rspa.royalsocietypublishing.org

ResearchCite this article: Gawthrop PJ, Crampin EJ.2017 Energy-based analysis of biomolecularpathways. Proc. R. Soc. A 473: 20160825.http://dx.doi.org/10.1098/rspa.2016.0825

Received: 8 November 2016Accepted: 26 May 2017

Subject Areas:mathematical modelling, biochemistry,biomedical engineering

Keywords:network thermodynamics, biomolecularsystems, bond graph, reaction kinetics

Author for correspondence:Peter J. Gawthrope-mail: peter.gawthrop@unimelb.edu.au

Electronic supplementary material is availableonline at https://doi.org/10.6084/m9.figshare.c.3799426.

Energy-based analysis ofbiomolecular pathwaysPeter J. Gawthrop1 and Edmund J. Crampin1,2,3,4

1Systems Biology Laboratory, 2School of Mathematics and Statistics,3School of Medicine, and 4ARC Centre of Excellence in ConvergentBio-Nano Science, Melbourne School of Engineering, University ofMelbourne, Victoria 3010, Australia

PJG, 0000-0002-6029-515X

Decomposition of biomolecular reaction networksinto pathways is a powerful approach to the analysisof metabolic and signalling networks. Currentapproaches based on analysis of the stoichiometricmatrix reveal information about steady-state massflows (reaction rates) through the network. Inthis work, we show how pathway analysis ofbiomolecular networks can be extended using anenergy-based approach to provide informationabout energy flows through the network. Thisenergy-based approach is developed using theengineering-inspired bond graph methodologyto represent biomolecular reaction networks. Theapproach is introduced using glycolysis as anexemplar; and is then applied to analyse the efficiencyof free energy transduction in a biomolecular cyclemodel of a transporter protein [sodium-glucosetransport protein 1 (SGLT1)]. The overall aim of ourwork is to present a framework for modelling andanalysis of biomolecular reactions and processeswhich considers energy flows and losses as well asmass transport.

1. IntroductionThe term ‘pathway analysis’ is used very broadlyin systems biology to describe several quite distinctapproaches to the analysis of biomolecular networks[1–3] and often the definition of a ‘pathway’ is somewhatnebulous. In this work, we are concerned with thosepathways defined in terms of the stoichiometric analysisof biomolecular reaction networks [4–8]; in particular,the null space1 of an appropriate stoichiometric matrix

1The mathematical concept of a null space is given a biologicalinterpretation in appendix B.

2017 The Author(s) Published by the Royal Society. All rights reserved.

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is used to identify the pathways of a biomolecular network. Two alternative concepts ofpathways: elementary modes and extreme pathways are compared and contrasted by Papin et al. [3].Computational issues are considered by Schuster & Hilgetag [9], Pfeiffer et al. [10], Schilling et al.[11] and Schuster et al. [12]. A brief introduction to the relevant concepts appears in appendix B.

These approaches have proven to be very useful in determining network properties andemergent behaviour of biomolecular reaction networks in terms of the pathway ‘building blocks’of these networks. However, these approaches are solely focused on mass flows of biomolecularreaction networks. However, to date, little attention has been given to the identification andanalysis of pathways in the context of energy flows in biomolecular reaction networks. This paperextends the pathway concept to include such energy flows using an engineering-inspired method:the bond graph.

Like engineering systems, living systems are subject to the laws of physics in general andthe laws of thermodynamics in particular [13–17]. This fact gives the opportunity of applyingengineering science to the modelling, analysis and understanding of living systems. The bondgraph method of Paynter [18] is one such energy-based engineering approach [19–24] whichhas been extended to include chemical systems [20,25], biological systems [26] and biomolecularsystems [27–32]. A brief introduction to the bond graph approach appears in appendix A.

Applying the bond graph method to modelling biomolecular pathways moves the focus frommass flow to energy flow. Hence this paper brings together stoichiometric pathway analysiswith energy-based bond graph analysis to identify the pathways of steady-state free energytransduction in biomolecular networks. Although the bond graph approach is well establishedin the field of engineering, and the stoichiometric pathway analysis approach well established inthe field of biochemical analysis, they have not hitherto been brought together. As illustrated inthe sodium-glucose transport protein example of §6, this interdisciplinary synthesis gives newinsight into the energetic behaviour of living systems.

Section 2 introduces the bond graph approach to pathway analysis using glycolysis asan exemplar. As a preliminary to the energy-based approach, §3 discusses the stoichiometricapproach to pathway analysis. Section 4 combines the energy-based reaction analysis of §2with the stoichiometric pathway analysis of §3 to give an energy-based pathway analysis ofbiomolecular systems. Section 5 looks at the generic biomolecular cycle transporter model of Hill[14] focusing on the energy transduction aspects; in particular, a two-pathway decompositionprovides insight into free-energy dissipation and efficiency. Section 6 uses this two-pathwaydecomposition to look at a particular transporter—the sodium-glucose transport protein 1(SGLT1)—using parameters drawn from the experimental results of Eskandari et al. [33]. Section7 draws some conclusions and indicates future research directions. Appendix A provides a shortintroduction to those features of the bond graph approach necessary to understand this paperand appendix B similarly introduces systems biology. The complete equations describing theexamples are given in the electronic supplementary material.

2. Energy-based reaction analysisAs reviewed in the Introduction, and as discussed by Gawthrop & Crampin [29,31] andappendix A, chemical equations can be written in the form of bond graphs to enable energy-basedanalysis. As an introductory and illustrative example, the glycolysis example from the seminalbook of Heinrich & Schuster [5, fig. 3.4] has the following chemical equations:

ATPr1−⇀↽− ADP ATP + F6P

r2−⇀↽− ADP + F2P F2Pr3−⇀↽− F6P

GLC + ATPr4−⇀↽− ADP + G6P G6P

r5−⇀↽− F6P ATP + F6Pr6−⇀↽− ADP + 2TP

2ADP + TPr7−⇀↽− Pyr + 2ATP ATP + AMP

r8−⇀↽− 2ADP G6Pr9−⇀↽− G1P

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r4

ATP ADP

r5

r2 r3

r6

ADPATP

r7

2ADP 2ATP

r9

GLC Pyr

G1P

G6P

TP

TPF6P

F2PADP

ATP

r1

ADPAMP

r8

0 1

1

Re:r2 Re:r3

1

0

1 0

0

010

Re:r9

0 0 0

0 00 0

Re:r5

0 01

Re:r4

Re:r8

0 1

C:F6P

Re:r6

C:PyrC:TPC:G6PC:GLC

Re:r1

C:ADP C:ATP

1

C:F2P

Re:r7

C:G1P

1 10

C:AMP(b)

(a)

Figure 1. Glycolysis example from Heinrich & Schuster [5]. (a) Biomolecular reaction diagram of the glycolysis pathway.(b) Bond graph representation of the biomolecular pathway.

These equations can be represented by the biomolecular reaction diagram of figure 1a or theequivalent bond graph of figure 1b. Briefly, each reaction is represented by an Re component andeach species by a C component. The (Gibbs) energy flows are represented by bonds ⇁ whichcarry both chemical potential and molar flow. Bonds are connected by 0 junctions which imply acommon chemical potential on each bond and 1 junctions which imply a common molar flow oneach bond. C components store, Re components dissipate and bonds and junctions transmit Gibbsenergy. Further details are given in appendix A and references [29–32].

Closed biomolecular systems are described in stoichiometric form as:

X = NV, (2.1)

where X is the nX × 1 system state, V the nV × 1 vector of reaction flows and N is the nX × nV

stoichiometric matrix which can be derived from the bond graph representation [29,32]. In the

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case of the example of figure 1:

X =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xGLCxPyrxATPxADPxG1PxG6PxF6PxTPxF2P

xAMP

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, N =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 −1 0 0 0 0 00 0 0 0 0 0 1 0 0

−1 −1 0 −1 0 −1 2 −1 01 1 0 1 0 1 −2 2 00 0 0 0 0 0 0 0 10 0 0 1 −1 0 0 0 −10 −1 1 0 1 −1 0 0 00 0 0 0 0 2 −1 0 00 1 −1 0 0 0 0 0 00 0 0 0 0 0 0 −1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and V =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vr1vr2vr3vr4vr5vr6vr7vr8vr9

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(2.2)

In the case of mass-action kinetics [8], the reaction flows are generated by the formula:

V = κ

(exp

Af

RT− exp

Ar

RT

), (2.3)

where Af and Ar are the forward and reverse2 reaction affinities. As discussed by Gawthrop &Crampin [29,31], these affinities are given by:

Af = NfTμ and Ar = NrTμ (2.4)

where superscript ‘T’ indicates matrix transpose and μ is the (vector of) chemical potentials wherethe ith element is a logarithmic function of the ith element xi of X:

μi = RT ln Kixi (2.5)

where Ki is a species-specific positive constant. Nf and Nr are the forward and reversestoichiometric matrices, respectively, and conservation of energy requires that:

− Nf + Nr = N. (2.6)

By definition, all stoichiometric parameters, that is the elements Nfij of Nf and the elements Nr

ij ofNr have the following properties:

Nfij is integer Nf

ij ≥ 0 Nrij is integer Nr

ij ≥ 0 (2.7)

As discussed by Gawthrop & Crampin [31], open biomolecular systems can be describedand analysed using the notion of chemostats [31,34]. Chemostats have two biomolecularinterpretations:

(i) one or more species are fixed to give a constant concentration (for example, under aspecific experimental protocol); this implies that an appropriate external flow is appliedto balance the internal flow of the species,

(ii) as a C component with a fixed state imposed on a model in order to analyse itsproperties [30].

Additionally, in the context of a control systems analysis, the chemostat can be used as an idealfeedback controller, applied to species to be fixed with setpoint as the fixed concentration andcontrol signal an external flow.

When chemostats are present, the reaction flows are determined by the dynamic part of thestoichiometric matrix. In this case, the stoichiometric matrix N can be decomposed as the sum oftwo matrices [31]: the chemostatic stoichiometric matrix Ncs and the chemodynamic stoichiometric

2The terms ‘forward’ and ‘reverse’ often correspond to ‘substrate’ and ‘product’ or ‘input’ and ‘output’, respectively; they areused here for consistency with previous work, to avoid ambiguity and to recognize that reactions may be reversible.

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matrix Ncd, where N = Ncs + Ncd and:

Ncs = IcsN Ncd = IcdN Icsii =

{1 if i ∈ Ics

0 if i �∈ IcsIcdii =

{0 if i ∈ Ics

1 if i �∈ Ics.(2.8)

Note that Ncd is the same as N except that the rows corresponding to the chemostat variables areset to zero. In this case, Ncd is given by N of equation (2.2) with rows 1–4 set to zero.

With these definitions, an open system can be expressed as

X = NcdV. (2.9)

The stoichiometric properties of Ncd, rather than N, determine system properties whenchemostats are present. Using equation (2.8), X in equation (2.9) can also be written as

X = NV − NcsV = NV + Vs, where Vs = −NcsV (2.10)

Vs can be interpreted as the external flows required to hold the chemostat states constant. Thereaction flows are given by the same formulae (2.3) and (2.4) as closed systems.

3. Stoichiometric pathway analysisAs discussed in the textbooks [4–8] and appendix B, the (non-unique) nV × nP null-space matrixKp of the open system stoichiometric matrix Ncd has the property that

NcdKp = 0, where nP = nV − r (3.1)

and r is the rank of N. Furthermore, if the reaction flows V are constrained in terms of the nP

pathway flows Vp as

V = KpVp, (3.2)

then substituting equation (3.2) into equation (2.1) and using equation (3.1) implies that X = 0.This is significant because the biomolecular system of equation (2.1) may be in a steady state forany choice of Vp.

As mentioned above, Kp is not unique: there are many possible approaches to choosing Kp suchthat equation (3.1) holds. As discussed by, for example, Pfeiffer et al. [10], Kp can be computedin such a way as to give useful features such as integer entities and maximal number of zeroelements; moreover, if all reactions are irreversible, the columns of Kp must correspond to aconvex space.3 As discussed in §4, the analysis of this paper requires that all elements Kpij ofKp must satisfy the same conditions as those on Nf

ij and Nrij (2.7), namely:

Kpij is integer Kpij ≥ 0. (3.3)

For this reason, Kp is referred to as the positive-pathway matrix (PPM) in the sequel. This does notimply that reactions are assumed to be irreversible; however, one approach to generating such aKp is using software such as metatool [10] as if all reactions were irreversible.

In the case of the glycolysis system of figure 1, Heinrich & Schuster [5] choose the chemostatsto be: GLC, Pyr, ATP and ADP. Using the algorithm of Pfeiffer et al. [10],4 Kp for the glycolysis

3This is related to the classical Cone lemma [35, ch. 10].

4The Octave [36] version of metatool from http://pinguin.biologie.uni-jena.de/bioinformatik/networks/metatool/metatool5.0/metatool5.0.html was used.

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system of figure 1 was computed as

Kp =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 00 1 00 1 00 0 10 0 10 0 10 0 20 0 00 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (3.4)

The pathways corresponding to the three columns of Kp, (referring to figure 1a), are: {r1}, {r2 r3}and {r4 r5 r6 2r7}. The latter pathway is marked on the bond graph of 1b using bold bonds. Aspointed out by Heinrich & Schuster [5], these three pathways are independent insofar as theyhave no reactions in common; as will be seen in the rest of this section and in the examples of §§5and 6, this is not always the case.

The number of pathways, and their independence, not only depend on the network structure,but also on the choice of chemostats. To illustrate this point, consider the same glycolysis systemof figure 1, but choose an additional chemostat to be G1P. In this case, Kp is given by

Kp =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 00 1 0 00 1 0 00 0 1 10 0 1 00 0 1 00 0 2 00 0 0 00 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.5)

The first three columns of equation (3.5) are identical to the three columns of equation (3.4). Thefourth column corresponds to the additional pathway {r4 r9}, which shares reaction r4 with thepathway {r4 r5 r6 2r7} corresponding to the third column of equations (3.5) and (3.4).

4. Energy-based pathway analysisThis section combines the energy-based reaction analysis of §2 with the stoichiometric pathwayanalysis of §3 to give an energy-based pathway analysis of biomolecular systems.

As discussed by Gawthrop & Crampin [29,31], equations (2.1), (2.4) and (2.6) can besummarized by the diagram of figure 2a. The key point here is that the energy flow from thereaction components represented by the vectors of covariables Ar and V is transformed withoutenergy loss by T F : Nr into the energy flow to the C components represented by the vectors ofcovariables μ and Xr and the energy flow from the C components represented by the vectorsof covariables μ and Xf is transformed without energy loss by T F : Nf into the energy flow toreaction components represented by the vectors of covariables Af and V. Dissipation of energyoccurs at the reaction components Re. The challenge for an energy-based pathway analysis is todetermine the energy flow associated with each pathway.

The key result of the stoichiometric pathway analysis summarized in §3 is equation (3.2)relating the pathway flow vector Vp to the reaction flow vector V by V = KpVp. To bring thisresult into the energy domain, we define the forward and reverse pathway affinities Af

p and Arp,

respectively, and the pathway affinity Ap, as

Afp = Kp

TAf Arp = Kp

TAr Ap = Afp − Ar

p = KpT(Af − Ar) = Kp

TA. (4.1)

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O

m

mm

m

mm

mX

m m

X fX r

X r

X

X f

Xr Xf

X

T F:N r T F:N f

T F:N r T F:N f

T F:Kp T F:KpT F:N r

pT F:N f

p

V

Af

V

Ar

V s

O

V s

Af

VsAr VV

Vp Vp VpVp

Arp

ArpAf

pAf

p

O

Re

(b)(a) (c)

C

C

C

Figure 2. Energy-based analysis. (a) Reaction-based system [31]. The bond symbols ⇁ correspond to vectors of bonds; C,Re andO correspond to arrays of C, Re and 0 components; the two T F components represent the intervening junctionstructure comprising bonds, 0 and 1 junctions and TF components. Nf and Nr are the forward and reverse stoichiometricmatrices, respectively. Vs represents the chemostatic flows. (b) The positive-pathway matrix Kp maps the nP positive-pathwayflows Vp onto the nV reaction flows V and also maps the nV reaction affinities onto the nP positive-pathway affinities. PWconceptually represents the non-linear function generating the steady-state positive-pathway flows in terms of the positive-pathway affinities; unlikeRe, it does not have a diagonal structure. (c) T F components have been merged using Nfp = NfKpand Nrp = NrKp.

With these definitions, we can define powers (energy flows) associated with pathways:

Pfp = Af

pTVp = (Kp

TAf)TVp = AfTKpVp = AfTV = Pf (4.2)

similarly Pfp = Ar

pTVp = (Kp

TAr)TVp = ArTKpVp = ArTV = Pr. (4.3)

Thus, the net forward pathway energy flow Pfp equals the net forward reaction energy flow Pf and

the net reverse pathway energy flow Pfp equals the net reverse reaction energy flow Pr. Hence Kp

can be considered as an energy transmitting transformer for both forward and backward pathwayenergies and is represented in figure 2b by the symbol T F : Kp.

Figure 2c can be obtained from figure 2b by combining equations (2.1) and (2.6) with equation(3.2), and equations (4.1) with equations (2.4) to give:

X = NpVp Afp= Nf

pTμ Ar

p = Nrp

Tμ, (4.4)

where Np = NKp Nfp = NfKp Nr

p = NrKp. (4.5)

Equation (4.5) defines the forward Nfp and reverse Nr

p pathway stoichiometric matrices; using

conditions (2.7) and (3.3), it follows that Nfp and Nr

p have positive integer elements:

Nfpij

is integer Nfpij

≥ 0 Nrpij

is integer Nrpij

≥ 0. (4.6)

A property of Np follows from combining equations (4.5), (2.8) and (3.1); in particular, Np maybe rewritten as

Np = (Ncs + Ncd)Kp = NcdKp = IcsNKp (4.7)

Hence Np has the property that the only non-zero rows correspond to the chemostats.

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In the case of the example of figure 1:

Np =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 −10 0 2

−1 −1 21 1 −20 0 00 0 00 0 00 0 00 0 00 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, Nfp =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 10 0 01 1 20 0 40 0 00 0 10 1 10 0 20 1 00 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and Nrp =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 00 0 20 0 41 1 20 0 00 0 10 1 10 0 20 1 00 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(4.8)

Using equation (4.4) and Np from equation (4.8), the affinity associated with each of the threepathways is, as expected:

Ap1 = Ap2 = μATP − μADP Ap3 = (μGLC + 2μADP) − (2μPyr + 2μATP) (4.9)

Because conditions (4.6) agree with conditions (2.7), Nfp and Nr

p could correspond to theforward and reverse stoichiometric matrices of the following three reactions:

ATPp1−⇀↽−PP

ADP

ATP + CS2p2−⇀↽−PP

ADP + CS2

GLC + 2ADP + CS3p3−⇀↽−PP

2Pyr + 2ATP + CS3,

where CS2 = F6P + F2P and CS3 = 2ATP + 2ADP + G6P + F6P + 2TP. However, some care mustbe taken in this interpretation. Firstly, the kinetics are not mass-action even if the originalequations correspond to mass-action kinetics. Secondly, and unlike conventional reactions,these three reactions are not independent; the consequences of this fact are discussed in §5.Nevertheless, such a pathway decomposition provides insight into the energy flows in a reactionnetwork. This is illustrated using a generic example of Hill [14] in §5 and using a specific example,the sodium-glucose transport protein of Eskandari et al. [33], in §6.

5. Example: free energy transduction and biomolecular cyclesIn his classic monograph, ‘Free energy transduction and biomolecular cycle kinetics’ Hill [14]discusses how the difference between the concentration of a species Mo outside a membrane andthe concentration of the same species Mi inside the membrane can be used to transport anotherspecies Li inside the membrane to outside the membrane Lo via a large protein molecule with twoconformations E and E� the former allowing successive binding to Mi and Li and the latter to Mo

and Lo. This biomolecular cycle is represented by the diagram of figure 3a which corresponds tothat of Hill [14, fig. 1.2(a)]. There are seven reactions

Mi + Eem−⇀↽− EM Li + EM

lem−−⇀↽−− LEM LEMlesm−−⇀↽−− LE�M LE�M

esm−−⇀↽−− Lo + E�M

E�Mes−⇀↽− Mo + E� E�

e−⇀↽− E EMslip−−⇀↽−− E�M

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Mi

Li

E E

Mo

Lo

E MEM

LEM LE M

slip

loop

Re:

e

0

1

Re:em

0

1

Re:lem

0 0

0

0

Re:esm

1

Re:es

1

Re:

slip

Re:

lesm

C:Mo

C:Lo

C:Mi

C:Li

C:E

C:LEM

C:EM

C:LEsM

C:EsM

C:Es

0

00

0

(b)(a)

Figure 3. Free energy transduction and biomolecular cycles: model. (a) As discussed in the text, this is a generic model of atransmembrane transporter due to Hill [14] and based on the conformations of the protein E which uses the chemical gradientof M to pump L across the membrane against an adverse gradient. An ideal cycle would have no ‘slippage’: the link from EM toE�Mwouldnot exist. As discussed in the text, twopathways: loop and sliphavebeenmarked. (b) Thebondgraph is geometricallythe same as (a) but gives a precise description. (Es is used in place of E� for syntactical reasons).

where the last reaction is the so-called slippage5 term in which the enzyme changesconformation without transporting species L. The original Hill diagram does not name the sevenreactions, they are named here to provide a link to the bond graph which explicitly namesreactions.

The corresponding bond graph appears in figure 3b where E� is replaced by Es for syntacticalreasons. The bond graph clearly shows the cyclic structure of the chemical reactions and isgeometrically similar to the diagram of figure 3a. As discussed by Hill [14], the four species Mo,Mi, Lo and Li, are assumed to have constant concentration: therefore, they are modelled by fourchemostats.

The states X, stoichiometric matrix N and reaction flows V of this system are

X =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xLi

xLo

xMi

xMo

xE

xEM

xLEM

xE�

xE�M

xLE�M

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, N =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 −1 0 0 0 0 00 0 0 1 0 0 0

−1 0 0 0 0 0 00 0 0 0 1 0 0

−1 0 0 0 0 1 01 −1 0 0 0 0 −10 1 −1 0 0 0 00 0 0 0 1 −1 00 0 0 1 −1 0 10 0 1 −1 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

and V =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vem

vlemvlesmvesm

ves

ve

vslip

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (5.1)

5The term ‘slippage’ was used in this context by Terrell L. Hill in his seminal book [14]. An ideal cycle would have no slippageand the link from EM to E�M would not exist in figure 3a.

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The matrix Ncd is the same as N but with the first four rows (corresponding to the fourchemostats) deleted. The rank of Ncd is r = 5 corresponding to the four chemostats and theconserved moiety of the remaining states; thus the null space has dimension nV − r = 2.

As discussed in §4, two positive pathways were identified both by bond graph pathwayanalysis and using metatool.6 The corresponding PPM Kp is

KTp =

(1 1 1 1 1 1 01 0 0 0 1 1 1

). (5.2)

Apart from the ordering of the two columns, the PPM for this system is unique. These twocolumns correspond to the path involving the six reactions: {e em lem es esm lesm} and the upperloop involving the four reactions: {e em es slip}. These are the only positive pathways for thissystem. For convenience, the first (outer) pathway will be named loop and the second pathway(involving the ‘slip’ reaction) will be named slip; this nomenclature is indicated in figure 3a.

For the purposes of illustration, the thermodynamic constants of the 10 species are taken tobe unity, the total amount of E, EM, LEM, E�, E�M and LE�M is taken as Etot = 10, the amount ofMo and Li is taken as unity, the amount of Lo as two and Mi is variable. The rate constant of theslippage reaction is varied in the sequel, and that of the other reactions is 10.

The system was simulated with a slow change of the amount of Mi from 2 to 100 such thatthe system was effectively in a steady state for each value of the amount of Mi; these values werethen refined using a steady-state finder. The conserved moiety was automatically accounted forusing the method of Gawthrop & Crampin [29, §3(c)]. The results were checked using explicitexpressions for the steady-state values using the software of Qi et al. [37], which is based onthe method of King & Altman. The system equations are given in the electronic supplementarymaterial.

Figure 4 shows the two pathway flows vP1 and vP

2 against the amount xMi of Mi for four valuesof κslip. As derived in the electronic supplementary material, the maximum flow when the κslip = 0is given by vmax = 100/21 = 4.76 and this value is also plotted on each graph.

The purpose of this cycle is to use the excess of M to pump L against the concentration gradient.In addition to viewing the cycle as a mass transport system, it may also be viewed as a system fortransducing free energy from M to L. From this point of view, it is interesting to investigate the(steady-state) efficiency η of the energy transduction [14, §1.4], which we define as

η = PL

PMwhere PL = (μLo − μLi)vL and PM = (μMi − μMo)vM (5.3)

The simulated η is plotted against normalized flow vL = vLo/vmax in figure 5a for the four valuesof κslip and reveals three features:

(i) with negligible slip, the maximum efficiency occurs at low flow rates.(ii) the efficiency decreases as κslip increases

(iii) the flow rate corresponding to maximum efficiency increases with κslip.

To further investigate the source of the inefficiency, the pathway dissipation Pp was computedand normalized to give Pp = Pp/PM. This is plotted for each pathway, together with the totalnormalized dissipation, against the normalized flow vL of L in figure 5b,d for three values of κslip.As can be seen, the main loop pathway dissipation increases with flow of L with all of the freeenergy associated with M dissipated at the maximum flow rate. By contrast, the slippage pathwaydissipation decreases with flow of L with all of the free energy associated with M dissipated at zeroflow rate. The combined normalized dissipation thus has a minimum (i.e. energy transduction ismost efficient) at an intermediate flow rate dependant on κslip.

Figure 6 is similar to figure 5 except that the efficiency η and normalized energy flows Pp

are plotted against Mi in the positive flow regime. The minima occur at values of Mi towards

6The Octave [36] version of metatool from http://pinguin.biologie.uni-jena.de/bioinformatik/networks/metatool/metatool5.0/metatool5.0.html was used.

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10xMi

10xMi

10xMi

10xMi

loop

slipvmax

kslip = 10–3 kslip = 10–2

kslip = 10–1 kslip = 1

1

10–1

10–2

10–3

1

10–1

10–2

10–3

VP

1

10–1

10–2

10–3

1

10–1

10–2

10–3

VP

(b)(a)

(c) (d )

Figure 4. Free energy transduction and biomolecular cycles: positive flow regime. (a) The flow in the two pathways labelledloop and slip in figure 3a are shown together with the maximum flow vmax for slippage coefficient κslip = 10−3. (b)–(d) Thesame information for three further values of the slippage coefficient κslip.

the lower end of the positive flow regime. This reflects the fact that, as shown in figure 4, largechanges in flow vP correspond to small changes in concentration xMi towards the lower end ofthe positive flow regime; for larger values of xMi , the effect of changes in xMi on vP is smaller.

As mentioned in §4, the reactions corresponding to the two pathways of this exampleare not independent. To examine the consequences of this interaction, figure 7 focuses onthe system behaviour in the crossover region where the two pathway flows have oppositedirections. Figure 7a shows the two mass flows which have opposite sign in the crossover region0.42 < Mi < 2.62 delineated by the two vertical lines. Within this region, figure 7b shows that theindividual pathway energy dissipations may be negative; of course, as indicated in figure 7b, thetotal dissipation remains positive. This behaviour is a consequence of the intersection of the twopathways leading to a non-diagonal PW (figure 2). Furthermore, the sum of the two pathwayflows, corresponding to the flow of Mi and Mo, also changes sign thus causing the associatedenergy flow PM (5.3) to also change sign. Thus normalizing the pathway energy transduction inthe crossover region using equation (5.3) is not helpful. However, the normal operating regionof such a biomolecular cycle is outside the crossover region and hence this situation does notnormally arise.

6. Example: the sodium-glucose transport protein 1The SGLT1 (also known as the Na+/glucose transporter) was studied experimentally by Parentet al. [38] and explained by a biophysical model [39]; further experiments and modelling were

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0

0.2

0.4

0.6

0.8

1.00.0010.010.1

loopsliptotal

vLvL

Pp

PpPp

loopsliptotal

10–3 10–2 10–1 1

0.2 0.4 0.6 0.8 1

10–3 10–2 10–1 1

10–3 10–2 10–1 1

1

10–1

10–2

1

10–1

10–2

1

10–1

10–2

(b)(a)

(c) (d )

h

loopsliptotal

Figure 5. Free energy transduction and biomolecular cycles: energy-based analysis. (a) The free energy transduction efficiencyη is plotted against normalized flow vL of L for the three values of the slippage coefficient κslip used in (b–d). (b–d) Thenormalized pathway free energy dissipation Pp is plotted against normalized flow VL of L for the two pathways loop and slipmarked on figure 3a together with the total normalized free energy dissipation. (a) Efficiency η for different κslip; normalizeddissipation (b) Pp: κslip = 10−3, (c) Pp: κslip = 10−2 and (d) Pp: κslip = 10−1.

conducted by Chen et al. [40]. Eskandari et al. [33] examined the kinetics of the reverse mode usingsimilar experiments and analysis to Parent et al. [38,39] but with reverse transport and currents.This example looks at a bond graph-based model of SGLT1 based on the model of Eskandariet al. [33]. For simplicity, it is assumed that the membrane potential is zero and thus there are noelectrogenic effects.

The model of Eskandari et al. [33, fig. 6B], given in diagrammatic form in figure 8a is basedon the six-state biomolecular cycle of fig. 2 of Parent et al. [39]. When operating normally, sugaris transported from the outside to the inside of the membrane driven against a possibly adversegradient by the concentration gradient of Na+. The diagram of figure 8a is similar to the Hillmodel of figure 3a. Apart from the renaming of components, and the reversal of inside andoutside, the major difference is that the single driving molecule Mi and Mo replaced by two drivingmolecules 2Na+

o and 2Na+i . This change is reflected in the bond graph of 8b by the double bonds.

This also means that the stoichiometric matrix N is the same as that of equation (5.1) except thatthe rows corresponding to Na+

o and Na+i are multiplied by 2. However, as Na+

o and Na+i are

chemostats, this change does not affect the PPM Kp which is still given by (5.2). The actual matricesare given in the electronic supplementary material.

The seven sets of reaction kinetic parameters are given in Eskandari et al. [33, fig. 6B] andlisted in the first two columns of table S1 of the electronic supplementary material and the thirdcolumn gives the corresponding equilibrium constants. The vector of seven equilibrium constantsKeq is converted into the vector of 10 thermodynamic constants K of table S2 of the electronicsupplementary material using the formula of Gawthrop et al. [30]:

Keq = Exp (−NTLn K), (6.1)

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0

0.2

0.4

0.6

0.8

1.00.001

0.01

0.1

loop

slip

total

2.0 2.5 3.0 3.5 4.0xMi

2.0 2.5 3.0 3.5 4.0xMi

xMi

2.0 2.5 3.0 3.5 4.0

2.0 2.5 3.0 3.5 4.0

xMi

1

10–1

10–2

1

10–1

10–2

1

10–1

10–2

Pp

Pp

Pp

(a)

(c) (d )

(b)

h

loop

slip

total

loop

slip

total

Figure 6. Free energy transduction and biomolecular cycles: energy-based analysis. Figure 5 is replotted with xMi replacing VL.(a) Efficiency η for different κslip; normalized dissipation (b) Pp: κslip = 10−3, (c) Pp: κslip = 10−2 and (d) Pp: κslip = 10−1.

–30 1 2 3 4 0 1 2 3 4

–2

–1

0

1

2

3loopslip

total

–0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

P

xMixMi

VP

(b)(a)

Figure 7. Mass and energy flows in the crossover region. For sufficiently small Mi , the cycle no longer acts as a transporter asthe flows in the two pathways labelled loop and slip in figure 3a become negative. This figure examines the crossover regime(delineated by the two vertical lines)where one pathway flow is positive and the other negative—with a large value (κslip = 1)of slippage to exaggerate the effect. Note the negative pathway dissipations when the sign of the pathway flows is different—this is due to the intersection of the two pathways leading to a non-diagonalPW . (a) Mass flows and (b) energy flows.

where N is the 10 × 7 stoichiometric matrix. The corresponding rate constants κ are thencomputed as discussed by Gawthrop et al. [30] and listed in the final column of table S1 of theelectronic supplementary material. The system was simulated as described in §5 and the systemequations are given in the electronic supplementary material.

As in figure 4, figure 9a shows the two pathway flows and, as in figure 5, figure 9bshows the corresponding pathway dissipation. As in §5, but using parameters corresponding to

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CNao CNai

CiCo

2Nai2Nao

SiSo

1

(a) (b)

2

3 4

6

5

Re:

r61

0

1

Re:r12

0

1

Re:r23

0 0

0

0

Re:r45

1

Re:r56

1

Re:

r34

Re:

r25

C:Nai

C:Si

C:Nao

C:So

C:Co

C:SCNao

C:CNao

C:SCNai

C:CNai

C:Ci

0

00

0

1

2

3 4

5

6

SCNao SCNai

Figure 8. The SGLT1. (a) Diagram with the six states marked. (b) The corresponding bond graph.

xNao VS

VP Pp

1

1

10–1

10–1

10–2

1

10–1

10–2

10–4 10–3 10–2 10–1 110–3

(b)(a)

loopsliptotal

loopslipvmax

Figure9. TheSGLT1. (a) Simulatedflows (normalized) in the twopathways labelled loop and slip in figure 3a. (b) Thenormalizedpathway dissipations as a function of normalized flows.

experimental data, the combined normalized dissipation has a minimum (i.e. energy transductionis most efficient) at an intermediate flow rate.

7. ConclusionIt has been shown that standard methods of mass-flow pathway analysis can be extended toenergy-flow pathway analysis making use of the bond graph method arising from engineeringscience. The method has been applied to a glycolysis example of Heinrich & Schuster [5, fig. 3.4]and the biomolecular cycle of Hill [14, fig. 1.2(a)] to enable comparison with standard approaches.

The analysis of the biomolecular transporter cycle was shown to apply to a model of the SGLT1based on the experimentally determined parameters of Eskandari et al. [33, fig. 6B]. Intriguingly,it was found that the rate of energy dissipation has a minimum value at a particular normalizedflow rate which in turn corresponds to a particular driving concentration. This minimum is dueto the interaction of a number of factors including the system parameters, the presence of twointeracting pathways and the concentration of Na+ (or M in the case of §5) needed to generate

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the transporter flow. It would be interesting to compare this theoretical flow rate to that foundin nature.

The bond graph approach can be used to decompose complex systems into computationalmodules [30,31]. Combining such modularity with the energy-based pathway analysis approachof this paper would provide an approach to analysing and understanding energy flows incomplex biomolecular systems, for example, those within the Physiome Project [41], which isthe subject of current research.

Although this paper is restricted to flows of chemical energy, the bond graph approach enablesmodels to be built across multiple energy domains including chemoelectrical transduction [42,43].Hence the pathway approach can be equally well applied to systems with electrogenic featuressuch as excitable membranes [44] and the mitochondrial electron transport chain [45].

The effective use of energy is an important determinant of evolution [13,17,46,47]. Therefore,the energy-based pathway analysis of this paper is potentially relevant to investigating why livingsystems have evolved as they have. For example, do real SGLT1 transporters operate near thepoint of minimal energy dissipation?

The supply of energy is essential to life and disruption of energy supply has been implicatedin many diseases such as cardiac failure [48,49], Parkinson’s disease [50–54] and cancer [55–57].Therefore, it seems natural to apply the energy-based methods of this paper to investigate suchsystems. In particular, mitochondria are important for energy transduction in living systems andmitochondrial dysfunction is hypothesized to be the source of ageing [58, ch. 14], cancer [59,60]and other diseases [61]. Mathematical models of mitochondria exist already [62–65], and it ishoped that the energy-based pathway analysis of this paper will shed further light on the functionand dysfunction of mitochondria, and this is the subject of current research.

Data accessibility. A virtual reference environment [66] is available for this paper at http://dx.doi.org/10.5281/zenodo.165180. The simulation parameters are listed in the electronic supplementary material.Authors’ contributions. All authors contributed to drafting and revising the paper, and they affirm that they haveapproved the final version of the manuscript.Competing interests. We declare we have no competing interests.Funding. This research was in part conducted and funded by the Australian Research Council Centre ofExcellence in Convergent Bio-Nano Science and Technology (project no. CE140100036).Acknowledgements. P.J.G. thanks the Melbourne School of Engineering for its support via a ProfessorialFellowship. Both authors thank Dr Ivo Siekmann for discussions relating to conic spaces and Dr DanielHurley for help with the virtual reference environment. We also thank the reviewers for their suggestionsfor improving the paper and Professor Peter Hunter for pointing out some errors in the draft manuscript.

Appendix A. A short introduction to bond graph modellingThe purpose of this section is to provide the bond graph background necessary to understand thepaper itself. More information is to be found in references [29–32].

Bond graphs unify the modelling of energy within and across multiple physical domains usingthe concepts of effort and flow variables whose product is power. Thus, in the electrical domainthe effort variable voltage V with units of V and flow variable current i with units of A havethe product P = Vi with units of J s−1. In the biomolecular domain, the effort variable is chemicalpotential μ with units of J mol−1 and the flow variable is reaction flow rate v with units of mol s−1;μv also has units of J s−1.

The bond graph C component is a generic energy storage component corresponding to acapacitor in the electrical domain and the amount of a chemical species in the biomoleculardomain. If q is the time integral of the flow variable, the electrical capacitor generates voltageV according to the linear relationship: V = q/C, where C is the capacitance in Farads and q isthe charge in Coulombs. By contrast, the chemical species generates chemical potential µJ mol−1

according to the nonlinear relationship: μ = RT ln Kx, where R is the universal gas constant withunits of J mol−1 K−1, T is the absolute temperature with units of K, x = q/q0, where q0 is the

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ri

A C

B D

0

C:B C:D

1

00

1

C:A(a)

(b) R:r C:C

10

−i i

−iVA

iVC

VB

VD

VC

VD

VA

VB

DVDV i

Figure 10. Modelling electrical systems. (a) Electrical schematic diagram. (b) Bond graph:�V = (VA + VB) − (VC + VD).

reference quantity and K is a species-dependent constant which is dimensionless and dependson the reference quantity.

The bond graph R component is a generic energy dissipation component corresponding toa resistor in the electrical domain. In the linear case, it generates current i according to thelinear relationship: i = κ�V, where κ is the conductance and �V is the net voltage acrossthe resistor. The R component corresponds to a chemical reaction in the biomolecular domain.Assuming mass-action kinetics it generates molar flow v according to the nonlinear relationship:v = κ(exp Af/RT − exp Ar/RT), where Af (the forward affinity) and Ar (the reverse affinity) are thenet chemical potentials (with units of J mol−1) of the reactants and products of the reaction,respectively; κ is the reaction rate constant with units of mol s−1. Because this relationship treatsthe forward and reverse affinities differently, this reaction version of the R component is given aspecial symbol: Re.

Bond graphs represent the flow of energy between components by the bond symbol ⇁ wherethe direction of the harpoon corresponds to the direction of positive energy flow; this is a signconvention. The bonds connect components via junctions which transmit, but do not store ordissipate energy. There are two junctions: the 0 junction where all impinging bonds have the sameeffort and the 1 junction where all impinging bonds have the same flow. The expression for theflows associated with the 0 junction, and the efforts associated with the 1 junction, are determinedby the energy conservation requirement.

Figure 10 illustrates bond graph modelling in the electrical domain. The four capacitors andone resistor are connected as in the schematic diagram of figure 10a. In bond graph colon notation,the symbol before the colon indicates the component type and the symbol after the colon indicatesthe component name. Thus, the bond graph of figure 10b has four C components named A–D torepresent the four capacitors and a single R component named r to represent the resistor. The two0 junctions on the left reverse the sign of the flow associated with C:A and C:B. The two 0 junctionson the right are not necessary, but are convenient for connecting to others via bonds componentsto form a larger system. The left 1 junction corresponds to Vf = VA + VB and the right 1 junctioncorresponds to Vr = VC + VD; the centre 1 junction corresponds to �V = Vf − Vr = (VA + VB) −(VC + VD). Given the constitutive relations for the C and R component, the dynamical equationsdescribing the system can be automatically generated from the bond graph of figure 10b.

Figure 11 illustrates bond graph modelling in the chemical domain; the corresponding reaction

is A + Br−⇀↽− C + D. In a similar manner to the electrical system of figure 10, the bond graph

of figure 11b has four C components named A–D to represent the four species and a singleRe component named r to represent the reaction. The major difference is the use of the Recomponent to replace the 1 and R combination within the dotted box of figure 10b; as mentionedpreviously this is because the forward and reverse affinities are treated differently. In particular,with reference to the example of figure 11, Af = μA + μB, Ar = μC + μD which, with reference toequation (2.4) corresponds to

Nf = (1 1 0 0n)T Nr = (0 0 1 1)T.

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u

u

uu

DB

A Cr 0

Re:r

C:B C:D

1

00

C:C

1(a)

(b)

C:A

0

−u

−u

mC

mA

µB

µD

Af Ar

Figure 11. Modelling biomolecular systems: A + Br−⇀↽− C + D. (a) Biomolecular schematic diagram. (b) Bond graph:

Af = μA + μB, Ar = μC + μD.

Using equations (2.3) and (2.5), it follows that:

Af = RT(ln KAxA + ln KBxB) = RT ln KAxAKBxB,

Ar = RT(ln KCxC + ln KDxD) = RT ln KCxCKDxDxD

and V = κr(eAf/RT − eAr/RT) = κr(KAxAKBxB − KCxCKDxD).

This is a form of the mass-action kinetics equation discussed in the textbooks [8]. In particular, theequation can be rewritten as follows:

V = kfxAxB − krxCxD,

wherekf = κKAKB and kr = κKDKC.

As mentioned above, the K constants are dimensionless and thus kf and kr are dimensionless. Thedimensionless equilibrium constant Keq is defined as

Keq = kf

kr.

As discussed by Gawthrop et al. [30], Keq and KA, KB, KC and KD are related by the generalformula (6.1).

In both the electrical and biomolecular systems, there is only one flow: the current i in theelectrical case and the molar flow v in the biomolecular case. This flow f is out of C:A and C:Band into C:C and C:D. Hence, in the biomolecular case, −xA = −xB = xC = xD = v. This impliesthat xA + xC = xAC, xA + xD = xAD and xB + xC = xBC where xAC, xAD and xBC are constant. Eachof these three equations represents a conserved moiety: the total amount of the charge or speciesinvolved remains constant. The choice of conserved moieties in a given situation is generally notunique; this is discussed further in appendix B.

Replacing A,B,C and D by GLC, G6P, ATP and ADP, respectively, and r by r4 in figure 11corresponds to the reaction to the left of the diagram in figure 1. The apparently redundant0 junctions in figure 11 are used to provide connections between this particular reaction andthe overall reaction network of figure 1. The reaction diagram of figure 1a has the entities ATPand ADP represented more than once. Although this enhances clarity by removing the need forintersecting lines on the diagram, it reduces clarity by having each entity appear in multiplelocations. By contrast, the bond graph of figure 1b represents ATP and ADP each by a singleC component (ATP and ADP, respectively) with appropriate connections to each of the reactionsrepresented by Re:r1–Re:r9.

This idea of representing biomolecular reaction networks by C components (representingspecies) and Re components (representing reactions) connected by bonds (⇁) and 0 and 1junctions is summarized and formalized in figure 2a. This representation is summarized in thecaption to figure 2 and is discussed in detail by Gawthrop & Crampin [29,31].

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Bond graphs provide one foundation for this paper, the other is the systems biology conceptof pathways outlined in the next section.

Appendix B. A short introduction to systems biologyThe purpose of this section is to provide the systems biology background necessary to understandthe paper itself.

As discussed in the basic textbooks (for example, that of [8]), the stoichiometric matrix N isa fundamental construct in describing and understanding biomolecular systems. In particular,given nX chemical species with molar amounts contained in the column vector X and nV reactionswith molar flow rates contained in the column vector V; the nX × nV stoichiometric matrix Nrelates the rate of change X of X to V by equation (2.1) repeated here:

X = NV. (2.1)

The elements of N are integers that determine the amount of each species participating in the

reaction. For example, the reaction A + Br−⇀↽− C + D of figure 11 can be represented by equation

(2.1) where:X = (xA xB xC xD)T N = (−1 − 1 1 1)T V = vr,

where xA is the molar amount of substance A, etc., and vr is the reaction flow.As discussed in the basic textbooks (for example, that of [8]), the mathematical concept of a

null space of N gives useful information about the fundamental properties of the biomolecularsystem described by N. These are two null spaces: the left null space described by the matrix Gwhere GN = 0 and the right null space described by the matrix Kp where NKp = 0. In the case ofthe open systems discussed in §2, N is replaced by Ncd.

The significance of G is that pre-multiplying both sides of equation (2.1) by G gives

GX = GNV = 0.

Hence the linear combinations of X implied by the rows of G are constant; in systems biologynomenclature such combinations are known as conserved moieties. In the case of the examplesystem, one possible matrix G is

G =

⎛⎜⎝1 0 1 0

1 0 0 10 1 1 0

⎞⎟⎠ .

It can be verified that GN = 0 and the corresponding three conserved moieties are

xA + xC xA + xD xB + xC.

The significance of Kp is that if the elements of V are not independent but rather defined byV = Kpv then equation (2.1) gives

X = NKpv = 0.

Hence the columns of Kp determine pathways: combinations of non-zero flows which lead toconstant species amounts. In this example, there is no Kp such that NKp = 0. However, in thespecial case that the four species A–D are chemostats (their values are held constant by someexternal flows) then N now has four zero entries and Kp = 1 and thus the four species have constantamounts for any v and thus the single reaction r trivially becomes a pathway. As discussed in thepaper, the more complex systems of figures 1, 3 and 8 have non-trivial pathways.

Neither G nor Kp are unique. The seminal contribution discussed by Heinrich & Schuster [5]was to examine the case where the entries of Kp are both integer and non-negative thus definingpathways where all reaction flows have the same sign and are integer multiples of some baseflows. This idea forms the basis of analysing large-scale biomolecular systems by flux-balanceanalysis (FBA) [8]. Together with the bond graph concepts outlined in appendix A, this ideaprovides the foundation for this paper.

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