Network Thermodynamical Modelling ofBioelectrical Systems: A Bond Graph Approach

Peter J. Gawthrop1,2,4 and Michael Pan1,2,3

1 Systems Biology Laboratory, Department of Biomedical Engineering,Melbourne School of Engineering, University of Melbourne, Victoria 3010,

Australia.2Systems Biology Laboratory, School of Mathematics and Statistics, University

of Melbourne University of Melbourne, Victoria 30103ARC Centre of Excellence in Convergent Bio-Nano Science and Technology,

Melbourne School of Engineering, University of Melbourne, Parkville, Victoria3010, Australia

4Corresponding author peter.gawthrop@unimelb.edu.au

October 28, 2020

Abstract

Interactions between biomolecules, electrons and protons are essential to many funda-mental processes sustaining life. It is therefore of interest to build mathematical models ofthese bioelectrical processes not only to enhance understanding but also to enable computermodels to complement in vitro and in vivo experiments. Such models can never be entirelyaccurate; it is nevertheless important that the models are compatible with physical principles.Network Thermodynamics, as implemented with bond graphs, provide one approach to cre-ating physically compatible mathematical models of bioelectrical systems. This is illustratedusing simple models of ion channels, redox reactions, proton pumps and electrogenic mem-brane transporters thus demonstrating that the approach can be used to build mathematicaland computer models of a wide range of bioelectrical systems.

Keywords Biological system modelling; bioelectricity; redox reactions; network thermody-namics; computational systems biology.

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IntroductionIn recent years, it is becoming increasingly clear that bioelectricity is involved in several keyprocesses in cell biology, from development to signalling to proliferation1,2. As such, it playsan important role in many diseases and also forms the underlying basis of several promisingtreatments3,4,5. Moreover, there is a strong current interest in using, or modifying, microbes forgenerating energy in the form of electricity, hydrogen and biofuels6,7,8. Mathematical modelscan help to provide a mechanistic understanding of bioelectrical systems and also to test syn-thetic biology constructs in silico9. To avoid false conclusions, it is vital that such models areconstrained by physical laws such as energy conservation. To allow the construction of largemodels, a modular approach based on interconnection of reusable components is essential. Toaid both understanding and computer modelling, a graphical representation is helpful. The bondgraph implementation of Network Thermodynamics outlined in this paper provides a system-atic way of constructing mathematical models of living systems which are both modular andphysically constrained and have an underlying graphical representation.

Bioelectrical systems involve two physical domains: chemistry and electricity. However,as physical domains, they are linked by the laws of physics, and in particular by the commoncurrency of energy. This energy-based approach provides a unifying framework to model bio-electrical systems. Engineering has several ways of modelling multi-physics energetic systems;one such method is the bond graph methodology10,11,12,13. This approach was extended some50 years ago as an energy-based approach to modelling biomolecular systems14,15,16. As notedby Perelson 16 , the bond graph approach combines multi-physics modelling with graphical ap-proaches from electrical circuit theory: “Graphical representations similar to engineering circuitdiagrams can be constructed for thermodynamic systems. Although the proverb that a pictureis worth a thousand words may not be completely applicable, such diagrams do increase one’sintuition about system behaviour. Moreover, as in circuit theory, one can algorithmically readthe algebraic and differential equations that describe the system directly from the diagram muchmore easily than they can be constructed directly.”.

The Network Thermodynamic approach can be applied not only to the chemical propertiesbut also to the electrical properties of biomolecular systems in a uniform manner. Indeed, thebond graph approach can be seen as a formal approach to the concept of physical analogiesintroduced by James Clerk Maxwell17 who pointed out that analogies are central to scientificthinking and allow mathematical results and intuition from one domain to be transferred to an-other – see Table 1. An issue that arises in multi-physics models is the different dimensions andunits used in each domain. For example, mechanical systems use force (N) and velocity (m s−1),electrical systems use voltage (V) and current (A) and chemical systems use chemical potential(µ J mol−1) and molar flow (v mol s−1).

Despite these differences, the product of the two physical covariables in each of these threeexamples is energy flow (J s−1) . It is therefore clear that the common quantity among allphysical domains is energy (J). The commonality of energy over different physical domainsmakes the bond graph approach appropriate for modelling multi-domain systems, in particularbioelectrical systems18,19. Noting that the conversion factor relating the electrical and chemicaldomains is Faraday’s constant F ≈ 96 485 C mol−1, the chemical covariables µ and v may

2

Electrical ChemicalPotential Voltage Gibbs energy

V J C−1 or V µ J mol−1

Flow Current Molar flowi C s−1 or A v mol s−1

Quantity Charge MolesQ C where i = Q q mol where v = q

Capacitance C V =Q

Cµ = µ +RT ln q

q

Resistance Re i =Vf − Vr

ρv = κ

(eµfRT − e µrRT

)Table 1: Analogies and Energy-based modelling. In both electrical and chemical domains:potential × flow = power J s−1. µ and q refer to standard conditions, RT is the product ofthe gas constant and temperature and subscripts f and r abbreviate forward and reverse.

be scaled by F to give the pair of covariables φ (Faraday-equivalent chemical potential) and f(Faraday-equivalent flow) in electrical units19:

φ =µ

F(V) (1)

and f = Fv (A) (2)

This reformulation is frequently used in the bioelectrical disciplines of electrophysiology andmitochondrial energetics where it is convenient to represent quantities in terms of voltages andcurrents7,20,21,22.

Because energy is the common quantity across physical domains, this approach allows mod-ular construction of large multi-physical models from reusable components23,24. Moreover, com-ponents can be replaced by simpler, but physically plausible, alternatives to aid understandingand reduce computational load25.

Equations (1)–(2) unify the description of the electrical and chemical domains of bioelectricalsystems including the ion channel and redox examples treated in the sequel. These conceptsare explored in the following section using electrodiffusion and ion channels as an introductoryexample. The potential of bond graphs in representing bioelectrical systems in a modular manneris then illustrated using a model of redox reactions and electron-driven proton pumps in thecontext of the mitochondrial electron transport chain. Finally, we consider the thermodynamicsof electrogenic transport in a model of the sodium-glucose symporter, which is compared toexperimental data.

Energy-based modelling – a brief introductionAs noted in Table 1, both the electrical and chemical physical domains have energy (measuredin Joules (J) ) in common. This is the fundamental idea of energy based modelling and meansthat both domains can be modelled within a common framework. The bond graph framework for

3

Component Electrical ChemicalC capacitor speciesRe resistor reaction0 parallel connection common potential1 seriesl connection common flow

Table 2: Bond graph components express the analogies of Table 1.

energy-based modelling is summarised in this section and the next section takes a closer look.The four relevant bond graph components are listed in Table 2 and express the analogies of Table1. Figure 1 shows two analogous systems, a chemical reaction system and an electrical circuit,together with the corresponding bond graph.

4

r1CA CB

r2CC

CD

(a) Electrical Circuit

Re:r

1

10

C:B C:C

C:D

C:A

Re:r

2

(b) Bond Graph

Figure 1: Analogous systems: (a) A reaction system Ar1 B

r2 C + D comprises two re-actions r1 and r2 and four species A, D, C and D. Using the analogies of Table 1, this is analogousto the circuit diagram shown comprising two resistors r1 and r2 and four capacitors. (b) Usingthe components of Table 2, both systems have the bond graph representation shown where thetwo reactions or resistors are modelled by Re:r1 and Re:r2 and the four species or capacitorsare represented by C:A, C:B, C:C and C:D. The difference between the two domains lies inthe different equations listed in Table 1. The ⇁ indicate energy flows of either electrical energyor chemical energy and the 0 and 1 junctions of Table 2 provide the connections.

5

Bond Graph Modelling of Electrogenic SystemsThe bond graph representation of the network thermodynamics of chemical systems was intro-duced Katchalsky and coworkers14,15 and summarised by Perelson 16 . In 1993, the inventor ofbond graphs, Henry Paynter, said26:

Katchalsky’s breakthroughs in extending bond graphs to biochemistry are verymuch on my own mind. I remain convinced that BG models will play an increasinglyimportant role in the upcoming century, applied to chemistry, electrochemistry andbiochemistry, fields whose practical consequences will have a significance compa-rable to that of electronics in this century. This will occur both in device form, sayas chemfets, biochips, etc, as well as in the basic sciences of biology, genetics, etc.

This challenge was largely ignored until recently when the bond graph approach was extendedby Gawthrop and Crampin 27 .

The approach is introduced here in a tutorial fashion with reference to a basic bioelectricalentity: electrodiffusion though a membrane pore (Figure 2(a)). A bond graph representationappears in Figure 2(b) and explained in the following sections.

The energy bondThe ⇁ symbol indicates an energy bond: an energetic connection between two subsystems;the half-arrow indicates the direction corresponding to positive energy flow. In the contextof this paper, the bonds carry the Faraday-equivalent covariables φ and f . These bonds con-nect four types of component: the four C components representing accumulation of chemicalspecies (C:Ie,C:Ii) or electrical charge (C:Ee,C:Ei); an Re component representing the mem-brane pore; and the 1 junction connects the components via the bonds. These four componentsare now discussed in detail.

C componentWe consider two types of C components in this paper: those corresponding to chemical speciesand those corresponding to electrical charge. The C components representing chemical speciesgenerate the Faraday-equivalent potential φ of Equation (1) in terms of the amount of species xas19:

φ = φ + φN lnx

x(3)

where φN =RT

F≈ 26 mV (4)

φ is the standard potential at the standard amount x. R = 8.314 J K−1 mol−1 is the universalgas constant and F is Faraday’s constant. The amount of species x is the integral of the speciesFaraday-equivalent flow f :

x =

∫ t

f(τ)dτ (5)

6

I+

I+

I+

I+I+✕

++−++

−−+−−

Ee

Ei

ΔE−

+

Ion pore Ion channel

External

Internal

(a) Schematic

External

Chemical Electrical

Internal

Re:r

1

C:Ie C:Ee

1

C:EiC:Ii

(b) Electrodiffusion

External

Chemical Electrical

Internal

Re:r0

1

C:Ie C:Ee

1

C:EiC:Ii

C:G

(c) Ion Channel

Figure 2: Bond Graph Modelling of Electrogenic Systems. (a) A schematic of an ion transportthrough a charged membrane with differential voltage ∆E = Ei − Ee. A monovalent ion I+

(which could be, for example, Na+ or K+) passes though either a membrane pore (left) or anion channel (middle and right). While the membrane pore is always open, the ion channel canswitch between closed (middle) and open (right) states under the influence of a gating variablewhich may be either a ligand or a voltage. (b) Electrodiffusion. A monovalent ion I+ passesthough a membrane pore where a flow from interior to exterior is regarded as positive. C:Iirepresents the accumulation of the ion, and C:Ei represents the accumulation of charge, insidethe membrane. Similarly C:Ie and C:Ee represent these quantities outside the membrane. Re:rrepresents the net effect of the pores in a given area of the membrane. The 0 and 1 componentsare junctions connecting two or more bonds. The 1 junction connects so that the flow is common;the 0 junction connects so that the potential is common. C components store, but do not dissi-pate, energy; Re components dissipate, but do not store, energy; 0 and 1 junctions transmit, butneither dissipate nor store energy. (c) Gated ion channel. The model is identical to that of (b)except the flow is gated by the potential of the gate represented by C:G. Depending on whetherthe channel is ligand-gated or voltage-gated, C:G can represent either an accumulation of ligandor an accumulation of charge. 7

In some cases, it is convenient to express the potential in terms of concentration c as

φ = φ + φN lnc

c(6)

where φ is the standard potential referenced to a standard concentration c.In contrast to the nonlinear chemical components, the C components representing electrical

capacitance are linear and generate electrical potential φ (Volts) in terms of the charge x =∫ tf(τ)dτ and electrical capacitance C as:

φ =x

C(7)

Open systems & ChemostatsThe concept of a chemostat 28 provides a way of converting a closed system to an open systemwhilst retaining the basic closed system bond graph formulation23. The chemostat has a numberof interpretations29:

1. one or more species is fixed to give a constant concentration; this implies that an appropri-ate external flow is applied to balance the internal flow of the species.

2. an ideal feedback controller is applied to species to be fixed with setpoint as the fixedconcentration and control signal an external flow.

3. as a C component with a fixed state.

4. as a concentration clamp or fixed boundary condition.

The chemostatted species can be thought of as external connections turning a closed systeminto an open system; as such, they can be also thought of as system ports providing a point ofinterconnection with other systems and thus leading to a modular modelling paradigm19,23,30.

Re componentThe R component is the bond graph abstraction of an electrical resistor. In the chemical context,a two-port R component represents a chemical reaction with chemical affinity (net chemicalpotential) replacing voltage and molar flow replacing current14,15. As it is so fundamental, thistwo port R component is given a special symbol: Re 27.

Again, there are two versions: a non linear version corresponding to chemical systems and alinear version corresponding to electrical systems. In particular, the Re component determines areaction flow f in terms of forward and reverse reaction potentials Φf and Φr as the Marcelin –de Donder formula31 rewritten in Faraday-equivalent form:

f = κ

(exp

Φf

φN

− expΦr

φN

)(8)

8

φN is given by Equation (4) and κ is the reaction rate constant. This formulation corresponds tothe mass-action formulation; other formulations are possible within this framework25,27,30.

The net reaction potential Φ is given in terms of the forward and reverse reaction potentialsby:

Φ = Φf − Φr (9)

When Φ = 0, the reaction is said to be at equilibrium, which also implies via Equation (8) thatthe flow through the reaction f is zero.

0 and 1 junctionsElectrical components may be connected in parallel (where the voltage is common) and series(where the current is common). These two concepts are generalised in the bond graph notationas the 0 junction which implies that all impinging bonds have the same potential (but differentflows) and the 1 junction which implies that all impinging bonds have the same flow (but differentpotentials). The direction of positive energy transmission is determined by the bond half arrow.As all bonds impinging on a 0 junction have the same potential, the half arrow implies the signof the flows for each impinging bond. The reverse is true for 1 junctions, where the half arrowimplies the signs of the potentials.

These components therefore describe the network topology of the system.

Electrodiffusion and the Nernst PotentialFigure 2(b) combines these bond graph components into a model of electrodiffusion: the flowof charged ions though a membrane pore. The italicised text and the dashed lines are not part ofthe bond graph but add the clarity of the diagram; in particular they emphasise the two divisions:internal/external and chemical/electrical.

The external compartment is modelled by two C components: C:Ie accumulating the ex-ternal ion species as a chemical entity and C:Ee accumulating the external ion species as anelectrical entity. The internal compartment is modelled in a similar fashion. The flow f betweenthe compartments is modelled by a single Re component; this is taken as positive if the flow isfrom the interior to the exterior. The upper 1 junction ensures that the flow in to both C:Ie andC:Ee is the same as that through Re:r; the lower 1 junction ensures that the flow from both C:Ieand C:Ee is also the same as that through Re:r.

The 0 and 1 junctions contribute to the network topology of the bond graph, which simul-taneously (via 1 junctions) distributes the flow from reactions to species and potentials fromspecies to reactions. This can be summarised in terms of the stoichiometric matrix N as27:

d

dtX = Nf Φ = −NTφ (10)

9

where, in this case:

N =

1−11−1

X =

xEe

xEi

xIexIi

φ =

φEe

φEi

φIe

φIi

(11)

and Φ is the reaction potential (9). In this case, Φ is given by:

Φ = φEi − φEe + φIi − φIe (12)

Defining the membrane voltage as ∆E = φEi − φEe and using equation (6):

Φ = ∆E + φIi + φN

(lncici

)− φ

Ie − φN

(lncece

)(13)

= ∆E − φN lnceci

(14)

where we have used the equalities φIi = φ

Ie and ci = ce .The equilibrium Φ = 0 occurs when the membrane potential is given by:

∆E = φN lnceci

(15)

This is the well-known formula for the Nernst potential32.Away from equilibrium, the flow f depends on the channel characteristics and is typically

modelled by the Goldman-Hodgkin-Katz equation32 and can be implemented by suitably modi-fying the basic Re flow equation (8)18.

Membrane Ion channels can be modulated (or gated) by ligands or by voltages33. Figure2(c) indicates how the electrodiffusion model can be extended by adding an additional chemicalC component C:G acting as a gate; this is directly analogous to modulating a chemical reactionwith an enzyme27. The dynamics of such gates may also be modelled using the bond graphapproach18,34.

The interaction of two or more different ion channels sharing the same membrane potentialleads to action potentials; the following section shows how a modular approach can be used tocombine gated ion channels.

Modularity: Na+ and K+ ion channelsA strength of the graphical nature of the bond graph approach is that individual modules can beduplicated and connected. In this example, two instances of the ion channel module (Figure 2(c))are combined in Figure 3(b) to model the action potential (Figure 3(a)). The species concentra-tions are encapsulated in the individual modules, but the electrical C components are shared viathe ports. This is a simplified version of the Hodgkin-Huxley35 model of the squid giant axonand the corresponding Na+ and K+ concentrations are used.

10

++−++

−−+−−

Na+

ΔE−

+

External

Internal

K+

Ee

Ei

(a) Schematic of interacting ion channels

0

C:Ee

C:Ei

0

IonChannel:Na IonChannel:K

[Ei]

[Ee][Ee]

[Ei]

(b) Interacting Ion Channels

0.0 0.1 0.2 0.3 0.4 0.5 0.6t

60

40

20

0

20

40

E m

V

(c) Action potential ∆E

Figure 3: Coupled Na+ and K+ ion channels motivated by the Hodgkin-Huxley model of thesquid giant axon18,35. (a) A schematic of a Na+ channel and K+ channel located on the samecharged membrane. (b) A model of this membrane can be created by coupling together twoinstances of the ion channel module (IonChannel) of Figure 2(c); one instance correspondsto the Na+ channel (IonChannel:Na) and one to K+ channel (IonChannel:K). The electricalcomponents C:Ee and C:Ei, corresponding to the external and internal membrane voltages Ee

and Ei within the ion channel module, become the ports [Ee] and [Ei]. The 0 junctions ensurethat the external and internal membrane voltagesEe andEi are shared with the corresponding ionchannel voltages. (c) Action potential. Concentrations (mM) are taken from Table 2.1 of Keenerand Sneyd 20: External. cNa = 437 mM, cK = 20 mM; Internal. cNa = 50 mM, cK = 397 mMand correspond to the model of Hodgkin and Huxley 35 .

11

In the Hodgkin and Huxley 35 model and in reality, the gating variables are dynamically mod-ulated by the membrane potential ∆E. However, for simplicity, we model the gating variables asexternally controlled variables in this example. Accordingly, gating variables GNa and GK arepiecewise constant functions of time:

GK =

{10−6 for 0.3 < t < 0.35

1 otherwise(16)

GNa =

{1 for 0.3 < t < 0.35

4.3× 10−3 otherwise(17)

The time course of the membrane potential ∆E is shown in Figure 3(c) and can be explainedas follows:

t < 0.3: ∆E moves from the initial condition of zero to a resting potential of about −65mV.This corresponds to the value in Table 2.1 of Keener and Sneyd 20; the resting potential dependsnot only on Nernst potentials of Na+ and K+ but also on the values of the gating potential (i.e.their relative permeability).

0.3 < t < 0.35: The Na+ gate opens, causing the membrane potential ∆E to move towardsthe Nernst potential for Na+. This results in the initial depolarisation phase of the action potential.

t > 0.35 The Na+ gate closes and the K+ gate opens, causing ∆E to return to the restingpotential. This completes the repolarisation phase of the action potential.

Redox Reactions and Proton PumpsIn his book Power, Sex, Suicide: Mitochondria and the Meaning of Life36, Nick Lane pointstowards the fundamental role that redox reactions play in biology, stating that “essentially all ofthe energy-generating reactions of life are redox reactions”. One such energy-generating reactionthat within the mitochondrial respiratory chain is

NADH + Q + H+ r NAD+ + QH2 (18)

As noted by Nicholls and Ferguson 22 , “The additional facility afforded by an electrochemicaltreatment of a redox reaction is the ability to dissect the overall electron transfer into two half-reactions involving the donation and acceptance of electrons respectively”. With this in mind,reaction (18) can be divided into the reactions:

NADHr1 2 e –

1 + H+ + NAD+ (19)

2 e –2 + 2 H+ + Q

r2 QH2 (20)

A bond graph representation of this decomposition is given in Figure 4(b).From the bond graph, the reaction potentials of the two half-reactions are:

Φ1 = φNADH − (2E1 + φH + φNAD) (21)Φ2 = 2E2 + 2φH + φQ − φQH2

(22)

12

External

Internal

NADH

NAD+

e-

H+

Q

QH2

H+Pump CoQ10

e−1 e−2

(a) Redox reaction schematic

Re:r11 1 0

0

C:H

0 1 Re:r2 1

0

C:NADH C:NAD C:E1 C:E2 C:Q C:QH2

(b) Redox reaction

Chemical Electrical

External

Internal

Re:rp

10 0

10 0

0

C:E1

0

C:E2

C:He

C:Hi C:Ei

C:Ee

(c) Proton Pump

C:E2

0

0

C:E1

[E1]

[E2]

Redox:rr ProtonPump:pp[E2]

[E1]

(d) Complex I

Figure 4: Redox Reactions and Proton Pump. (a) A schematic of the redox reaction and protonpump. The electrons generated from the redox reaction (e−1 ) are shuttled through the protonpump to pump hydrogen into the intermembrane space. Following this, the electron (e−2 ) is thendelivered to Coenzyme Q10. (b) A bond graph of the redox reaction. Half-reactions (19) and(20) are represented by the reaction components Re:r1 and Re:r2 respectively. The dashedbox corresponds to the electrical part of the bond graph with C:E1 and C:E2 representing theaccumulations of electrons e –

1 and e –2 . The double bonds correspond to a stoichiometry of 2.

(c) Bond graph of the proton pump. Re:rp corresponds to reaction (25). Internal and externalprotons H +

i and H +e have chemical properties represented by C:Hi and C:He respectively and

electrical properties represented by C:Ei and C:Ee respectively. The redox potential differenceE1−E2 drives the protons across the membrane. (d) Bond graph of the redox reaction and protonpump coupled together.

13

Species φ (mV)19 Conc. φ (mV)NAD 188 5.02e-0437 -15NADH 407 7.50e-0537 154Q 675 1.00e-0238 552QH2 -241 1.00e-0238 -365H 0 1.00e-07 -431

Table 3: Faraday-equivalent potentials with pH = 7 and equal concentrations of Q and QH2.The potentials φ are computed from Equation (6).

where E1 and E2 are the potentials associated the electrical capacitors C:E1 and C:E2 and theassociated electrons e –

1 and e –2 . At equilibrium Φ1 = Φ2 = 0. Hence, using the values in Table

3, the voltages corresponding to the capacitors C:E1 and C:E2 are

E1 =1

2(154 + 15 + 431) = 300 mV (23)

E2 =1

2(−365− 552 + 2× 431) = −28 mV (24)

Hence the net electronic potential ∆E = E1 − E2 = 328 mV is available to power a protonpump (note the distinction from the membrane potential ∆E = Ei − Ee in earlier sections).Indeed, reaction (18) occurs within complex I of the mitochondrial respiratory chain (Figure4(a)). Complex I is a “giant molecular proton pump”39 which uses the high-energy electron (e –

1 )from Reaction (19) to generate a proton gradient across the mitochondrial membrane before thelower-energy electron (e –

2 ) is consumed by Reaction (20).In this case, two protons are pumped across the membrane for each electron. Such an

electron-driven proton pump can be modelled by the bond graph of Figure 4(c); the correspond-ing equation is

e –1 + 2 e –

i + 2 H +i

rpe –

2 + 2 e –e + 2 H +

e (25)

The reaction affinity is

Φrp = ∆E −∆p (26)

where we have defined the proton-motive force ∆p to be the sum of the electrical and chemicalpotentials arising from the proton (H+) gradient across the membrane22:

∆p = φe − φi −∆E (27)

where φe and φi are the chemical potentials of the external and internal protons H+e and H+

i . Atequilibrium, Φrp = 0 and it follows that

∆p =∆E2

= 164 mV (28)

14

The PMF of Equation (28) corresponds to the equilibrium value with standard concentrationsof the species in Table 3. This value will change if the concentrations of the species change ordue to potential losses in the Re components when the flows are non zero. In fact the exchangeof species between complexes CI, CIII and CIV, in particular Q and QH2, leads to an “affin-ity equalisation” of PMFs between the three complexes with a typical PMF of approximately230 mV19.

Example: Sodium-Glucose Symporter

Co

Co−S−2Na

Co−2Na

Ci

Ci−S−2Na

Ci−2Na

2Na+2Na+

S S

Out InMembrane (transporter)

ΔE +−

(a) Reaction network

OUT IN

Re:r

61

1 0

Re:r56

1

C:Ci10

Re:r12

0

1 1

Re:r

34

0

0

Re:r45

1

C:SCNai

C:CNai1

1

1

1

0

1

Re:r23

0C:SCNao

C:CNao

Re:r

25

0

C:Nao

0

C:So

0

C:Si

0

C:NaiC:E

1

C:Co

(b) Bond graph

Figure 5: Sodium-Glucose Symporter. (a) A schematic of the reaction network of the transporter.Under physiological conditions, the symporter uses the natural inward electrochemical gradientof Na+ to drive transport of the sugar glucose (S) into the cell, against a chemical gradient. Thetransporter operates via a six-state mechanism (outer loop), with a leakage reaction that causesNa+ to be translocated without coupling to the transport of glucose. (b) A bond graph of thetranspoter. The cycle formed by the reactions in the outer loop (Re:r12, Re:r23, Re:r34,Re:r45, Re:r56 and Re:r61) is driven by the outside and inside concentrations of sodium(C:Nao and C:Nai respectively) and glucose (C:So and C:Si). Re:r25 represents the leakagereaction. The contribution of the membrane potential is given by the C:E component in thedashed red box.

The transport of substrates across a membrane is vital to cellular homeostasis. It is widelyacknowledged that the laws of thermodynamics constrain the direction of flux through a trans-porter; a thermodynamic treatment is given in Hill 41 and a bond graph approaches are devel-oped in Gawthrop and Crampin 42 and Pan et al. 43 . In the case of electrogenic transporters that

15

150 100 50 0 50E mV

1

2

3

4

5f p

A

ModelExperimental

Figure 6: Theoretical and experimental results. The experimental results of Eskandari et al. 40

are compared with the bond graph model with the parameters of Tables 4 and 5 and total numberof transporters NC = 7.5× 107.

translocates a net charge across a membrane, the effects of membrane potential must be consid-ered. This section looks at a particular electrogenic transporter: the Sodium-Glucose TransportProtein 1 (SGLT1) (also known as the Na+/glucose symporter) which was studied experimentallyand explained by a biophysical model40,44,45,46. When operating normally, sugar is transportedfrom the outside to the inside of the membrane driven against a possibly adverse gradient by theconcentration gradient of Na+ (Figure 5(a)).

According to Parent et al. 45, p.64, “The carrier is assumed to bear a charge z = −2 withthe result that the Na+ loaded carrier is electroneutral.” Hence, in terms of Figure 5(b), the twocomplexes corresponding to C:Ci and C:Co bear a charge of −2 whereas the loaded complexesC:CNao, C:SCNao, C:SCNai and C:CNai are electrically neutral.

With reference to the bond graph of Figure 5(b), the only reactions with electrogenic proper-ties are those corresponding to: Re:r61 where the unloaded complex passes from the inside tothe outside of the membrane; Re:r12 where the external Na+ of C:CNao binds/unbinds to thecomplex and Re:r56 where the internal Na+ of C:CNai binds/unbinds to the complex. As dis-cussed by Parent et al. 45 their experimental results suggest the “absence of voltage dependencefor internal Na+ binding, and thus asymmetrical binding sites at the external and internal face ofthe membrane.” In the context of the bond graph model of Figure 5(b), this implies that the reac-tion corresponding to Re:r56 is not electrogenic and thus electrogenic effects are confined to thereactions corresponding to Re:r61 and Re:r12. These considerations give rise to Figure 5(b)where the portion within the dashed box corresponds to electrogenic effects. A non-electrogenicversion of this model is analysed in § 1.1 of Hill 41 and in Gawthrop and Crampin 42 .

16

Parameters

Reac. kf kr keq =kfkr

κ

r12 80000 500 160 10.1796r23 100000 20 5000 202.023r34 50 50 1 505.058r45 800 12190 0.0656276 8080.93r56 10 4500 0.00222222 67.1184r61 3 350 0.00857143 8.67804r25 0.3 0.00091 329.67 0.00610777

Table 4: Reaction Parameters

Species φ (mV)So 61.71Si 61.84Nao 70.42Nai 70.36Co 98.76CNao 104.03SCNao -61.78Ci -28.37CNai -50.86SCNai -61.78

Table 5: Species Parameters, referenced to a standard concentration of 1 M for the substrates(So, Si, Nao, Nai) and to a standard amount of 1 mol for transporter states (Co, CNao, SCNao,Ci, CNai, SCNai)

As is common in the literature, the experimentally derived reaction parameters are expressedin terms of forward kf and reverse kr rate constants whereas bond graph models are parame-terised in terms of the standard potential φ (3) and reaction rate constant κ (8). Thus theseparameters need to be converted into equivalent bond graph parameters. The thermodynamicallyconsistent reaction kinetic parameters listed in the first two columns of Table 4 (given in Figure6B of Eskandari et al. 40), and are converted into ten equivalent thermodynamic constants usingthe methods described in Gawthrop and Crampin 42 .

The total number of transportersNC per unit area corresponding to the experimental situationreported by Eskandari et al. 40 is a key parameter. It not only determines the steady state flowsbut also the transient time constants; here it is adjusted to fit the data shown in Figure 6. Thefitted theoretical results are compared with experimental values in Figure 6.

No model is definitive but rather is a step towards understanding a system and suggestingfurther experiments. For example, the model presented here assumes that the transmembrane

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current is entirely due to tranlocated charge. Charge leakage could be included as an ohmicresistance associated with the electrical C:E component and the corresponding resistance fittedto the model. As well as potentially allowing a better fit to the data, such a refined model wouldsuggest further experimental verification.

The inhibition of the Na+/glucose symporter has been suggested as a treatment for type 2diabetes mellitus47. It is hoped that physically based models, such as that developed here, willeventually lead to computational approaches to understanding such diseases and evaluating treat-ment strategies.

ConclusionThe study of bioelectrical systems can be greatly facilitated by a modelling framework that isboth graphical and physically consistent. In this perspective, a series of examples has illustratedhow Network Thermodynamics, as implemented with bond graphs, provides a graphical frame-work for modelling bioelectrical systems while ensuring thermodynamic consistency. This ismade possible by treating electricity and chemistry in a unified fashion where potentials andflows are given in the same electrical units.

The method is naturally modular and any C component within a system can be used as aport to connect to other subsystems. This provides a flexible hierarchical approach to creatinglarge models of bioelectrical systems. Moreover, the use of energy ports allows one particularmodel of a subsystem to be replaced by another with the same ports, allowing one to swapsubmodels within a large model to allow high detail in one part whilst simplifying the rest. Thisapproach is made possible by the fact that even low-resolution submodels are thermodynamicallyconsistent; for example it is possible to build a low resolution but physically plausible model ofthe mitochondrial electron transport chain25.

Energy flow in the form of molecules, protons and electrons shapes cell evolution48,49; inthe same vein, energy is central to synthetic biology 50,51,52,53. Hence it is appropriate to use theenergy-based modelling approach to bioelectricity to not only evaluate natural systems from anevolutionary point of view but also to evaluate novel bioelectrical approaches to synthetic biologysuch as, for example, the generation of electricity from waste and making microbial communitiesform specific patterns54.

A set of Python-based tools https://pypi.org/project/BondGraphTools/ isavailable to assist model building55 and to provide avenues for the automation of model devel-opment.

AcknowledgementsPeter Gawthrop would like to thank the Melbourne School of Engineering for its support via aProfessorial Fellowship, and both authors would like to thank Edmund Crampin for help, adviceand encouragement, and Peter Cudmore for developing the BondGraphTools software package.This research was in part conducted and funded by the Australian Research Council Centre of

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Excellence in Convergent Bio-Nano Science and Technology (project number CE140100036).The authors would like to thank the reviewers and editor for helpful comments which improvedthe paper.

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