond graph representa�on in place, we can derive the associated system of equa�ons. If the system is opera�ng at constant temperature and pressure , the Gibbs free energy can be used to characterise the poten�als on either side of the reac�on, and for a dilute system the chemical poten�al (the Gibbs free energy per mole) is given by the Boltzmann thermodynamic rela�on [12], 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑢𝑢0 𝑖𝑖+ 𝑅𝑅𝑅𝑅ln 𝑑𝑑𝑐𝑐𝑖𝑖 𝑑𝑑𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡 (J.mol-1) w here 𝑞𝑞𝑐𝑐𝑖𝑖 is the number of moles of chemical species 𝑖𝑖, 𝑞𝑞𝑐𝑐𝑑𝑑𝑡𝑡𝑑𝑑 is the total number of moles of all substances in the mixture in compartment 𝑐𝑐. 𝑢𝑢0 𝑖𝑖 is the (reference) poten�al when 𝑞𝑞𝑐𝑐𝑖𝑖= 𝑞𝑞𝑐𝑐𝑑𝑑𝑡𝑡 𝑑𝑑. M ore compactly, 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑅𝑅𝑅𝑅ln 𝐾𝐾𝑐𝑐𝑖𝑖𝑞𝑞𝑐𝑐𝑖𝑖 (J.mol-1), where 𝐾𝐾𝑐𝑐𝑖𝑖= 1 𝑑𝑑𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑒𝑒𝑢𝑢0 𝑖𝑖𝑅𝑅𝑅𝑅⁄ (mol-1), is the cons�tu�ve law for biochemical energy storage, and 𝐾𝐾𝑐𝑐𝑖𝑖 (mol-1) is a thermodynamic parameter. To simplify the equa�ons, we introduce the non-dimensional quan�ty 𝑞𝑞 �𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖𝑞𝑞𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖. 𝑉𝑉𝑐𝑐. 𝑐𝑐𝑐𝑐𝑖𝑖, (1) where 𝑐𝑐𝑐𝑐𝑖𝑖 is the concentra�on of chemical species 𝑖𝑖, 𝑉𝑉𝑐𝑐 is the volume of 𝑐𝑐. The chemical poten�al is then 𝑢𝑢𝑐𝑐𝑖𝑖= 𝑅𝑅𝑅𝑅ln 𝑞𝑞 �𝑐𝑐𝑖𝑖 (J.mol-1). (2) Energy balance at the 1:nodes is 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓= 𝑢𝑢𝑐𝑐1 and 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟= 𝑢𝑢𝑐𝑐2, where 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 are the forward and reverse poten�als for reac�on 𝑅𝑅𝑅𝑅. In this case, with only one chemical species entering the reac�on, the energy balance is trivial, but i f there were mul�ple reactants and/or mul�ple products, the 1:node energy balance ensures the appropriate chemical stoichiometry for the reac�on. The molar flow for the reac�on 𝑅𝑅𝑅𝑅 can be given by the Marcelin-de Donder formula [4] (a cons�tu�ve rela�on but one that, like the Boltzmann rela�on, can be derived from assump�ons about the distribu�on of par�cle veloci�es): 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟𝑅𝑅𝑅𝑅⁄ � (mol.s-1). where 𝜅𝜅𝑅𝑅𝑅𝑅 (mol.s-1) is the e xperimentally determined reac�on rate constant (a cons�tu�ve parameter). Subs�tu�ng for the poten�als gives .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 6 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅� 𝑒𝑒𝑢𝑢𝑐𝑐1 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑐𝑐2 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅(𝑞𝑞 �𝑐𝑐1 − 𝑞𝑞 �𝑐𝑐2), (mol.s-1) (3) with equilibrium (zero flow) achieved when 𝑞𝑞 �𝑐𝑐1 = 𝑞𝑞 �𝑐𝑐2. The change in Gibbs free energy per mole for the reac�on is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓, which must be nega�ve for the reac�on to proceed (the second law of thermodynamics ), with heat output rate 𝑣𝑣𝑅𝑅𝑅𝑅. � 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟� . Note that heat output rate being the poten�al difference �mes the flow is the same in a biochemical reac�on as it is in an electrical resistance or a mechanical damper but the flux for a biochemical reac�on depends on the difference in the exponentials of 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 (the Marcelin-de Donder formula above) , whereas the flux through an electrical resistor or mechanical damper is dependent only on the poten�al difference across the resistor or damper (the voltage drop or net force, respec�vely). At equilibrium, the change in Gibbs free energy is zero. Note that, from now on, every reac�on is assumed to have associated poten�als 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 driving the forward and reverse reac�ons, respec�vely, and these will not be explicitly labelled on the bond graph diagram. In summary, the poten�als 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 and 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 that drive the reac�on in the forward and reverse direc�ons are obtained (via energy balance rela�ons) from the chemical solute poten�als 𝑢𝑢𝑐𝑐𝑖𝑖 that depend (via Boltzmann’s cons�tu�ve equa�on) on the nondimensional terms 𝑞𝑞 �𝑐𝑐𝑖𝑖 (which include the thermodynamic constants 𝐾𝐾𝑐𝑐𝑖𝑖). The nondimensional solute quan�ty 𝑞𝑞 �𝑐𝑐𝑖𝑖 is expressed in terms of the solute concentra�on 𝑐𝑐𝑐𝑐𝑖𝑖 (in compartment c with volume 𝑉𝑉𝑐𝑐) by 𝑞𝑞 �𝑐𝑐𝑖𝑖= 𝐾𝐾𝑐𝑐𝑖𝑖. 𝑉𝑉𝑐𝑐. 𝑐𝑐𝑐𝑐𝑖𝑖. The change in Gibbs free energy occurring in the reac�on is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓≤ 0 (∆𝐺𝐺𝑅𝑅𝑅𝑅= 0 at equilibrium). The reac�on flux 𝑣𝑣𝑅𝑅𝑅𝑅 is given (via the Marcelin-de Donder formula) by equa�on (3) and the heat output is −∆𝐺𝐺𝑅𝑅𝑅𝑅. 𝑣𝑣𝑅𝑅𝑅𝑅. (ii) A reaction with multiple reactants and products Most reac�ons involve mul�ple reactants and products, o�en with variable stoichiometry. These can all be represented as illustrated in Figure 3 (including mul�ple instances of one chemical species, where, for example, 2 moles of one species combine with 1 mole of another species). (a) (b) Figure 3. (a) A reac�on with mul�ple reactants and products. ∑ . .𝑚𝑚 𝑖𝑖=1 and ∏ . .𝑚𝑚 𝑖𝑖=1 imply a sum and product, respec�vely, over the 𝑚𝑚 reactants, and ∑ . .𝑛𝑛 𝑖𝑖=1 and ∏ . .𝑛𝑛 𝑖𝑖=1 imply a sum and product over the 𝑛𝑛 reac�on products. (b) The corresponding bond graph diagram. The reac�on flux is 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅� ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 − ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 � , (4) and the change in Gibbs free energy is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� ∏ 𝑞𝑞� 𝑐𝑐 2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝑞𝑞� 𝑐𝑐 1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � . In terms of solute concentra�ons (using equa�on 1), ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖.𝑛𝑛 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖.𝑚𝑚 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛�(𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖.𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖.𝑚𝑚 𝑖𝑖=1 � + 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � . This can be writen 𝒗𝒗𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 𝒒𝒒𝒄𝒄𝟏𝟏𝟐𝟐 𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 … 𝒒𝒒𝒄𝒄𝟐𝟐𝟏𝟏 𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 … 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 𝑘𝑘𝑓𝑓 𝑘𝑘𝑟𝑟 � 𝒒𝒒 𝒄𝒄𝟏𝟏𝒊𝒊 𝑚𝑚 𝑖𝑖=1 𝒒𝒒𝒄𝒄 𝟏𝟏𝟏𝟏+ 𝒒𝒒𝒄𝒄 𝟏𝟏𝟐𝟐+ . . +𝒒𝒒𝒄𝒄 𝟏𝟏𝟏𝟏 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓= � 𝒖𝒖 𝒄𝒄 𝟏𝟏 𝒊𝒊 𝑚𝑚 𝑖𝑖=1 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�� 𝑞𝑞 � 𝑐𝑐 1𝑖𝑖 𝑚𝑚 𝑖𝑖=1 � � 𝒒𝒒 𝒄𝒄𝟐𝟐𝒊𝒊 𝑛𝑛 𝑖𝑖=1 𝒒𝒒𝒄𝒄 𝟐𝟐𝟏𝟏+ 𝒒𝒒𝒄𝒄 𝟐𝟐𝟐𝟐+ . . +𝒒𝒒𝒄𝒄 𝟐𝟐𝟐𝟐 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟= � 𝒖𝒖 𝒄𝒄 𝟏𝟏 𝒊𝒊 𝑛𝑛 𝑖𝑖=1 = 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛�� 𝑞𝑞 � 𝑐𝑐 1𝑖𝑖 𝑛𝑛 𝑖𝑖=1 � .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 7 ∆𝐺𝐺𝑅𝑅𝑅𝑅= ∆𝐺𝐺𝑅𝑅𝑅𝑅 𝑜𝑜− 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅 or ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛� 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑞𝑞� − 𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑞𝑞� , (5) where 𝑄𝑄𝑅𝑅𝑅𝑅= ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 (6) is the ra�o of reactant concentra�ons to product concentra�ons, 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑= (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 , (7) is the equilibrium constant (i.e., when 𝑄𝑄𝑅𝑅𝑅𝑅= 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑, then 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒= 1 and ∆𝐺𝐺𝑅𝑅𝑅𝑅= 0), and ∆𝐺𝐺𝑅𝑅𝑅𝑅 𝑡𝑡= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑� i s the free energy needed to move the system from the reference state (all quan��es at the reference concentra�on of 1M and hence 𝑄𝑄𝑅𝑅𝑅𝑅= 1 or 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛𝑄𝑄𝑅𝑅𝑅𝑅= 0) to the equilibrium state where 𝑄𝑄𝑅𝑅𝑅𝑅= 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑 and ∆𝐺𝐺𝑅𝑅𝑅𝑅= 0. N ote that the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑, obtained from the measured ra�o of reactant concentra�ons to product concentra�ons at equilibrium (when 𝑣𝑣𝑅𝑅𝑅𝑅 and ∆𝐺𝐺𝑅𝑅𝑅𝑅 are zero), along with the reac�on rate (see below), is an important parameter characterising a reac�on. It has a value of 1 for a simple diffusive process, since the flow is zero when the reactant and product concentra�ons are equal. When the flow 𝑣𝑣𝑅𝑅𝑅𝑅 is non-zero, the ra�o 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄�𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒 (with a corresponding ∆𝐺𝐺𝑅𝑅𝑅𝑅= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄�𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒) characterizes how far the reac�on is from equilibrium under those condi�ons. Summarising, the change in Gibbs free energy (the free energy of the reaction) is ∆𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓= 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� ∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒.∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � = −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑞𝑞 , (8) a r ela�onship we will make much use of below. The thermodynamic constants for each chemical species have been combined into just one constant 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑, whose value can be established simply by measuring the ra�o 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑= 𝑄𝑄𝑅𝑅𝑅𝑅�𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚= � ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 �� 𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚 . For a reac�on t o proceed in the forward direc�on requires ∆𝐺𝐺𝑅𝑅𝑅𝑅 1 (the second law of thermodynamics). Note that the reac�on flux given by (4) can also be expressed in terms of concentra�ons: 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�(𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 . ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 − (𝑉𝑉𝑐𝑐)𝑛𝑛. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 . ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 � , or 𝑣𝑣𝑅𝑅𝑅𝑅= 𝑘𝑘𝑓𝑓. � ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 � − 𝑘𝑘𝑟𝑟. � ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 � , where 𝑘𝑘𝑓𝑓= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 and 𝑘𝑘𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑛𝑛. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 . This is the “mass ac�on ” equa�on widely used in the biochemical literature but note that the parameters 𝑘𝑘𝑓𝑓 and 𝑘𝑘𝑟𝑟 combine two completely different material quan��es: the thermodynamic Boltzmann coefficient 𝐾𝐾𝑐𝑐𝑖𝑖 associated with capaci�ve storage of solute 𝑖𝑖 (which generates the chemical poten�al 𝑢𝑢𝑐𝑐𝑖𝑖 via the empirical Boltzmann equa�on), and the reaction rate constant 𝜅𝜅𝑅𝑅𝑅𝑅, which is a

property of the reac�on. A preferable way of represen�ng the reac�on is to define 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 (which depends on both the actual reac�on rate and the thermodynamic constants of the reactants) and then 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅. �∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 − 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑. ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 �, (9) .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 8 highligh�ng the fact that only two parameters are needed to describe the reac�on, one that can be obtained from steady state concentra�ons, and one that specifies how quickly the reac�on occurs. We illustrate these concepts with a simple sodium-chloride reac�on in Figure 4. (a) (b) Figure 4. (a) A sodium-chloride reac�on. (b) The bond graph model in simplified form. The 0:node mass balance equa�ons are: 𝑐𝑐𝑑𝑑𝑐𝑐𝑁𝑁𝑁𝑁+ 𝑐𝑐𝑑𝑑= −𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅, 𝑐𝑐𝑑𝑑𝑐𝑐𝐶𝐶𝐶𝐶− 𝑐𝑐𝑑𝑑= −𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅, 𝑐𝑐𝑑𝑑𝑐𝑐𝑁𝑁𝑁𝑁𝐶𝐶 𝐶𝐶 𝑐𝑐𝑑𝑑= 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 The reac�on flux (inser�ng the Boltzmann equa�ons for each chemical species) is 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅�𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓/𝑅𝑅𝑅𝑅− 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟/𝑅𝑅𝑅𝑅� = 𝜅𝜅𝑅𝑅𝑅𝑅� 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+ . 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎+ . 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒− . 𝑞𝑞𝑐𝑐𝐶𝐶 𝑒𝑒− − 𝐾𝐾𝑐𝑐𝑁𝑁𝑎𝑎𝐶𝐶 𝑒𝑒. 𝑞𝑞𝑐𝑐𝑁𝑁𝑎𝑎 𝐶𝐶𝑒𝑒� . (10) W ith 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎+ = 𝑉𝑉𝑐𝑐. [𝑁𝑁𝑁𝑁+]𝑐𝑐, 𝑞𝑞𝑐𝑐𝐶𝐶 𝑒𝑒− = 𝑉𝑉𝑐𝑐. [𝐶𝐶 𝑅𝑅−]𝑐𝑐, and 𝑞𝑞𝑐𝑐𝑁𝑁 𝑎𝑎𝐶𝐶𝑒𝑒= 𝑉𝑉𝑐𝑐. [𝑁𝑁𝑁𝑁𝐶𝐶 𝑅𝑅 ]𝑐𝑐 for compartment c of volume 𝑉𝑉𝑐𝑐, equa�on (10) becomes 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅�[𝑁𝑁𝑁𝑁+]𝑐𝑐. [𝐶𝐶 𝑅𝑅−]𝑐𝑐− 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑. [𝑁𝑁𝑁𝑁𝐶𝐶 𝑅𝑅 ]𝑐𝑐� (11) wh ere 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+ . 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒− . 𝑉𝑉𝑐𝑐2 and 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑= 𝐾𝐾𝑐𝑐𝑁𝑁𝑁𝑁𝐶𝐶 𝐶𝐶 𝐾𝐾𝑐𝑐𝑁𝑁𝑁𝑁+.𝐾𝐾𝑐𝑐𝐶𝐶𝐶𝐶−.𝑉𝑉𝑐𝑐 = � [𝑁𝑁 𝑎𝑎+]𝑐𝑐.[𝐶𝐶 𝑒𝑒−]𝑐𝑐 [𝑁𝑁𝑎𝑎𝐶𝐶 𝑒𝑒]𝑐𝑐 �� 𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚 . T his equa�on, now using concentra�ons, is the most useful form of the reac�on flux, but it is important to remember that the ‘reac�on rate’ constant 𝜅𝜅̂𝑅𝑅𝑅𝑅 is now a combina�on of the kine�c parameter 𝜅𝜅𝑅𝑅𝑅𝑅 and the thermodynamic parameters 𝐾𝐾𝑐𝑐𝑁𝑁 𝑎𝑎+ , 𝐾𝐾𝑐𝑐𝐶𝐶 𝑒𝑒− . In prac�ce 𝜅𝜅̂𝑅𝑅𝑅𝑅 is determined experimentally by measuring the non-equilibrium flux, and the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑 is determined by measuring the concentra�ons of Na+, Cl- and NaCl at equilibrium. ( iii) Enzyme-catalysed reactions An enzyme-catalysed reac�on is illustrated in Figure 5. (a) (b) (c) Figure 5. (a) An enzyme -catalysed reac�on in which a solute 𝑞𝑞𝑐𝑐 1 combines (reac�on 1) with the enzyme 𝑞𝑞𝑐𝑐 3 to form an intermediate 𝑞𝑞𝑐𝑐 4 which then dissociates (reac�on 2) to form the solute product 𝑞𝑞𝑐𝑐 2 and recycled enzyme 𝑞𝑞𝑐𝑐 3. (b) The enzyme-catalysed reac�on shown as a bond graph with the four red-bordered symbols represen�ng both mass balance and storage, and the two reac�ons (with 1:nodes incorporated as in Figure 2) represen�ng the points of energy balance (chemical stoichiometry). (b) A further simplifica�on of the bond graph model in which the steady-state flux 𝑣𝑣𝑐𝑐 𝑅𝑅𝑅𝑅 is the final net flux, computed from (19), for the whole reac�on. The flux balance equa�ons, defined at the four 0:nodes, are (i) 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐1 = −𝑣𝑣𝑅𝑅𝑅𝑅 1 ; (ii) 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐2 = 𝑣𝑣𝑅𝑅𝑅𝑅 2 ; (iii) 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐3 = −𝑣𝑣𝑅𝑅𝑅𝑅 1 + 𝑣𝑣𝑅𝑅𝑅𝑅 2 ; (iv) 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐4 = 𝑣𝑣𝑅𝑅𝑅𝑅 1 − 𝑣𝑣𝑅𝑅𝑅𝑅 2 . N ote that, since 𝑐𝑐 𝑐𝑐 𝑑𝑑(𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4) = 0, the total amount of enzyme is constant. i.e., 𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4 = 𝐸𝐸0, (12) 𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵𝑵𝑵𝑵𝑵 𝒗𝒗𝒄𝒄𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵+ 𝒒𝒒𝒄𝒄𝑵𝑵𝑵𝑵− 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 𝑁𝑁𝑁𝑁+ + 𝐶𝐶𝑅𝑅− 𝑁𝑁𝑁𝑁𝐶𝐶𝑅𝑅 𝐴𝐴𝑓𝑓 𝐴𝐴𝑟𝑟 𝑞𝑞𝑐𝑐 1 + 𝑞𝑞𝑐𝑐 3 𝑞𝑞𝑐𝑐 4 𝑞𝑞𝑐𝑐 2 + 𝑞𝑞𝑐𝑐 3 𝑘𝑘1 𝑓𝑓 𝑘𝑘1 𝑟𝑟 𝑘𝑘2 𝑓𝑓 𝑘𝑘2 𝑟𝑟 𝒒𝒒𝒄𝒄𝟑𝟑 𝒒𝒒𝒄𝒄𝟏𝟏 𝒒𝒒𝒄𝒄𝟒𝟒 𝒒𝒒𝒄𝒄𝟐𝟐 𝒗𝒗𝑹𝑹𝑹𝑹 𝟏𝟏 𝒗𝒗𝑹𝑹𝑹𝑹 𝟐𝟐 𝒒𝒒𝒄𝒄𝟏𝟏 𝒒𝒒𝒄𝒄𝟐𝟐 𝒗𝒗𝒄𝒄𝑹𝑹𝑹𝑹 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 9 where 𝐸𝐸0 is the ini�al quan�ty of enzyme. The energy balance equa�ons, defined at the four 1:nodes, are (i) 𝑢𝑢𝑅𝑅𝑅𝑅1 𝑓𝑓 = 𝑢𝑢𝑐𝑐1 + 𝑢𝑢𝑐𝑐3; (ii) 𝑢𝑢𝑅𝑅𝑅𝑅1 𝑟𝑟 = 𝑢𝑢𝑐𝑐4; (iii) 𝑢𝑢𝑅𝑅𝑅𝑅2 𝑓𝑓 = 𝑢𝑢𝑐𝑐4; (iv) 𝑢𝑢𝑅𝑅𝑅𝑅2 𝑟𝑟 = 𝑢𝑢𝑐𝑐2 + 𝑢𝑢𝑐𝑐3. Using these poten�als, the two reac�ons are 𝑣𝑣𝑅𝑅𝑅𝑅 1 = 𝜅𝜅𝑅𝑅𝑅𝑅 1 �𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅1 𝑓𝑓 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅1 𝑟𝑟 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅 1 (𝑞𝑞 �𝑐𝑐1𝑞𝑞 �𝑐𝑐3 − 𝑞𝑞 �𝑐𝑐4), (13) 𝑣𝑣𝑅𝑅𝑅𝑅 2 = 𝜅𝜅𝑅𝑅𝑅𝑅 2 �𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅2 𝑓𝑓 𝑅𝑅𝑅𝑅⁄ − 𝑒𝑒𝑢𝑢𝑅𝑅𝑅𝑅2 𝑟𝑟 𝑅𝑅𝑅𝑅⁄ � = 𝜅𝜅𝑅𝑅𝑅𝑅 2 (𝑞𝑞 �𝑐𝑐4 − 𝑞𝑞 �𝑐𝑐2𝑞𝑞 �𝑐𝑐3). (14) We can also express these fluxes in “mass ac�on” form as 𝑣𝑣𝑅𝑅𝑅𝑅 1 = 𝑘𝑘1 𝑓𝑓𝑐𝑐𝑐𝑐1𝑐𝑐𝑐𝑐3 − 𝑘𝑘1 𝑟𝑟𝑐𝑐𝑐𝑐4 and 𝑣𝑣𝑅𝑅𝑅𝑅 2 = 𝑘𝑘2 𝑓𝑓𝑐𝑐𝑐𝑐4 − 𝑘𝑘2 𝑟𝑟𝑐𝑐𝑐𝑐2𝑐𝑐𝑐𝑐3, where 𝑘𝑘1 𝑓𝑓 = 𝜅𝜅𝑅𝑅𝑅𝑅 1 . 𝐾𝐾𝑐𝑐1𝐾𝐾𝑐𝑐3. (𝑉𝑉𝑐𝑐)2 (m6.mol-1.s-1), 𝑘𝑘1 𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅 2 . 𝐾𝐾𝑐𝑐4. 𝑉𝑉𝑐𝑐 (m3.s-1), and 𝑘𝑘2 𝑓𝑓 = 𝜅𝜅𝑅𝑅𝑅𝑅 2 . 𝐾𝐾𝑐𝑐4. 𝑉𝑉𝑐𝑐 (m3.s-1), 𝑘𝑘2 𝑟𝑟= 𝜅𝜅𝑅𝑅𝑅𝑅 2 . 𝐾𝐾𝑐𝑐2𝐾𝐾𝑐𝑐3. (𝑉𝑉𝑐𝑐)2 (m6.mol-1.s-1). Note the inconsistent units that result from combining reac�on rate constants (mol.s-1) with thermodynamic constants (mol-1). The Briggs-Haldane assumption [14] is that the amount of enzyme (𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4) is much less than the amounts of substrate (𝑞𝑞𝑐𝑐1 and 𝑞𝑞𝑐𝑐2), so that the unbound enzyme 𝑞𝑞𝑐𝑐3 and the complex 𝑞𝑞𝑐𝑐4 quickly reach a steady state ( 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐3 = 0 and 𝑐𝑐 𝑐𝑐 𝑑𝑑𝑞𝑞𝑐𝑐4 = 0 ). The second and third equa�ons in (4) then give 𝑣𝑣𝑅𝑅𝑅𝑅 1 = 𝑣𝑣𝑅𝑅𝑅𝑅 2 = 𝑣𝑣𝑅𝑅𝑅𝑅 and hence, from (13) and (14), 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅 1 (𝑞𝑞 �𝑐𝑐1𝑞𝑞 �𝑐𝑐3 − 𝑞𝑞 �𝑐𝑐4) = 𝜅𝜅𝑅𝑅𝑅𝑅 2 (𝑞𝑞 �𝑐𝑐4 − 𝑞𝑞 �𝑐𝑐2𝑞𝑞 �𝑐𝑐3) (15) or 𝑞𝑞 �𝑐𝑐4 = � 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� 𝑑𝑑� 𝑐𝑐3 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 . (16) Also, from (12), 𝑞𝑞𝑐𝑐3 + 𝑞𝑞𝑐𝑐4 = 𝐸𝐸0 or 𝑑𝑑� 𝑐𝑐3 𝐾𝐾𝑐𝑐3 + 𝑑𝑑� 𝑐𝑐4 𝐾𝐾𝑐𝑐4 = 𝐸𝐸0 or 𝑞𝑞 �𝑐𝑐3 = 𝐾𝐾𝑐𝑐3𝐸𝐸0 − 𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4 𝑞𝑞 �𝑐𝑐4. (17) Subs�tu�ng (17) into (16), 𝑞𝑞 �𝑐𝑐4 = � 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 �𝐾𝐾𝑐𝑐3. 𝐸𝐸0 − 𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4 𝑞𝑞 �𝑐𝑐4� , or 𝑞𝑞 �𝑐𝑐4 = 𝐾𝐾𝑐𝑐3. 𝐸𝐸0. 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� (18) and hence, from (17), 𝑞𝑞 �𝑐𝑐3 = 𝐾𝐾𝑐𝑐3𝐸𝐸0 � 1 − 𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4 . 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� � = 𝐾𝐾𝑐𝑐3𝐸𝐸0 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� . (19) Subs�tu�ng (18) and (19) back into the first equa�on in (15), 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝐾𝐾𝑐𝑐3𝐸𝐸0 � 𝑑𝑑� 𝑐𝑐1� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� − 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� � , which simplifies to .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 10 𝑣𝑣𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝜅𝜅𝑅𝑅𝑅𝑅 2 𝐾𝐾𝑐𝑐3𝐸𝐸0. 𝑑𝑑� 𝑐𝑐1−𝑑𝑑� 𝑐𝑐2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 +𝐾𝐾𝑐𝑐3 𝐾𝐾𝑐𝑐4.� 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝑑𝑑� 𝑐𝑐1+𝜅𝜅𝑅𝑅𝑅𝑅 2 𝑑𝑑� 𝑐𝑐2� . A convenient way of expressing this enzyme-catalysed reac�on flux is 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. 𝑑𝑑� 𝑐𝑐1−𝑑𝑑� 𝑐𝑐2 1+ 𝑒𝑒� 𝑐𝑐 1 𝑘𝑘𝑅𝑅𝑅𝑅 1 + 𝑒𝑒� 𝑐𝑐2 𝑘𝑘𝑅𝑅𝑅𝑅 2 , (20) where the three parameters 𝜅𝜅𝑅𝑅𝑅𝑅= 𝐾𝐾𝑐𝑐3𝐸𝐸0. 𝜅𝜅𝑅𝑅𝑅𝑅 1 𝜅𝜅𝑅𝑅𝑅𝑅 2 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 (mol.s-1); 𝑘𝑘𝑅𝑅𝑅𝑅 1 = 𝐾𝐾𝑐𝑐4 𝐾𝐾𝑐𝑐3 . 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 𝜅𝜅𝑅𝑅𝑅𝑅 1 ; 𝑘𝑘𝑅𝑅𝑅𝑅 2 = 𝐾𝐾𝑐𝑐4 𝐾𝐾𝑐𝑐3 . 𝜅𝜅𝑅𝑅𝑅𝑅 1 +𝜅𝜅𝑅𝑅𝑅𝑅 2 𝜅𝜅𝑅𝑅𝑅𝑅 2 (21) characterise the maximum flux, the value of 𝑞𝑞 �𝑐𝑐1 that corresponds to half that maximum when 𝑞𝑞 �𝑐𝑐2 = 0, and the value of 𝑞𝑞 �𝑐𝑐2 that corresponds to half that maximum when 𝑞𝑞 �𝑐𝑐1 = 0. The flux term (20) shows satura�on at high values of 𝑞𝑞 �𝑐𝑐1 or 𝑞𝑞 �𝑐𝑐2. Note that we use 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 in (20) and in the bond graph diagrams in Figure 6, with 𝑅𝑅𝑅𝑅 as a superscript to indicate that this is the steady-state flux for the whole enzyme- catalysed reac�on. This conven�on greatly simplifies subsequent mul�-enzyme reac�ons (e.g., for membrane transporters). If 𝑞𝑞 �𝑐𝑐2 is assumed to be much smaller than 𝑞𝑞 �𝑐𝑐1 and is set to zero, equa�on (20) becomes the familiar Michaelis-Menten equa�on [14], but the reac�on is then irreversible. The validity of the Briggs- Haldane assump�on can be tested by comparing the solu�on of the full bond graph model with the reduced model given by (20). Figure 6b shows the bond graph for an enzyme-catalysed reac�on which has 𝑚𝑚 reactants (assumed to bind simultaneously) and 𝑛𝑛 products (assumed to unbind simultaneously). (a) (b) Figure 6. (a) The bond graph diagram for an enzyme-catalysed reac�on under the Briggs-Haldane assump�on where the flux term 𝑣𝑣𝑐𝑐 𝑅𝑅𝑅𝑅 (given by equa�on (19)) now incorporates the ac�on of the enzyme. (b) An enzyme- catalysed reac�on with 𝑚𝑚 reactants and 𝑛𝑛 products. Following the same process described above for the deriva�on of the full bond graph model, and then the corresponding reduced Briggs-Haldane model, 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. ∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 − ∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 1+ ∏ 𝑒𝑒� 𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 𝑘𝑘𝑅𝑅𝑅𝑅 1 + ∏ 𝑒𝑒� 𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 𝑘𝑘𝑅𝑅𝑅𝑅 2 , (22) where the parameters 𝜅𝜅𝑅𝑅𝑅𝑅, 𝑘𝑘𝑅𝑅𝑅𝑅 1 and 𝑘𝑘𝑅𝑅𝑅𝑅 2 , which characterise the catalysed reac�on are given by (22). The zero-flux state (𝑣𝑣𝑅𝑅𝑅𝑅= 0) is reached when ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 = ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 . The free energy of the reac�on is 𝛥𝛥𝐺𝐺𝑅𝑅𝑅𝑅= 𝑢𝑢𝑐𝑐𝑟𝑟− 𝑢𝑢𝑐𝑐 𝑓𝑓= 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛� ∏ 𝑞𝑞 �𝑐𝑐2𝑖𝑖𝑚𝑚 𝑖𝑖=1 � − 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛� ∏ 𝑞𝑞 �𝑐𝑐1𝑖𝑖𝑛𝑛 𝑖𝑖=1 � = 𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛� ∏ 𝑑𝑑� 𝑐𝑐2𝑖𝑖𝑚𝑚 𝑖𝑖=1 ∏ 𝑑𝑑� 𝑐𝑐1𝑖𝑖𝑛𝑛 𝑖𝑖=1 � = −𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒, where 𝑄𝑄𝑅𝑅𝑅𝑅= ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 , and 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑= (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 = 𝑄𝑄𝑅𝑅𝑅𝑅|𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚. (23) 𝒗𝒗𝒄𝒄 𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 𝒒𝒒𝒄𝒄𝟏𝟏𝟐𝟐 𝒒𝒒𝒄𝒄𝟏𝟏𝟏𝟏 … 𝒒𝒒𝒄𝒄𝟐𝟐𝟏𝟏 𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 𝒒𝒒𝒄𝒄𝟐𝟐𝟐𝟐 … 𝒒𝒒𝒄𝒄𝟏𝟏 𝒗𝒗𝒄𝒄 𝑹𝑹𝑹𝑹 𝒒𝒒𝒄𝒄𝟐𝟐 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 11 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑 is the equilibrium constant for the reac�on, which replaces the 𝑚𝑚+ 𝑛𝑛 thermodynamic coefficients. Th e zero-flux state for the reac�on corresponds to 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒= 1 and 𝛥𝛥𝐺𝐺𝑅𝑅𝑅𝑅= 0. Using (1) and (21), the flux expression (20) can be rewriten in terms of concentra�ons: 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅= 𝜅𝜅̂𝑅𝑅𝑅𝑅. ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 −𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒.∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 1+ ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 𝑘𝑘� 𝑅𝑅𝑅𝑅 1 + ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 𝑘𝑘� 𝑅𝑅𝑅𝑅 2 , (24) where 𝜅𝜅̂𝑅𝑅𝑅𝑅= 𝜅𝜅𝑅𝑅𝑅𝑅. (𝑉𝑉𝑐𝑐)𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 , 𝑘𝑘� 𝑅𝑅𝑅𝑅 1 = 𝑘𝑘𝑅𝑅𝑅𝑅 1 (𝑉𝑉𝑐𝑐)𝑚𝑚.∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 , 𝑘𝑘� 𝑅𝑅𝑅𝑅 2 = 𝑘𝑘𝑅𝑅𝑅𝑅 2 (𝑉𝑉𝑐𝑐)𝑛𝑛.∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 . (25) Th ere are now four constants available ( 𝜅𝜅̂𝑅𝑅𝑅𝑅, 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑, 𝑘𝑘� 𝑅𝑅𝑅𝑅 1 , 𝑘𝑘� 𝑅𝑅𝑅𝑅 2 ) to fit the general enzyme catalysed reac�on (24) to experimental data on the rela�onship between flux and the reactant and product concentra�ons. Fo r a very fast reac�on (when the reac�on rate constant 𝜅𝜅𝑅𝑅𝑅𝑅→ ∞), a finite (non-zero) value for the steady enzyme turn-over flux 𝑣𝑣𝑐𝑐𝑅𝑅𝑅𝑅 implies that the reac�on is close to equilibrium ( 𝑄𝑄𝑅𝑅𝑅𝑅 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒→ 1). (iv) ATP hydrolysis ATP hydrolysis supplies the energy (usually via the ‘sodium batery’) to drive nearly all physiological processes [13]. The hydrolysis of ATP to ADP, Pi (inorganic phosphate), and H+ is represented chemically in Figure 7a and as an enzyme-catalysed bond graph process in Figure 7b. (a) (b) F igure 7. The bond graph representa�on of ATP hydrolysis. At pH 7, inorganic phosphate (𝑃𝑃𝑖𝑖) is a mixture of the protonated forms of phosphate: 𝐻𝐻2𝑃𝑃𝑃𝑃4 − and 𝐻𝐻𝑃𝑃𝑃𝑃4 2−. Here we assume 𝐻𝐻2𝑃𝑃𝑃𝑃4 − only. This is an enzyme-catalysed reac�on with an equilibrium constant, from (25), defined as 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑= (𝑉𝑉𝑐𝑐)3−2. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖3 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖2 𝑖𝑖=1 = 𝑉𝑉𝑐𝑐. 𝐾𝐾𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴.𝐾𝐾𝑐𝑐 𝑃𝑃𝑖𝑖.𝐾𝐾𝑐𝑐𝐻𝐻+ 𝐾𝐾𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴.𝐾𝐾𝑐𝑐 𝐻𝐻2𝑃𝑃, but determined from the experimentally measured concentra�ons at equilibrium: 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑= 𝑄𝑄𝐴𝐴𝑅𝑅𝐴𝐴|𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒= � 𝑅𝑅𝑒𝑒 𝑎𝑎𝑐𝑐𝑑𝑑𝑎𝑎𝑛𝑛𝑑𝑑 𝑐𝑐𝑡𝑡𝑛𝑛𝑐𝑐𝑒𝑒 𝑛𝑛𝑑𝑑𝑟𝑟𝑎𝑎𝑑𝑑𝑖𝑖𝑡𝑡𝑛𝑛𝐶𝐶 𝐴𝐴𝑟𝑟𝑡𝑡𝑐𝑐𝑢𝑢𝑐𝑐𝑑𝑑 𝑐𝑐𝑡𝑡𝑛𝑛𝑐𝑐𝑒𝑒 𝑛𝑛𝑑𝑑𝑟𝑟𝑎𝑎𝑑𝑑𝑖𝑖𝑡𝑡𝑛𝑛𝐶𝐶�� 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒 = � ∏ 𝑐𝑐𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 ∏ 𝑐𝑐𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 �� 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒 = � [𝐴𝐴𝑅𝑅𝑃𝑃].[𝐻𝐻2𝑂𝑂] [𝐴𝐴𝐴𝐴𝑃𝑃].[𝑃𝑃𝑖𝑖].[𝐻𝐻+]�� 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒 , wh ere the concentra�ons [. . ] are defined in units of mol.m-3, which is the same as millimol.L-1 or mM. The concentra�on of water is 55.5 mM (the density 1 kg.L-1 divided by 0.018 kg.mol-1), but since the amount of water absorbed by the reac�on is small compared to the quan�ty of water in which the reac�on is taking place, a modified equilibrium constant is measured using 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑= � [𝐴𝐴𝑅𝑅𝑃𝑃] [𝐴𝐴𝐴𝐴𝑃𝑃].[𝑃𝑃𝑖𝑖].[𝐻𝐻+]�� 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒 . Wh en measured at 298.15 K (25 oC) at pH 7 ([𝐻𝐻+] = 10-4 mM and 1 mM [𝑀𝑀𝑀𝑀2+] [18]), the equilibrium constant has the value 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑≈ 0.5x10-5 (assuming mM units) or, with 𝑅𝑅𝑅𝑅 = 2.5 kJ.mol-1, 𝐴𝐴𝑅𝑅𝑃𝑃4− + 𝐻𝐻2𝑃𝑃 ⇄ 𝐴𝐴𝐴𝐴𝑃𝑃4− + 𝐻𝐻2𝑃𝑃𝑃𝑃4 − + 𝐻𝐻+ where ATP is C₁₀H₁₆N₅O₁₃P₃ and ADP is C₁₀H₁₅N₅O₁₀P₂ 𝑘𝑘𝑓𝑓 𝑘𝑘𝑟𝑟 𝒗𝒗𝒄𝒄 𝑹𝑹𝑹𝑹 𝑢𝑢𝑅𝑅𝑅𝑅 𝑓𝑓 𝑢𝑢𝑅𝑅𝑅𝑅 𝑟𝑟 𝑞𝑞𝑐𝑐𝐴𝐴𝑅𝑅 𝐴𝐴 𝑞𝑞𝑐𝑐 𝐻𝐻2𝑂𝑂 𝑞𝑞𝑐𝑐 𝐴𝐴𝑖𝑖 𝑞𝑞𝑐𝑐𝐴𝐴𝐴𝐴𝐴𝐴 𝑞𝑞𝑐𝑐𝐻𝐻+ .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 12 ∆𝐺𝐺𝐴𝐴𝑅𝑅𝐴𝐴 𝑡𝑡 = 𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑� = −2.5 𝑅𝑅𝑛𝑛(2 . 105)= -30.5 kJ.mol-1. As we show below for the ATPase-driven NKA pump, in many situa�ons ATP hydrolysis operates at ∆𝐺𝐺s that are well in excess of this equilibrium. (v) Glucose transporter (GLUT2) Us ing a similar analysis (see [7] for details), the bond graph model of facilitated transport of glucose (‘Glc’) through the SLC2A2-encoded membrane transporter GLUT2 (see Figure 8), under the Briggs- Haldane assump�on of steady state enzyme cycling and fast binding and unbinding, is given in terms of the non-dimensional intracellular and extracellular glucose amounts (𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒𝑐𝑐, 𝑞𝑞 �𝑡𝑡 𝐺𝐺𝑒𝑒𝑐𝑐) by 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 = 𝜅𝜅𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2. 𝑞𝑞� 𝑖𝑖 𝐺𝐺 𝑅𝑅𝑐𝑐−𝑞𝑞� 𝑜𝑜 𝐺𝐺 𝑅𝑅𝑐𝑐 1+ 𝑞𝑞� 𝑖𝑖 𝐺𝐺 𝑅𝑅𝑐𝑐 𝜅𝜅𝑚𝑚1 + 𝑞𝑞� 𝑜𝑜 𝐺𝐺 𝑅𝑅𝑐𝑐 𝜅𝜅𝑚𝑚2 + 𝑞𝑞� 𝑖𝑖 𝐺𝐺 𝑅𝑅𝑐𝑐.𝑞𝑞� 𝑜𝑜 𝐺𝐺 𝑅𝑅𝑐𝑐 𝜅𝜅𝑚𝑚3 (26) a nd, in terms of the intracellular and extracellular glucose concentra�ons ([𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡), 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 = 𝜅𝜅̂𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2. [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝐴𝐴2 𝑒𝑒𝑒𝑒.[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑡𝑡 1+ [𝐺𝐺 𝐶𝐶𝑐𝑐]𝑖𝑖 𝑘𝑘𝑖𝑖 𝐺𝐺𝐺𝐺 𝐺𝐺𝐴𝐴2 + [𝐺𝐺 𝐶𝐶𝑐𝑐]𝑡𝑡 𝑘𝑘𝑡𝑡𝐺𝐺𝐺𝐺 𝐺𝐺𝐴𝐴2 + [𝐺𝐺 𝐶𝐶𝑐𝑐]𝑖𝑖.[𝐺𝐺 𝐶𝐶𝑐𝑐]𝑡𝑡 𝑘𝑘𝑖𝑖 𝑡𝑡 𝐺𝐺𝐺𝐺 𝐺𝐺𝐴𝐴2 . fmol.mm-2.s-1 (27) F or the analysis of cellular processes, we use units fmol (10-15 mol) for molar quan�ty, mm2 (10-6 m2) for 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 transmembrane flux, it is convenient to express as a molar flux per unit area of membrane , and therefore to use units of fmol.mm-2.s-1. S ince the solute concentra�ons are in mM (mol.m-3), the reac�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 cell data, is therefore a combina�on of reac�on rate for the individual membrane protein transporter and a scaling constant that takes into account the area l protein density. When we later place this transporter into the whole cell model, we introduce an area -to-volume ra�o for the cell that links concentra�ons and per-unit-area fluxes to changes in molar amounts. (a) (b) (c) (d) F igure 8. The GLUT2 transporter. (a) Facilitated transport of glucose across the cell membrane, shown by the double line ( green background is inside the cell, blue outside); (b) Full bond graph model with reac�ons for 1. glucose binding, 2. transloca�on of the ligand-bound enzyme from inward-facing to outward-facing, 3. release of glucose, and 4. return of the enzyme to inward facing; (c) Reduced model where the flux 𝑣𝑣𝑚𝑚 𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 through the membrane is an algebraic func�on (equa�on 24) of the quan��es of external and internal glucose; (d) the simpler representa�on in which the 1:nodes are included in the reac�on. Note that, from (7), the equilibrium constant is 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 𝑒𝑒 𝑑𝑑 = (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 = 𝐾𝐾𝑐𝑐 𝐺𝐺 𝐶𝐶𝑐𝑐𝑡𝑡 𝐾𝐾𝑐𝑐 𝐺𝐺 𝐶𝐶𝑐𝑐𝑖𝑖= 1, since the Boltzmann thermodynamic constant 𝐾𝐾𝑐𝑐 is the same for internal and external glucose. T he parameters 𝑘𝑘� 𝑚𝑚 1 , 𝑘𝑘� 𝑚𝑚 2 and 𝑘𝑘� 𝑚𝑚 3 scale the rela�ve contribu�ons of extracellular and intracellular glucose and their product, respec�vely (see Table 1 for the value of these parameters fited to data from Lowe and Walmsley [15]). 𝑘𝑘� 𝑚𝑚1 is the value of [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 that reduces the flux to 50% of its maximum value when [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡 is zero, and 𝑘𝑘� 𝑚𝑚2 is the value of [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡 that reduces the flux to 50% of its maximum value when [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 is zero. Note that 𝑘𝑘� 𝑚𝑚 2 is an order of magnitude larger than 𝑘𝑘� 𝑚𝑚 1 and that 𝑘𝑘� 𝑚𝑚 3 is a further .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 13 order of magnitude larger than 𝑘𝑘� 𝑚𝑚 2 . These parameters are expressed in terms of the thermodynamic and kine�c constants for the transporter as Parameters Value Unit Parameters Value Unit 𝐾𝐾𝐺𝐺𝑒𝑒𝑐𝑐 ? fmol-1 𝑘𝑘𝑚𝑚1 1.474 dimensionless 𝑘𝑘𝑖𝑖 𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.1094 mM 𝑘𝑘𝑚𝑚2 21.67 dimensionless 𝑘𝑘𝑡𝑡𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 1.609 mM 𝑘𝑘𝑚𝑚3 235.1 dimensionless 𝑘𝑘𝑖𝑖𝑡𝑡 𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 1.296 (mM)2 𝜅𝜅𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.003284 fmol.s-1 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 𝑒𝑒𝑑𝑑 1 dimensionless 𝜅𝜅̂𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 0.0442 pL.s-1 Table 1. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚 𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 through the GLUT2 transporter. Values on the le� are obtained by fi�ng (24) to data from Lowe and Walmsley [19]. Values on the right have been adjusted to reflect use of concentra�ons in (25). Figure 9 illustrates the flux through the steady-state (SS) bond graph model as a func�on of intracellular concentra�on, using equa�on (27) with the parameters given in Table 1. Figure 9. The GLUT2 bond graph model fited to flux data from [19]. Reproduced from [7] with permission). (vi) Electrogenic sodium-glucose cotransporter (SGLT1) Glucose transport through the SLC5A1-encoded membrane co -transporter SGLT1 (see Figure 10), under the same assump�ons used above (Briggs-Haldane and fast binding/unbinding) together with the assump�on of no slippage (see [7]), is given in terms of non-dimensional quan��es by 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝜅𝜅6𝐾𝐾1𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑�� 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐− �𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒 𝑐𝑐. 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� /𝐵𝐵 , (28) wh ere 𝐵𝐵= �𝑒𝑒2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ � 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ � 2 � 𝐾𝐾1 𝐾𝐾2 + 𝐾𝐾1 𝐾𝐾3 . 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐� � ��𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒 𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 � + � 𝐾𝐾1 𝐾𝐾6 + �𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 � 𝐾𝐾1 𝐾𝐾5 + 𝐾𝐾1 𝐾𝐾4 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒 𝑐𝑐� � �� 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑡𝑡𝐺𝐺𝑒𝑒 𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� and 𝑧𝑧1 = 𝑧𝑧2 = 0.5. D ividing numerator and denominator in (2 8) by �𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒𝑐𝑐, using concentra�ons rather than dimensionless molar quan��es, and iden�fying the external compartment as ‘gut lumen’, (28) becomes 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝜅𝜅𝑚𝑚𝑆𝑆 𝐺𝐺𝐺𝐺𝐴𝐴1 Φ𝑆𝑆𝐺𝐺𝐺𝐺𝐴𝐴1 � � [𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 2 . [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡 [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 𝑒𝑒 𝑑𝑑 . 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� fmol.mm-2.s-1 (29) .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 14 where 𝑄𝑄𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 𝑒𝑒 𝑑𝑑 = (𝑉𝑉𝑐𝑐)𝑛𝑛−𝑚𝑚. ∏ 𝐾𝐾𝑐𝑐2𝑖𝑖𝑛𝑛 𝑖𝑖=1 ∏ 𝐾𝐾𝑐𝑐1𝑖𝑖𝑚𝑚 𝑖𝑖=1 = � 𝐾𝐾𝑐𝑐 𝑁𝑁𝑁𝑁𝑡𝑡+ 𝐾𝐾𝑐𝑐 𝑁𝑁𝑁𝑁𝑖𝑖 + � 2 . 𝐾𝐾𝑐𝑐 𝐺𝐺 𝐶𝐶𝑐𝑐𝑡𝑡 𝐾𝐾𝑐𝑐 𝐺𝐺 𝐶𝐶𝑐𝑐𝑖𝑖= 1. (30) and the denominator term is Φ𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = �� 𝑒𝑒2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ � 𝑞𝑞� 𝑀𝑀𝑢𝑢𝑔𝑔 𝑁𝑁𝑁𝑁+ � 2 � 𝐾𝐾1 𝐾𝐾2 + 𝐾𝐾1 𝐾𝐾3 . 𝑞𝑞� 𝑀𝑀𝑢𝑢𝑔𝑔 𝐺𝐺𝑅𝑅𝑐𝑐� � �� 𝑞𝑞� 𝑖𝑖 𝑁𝑁𝑁𝑁+ � 2 . 𝑞𝑞� 𝑖𝑖 𝐺𝐺 𝑅𝑅𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 � + � 𝐾𝐾1 𝐾𝐾6 + �𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 � 𝐾𝐾1 𝐾𝐾5 + 𝐾𝐾1 𝐾𝐾4 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒𝑐𝑐�� ��𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝐺𝐺𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�� /�𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺𝑒𝑒𝑐𝑐 (31) Note that 𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝜅𝜅6𝐾𝐾1𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑 (fmol.mm-2.s-1) is a rate parameter also reflec�ng the overall expression level for the transporter and its reac�on rate. (a) (b) (c) (d) F igure 10. The sodium-glucose transporter SGLT1. (a) Symbolic representa�on for the transport of two moles of sodium for every one mole of glucose; (b) Full bond graph model; and (c) Reduced model where the flux 𝑣𝑣𝑚𝑚 𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1 is an algebraic func�on (27) of the quan��es of sodium and glucose either side of the membrane (see [7]); (d) the alterna�ve representa�on in which the 1:nodes are included in the reac�on. Parameters Value Unit Solute & enzyme states 𝐾𝐾𝑁𝑁𝑎𝑎+ 0. 0159 fmol-1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑 𝑁𝑁 𝑎𝑎+ , 𝑞𝑞𝑖𝑖 𝑁𝑁 𝑎𝑎+ 𝐾𝐾𝐺𝐺𝑒𝑒𝑐𝑐 ? f mol-1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑 𝐺𝐺 𝑒𝑒𝑐𝑐, 𝑞𝑞𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐 𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 3. 5e6 fmol.mm-2. s-1 𝐾𝐾1 3.66 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡 𝐾𝐾2 330 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+ 𝐾𝐾3 321 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐 𝐾𝐾4 321 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐 𝐾𝐾5 905 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+ 𝐾𝐾6 0.314 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖 𝜅𝜅3 0.156 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐→ 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖2𝑁𝑁 𝑎𝑎+𝐺𝐺𝑒𝑒𝑐𝑐 𝜅𝜅6 9.562 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖→ 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡 Table 2. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚 𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1 through the SGLT1 transporter. Values obtained by fi�ng (29) and (31) to data from Lowe and Walmsley [19] (see [7]}. The enzyme states corresponding to the thermodynamic parameters labelled 𝐾𝐾1.. 𝐾𝐾6, and the reac�ons corresponding to the kine�c parameters 𝜅𝜅3 and 𝜅𝜅6, are shown in the right-hand column. Figure 11 shows the flux through the steady-state (SS) bond graph model using the parameters given in Table 2. 𝒗𝒗𝟏𝟏𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝟏𝟏 𝒒𝒒𝒐𝒐 𝑺𝑺𝑵𝑵𝒄𝒄 𝒒𝒒𝒊𝒊 𝑺𝑺𝑵𝑵𝒄𝒄 𝒒𝒒𝒐𝒐 𝑵𝑵𝑵𝑵+ 𝒒𝒒𝒊𝒊 𝑵𝑵𝑵𝑵+ 2 2 𝑢𝑢𝑚𝑚 𝑒𝑒 2F .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 15 Figure 11. Shows SGLT1 model fited to data (normalised flux). Reproduced from [7] with permission. From (29) the reversal poten�al for SGLT1 (i.e. when 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1= 0) is given by 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 = 𝑅𝑅𝑅𝑅 2𝐹𝐹𝑅𝑅𝑛𝑛�� [𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 2 . [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡 [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 � (32) S ubs�tu�ng typical values (following a carbohydrate-rich meal) of [𝑁𝑁𝑁𝑁+]𝑔𝑔𝑢𝑢𝑑𝑑= 140 mM, [𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑔𝑔𝑢𝑢𝑑𝑑= 40 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖= 1 mM (with 𝑅𝑅𝑅𝑅/𝐹𝐹= 25 mV) gives 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1= 102 mV. Given a typical epithelial cell membrane poten�al of about –50 mV, the SGLT1 cotransporter operates well away from the reversal poten�al. Note that at 𝑢𝑢𝑚𝑚𝑒𝑒= -50 mV, and with these sodium and glucose concentra�ons, the ra�o of forward to reverse flux is � [𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 2 . [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡 [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 / 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅= 1.9x105, which means that 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 is not impeded by the rising [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖 (which is also rapidly transferred to the blood by GLUT2). ( vii) Sodium-potassium ATPase pump (NKA) The Na+-K+ ATPase NKA (a P-type ATPase) pumps 3 sodium ions out of the cell and 2 potassium ions in against their concentra�on gradients, using the energy supplied by the enzyme-catalysed hydrolysis of ATP [16]. The pump is electrogenic since there is a net movement of 1 charge for every ATP hydrolysed. By maintaining the sodium gradient across cell membranes, the NKA pump maintains the energy gradient used by many SLC transporters and is responsible for nearly 30% of the body’s res�ng metabolism. We have developed a 6-state model of NKA following the Post-Albers scheme [ 17], as illustrated in Figure 12 (see [9]). The 6 states of the NKA protein, each followed by a reac�on (defining the state transi�on), are (in clockwise order star�ng at the top right): (1) The first is the unbound inward-facing state 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖 is followed by the membrane reac�on 𝑅𝑅𝑅𝑅𝑚𝑚1 in which three intracellular sodium ions and one ATP molecule bind. Note that sodium binding is coopera�ve and that A TP binds to a cataly�c site on the enzyme’s cytoplasmic domain with high affinity in the presence of bound sodium. The voltage sensi�vity is minimal at this stage, as the binding of the three charged 𝑁𝑁𝑁𝑁+s occurs on the cytoplasmic side without crossing the membrane’s electric field. (2 ) The second (inward facing) state 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.3𝑁𝑁 𝑎𝑎+.𝐴𝐴 𝑅𝑅𝐴𝐴 is followed by the 𝑅𝑅𝑅𝑅𝑚𝑚2 reac�on in which bound ATP is hydrolysed and ADP is ejected. The other products of ATP hydrolysis, inorganic phosphate 𝑃𝑃𝑖𝑖 and a proton 𝐻𝐻+, remain bound. The phosphoryla�on associated with ATP hydrolysis is thought to occlude the sodium ions within the membrane. .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 16 (3) The third (inward facing) state is therefore 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.3𝑁𝑁 𝑎𝑎+.𝐴𝐴𝑖𝑖.𝐻𝐻+ and is followed by reac�on 𝑅𝑅𝑅𝑅𝑚𝑚3 in which the protein transi�ons to outward facing and the three sodium ions are ejected (since the affinity for 𝑁𝑁𝑁𝑁+ is greatly reduced by the inward-facing to outward-facing transi�on). This transi�on is highly voltage dependent as the three charged sodium ions are being transported outward against the membrane’s electric field gradient (and against a large 𝑁𝑁𝑁𝑁+ concentra�on gradient). A more nega�ve highly charged membrane (i.e., more nega�ve membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒) facilitates the reac�on. (4 ) The fourth (now outward facing) state 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+ is followed by reac�on 𝑅𝑅𝑅𝑅𝑚𝑚4 , which binds two extracellular potassium ions. Note that cardiac glycosides, such as ouabain, bind to the 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+ state, stabilizing it and blocking 𝐾𝐾+ binding. (5 ) The fi�h (outward facing) state is therefore 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+ . Potassium binding triggers dephosphoryla�on (𝑅𝑅𝑅𝑅𝑚𝑚5 ) which occludes the bound 𝐾𝐾+ ions and ejects the remaining hydrolysis products (𝑃𝑃𝑖𝑖, 𝐻𝐻+). There is minimal voltage sensi�vity. (6 ) The sixth and final (outward facing) state is 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.2𝐾𝐾+ . In the final reac�on 𝑅𝑅𝑅𝑅𝑚𝑚6 , the two potassium ions are ejected into the intracellular space leaving the unbound protein ready to begin the next cycle. This 𝐾𝐾 + release involves moving 2 posi�ve charges inward, facilitated by the membrane’s electric field gradient. (a) (b) (c) F igure 12. The sodium-potassium ATPase pump. (a) Symbolic representa�on for the transport of three moles of sodium outwards and two moles of potassium inwards for every mole of ATP hydrolysed; (b) The 6-state bond graph model; and (c) the reduced model for the flux 𝑣𝑣𝑚𝑚 𝑁𝑁𝐾𝐾𝐴𝐴. The expression for NKA flux [9] is 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 𝜅𝜅6𝐾𝐾6𝑑𝑑𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶−𝐴𝐴𝐴𝐴�� 𝑑𝑑� 𝑖𝑖 𝑁𝑁𝑁𝑁+ 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+ � 3 . 𝑑𝑑� 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑� 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴.𝑑𝑑� 𝑖𝑖 𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖 𝐻𝐻+ − � 𝑑𝑑� 𝑖𝑖 𝐾𝐾+ 𝑑𝑑� 𝑡𝑡𝐾𝐾+ � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴� (33) w here the flux rate depends on the non-dimensional quan��es 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, 𝐴𝐴 g iven in Table 4. See [9] for the meaning of each variable and parameter, and the deriva�on of these expressions. Model parameters are given in Table 3. S ubs�tu�ng for the concentra�ons using (1), equa�on (33) becomes 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴 Φ𝑁𝑁𝐾𝐾𝐴𝐴. �� [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . [𝐴𝐴 𝑅𝑅𝐴𝐴] [𝐴𝐴𝐴𝐴 𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+] − 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑. � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� fmol.mm-2.s-1 or 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴 Φ𝑁𝑁𝐾𝐾𝐴𝐴. �� [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴− 𝑄𝑄�𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑. � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� fmol.mm-2.s-1 (34) .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 17 where 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴= [𝐴𝐴 𝑅𝑅𝐴𝐴] [𝐴𝐴𝐴𝐴 𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+] is the ra�o of reactants to products for the intracellular components of ATP hydrolysis (assuming [𝐻𝐻2𝑃𝑃] c onstant) and 𝑄𝑄�𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑= 𝑄𝑄�𝐴𝐴𝑅𝑅𝐴𝐴 𝑒𝑒 𝑑𝑑 is the corresponding ra�o of these components that atains equilibrium for ATP hydrolysis (i.e., the equilibrium constant for ATP h 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 𝑒𝑒 𝑑𝑑 because the Boltzmann terms 𝐾𝐾𝑁𝑁𝑎𝑎+ and 𝐾𝐾𝐾𝐾+ cancel out in the concentra�on ra�o terms. The dimensionless ra�o 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒 provides the ac�ve driving force for the reac�on. 𝜅𝜅𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 𝜅𝜅6𝐾𝐾6𝑞𝑞𝑑𝑑𝑡𝑡𝑑𝑑 (fmol.mm-2.s-1) is a reac�on rate constant, and Φ𝑁𝑁𝐾𝐾𝐴𝐴= 𝐶𝐶 −𝐴𝐴𝐴𝐴 is a nondimensional scaling factor for flux that is dependent on the concentra�ons of all solutes involved in the reac�on (and is given by the expressions for 𝐴𝐴, 𝐵𝐵, 𝐶𝐶, & 𝐴𝐴 in Table 4). Parameters Value Unit Solute & enzyme states 𝐾𝐾𝐾𝐾+ 0.0239 fmol-1 𝑞𝑞𝑖𝑖 𝐾𝐾+ , 𝑞𝑞𝑡𝑡𝐾𝐾+ 𝐾𝐾𝑁𝑁𝑎𝑎+ 0. 0159 fmol-1 𝑞𝑞𝑖𝑖 𝑁𝑁𝑎𝑎+ , 𝑞𝑞𝑡𝑡𝑁𝑁𝑎𝑎+ 𝐾𝐾𝐴𝐴𝑅𝑅 𝐴𝐴 45. 06 fmol-1 𝑞𝑞𝑖𝑖 𝐴𝐴𝑅𝑅 𝐴𝐴 𝐾𝐾𝐴𝐴𝐴𝐴𝐴𝐴 0.0015 fmol-1 𝑞𝑞𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴 𝐾𝐾𝐴𝐴𝑖𝑖 0. 8673 fmol-1 𝑞𝑞𝑖𝑖 𝐴𝐴𝑖𝑖 𝐾𝐾𝐻𝐻+ 0.8673 fmol-1 𝑞𝑞𝑖𝑖 𝐻𝐻 𝐾𝐾1 0.01673 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖 𝐾𝐾2 4.477e3 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+ 𝐾𝐾3 46.64e3 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁𝑎𝑎+) 𝐾𝐾4 720.6 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+ 𝐾𝐾5 1.607 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+ 𝐾𝐾6 30.25e3 fmol-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡. (2𝐾𝐾+) 𝜅𝜅1 2.839e3 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖 → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+ 𝜅𝜅2 51.10 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑅𝑅 𝐴𝐴.3𝑁𝑁 𝑎𝑎+ → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁 𝑎𝑎+) 𝜅𝜅3 469.3 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖.𝐴𝐴𝑖𝑖.𝐻𝐻+.(3𝑁𝑁𝑎𝑎+) → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+ 𝜅𝜅4 619.1e3 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+ → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+ 𝜅𝜅5 35.87 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡.𝐴𝐴𝑖𝑖.𝐻𝐻+.2𝐾𝐾+ → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡. (2𝐾𝐾+) 𝜅𝜅6 868.4 fmol.s-1 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡. (2𝐾𝐾+) → 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖 𝑧𝑧1 0.9450 dim 𝑧𝑧2 𝑧𝑧1 − 1 dim Table 3. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴 through the NKA transporter. Values obtained by fi�ng (34) to data from [18]. The enzyme states corresponding to the thermodynamic parameters labelled 𝐾𝐾1.. 𝐾𝐾6, and the reac�ons corresponding to the kine�c parameters 𝜅𝜅1.. 𝜅𝜅6, are shown in the right-hand column. Figure 13 illustrates the dependence of flux (34) on 𝑢𝑢𝑚𝑚𝑒𝑒 for varying values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 between 10 mM and 20 mM, with fixed values of [𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 145 mM, [𝐾𝐾+]𝑡𝑡 = 4.5 mM and a Gibbs free energy change of ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� = -61 kJ.mol-1. .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 18 Figure 13. NKA transporter flux 𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴 ploted against 𝑢𝑢𝑚𝑚 𝑒𝑒 for three values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 (10 mM, 15 mM and 20 mM). Other concentra�ons are [𝑁𝑁𝑁𝑁+]𝑡𝑡 = 140 mM, [𝐾𝐾+]𝑖𝑖 = 140 mM, [𝐾𝐾+]𝑡𝑡= 5 mM . T he Gibbs free energy made available by ATP hydrolysis to drive the pump is ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛� 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� = -61 kJ.mol-1. The reversal poten�als corresponding to the intracellular sodium concentra�ons are (-238 mV, -271 mV, and -293 mV, respec�vely). At equilibrium (𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴= 0), the ra�o � 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� � 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒 needed to maintain transmembrane ion gradients is � 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� � 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖 𝑒𝑒 = � [𝑁𝑁 𝑎𝑎+]𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 3 . � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 . (35) Applying − 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛 .. t o both sides gives ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒𝑒𝑒 𝐹𝐹 = −3. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎+ 2. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾+ 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸 wh ere 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎= 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛 [𝑁𝑁 𝑎𝑎+]0 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 = 58 mV is the reversal (Nernst) poten�al for sodium (using 𝑅𝑅𝑅𝑅 𝐹𝐹= 26 mV), 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾= 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛 [𝐾𝐾+]0 [𝐾𝐾+]𝑖𝑖 = -87 mV is the reversal (Nernst) poten�al for potassium, ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒= -55 kJ.mol-1 is the Gibbs free energy driving the equilibrium reac�on, and 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴 is the reversal poten�al for the NKA pump obtained from � [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒= � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 or 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= − 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛� 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒. � [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . � [𝐾𝐾+]𝑡𝑡 [𝐾𝐾+]𝑖𝑖 � 2 � (36) Subs�tu�ng the standard concentra�ons into (36), 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= − 55𝑒𝑒3 0.96𝑒𝑒5+ 3x0.058 + 2x0.087 = −0.225 J.C-1 or 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= -225 mV. N ote that a Gibbs free energy of –55 kJ.mol-1 corresponds to 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒= 𝑒𝑒55/𝑅𝑅𝑅𝑅= 𝑒𝑒55/2.5 ≈ 3.6𝑒𝑒9, showing just how enormous the driving force is for maintaining intracellular sodium. Equa�on (34) also illustrates the par��oning of Gibbs free energy: ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑= −𝑅𝑅𝑅𝑅. 𝑅𝑅𝑛𝑛�� [𝑁𝑁 𝑎𝑎+]𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 3 . � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� = −3𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� [𝑁𝑁 𝑎𝑎+]𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � − 2𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛� [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � + 𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 or ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑= ∆𝐺𝐺𝑁𝑁𝑎𝑎+ ∆𝐺𝐺𝐾𝐾+ ∆𝐺𝐺𝑒𝑒, (37) w here ∆𝐺𝐺𝑁𝑁𝑎𝑎= −3𝐹𝐹. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁 𝑎𝑎 = -16.7 kJ.mol-1, ∆𝐺𝐺𝐾𝐾= −2𝐹𝐹. 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾 = -16.7 kJ.mol-1 and ∆𝐺𝐺𝑒𝑒 = -21.6 kJ.mol-1 (which add to give ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑 = -55 kJ.mol-1 as above). Note that 39% (21.6/55) of the Gibbs free energy [𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 𝑚𝑚𝑀𝑀 [𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 𝑚𝑚𝑀𝑀 [𝑁𝑁𝑁𝑁+]𝑖𝑖= 20 𝑚𝑚𝑀𝑀 𝑣𝑣𝑚𝑚 𝑁𝑁𝐾𝐾𝐴𝐴 (fmol.mm-2.s-1) .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 19 supplied by ATP hydrolysis at equilibrium goes into storing electrical energy in the cell membrane (by hyperpolarising it). F or this calcula�on of ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴 under equilibrium condi�ons, there is no heat loss because the NKA flux is zero; however, when the membrane voltage has the typically observed value of –50 mV, rather than being at the reversal poten�al of –225 mV, the free energy stored in the membrane capacitor drops to ∆𝐺𝐺𝑒𝑒= 𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 = -4.8 kJ.mol-1, with the remaining 16.8 kJ.mol-1 dissipated as heat. The ‘efficiency’ of the pump is now 16.7+16.7+4.8 55 ≈70%, but of course the heat is helping to maintain body temperature, so it is more accurate to say that of the 55 kJ.mol-1 available from ATP hydrolysis, 61% goes to maintaining the electrolyte gradients, 9% goes to electrical energy storage in the membrane capacitance and 30% goes to thermal energy storage in the heat capacity of the cell and its surroundings. (viii) The inwardly rectifying potassium ion channel (Kir) W e next consider the inwardly rec�fying ion channel Kir (e.g., the Kir7.1 subtype). Highly selective, low-conductance single -ion channels can be considered as three - or four-state transporters (depending on the number of necessary intermediate steps). As illustrated in Figure 14, we use a three-state model with a bound state 𝑞𝑞𝑚𝑚 𝐸𝐸𝑖𝑖𝐾𝐾+ (𝑞𝑞1), a meta-stable intermediate state 𝑞𝑞𝑚𝑚 𝐸𝐸𝑡𝑡𝐾𝐾+ (𝑞𝑞1), and an empty state 𝑞𝑞𝑚𝑚𝐸𝐸 (𝑞𝑞3). It is tempting to assume that the electrogenic step is associated with the transition between states 1 and 2 since this is where the ion transport occurs. But the bond graph model, while ensuring that physical conservation laws are satisfied for the protein as a whole, is limited in its ability to capture the details of the electrostatic field operating over the membrane and its interaction with the potassium ion moving through that field. The best we can do is to allow charge movement to be associated with all three reactions by including electrogenic effects via the unknown 𝑧𝑧𝑖𝑖 (for 𝑖𝑖=1..6) terms as shown in Figure 14. The parameter estimation process described in Supplementary Material is used to determine these parameters (subject to the requirement that for each ion that crosses the membrane, the overall charge displacement for the membrane is -1). Figure 14. A 3-state bond graph model of the Kir inward rec�fier with provision for charge movement by all reac�ons (represented by the purple transforming factors 𝑧𝑧𝑖𝑖𝐹𝐹). The binding and unbinding reac�ons (𝑅𝑅𝑅𝑅𝑚𝑚 1 and 𝑅𝑅𝑅𝑅𝑚𝑚 3 ) are assumed to be much faster than the protein transi�on from inward facing to outward facing (𝑅𝑅𝑅𝑅𝑚𝑚 2 ). From Figure 14, the flux equa�ons associated with the three reac�ons are 𝑣𝑣1 = 𝜅𝜅1 �𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾1𝑞𝑞1. 𝑒𝑒 𝑧𝑧2𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� (38) 𝑣𝑣2 = 𝜅𝜅2 �𝐾𝐾1𝑞𝑞1. 𝑒𝑒 𝑧𝑧3𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾2𝑞𝑞2. 𝑒𝑒 𝑧𝑧4𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� (39) 𝑣𝑣3 = 𝜅𝜅3 �𝐾𝐾2𝑞𝑞2. 𝑒𝑒 𝑧𝑧5𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+ . 𝑒𝑒 𝑧𝑧6𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴� (40) Assuming rapid binding and unbinding of K+ (𝜅𝜅1, 𝜅𝜅3 → ∞), for finite values of 𝑣𝑣1 and 𝑣𝑣3, the bracketed terms on the right in (38) and (39) must be zero, and hence .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 20 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾1𝑞𝑞1. 𝑒𝑒 𝑧𝑧2𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 = 0 (41) 𝐾𝐾2𝑞𝑞2. 𝑒𝑒 𝑧𝑧5𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+ . 𝑒𝑒 𝑧𝑧6𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 = 0 (42) If we also assume (Briggs-Haldane) that the enzyme states are at steady-state, 𝑣𝑣1 = 𝑣𝑣2 = 𝑣𝑣3 = 𝑣𝑣 and with subs�tu�ons for 𝐾𝐾1𝑞𝑞1 and 𝐾𝐾2𝑞𝑞2 from (41) and (42), respec�vely, (39) gives 𝑣𝑣= 𝑣𝑣2 = 𝜅𝜅2 �𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝐾𝐾3𝑞𝑞3𝑞𝑞 �𝑡𝑡K+ . 𝑒𝑒 (𝑧𝑧6−𝑧𝑧5+𝑧𝑧4)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 � or 𝑞𝑞3 = 𝑟𝑟 𝐾𝐾3𝜅𝜅2 / �𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝑞𝑞 �𝑡𝑡K+ . 𝑒𝑒 (𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 � (43) F or a fixed (and specified) number of ion channels, 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑, and using (41) and (42), 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑= 𝑞𝑞1 + 𝑞𝑞2 + 𝑞𝑞3 = 𝑞𝑞3. � 𝐾𝐾3 𝐾𝐾1 𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 + 𝐾𝐾3 𝐾𝐾2 𝑞𝑞 �𝑡𝑡K+ . 𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 + 1� or, with (43), 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑= 𝑟𝑟 𝐾𝐾3𝜅𝜅2 ⋅ � 𝐾𝐾3 𝐾𝐾1 𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 + 𝐾𝐾3 𝐾𝐾2 𝑞𝑞 �𝑡𝑡 K+ . 𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 + 1� / �𝑞𝑞 �𝑖𝑖 K+ . 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝑞𝑞 �𝑡𝑡 K+ . 𝑒𝑒 (𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 � Rearranging for 𝑣𝑣, 𝑣𝑣= 𝑞𝑞𝑑𝑑𝑡𝑡 𝑑𝑑𝐾𝐾3𝜅𝜅2. 𝑑𝑑� 𝑖𝑖 K+ .𝑒𝑒 (𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝑑𝑑� 𝑡𝑡K+ .𝑒𝑒 (𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 1 + 𝐾𝐾3 𝐾𝐾1 .𝑑𝑑� 𝑖𝑖 K+.𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 + 𝐾𝐾3 𝐾𝐾2 .𝑑𝑑� 𝑡𝑡K+.𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 . (44) F inally, with 𝑞𝑞 �𝑖𝑖 K+ = 𝐾𝐾𝑖𝑖 𝐾𝐾+ . 𝑉𝑉𝑖𝑖. [𝐾𝐾+]𝑖𝑖, 𝑞𝑞 �𝑡𝑡K+ = 𝐾𝐾𝑡𝑡𝐾𝐾+ . 𝑉𝑉𝑡𝑡. [𝐾𝐾+]𝑡𝑡, and defining 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖 𝑟𝑟= 𝑘𝑘𝑚𝑚1 𝐾𝐾𝑖𝑖 𝐾𝐾+ .𝑉𝑉𝑖𝑖 , 𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖 𝑟𝑟= 𝑘𝑘𝑚𝑚2 𝐾𝐾𝑡𝑡𝐾𝐾+.𝑉𝑉𝑡𝑡 , 𝑄𝑄𝐾𝐾𝑖𝑖 𝑟𝑟 𝑒𝑒 𝑑𝑑= 𝐾𝐾𝑡𝑡𝐾𝐾+ .𝑉𝑉𝑡𝑡 𝐾𝐾𝑖𝑖 𝐾𝐾+ .𝑉𝑉𝑖𝑖 and 𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟= 𝐾𝐾𝐾𝐾+ 𝑉𝑉𝑖𝑖. 𝜅𝜅𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟, (44) becomes 𝑣𝑣= 𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟. [𝐾𝐾+]𝑖𝑖.𝑒𝑒 (𝑧𝑧1−𝑧𝑧2+𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 − 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟 𝑒𝑒𝑒𝑒.[𝐾𝐾+]𝑡𝑡.𝑒𝑒 (𝑧𝑧4−𝑧𝑧5+𝑧𝑧6)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 1 + �𝐾𝐾+�𝑖𝑖 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖𝑟𝑟.𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴 + �𝐾𝐾+�𝑡𝑡 𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟.𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 fmol.mm-2.s-1 or, with −𝑧𝑧1 + 𝑧𝑧2 − 𝑧𝑧3 + 𝑧𝑧4 − 𝑧𝑧5 + 𝑧𝑧6 = −1 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟= 𝜅𝜅� 𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 Φ . �[𝐾𝐾+]𝑖𝑖− 𝑄𝑄𝐾𝐾𝑖𝑖 𝑟𝑟 𝑒𝑒 𝑑𝑑. [𝐾𝐾+]𝑡𝑡. 𝑒𝑒 −𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴�, fmol.mm-2.s-1 (45) where Φ = 𝑒𝑒 (−𝑧𝑧1+𝑧𝑧2−𝑧𝑧3)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 . �1 + [𝐾𝐾+]𝑖𝑖 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 + [𝐾𝐾+]𝑡𝑡 𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑔𝑔𝑚𝑚 𝑒𝑒 𝑅𝑅 𝐴𝐴 �. (46) Th e electrical current is 𝐼𝐼𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟= 𝐹𝐹𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟 (fC.mm-2.s-1) and the reversal or ‘Nernst’ poten�al is 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟= − 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛� �𝐾𝐾+�𝑖𝑖 �𝐾𝐾+�𝑜𝑜 � = −26.5𝑅𝑅𝑛𝑛� 140 4.5 � = -91 mV (47) Figure 15 shows the current-voltage rela�on for the Kir channel. The fluxes out of the intracellular solu�on and into the extracellular solu�on are 𝑐𝑐𝑑𝑑𝑖𝑖 K+ 𝑐𝑐𝑑𝑑= −𝑣𝑣1 = −𝑣𝑣, and 𝑐𝑐𝑑𝑑𝑖𝑖 K+ 𝑐𝑐𝑑𝑑= 𝑣𝑣3 = 𝑣𝑣. .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 21 Parameters Value Unit 𝐾𝐾𝐾𝐾+ 0.0239 fmol-1 𝜅𝜅̂𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 0.0442 pL.s-1 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟 𝑒𝑒𝑑𝑑 1 dimensionless 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖𝑟𝑟 1 mM 𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟 10 mM Table 1. Parameter values used in the expression for flux 𝑣𝑣𝑚𝑚 𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅2 through the GLUT2 transporter. Values on the le� are obtained by fi�ng (24) to data from Lowe and Walmsley [19]. Values on the right have been adjusted to reflect use of concentra�ons in (25). Figure 15. The current-voltage rela�on predicted for Kir by (45) and (46). Note the reversal poten �al at -91 mV and the inward rec�fica�on above this poten�al. Current-voltage curves are shown for 3 values of the parameter 𝑘𝑘𝑡𝑡 𝐾𝐾𝑖𝑖𝑟𝑟. The curve steepens as this parameter is increased from 1 mM to 10 mM and then hardly changes for higher values. Changing the parameter 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖𝑟𝑟 is equivalent to changes in 𝜅𝜅̂𝑚𝑚 𝐾𝐾𝑖𝑖𝑟𝑟 so this parameter is set to 1 mM. A COMMON FRAMEWORK FOR ALL TRANSPORTERS AND ATPase PUMPS To provide a single unifying framework for all enzyme -catalysed reac�ons (including all membrane transporters, pumps and ion channels), we will express all steady-state reac�on fluxes in the form: 𝑣𝑣𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚 𝑒𝑒) = 𝜅𝜅𝑅𝑅𝑅𝑅 Φ𝑅𝑅𝑅𝑅(𝒒𝒒,𝑢𝑢𝑚𝑚 𝑒𝑒) �𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) − 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑. 𝑃𝑃𝑅𝑅𝑅𝑅([𝒒𝒒]). 𝑒𝑒𝑧𝑧𝐹𝐹𝑢𝑢𝑚𝑚 𝑒𝑒/𝑅𝑅𝑅𝑅� (49) where 𝒒𝒒 is a state vector that includes all solutes (with concentra�ons [𝒒𝒒]) par�cipa�ng in the reac�on, 𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) and 𝑃𝑃𝑅𝑅𝑅𝑅([𝒒𝒒]) are products of reactant and product concentra�ons, 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑 is the equilibrium constant given by 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑= 𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) 𝐴𝐴𝑅𝑅𝑅𝑅([𝒒𝒒])� 𝑎𝑎𝑑𝑑 𝑒𝑒𝑑𝑑 𝑢𝑢𝑖𝑖𝑒𝑒𝑖𝑖𝑒𝑒𝑟𝑟𝑖𝑖 𝑢𝑢𝑚𝑚 , and Φ𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) is an algebraic expression that affects the speed of 𝑅𝑅𝑅𝑅 when the reac�on is not at equilibrium. The constant 𝜅𝜅𝑅𝑅𝑅𝑅 scales the reac�on flux and reflects the level of protein expression in a cell. Expressing all fluxes in this form clearly iden�fies: 1) th e {}-bracketed component that governs equilibrium, including the equilibrium constant 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒 𝑑𝑑, 2) th e explicit dependence on membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒 (for electrogenic reac�ons) that provides the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑅𝑅𝑅𝑅= 𝑅𝑅𝑅𝑅 𝑧𝑧 𝐹𝐹𝑅𝑅𝑛𝑛� 𝑅𝑅𝑅𝑅𝑅𝑅([𝒒𝒒]) 𝑄𝑄𝑅𝑅𝑅𝑅 𝑒𝑒𝑒𝑒.𝐴𝐴𝑅𝑅𝑅𝑅([𝒒𝒒])� , 3) the state-dependent term Φ𝑅𝑅𝑅𝑅(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) that governs the flux magnitude when not at equilibrium, 4) the scaling constant 𝜅𝜅𝑅𝑅𝑅𝑅 that reflects the expression levels for the protein. Table 4 summarises the transmembrane fluxes, all expressed in this form. 𝐼𝐼𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 (fC.mm-2.s-1) 𝑢𝑢𝑚𝑚 𝑒𝑒 (mV) 𝑘𝑘𝑡𝑡 𝐾𝐾𝑖𝑖𝑟𝑟= 1 mM 𝑘𝑘𝑡𝑡 𝐾𝐾𝑖𝑖𝑟𝑟= 10 mM 𝑘𝑘𝑡𝑡 𝐾𝐾𝑖𝑖𝑟𝑟= 100 mM Reversal poten�al for Kir .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 22 Protein 𝒒𝒒 Flux Eq # GLUT2 𝑞𝑞𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐 𝑞𝑞𝑡𝑡𝐺𝐺 𝑒𝑒𝑐𝑐 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2(𝒒𝒒) = 𝜅𝜅� 𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝐴𝐴2 Φ𝑅𝑅𝑅𝑅(𝒒𝒒) . �[𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 𝑒𝑒 𝑑𝑑 . [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡� fmol.mm-2.s-1 𝑄𝑄𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 𝑒𝑒 𝑑𝑑 = 1; Φ𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2(𝒒𝒒) = 1 + [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖 𝑘𝑘𝑖𝑖 𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2 + [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡 𝑘𝑘𝑡𝑡 𝐺𝐺 𝐺𝐺𝐺𝐺 𝑅𝑅2 + [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑖𝑖. [𝐺𝐺 𝑅𝑅𝑐𝑐]𝑡𝑡 𝑘𝑘𝑖𝑖𝑡𝑡 𝐺𝐺𝐺𝐺 𝐺𝐺𝑅𝑅2 27 SGLT1 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑 𝑁𝑁 𝑎𝑎+ 𝑞𝑞𝑖𝑖 𝑁𝑁 𝑎𝑎+ 𝑞𝑞𝑔𝑔𝑢𝑢𝑑𝑑 𝐺𝐺 𝑒𝑒𝑐𝑐 𝑞𝑞𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐 𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = 𝜅𝜅𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝐴𝐴1 Φ𝑆𝑆𝐺𝐺𝐺𝐺 𝐴𝐴 1(𝒒𝒒) �� [𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 2 . [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡 [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 − 𝑄𝑄𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1 𝑒𝑒 𝑑𝑑 . 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅� fmol.mm-2.s-1 𝑄𝑄𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1 𝑒𝑒 𝑑𝑑 = 1 Φ𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅 1(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = {�𝑒𝑒 2𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅+ �𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝑁𝑁 𝑎𝑎+ � 2 � 𝐾𝐾1 𝐾𝐾2 + 𝐾𝐾1 𝐾𝐾3 . 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝐺𝐺 𝑒𝑒𝑐𝑐�� . ��𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 � + � 𝐾𝐾1 𝐾𝐾6 + �𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 � 𝐾𝐾1 𝐾𝐾5 + 𝐾𝐾1 𝐾𝐾4 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐� �.��𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑔𝑔𝑢𝑢𝑑𝑑 𝐺𝐺 𝑒𝑒𝑐𝑐. 𝑒𝑒𝑧𝑧2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅+ 𝜅𝜅6 𝜅𝜅3 𝑒𝑒2𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒/𝑅𝑅𝑅𝑅�}/�𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 2 . 𝑞𝑞 �𝑖𝑖 𝐺𝐺 𝑒𝑒𝑐𝑐 R eversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 = 𝑅𝑅𝑅𝑅 2𝐹𝐹𝑅𝑅𝑛𝑛�� [𝑁𝑁 𝑎𝑎+]𝑔𝑔𝑔𝑔𝑡𝑡 [𝑁𝑁 𝑎𝑎+]𝑖𝑖 � 2 . [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑔𝑔𝑔𝑔𝑡𝑡 [𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 � = 102 mV 29 31 32 NKA 𝑞𝑞𝑖𝑖 𝑁𝑁 𝑎𝑎+ 𝑞𝑞𝑡𝑡𝑁𝑁 𝑎𝑎+ 𝑞𝑞𝑖𝑖 𝐾𝐾+ 𝑞𝑞𝑡𝑡𝐾𝐾+ 𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = 𝜅𝜅𝑚𝑚𝑁𝑁 𝐾𝐾𝑁𝑁 Φ𝑁𝑁𝐾𝐾𝑁𝑁(𝒒𝒒,𝑢𝑢𝑚𝑚𝑒𝑒) . �� [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . 𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴− 𝑄𝑄𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑. � [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � 2 . 𝑒𝑒−𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴� fmol.mm-2.s-1 𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴= [𝐴𝐴𝑅𝑅 𝐴𝐴] [𝐴𝐴𝐴𝐴𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+] and 𝑄𝑄𝑁𝑁𝐾𝐾𝐴𝐴 𝑒𝑒 𝑑𝑑= 𝑄𝑄𝐴𝐴𝑅𝑅 𝐴𝐴 𝑒𝑒 𝑑𝑑= [𝐴𝐴𝑅𝑅 𝐴𝐴] [𝐴𝐴𝐴𝐴𝐴𝐴].[𝐴𝐴𝑖𝑖].[𝐻𝐻+]� 𝑒𝑒𝑑𝑑𝑢𝑢 𝑖𝑖𝑒𝑒 ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛� 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� = -55 kJ/mol-1 Φ𝑁𝑁𝐾𝐾𝐸𝐸(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = C − AD, where 𝐴𝐴= � 𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ � 3 . 𝑞𝑞 �𝑖𝑖 𝐴𝐴 𝑅𝑅𝐴𝐴 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ − � 𝑞𝑞 �𝑖𝑖 𝐾𝐾+ 𝑞𝑞 �𝑡𝑡𝐾𝐾+ � 2 . 𝑒𝑒−𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅 𝐵𝐵= � 𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ 𝑞𝑞 �𝑡𝑡𝑁𝑁 𝑎𝑎+ � 3 . 𝑞𝑞 �𝑖𝑖 𝐴𝐴 𝑅𝑅𝐴𝐴 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ . 𝑒𝑒−𝑧𝑧1𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅�Λ4 + � Λ5 + 1 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ � . � 𝑞𝑞 �𝑡𝑡 𝐾𝐾+ � 2 � 𝐶𝐶= � Λ1 � 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+� 3 .𝑑𝑑� 𝑖𝑖 𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖 𝐻𝐻+ + (Λ2. 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴+ Λ3). � 𝑑𝑑� 𝑖𝑖 𝑁𝑁𝑁𝑁+ 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+� 3 . 𝑑𝑑� 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴 𝑑𝑑� 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴.𝑑𝑑� 𝑖𝑖 𝐴𝐴𝑖𝑖.𝑑𝑑� 𝑖𝑖 𝐻𝐻+ + 𝐵𝐵 � 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+� 3� x � 𝜆𝜆1+𝜆𝜆2 𝑑𝑑� 𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴+ 𝜆𝜆3 + � 𝜆𝜆4 + 𝜆𝜆5 + 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ � . � 𝑑𝑑� 𝑡𝑡𝑁𝑁𝑁𝑁+� 3 � 𝑑𝑑� 𝑡𝑡𝐾𝐾+� 2 . 𝑒𝑒 𝑧𝑧1𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴� 𝐴𝐴= Λ2. 𝜆𝜆1 + Λ5. 𝜆𝜆4 + 𝜆𝜆4 + 𝜆𝜆5 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ + Λ3. 𝜆𝜆1 + 𝜆𝜆2 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴+ �(𝜆𝜆1 + 𝜆𝜆2). 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ 𝑞𝑞 �𝑖𝑖 𝐴𝐴 𝑅𝑅𝐴𝐴+ 𝜆𝜆3 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝐴𝐴𝐴𝐴. 𝑞𝑞 �𝑖𝑖 𝐴𝐴𝑖𝑖. 𝑞𝑞 �𝑖𝑖 𝐻𝐻+ 𝑞𝑞 �𝑖𝑖 𝐴𝐴 𝑅𝑅𝐴𝐴 � . 𝐵𝐵 � 𝑞𝑞 �𝑖𝑖 𝑁𝑁 𝑎𝑎+ � 3 and 𝜆𝜆𝑖𝑖= 𝜅𝜅6 𝜅𝜅𝑖𝑖 , Λ𝑖𝑖= 𝐾𝐾6 𝐾𝐾𝑖𝑖 , for 𝑖𝑖= 1. .5. Reversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= − 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛� 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒. � [𝑁𝑁 𝑎𝑎+]𝑖𝑖 [𝑁𝑁 𝑎𝑎+]𝑡𝑡 � 3 . � [𝐾𝐾+]𝑡𝑡 [𝐾𝐾+]𝑖𝑖 � 2 � = -225 mV 34 36 Kir 𝑞𝑞𝑖𝑖 𝐾𝐾+ 𝑞𝑞𝑡𝑡𝐾𝐾+ 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = 𝜅𝜅� 𝑚𝑚𝐾𝐾𝑖𝑖𝑟𝑟 Φ𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒) . �[𝐾𝐾+]𝑖𝑖− 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟 𝑒𝑒 𝑑𝑑. [𝐾𝐾+]𝑡𝑡. 𝑒𝑒 −𝐹𝐹𝑔𝑔𝑚𝑚𝑒𝑒 𝑅𝑅 𝐴𝐴� fmol.mm-2.s-1 𝑄𝑄𝐾𝐾𝑖𝑖𝑟𝑟 𝑒𝑒 𝑑𝑑= 1 Φ𝐾𝐾𝑖𝑖𝑟𝑟(𝒒𝒒, 𝑢𝑢𝑚𝑚𝑒𝑒) = 𝑒𝑒 (−𝑧𝑧1+𝑧𝑧2−𝑧𝑧3)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅 � 1 + [𝐾𝐾+]𝑖𝑖 𝑘𝑘𝑖𝑖 𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒 (𝑧𝑧1−𝑧𝑧2)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅 + [𝐾𝐾+]𝑡𝑡 𝑘𝑘𝑡𝑡𝐾𝐾𝑖𝑖𝑟𝑟. 𝑒𝑒 (𝑧𝑧6−𝑧𝑧5)𝐹𝐹𝑢𝑢𝑚𝑚𝑒𝑒 𝑅𝑅𝑅𝑅� R eversal potential 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟= − 𝑅𝑅𝑅𝑅 𝐹𝐹𝑅𝑅𝑛𝑛� [𝐾𝐾+]𝑖𝑖 [𝐾𝐾+]𝑡𝑡 � = -91 mV 45 46 47 Table 4. A list of the flux expressions in standardised form for the SLC transporters GLUT2 and SGL T1, the NKA pump, and the ion channel/transporter Kir. The reversal poten�als shown for each membrane protein are calculated using values of [𝑁𝑁𝑁𝑁+]𝑖𝑖= 15 mM, [𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑖𝑖= 1 mM, [𝐺𝐺𝑅𝑅𝑐𝑐]𝑡𝑡= 40 mM, and 𝑅𝑅𝑅𝑅 𝐹𝐹=26.5 mV. .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 23 A MODEL OF THE ENTEROCYTE We now combine the GLUT2, SGLT1, NKA, and Kir models discussed above to create a model of homeostasis in the enterocyte, the absorp�ve cell of the epithelium in the small intes�nes. The NKA pump maintains a low intracellular [𝑁𝑁𝑁𝑁+]𝑖𝑖 such that the SGLT1 cotransporter can use the sodium gradient between the gut lumen and the intracellular space to bring glucose into the cell. Glucose is metabolised into the ATP used to drive NKA and also flows out of the cell via GLUT2 facilitated diffusion. The NKA-driven potassium flux into the cell is balanced by potassium efflux through the Kir ion channel, as illustrated in Figure 16. Figure 16. Equilibrium between the outward flow of 𝐾𝐾+ ions through Kir (when the membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 is above the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟 𝐾𝐾𝑖𝑖𝑟𝑟 = -87 mV) and the inward flow of 𝐾𝐾+ ions through NKA (when the membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 is above the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟 𝑁𝑁 𝐾𝐾𝐸𝐸 = -225 mV for NKA). Equilibrium is achieved when these two opposing currents are equal. The purple line shows the 𝐾𝐾+ ion efflux (𝑣𝑣𝑚𝑚 𝐾𝐾+(𝐾𝐾𝑖𝑖𝑟𝑟)) from Kir (equa�on [33]), and the dark brown line shows the influx 𝑣𝑣𝑚𝑚 𝐾𝐾+(𝑁𝑁 𝐾𝐾𝐴𝐴) associated with NKA (equa�on (33) with two 𝐾𝐾+ ions transported for each NKA cycle). 𝑢𝑢𝑒𝑒𝑑𝑑𝑢𝑢𝑖𝑖𝑒𝑒 𝐾𝐾 is membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 at this equilibrium. Figure 16 illustrates the balance of inward sodium flux from SGLT1 with outward flux by NKA. Figure 17. Equilibrium between the inward flow of 𝑁𝑁𝑁𝑁+ ions through SGLT1 (when the membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 is well below the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟 𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1 = 102 mV) and the outward flow of 𝑁𝑁𝑁𝑁+ ions through NKA (when the membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 is above the reversal poten�al 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟 𝑁𝑁 𝐾𝐾𝐸𝐸 = -225 mV for NKA). Equilibrium is achieved when these two opposing currents are equal. The green line shows the 𝑁𝑁𝑁𝑁+ ion influx (𝑣𝑣𝑚𝑚 𝑁𝑁 𝑎𝑎+(𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1)) from SGLT1 (equa�on 29 with two 𝑁𝑁𝑁𝑁+ ions transported for each SGLT1 cycle) and the dark brown line shows the effl ux 𝑣𝑣𝑚𝑚 𝑁𝑁 𝑎𝑎+(𝑁𝑁 𝐾𝐾𝐸𝐸) associated with NKA pump (equa�on (33) with three 𝑁𝑁𝑁𝑁+ ions transported for each NKA cycle). 𝑢𝑢𝑒𝑒𝑑𝑑𝑢𝑢𝑖𝑖𝑒𝑒 𝑁𝑁 𝑎𝑎 is the value of membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 at this equilibrium. Figure 18 shows the various membrane processes and the glycolysis pathway that regenerates the ATP used by ATP hydrolysis in the NKA pump. 𝒗𝒗𝟏𝟏 𝑲𝑲+(𝑲𝑲 𝒊𝒊𝑲𝑲) Opera�ng range for cell Outward 𝐾𝐾+current Inward 𝐾𝐾+current 𝒖𝒖𝟏𝟏 𝒆𝒆 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝐾𝐾𝑖𝑖𝑟𝑟=-87mV 𝒖𝒖𝒆𝒆𝒒𝒒 𝒖𝒖𝒊𝒊𝑵𝑵 𝑲𝑲 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸=-225mV 𝒗𝒗𝟏𝟏 𝑲𝑲+(𝑵𝑵𝑲𝑲 𝑵𝑵) = −𝟐𝟐𝒗𝒗𝟏𝟏 𝑵𝑵𝑲𝑲 𝑵𝑵 𝒖𝒖𝒆𝒆𝒒𝒒 𝒖𝒖𝒊𝒊𝑵𝑵 𝑵𝑵𝑵𝑵 𝒗𝒗𝟏𝟏 𝑵𝑵𝑵𝑵+(𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺 𝟏𝟏) = 𝟐𝟐𝒗𝒗𝟏𝟏 𝑺𝑺 𝑺𝑺𝑺𝑺𝑺𝑺 𝟏𝟏 Opera�ng range for cell Outward 𝑁𝑁𝑁𝑁+current Inward 𝑁𝑁𝑁𝑁+current 𝒖𝒖𝟏𝟏 𝒆𝒆 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑆𝑆𝐺𝐺𝐺𝐺𝑅𝑅1=102mV 𝑢𝑢𝑟𝑟𝑒𝑒 𝑟𝑟𝑁𝑁𝐾𝐾𝐸𝐸=-225mV 𝒗𝒗𝟏𝟏 𝑵𝑵𝑵𝑵+(𝑵𝑵𝑲𝑲 𝑵𝑵) = 𝟑𝟑𝒗𝒗𝟏𝟏 𝑵𝑵𝑲𝑲 𝑵𝑵 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 24 Figure 18. The enterocyte model includes NKA, Kir, and GLUT2 in the basolateral membrane and SGLT1 in the apical membrane. Energy is transferred between biochemical storage in ATP and solute concentra�ons, electrical storage in membrane capacitance, mechanical storage in the cell wall compliance (which determines the pressure-volume behaviour associated with water movement), and heat storage in the thermal capacity of water in the cell and its surroundings. One mole of glucose is assumed to instantaneously convert 2 moles of ADP to ATP via glycolysis with the genera�on of 2 moles of water. The membrane voltage is 𝑢𝑢𝑚𝑚𝑒𝑒= 𝑑𝑑𝑚𝑚𝑒𝑒 𝐶𝐶𝑚𝑚 is calculated from net influx of membrane charge: 𝑐𝑐𝑢𝑢𝑚𝑚𝑒𝑒 𝑐𝑐𝑑𝑑= 𝐹𝐹 𝐶𝐶𝑚𝑚 � 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴− 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟� , J.C-1.s-1 (50) wh ere 𝐶𝐶𝑚𝑚 is the membrane capacitance (assumed to be 1 µF.cm-2 or 10-8 C2.J-1.mm-2). Note that 𝐹𝐹 𝐶𝐶𝑚𝑚 = 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. Increases in intracellular potassium and sodium ion concentra�ons are given by 𝑐𝑐[𝐾𝐾+]𝑖𝑖 𝑐𝑐𝑑𝑑= Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. � 2𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴− 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟� mM.s-1 (51) a nd 𝑐𝑐[𝑁𝑁 𝑎𝑎+]𝑖𝑖 𝑐𝑐𝑑𝑑 = Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. (2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 3𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴), mM.s-1 (52) wh ere Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒 (m-1) is the ra�o of membrane area A𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒 to cell volume V𝑐𝑐𝑒𝑒 𝑒𝑒𝑒𝑒. Note that we are assuming the same areal density of the NKA pump, Kir channel, and GLUT2 transporter (all in the basolateral membrane in contact with inters��al fluid and effec�vely with blood) and the SGLT2 cotransporter (in the apical membrane in hence in contact with the intes�nal lumen). S ince 1 mol of glucose is assumed to replace 2 mol o f A D P w i t h 2 mol ATP instantaneously, the intracellular glucose is given by 𝑐𝑐[𝐺𝐺 𝑒𝑒𝑐𝑐]𝑖𝑖 𝑐𝑐𝑑𝑑= Γ𝑐𝑐𝑒𝑒𝑒𝑒 𝑒𝑒. (𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1 − 𝑣𝑣𝑚𝑚𝐺𝐺𝐺𝐺𝐺𝐺𝑅𝑅2) − V𝑐𝑐𝑒𝑒𝐶𝐶𝐶𝐶 32 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴. mM.s-1 (53) .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 25 To ex amine how these four membrane proteins achieve homeostasis of sodium, potassium and glucose in the enterocyte, we first consider the two transporters Kir and NKA. Figure 19(a) shows the reac�on fluxes 𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟 and 𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴 as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒 from -120 mV to 0 mV under physiologically normal condi�ons ([𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 mM, [𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 mM and ∆𝐺𝐺𝑁𝑁𝐾𝐾𝐴𝐴= −𝑅𝑅𝑅𝑅𝑅𝑅 𝑛𝑛� 𝑄𝑄� 𝐴𝐴𝐴𝐴𝐴𝐴 𝑄𝑄�𝐴𝐴𝐴𝐴𝐴𝐴 𝑒𝑒𝑒𝑒� = -61 kJ.mol-1). The reversal poten�als for the two membrane proteins are 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝐾𝐾𝑖𝑖 𝑟𝑟= -91 mV and 𝑢𝑢𝑟𝑟𝑒𝑒𝑟𝑟𝑁𝑁𝐾𝐾𝐴𝐴= -225 mV, respec�vely. The effect on 𝐾𝐾+ flux of these two transporters is shown in Figure 19(b) where the net 𝐾𝐾+ flux (𝑣𝑣𝑚𝑚𝐾𝐾𝑖𝑖 𝑟𝑟− 2𝑣𝑣𝑚𝑚𝑁𝑁 𝐾𝐾𝐴𝐴) is ploted against 𝑢𝑢𝑚𝑚𝑒𝑒 for different values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 from 0 mM to 12 mM. For [𝑁𝑁𝑁𝑁+]𝑖𝑖= 0 mM the net flux is zero at -91 mV and then occurs at progressively less nega�ve membrane poten�als as [𝑁𝑁𝑁𝑁+]𝑖𝑖 increases, up to a limit of litle over 10 mM. Beyond this there is no balance possible because the influx of 𝐾𝐾+ by NKA (associated with the high efflux of 𝑁𝑁𝑁𝑁+) exceeds the efflux of 𝐾𝐾+ achieved by Kir at all poten�als. (a) (b) F igure 19. (a) Fluxes 𝑣𝑣𝑚𝑚 𝐾𝐾 𝑖𝑖𝑟𝑟 and 𝑣𝑣𝑚𝑚 𝑁𝑁𝐾𝐾𝐴𝐴 through Kir and NKA as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒. The reversal poten�als for Kir and NKA under standard physiological condi�ons are -91 mV and -225 mV, respec�vely. (b) The net outward potassium flux 𝑣𝑣𝑚𝑚 𝐾𝐾𝑖𝑖𝑟𝑟− 2𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴 shown as a func�on of 𝑢𝑢𝑚𝑚 𝑒𝑒 for values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 from 0 mM to 12 mM. For the specified electrolyte concentra�ons ([𝑁𝑁𝑁𝑁+]𝑡𝑡= 140 mM, [𝐾𝐾+]𝑖𝑖= 140 mM, [𝐾𝐾+]𝑡𝑡= 4.5 mM) there is a well defined 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. The flux balance for 𝑁𝑁𝑁𝑁+ depends on both NKA and SGLT1. Figure 20 shows how the balance of sodium efflux 3𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴 and sodium influx 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 (i.e. a net efflux of 3𝑣𝑣𝑚𝑚𝑁𝑁𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚𝑆𝑆𝐺𝐺 𝐺𝐺𝑅𝑅1) depends on membrane poten�al 𝑢𝑢𝑚𝑚𝑒𝑒. (a) (b) F igure 20. (a) Fluxes 𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴 and 𝑣𝑣𝑚𝑚 𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1 through NKA and SGLT1 as a func�on of membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒. Both fluxes are computed for 3 values of [𝑁𝑁𝑁𝑁+]𝑖𝑖, but this has litle effect on 𝑣𝑣𝑚𝑚 𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1.(b) The corresponding net outward 𝑁𝑁𝑁𝑁+ flux (calculated from 3𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚 𝑆𝑆𝐺𝐺𝐺𝐺 𝑅𝑅1). Achieving zero net flux for both [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖 also ensures zero net flow of charge across the membrane since all charge transfer is associated with those ions. [𝑵𝑵𝑵𝑵+]𝒊𝒊 2 mM 10 mM 12 mM 0 mM Reversal poten�al for Kir Net outward 𝑲𝑲+ flux (fmol.mm-2.s-1) 𝒖𝒖𝟏𝟏𝒆𝒆 mV Physiological poten�al Reac�on fluxes for Kir & NKA at [𝑵𝑵𝑵𝑵+]𝒊𝒊= 10 mM (fmol.mm-2.s-1) 𝒖𝒖𝟏𝟏𝒆𝒆 mV 𝒗𝒗𝟏𝟏𝑵𝑵𝑲𝑲 𝑵𝑵 𝒗𝒗𝟏𝟏𝑲𝑲𝒊𝒊𝑲𝑲 Reversal poten�al for Kir [𝑵𝑵𝑵𝑵+]𝒊𝒊 5 mM 10 mM 12 mM Net outward 𝑵𝑵𝑵𝑵+ flux (fmol.mm-2.s-1) 𝒖𝒖𝟏𝟏𝒆𝒆 mV Reac�on fluxes for NKA & SGLT1 (fmol.mm-2.s-1) 𝒖𝒖𝟏𝟏𝒆𝒆 mV [𝑵𝑵𝑵𝑵+]𝒊𝒊 5 mM 10 mM 12 mM Physiological poten�al 𝒗𝒗𝟏𝟏𝑵𝑵𝑲𝑲 𝑵𝑵 𝒗𝒗𝟏𝟏𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝑺𝟏𝟏 .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 26 At [𝑁𝑁𝑁𝑁+]𝑖𝑖= 10 mM, both the net 𝐾𝐾+ flux and the net 𝑁𝑁𝑁𝑁+ flux are zero at the same poten�al (𝑢𝑢𝑚𝑚 𝑒𝑒= -60 mV). These values of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and 𝑢𝑢𝑚𝑚 𝑒𝑒 were calculated from (50) and (51) for a fixed values of 𝜅𝜅𝑚𝑚 𝑅𝑅𝑅𝑅 for each of the three transporters. An alterna�ve is to set the le� hand sides of (50) and (51) to zero for all values of 𝑢𝑢𝑚𝑚 𝑒𝑒 and compute the corresponding scaling on the 𝜅𝜅𝑚𝑚 𝑅𝑅𝑅𝑅 values (as a func�on of 𝑢𝑢𝑚𝑚 𝑒𝑒) in order to achieve zero net flux. The computed scale factors are shown as a func�on of 𝑢𝑢𝑚𝑚 𝑒𝑒 in the botom panel of Figure 21. The unscaled fluxes (𝑣𝑣𝑚𝑚 𝑅𝑅𝑅𝑅 for each of the transporters) are shown in the top panel. F igure 21. The transporter fluxes (top) and scaling on the reac�on rate constants (botom) needed for Kir and SGLT1 (rela�ve to that on NKA) to achieve homeostasis of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖. These rela�ve scaling values depend on the membrane poten�al 𝑢𝑢𝑚𝑚 𝑒𝑒 at which the net fluxes 𝑣𝑣𝑚𝑚 𝐾𝐾𝑖𝑖𝑟𝑟− 2𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴 and 3𝑣𝑣𝑚𝑚 𝑁𝑁 𝐾𝐾𝐴𝐴− 2𝑣𝑣𝑚𝑚 𝑆𝑆 𝐺𝐺𝐺𝐺𝑅𝑅1 are both zero (i.e. these net fluxes are zero for all values of 𝑢𝑢𝑚𝑚 𝑒𝑒 shown. Note the rapid increase in scaling on 𝜅𝜅𝑚𝑚 𝐾𝐾𝑖𝑖𝑟𝑟 in the botom graph as 𝑢𝑢𝑚𝑚 𝑒𝑒 approaches the reversal poten�al for Kir (i.e. 𝑣𝑣𝑚𝑚 𝐾𝐾𝑖𝑖𝑟𝑟 approaches zero on top graph). The rela�ve scale factors for SGLT1 and Kir shown in the botom panel of Figure 21 would correspond to either regula�on of the protein kine�cs (e.g., by phosphoryla�on) or to the protein expression levels (under transcrip�onal control).

Epithelial cells lining the small intes�nes take up glucose and sodium (delivered via ingested food) via the SGLT1 transporter, using solute gradients to drive the transmembrane flux (two sodium ions for each glucose molecule). Intracellular glucose is immediately converted to ATP via glycolysis (rather than oxida�ve metabolism), which then drives the NKA pump to maintain an intracellular sodium concentra�on of about 10 mM in the face of 140 mM sodium in the capillaries. The 140 mM/10 mM sodium difference provides a ‘batery’ that drives many other transmembrane transport processes. Note that a ‘batery’ is a source of poten�al energy and only loses energy when current flows (sodium ions in this case). The ‘sodium batery’ (i.e., the Gibbs free energy available from the sodium gradient) is therefore maintained by the Gibbs free energy of ATP hydrolysis – essen�ally the ’top-up batery’ needed to maintain the sodium batery. In this paper, we have explained how the physical principles of conserva�on of mass, conserva�on of charge, and conserva�on of energy are conveniently implemented in models of biological processes using bond graphs. The bond graph approach is well-known in the engineering literature for problems that involve energy transfer between the four available forms of energy storage – chemical, electrical, mechanical, and thermal. Physiological processes almost always involve energy transmission, exchange, and conversion (including dissipa�on) between all four physical domains, and bond graphs therefore provide an ideal framework for modelling these processes. We show how an appropriate Scale factors needed to achieve equilibrium 𝒗𝒗𝟏𝟏𝑹𝑹𝑹𝑹 for transporters (fmol.mm-2.s-1) (no scaling) SGLT1 NKA Kir SGLT1 NKA Kir 𝒖𝒖𝟏𝟏𝒆𝒆 mV 𝒖𝒖𝟏𝟏𝒆𝒆 mV .CC-BY 4.0 International licenseperpetuity. It is made available under a preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in The copyright holder for thisthis version posted January 30, 2026. ; https://doi.org/10.64898/2026.01.28.702213doi: bioRxiv preprint 27 choice of a small number of units (joules, meters, seconds, moles, coulombs, and entropy or Kelvin) provides a common founda�on for describing all physical processes, and we illustrate the applica�on of bond graphs to basic biochemical processes where Gibbs free energy is key to understanding and modelling reac�ons. We then derived biophysically based bond graph models of GLUT2, SGLT1, Kir, and the NKA ATPase pump and coupled NKA with the transport of glucose via SGLT1 and GLUT2. Glycolysis supplies the ATP needed by NKA for maintaining homeostasis of [𝑁𝑁𝑁𝑁 +]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖. Given the 3:2 ra�o of sodium efflux to potassium entry in NKA, the 2:1 ra�o of coupled sodium to glucose entry by SGLT1, and the single potassium efflux by Kir, the cycling rates of SGLT1 and Kir must be 1.5x and 2x that of NKA, respec�vely, in order to maintain homeostasis of [𝑁𝑁𝑁𝑁+]𝑖𝑖 and [𝐾𝐾+]𝑖𝑖, and hence also of the membrane res�ng poten�al 𝑢𝑢𝑚𝑚𝑒𝑒. This balance of the transporter fluxes can either be achieved by adjus�ng the levels of [𝑁𝑁𝑁𝑁+]𝑖𝑖, [𝐾𝐾+]𝑖𝑖, and 𝑢𝑢𝑚𝑚𝑒𝑒, or by altering either the phosphoryla�on status or expression levels of the transporters, as illustrated in Figure 21. Future work will explore the regula�on of these membrane transport proteins.

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