Bond Graphs Unify Stoichiometric Analysis and Thermodynamics
BBond Graphs Unify Stoichiometric Analysisand Thermodynamics
Peter J. Gawthrop *1,21
Systems Biology Laboratory, Department of Biomedical Engineering,Melbourne School of Engineering, University of Melbourne, Victoria 3010,Australia. Systems Biology Laboratory, School of Mathematics and Statistics, Universityof Melbourne University of Melbourne, Victoria 3010, AustraliaAugust 3, 2020
Abstract
Whole-cell modelling is constrained by the laws of nature in general and the laws ofthermodynamics in particular. This paper shows how one prolific source of information,stoichiometric models of biomolecular systems, can be integrated with thermodynamic prin-ciples using the bond graph approach to network thermodynamics. * Corresponding author. [email protected] a r X i v : . [ q - b i o . M N ] J u l ontents P & NADPH generation . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.3 R P generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.4 NADPH generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.5 NADPH & ATP generation . . . . . . . . . . . . . . . . . . . . . . . . . 13
B.1 Glycolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B.2 TCA cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27B.3 Electron Transport Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28B.4 ATPase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Introduction
Whole-cell modelling has the potential to “predict phenotype from genotype” (Karr et al., 2012;Covert, 2015) and has the potential to “transform bioscience and medicine” (Szigeti et al., 2018).However, there are currently significant issues in achieving reproducibility (Medley et al., 2016)and integrating disparate sources of information (Goldberg et al., 2018). However, whatever thesource of information, the whole-cell model is constrained by the laws of nature in general andthe laws of thermodynamics in particular. Unfortunately, “The requirement for thermodynamicconsistency, however, has not, in general, been adopted for whole-cell modelling” (Smith andCrampin, 2004). This paper shows how one prolific source of information, stoichiometric modelsof biomolecular systems, can be integrated with thermodynamic principles.Stoichiometric analysis of biomolecular systems has been developed over the years (Heinrichand Schuster, 1996; Palsson, 2006, 2011, 2015) has had notable successes including modellingand analysis of the E.coli genome-scale reconstruction (Orth et al., 2011; Thiele et al., 2013;Swainston et al., 2016). The basic idea is to describe a biomolecular system as a (sparse) integermatrix – the n X × n V stoichiometric matrix N connecting n X species and n V reactions. Asdiscussed by Palsson (2015) the stoichiometric approach has a number of advantages:1. The coefficients of N are integer; they can therefore be determined exactly.2. Mass balance of species is ensured and, with the inclusion of the elemental matrix (Palsson,2015, § 9.2.2), mass balance of elements is also ensured.3. The sparse integer matrix representation is scaleable to include large systems; for example,the iJO1366 genome-scale reconstruction of the metabolic network of Escherichia coli has2251 metabolic reactions, and 1136 unique metabolites (Orth et al., 2011).4. Standard linear algebraic concepts such as the null spaces of a matrix can be invoked toprovide precise and meaningful analysis of pathways and conserved moieties (Palsson,2006, 2011, 2015; Klipp et al., 2016).5. As discussed by Orth et al. (2011), the flux-balance analysis technique (Orth et al., 2010b)can be applied to predict metabolic flux distributions, growth rates, substrate uptake rates,and product secretion rates for large models.6. Because the enzymes catalysing the reactions are related to the genome, the stoichiometricapproach provides a bridge from genotype to phenotype (Palsson, 2015).7. Comprehensive software tools are readily available (Ebrahim et al., 2013; Heirendt et al.,2019).A number of works have discussed the fundamental significance of energy in the life sci-ences and evolution of living systems (Niven and Laughlin, 2008; Sousa et al., 2013; Martin The stoichiometric matrix has the symbol N is some works (Klipp et al., 2016) and S in others (Palsson, 2006,2011, 2015)
3t al., 2014; Lane, 2014, 2018; Dai and Locasale, 2018; Niebel et al., 2019). In particularly, the efficiency (Smith et al., 2005; Lopaschuk and Dhalla, 2014; Niven, 2016; Park et al., 2016; Larket al., 2016) of living systems is an evolutionary pressure. However, energy considerations arenot explicitly included in the stoichiometric approach. This can lead to mass flows that are notthermodynamically possible; such non-physical flows can be detected and eliminated by addingadditional thermodynamic constraints via
Energy Balance Analysis (EBA) (Beard et al., 2002;Qian et al., 2003; Noor et al., 2014; Noor, 2018).Like living systems, engineering systems are subject to the laws of physics in general and thelaws of thermodynamics in particular. This fact gives the opportunity of applying energy-basedengineering approaches to the modelling, analysis and understanding of living systems. Thebond graph method of Paynter (1961) is one such well-established engineering approach (Cel-lier, 1991; Gawthrop and Smith, 1996; Gawthrop and Bevan, 2007; Borutzky, 2010; Karnoppet al., 2012) which has been extended to include biomolecular systems (Oster et al., 1971, 1973;Gawthrop and Crampin, 2014). The stoichiometric matrix of a biomolecular network can bederived from the corresponding bond graph (Gawthrop and Crampin, 2014; Gawthrop et al.,2015); this paper shows that the converse is true: the bond graph of a biomolecular systemcan be deduced from the stoichiometric representation. Thus the large repository of models ofbiomolecular systems available in stoichiometric form can be automatically converted to bondgraph form.Once converted to bond graph form, the models are endowed with a number of additionalfeatures:1. They are thermodynamically compliant and thus subsume the EBA approach.2. As an energy based method, bond graphs can model multi-domain systems and thus read-ily incorporate charged species, electrons and protons in an integrated model (Gawthrop,2017; Gawthrop et al., 2017; Pan et al., 2018b,a).3. Bond graphs are modular (Gawthrop et al., 2015; Gawthrop and Crampin, 2016) a keyrequirement of any large-scale modelling endeavour.4. Bond graph models can be simplified in an energetically coherent fashion (Gawthrop andCrampin, 2014; Pan et al., 2017; Gawthrop et al., 2019).5. Bond graphs provide energy-based pathway analysis (Gawthrop and Crampin, 2017).The e.coli
Core Model (Orth et al., 2010a; Palsson, 2015) is a well-documented and readily-available stoichiometric model of a biomolecular system. This model is used in § 3.2 as anexemplar to illustrate how a bond graph can be automatically generated and to examine how itcan be used for the energetic analysis of pathways.
Bond graphs are, as the name implies, a graphical representation of a system. This has the ad-vantage of clear visual representation when dealing with small systems, but such visualisation4ecomes problematic for large systems. As meaningful biomolecular systems are large, this issuemust be addressed. There are two approaches to overcoming this issue: modularity and a non-graphical representation. This paper uses both approaches: a recent concept of bond graph mod-ularity (Gawthrop, 2017) is presented in § 4 and the recently developed BondGraphTools (Cud-more et al., 2019) ( https://pypi.org/project/BondGraphTools/ ) is used through-out as a non-graphical representation.The key concept is the energy bond represented by the (cid:43) symbol. This bond carries energyin the form of an effort/flow pair: in the case of biomolecular systems this pair is chemical freeenergy φ J mol − and molar flow v mol s − . Bonds transmit, but do not store or dissipate energy.Within this context, the bonds connect four bond graph components: & junctions Provide a method of connecting a two or more bonds. The bonds impingingon a junction share a common effort (chemical free energy ); the bonds impinging ona junction share a common flow. Both & junctions transmit, but do not store ordissipate energy. As discussed previously (Gawthrop and Crampin, 2014), the arrangementof bonds and junctions determines the stoichiometry of the corresponding biomolecularsystem and thus the relationship both between reaction and species flows and betweenspecies free energies and reaction forward and reverse free energies . As will be discussed,the reverse is also true: the stoichiometric matrix of a biomolecular system determines thebond graph. Ce Represents species . Thus species A is represented by Ce : A with the equations: x A ( t ) = (cid:90) t v A ( t (cid:48) ) dt (cid:48) + x A (0) (2.1) φ A = φ (cid:11) A + RT ln x A x (cid:11) A (2.2)Equation (2.1) accumulates the flow v A of species A. Equation (2.2) generates chemicalfree energy φ A in terms of the standard free energy φ (cid:11) A at standard conditions x (cid:11) A where R and T are the universal gas constant and temperature respectively Atkins and de Paula(2011). Ce components store, but do not dissipate, energy. Re Represents reactions . The flow associated with reaction 1 v is given by the Marcelin – deDonder formula (Van Rysselberghe, 1958): v = κ (cid:18) exp Φ f RT − exp Φ r RT (cid:19) (2.3)where Φ f and Φ r are the forward and reverse reaction free energies , or affinities. If κ isconstant, this represents the mass-action formula; in general, κ is a function of Φ f , Φ r and enzyme concentration (Gawthrop and Crampin, 2014). Re components dissipate, but The symbol φ is used for chemical free energy in place of µ .
5o not store, energy. In general V = V (Φ , φ ) (2.4)where Φ = Φ f − Φ r (2.5)where V () is dissipative in Φ for all φ : V i Φ i > (2.6)The key stoichiometric equations arising from bond graph analysis are (Gawthrop and Crampin,2014): ˙ X = N V (2.7)
Φ = − N T φ (2.8)where X , Φ and φ are the species amounts, reaction free energies and species free energies re-spectively. N is the system stoichiometric matrix. The network of bonds and junctions transmits,but does not dissipate or store, energy. As discussed by Gawthrop and Crampin (2014), this factcan be used to derive Equation (2.8) from (2.7).Moreover, the stoichiometric matrix N can be decomposed as (Gawthrop and Crampin,2014): N = N r − N f (2.9)where N r corresponds to the positive entries of N and N f to the negative entries. The forwardand reverse reaction free energies Φ f and Φ r are given by: Φ f = N f φ (2.10) Φ r = N r φ (2.11) R e : r (a) A r R e : r (b) B + C r D + E (Module M2)
Figure 1: Bond graphs of simple reactions.In other words, the stoichiometric matrix N can be derived from the system bond graph. Thissection shows that, conversely, the system bond graph can be derived from the stoichiometricmatrix N . The following constructive procedure is used:6. For each species create a Ce component with appropriate name and a junction; con-nect a bond from the junction to the Ce component.2. For each reaction create an Re component with appropriate name and two junctions;connect a bond from one junction to the forward port of the Re component and abond from the reverse port of the Re component to the other junction.3. For each negative entry N ij in the stoichiometric matrix, connect − N ij bonds fromthe zero junction connected to the i th species to the the one junction connected to theforward port of the j th reaction.4. For each positive entry N ij in the stoichiometric matrix, connect N ij bonds from theone junction connected to the reverse port of the j th reaction to the zero junction con-nected to the i th species.For example, the reaction A r N = (cid:18) − (cid:19) (2.12)and the bond graph of Figure 1(a). The reaction B + C r D + E has the stoichiometricmatrix N = − − (2.13)and has the bond graph of Figure 1(b). As discussed previously (Gawthrop and Crampin, 2016; Gawthrop, 2017), the notion of a chemo-stat (Polettini and Esposito, 2014) is useful in creating an open system from a closed system. Asdiscussed by Gawthrop (2017), the chemostat has a number of interpretations:1. one or more species are fixed to give a constant concentration (Gawthrop et al., 2015);this implies that an appropriate external flow is applied to balance the internal flow of thespecies.2. as a Ce component with a fixed state. 7. as an external port of a module which allows connection to other modules.In the context of stoichiometric analysis, the chemostat concept provides a flexible alternative tothe primary and currency exchange reactions (Schilling et al., 2000; Palsson, 2006, 2015).Gawthrop and Crampin (2016) discuss the dual concept of flowstats which again has a num-ber of interpretations:1. one or more reaction flows are fixed.2. as an Re component with a fixed flow.3. as an external port of a module which allows connection to other modules.In the context of stoichiometric analysis, the flowstat concept provides a way of isolating partsof a network by setting zero flow in the reactions connecting the parts. Such zero flow flowstatscan also be interpreted as removing the corresponding enzyme via gene knockout.In terms of stoichiometric analysis, the closed system equations (2.7) and (2.8) are replacedby: ˙ X = N cd V (3.1) Φ = − N T φ (3.2)where N cd is created from the stoichiometric matrix N by setting rows corresponding to chemostatsspecies and columns corresponding to flowstatted reactions to zero (Gawthrop and Crampin,2016). As discussed by Gawthrop and Crampin (2016), system pathways corresponding to(3.1) are defined by the right-null space of N cd that is the columns of the matrix K cd where N cd K cd = 0 . Further, then steady-state pathways are defined by: V = K cd v (3.3)were v is the pathway flow. It follows from Equation (3.1) that Equation (3.3) implies that ˙ X = 0 .Gawthrop and Crampin (2017) define the pathway stoichiometric matrix N p as: N p = N K cd (3.4)In a similar fashion to equation (3.2), the pathway reaction free energies Φ p are given by Φ p = − N Tp φ (3.5)In the same way as the stoichiometric matrix N relates reaction flows to species and thus repre-sents a set of reactions, the pathway stoichiometric matrix N p also represents a set of reactions:these reactions will be called the pathway reactions .Following Schilling et al. (2000), pathways can be divided into three categories accordingto the species corresponding to the non zero elements in the relevant column of the pathway stoichiometric matrix N p : I The species include primary metabolites; these pathways are of functional interest.8 I The species include currency metabolites only; these pathways dissipate energy without cre-ating or consuming primary metabolites. Schilling et al. (2000) call these pathways futilecycles . III
There are no species.Pathway reactions for type I pathways contain both primary and currency metabolites; pathwayreactions for type II pathways contain currency metabolites only; pathway reactions for type IIIpathways are empty.Pathways have an equivalent bond graph obtained by applying the conversion method of§ 2 to N p instead of N Gawthrop and Crampin (2017); this fact can be utilised to give simplephysically plausible models of complex systems Gawthrop et al. (2019). R e : r Re:r4 R e : r Re:r5 1Re:r20 Ce:ADPCe:CRe:r6Ce:E Ce:ATP (a) Bond graph R e : P R e : P (b) Pathway bond graph Figure 2: Bond graphs for illustrative example Noor (2018)Noor (2018) gives a simple illustrative example of the three types of pathway; Figure 2(a)gives the corresponding bond graph. the reactions are:A r B (3.6)ATP + B r ADP + C (3.7)C r D (3.8)D r A (3.9)A r C (3.10)C r E (3.11)9he there are seven species and six reactions giving states x and flows v : x = x A x ADP x AT P x B x C x D x E v = v r v r v r v r v r v r (3.12)The stoichiometric matrix is: N = − − − − − −
10 0 1 − (3.13)Setting A, E, ATP and ADP as chemostats, N cd is constructed by setting the correspondingrows of N to zero. The corresponding null space is three dimensional and corresponds to thethree pathways:1. r1 + r2 + r3 + r42. r3 + r4 + r53. r1 + r2 + r6Using (3.4), the pathway stoichiometric matrix N p is: N p = −
11 0 1 − −
10 0 00 0 00 0 00 0 1 (3.14)The three pathway reactions are: ATP P ADP (3.15) P (3.16)A + ATP P ADP + E (3.17)10athway reaction P1 corresponds to a type II pathway, pathway reaction P2 to a type III pathwayand pathway reaction P3 to a type I pathway where A is converted to E driven by the conversionof ATP to ADP. The example is extended by assigning a set of nominal chemical free energies φ (cid:11) to the species: φ (cid:11) A = 1 , φ AT P = 0 , φ ADP = 3 , φ (cid:11) B = 1 , φ (cid:11) C = 1 , φ (cid:11) D = 1 , φ (cid:11) E = 0 . Thepathway reaction free energies are then computed using (3.5) as Φ P = − , Φ P = 0 , Φ P = − .As the free energy for each pathway only depends on the species appearing in the pathwayreactions, the free energy of non-chemostatted species are irrelevant for this computation. In factthe free energies of the species will correspond to the steady-state values of concentrations ofthe non-chemostatted species arising from the flow patterns corresponding to the chemostat freeenergies (Gawthrop, 2018). The pathway bond graph appears in Figure 2(b). The combination of the Glycolysis & Pentose Phosphate networks provides a number of differ-ent products from the metabolism of glucose. This flexibility is adopted by proliferating cells,such as those associated with cancer, to adapt to changing requirements of biomass and energyproduction (Vander Heiden et al., 2009).The e.coli
Core Model (Orth et al., 2010a; Palsson, 2015) is used as the basis for the examplesin this section. In particular, the species, reactions and stoichiometric matrix were extracted fromthe spreadsheet ecoli core model.xlsx but with the biomass equations deleted and the thereaction CYTBD (containing O ) multiplied by 2 to give integer stoichiometry. The submodelcontaining the reactions of the combined Glycolysis & Pentose Phosphate pathways was thenextracted (see Appendix A for details) and converted to a bond graph in bond graph tools formatusing the algorithm of § 2. The following procedure was adopted to obtain physiologically-realistic values for the species free energies φ .1. The reaction free energies Φ were extracted from Table 4 provided by Park et al. (2016).2. A set of consistent species free energies φ was obtained from equation (3.2) using φ = − (cid:0) N T (cid:1) † Φ (3.18)where † denotes the pseudo inverse .The reaction free energies for each reaction are given in Appendix A and, because of theabove procedure, correspond to the reaction free energies listed by Park et al. (2016) Table 4.As discussed by Garrett and Grisham (2017, § 22.6d), it illuminating to pick out individualpaths through the network to see how these may be utilised to provide a variety of products. Thisis reproduced here by choosing appropriate chemostats and flowstats (§ 3) to give the resultslisted by Garrett and Grisham (2017, § 22.6d). In each case, the corresponding pathway reactionfree energy is given. For consistency with Garrett and Grisham (2017, § 22.6d), each pathwaystarts with Glucose 6-phosphate (G P).The following chemostat list is used (together with additional chemostats) in each of thefollowing sections: { ADP, ATP, CO , G P, H, H O, NAD, NADH, NADP, NADPH, PI, PYR } . The pseudo inverse was implemented using the python linear algebra package function linalg.pinv() .2.1 Glycolysis The glycolysis pathway is isolated from the pentose phosphate pathway by replacing the twoconnecting reactions (G6PDH2R and TKT2) by flowstats. This gives rise to the pathway:• PGI + PFK + FBA + TPI + 2GAPD - 2PGK - 2PGM + 2ENO + 2PYKThe corresponding pathway reaction is:3 ADP + G P + 2 NAD + 2 PI P O + 2 NADH + 2 PYR ( − .
22 kJ mol − ) The pathway reaction P is the overall glycolysis reaction Garrett and Grisham (2017, § 18.2).The negative reaction free energy indicates that the reaction proceeds in the forward direction. P & NADPH generation
This pathway is isolated by setting PGI and TKT2 as flowstats and the product R P is added tothe chemostat list. This gives rise to the pathway:• G6PDH2R + PGL + GND + RPIThe corresponding pathway reaction is:G P + H O + 2 NADP P CO + 2 H + 2 NADPH + R P ( − .
01 kJ mol − ) The pathway reaction P corresponds to the R P & NADPH synthesis discussed in comment 1of Garrett and Grisham (2017, § 22.6d). The negative reaction free energy indicates that thereaction proceeds in the forward direction. P generation
This pathway is isolated by setting GAPD and G6PDH2R as flowstats and the product R P isadded to the chemostat list. This gives rise to the pathway:• - 5PGI - PFK - FBA - TPI - 4RPI + 2TKT2 + 2TALA + 2TKT1 + 4RPEThe corresponding pathway reaction is:ADP + H + 6 R P P ATP + 5 G P (20 .
30 kJ mol − ) The pathway reaction P corresponds to the R P synthesis discussed in comment 2 of Garrett andGrisham (2017, § 22.6d). The positive reaction free energy indicates that the reaction proceedsin the reverse direction. 12 .2.4 NADPH generation
This pathway is isolated by setting GAPD as a flowstat. This gives rise to the pathway:• - 5PGI - PFK - FBA - TPI + 6G6PDH2R + 6PGL + 6GND + 2RPI + 2TKT2 + 2TALA +2TKT1 + 4RPEThe corresponding pathway reaction is:ADP + G P + 6 H O + 12 NADP P ATP + 6 CO + 11 H + 12 NADPH ( − .
79 kJ mol − ) The pathway reaction P corresponds to the NADPH synthesis discussed in comment 3 of Gar-rett and Grisham (2017, § 22.6d). The negative reaction free energy indicates that the reactionproceeds in the forward direction. This pathway is isolated by setting PGI as flowstat. This gives rise to the pathway:• 2PFK + 2FBA + 2TPI + 5GAPD - 5PGK - 5PGM + 5ENO + 5PYK + 3G6PDH2R + 3PGL+ 3GND + RPI + TKT2 + TALA + TKT1 + 2RPEThe corresponding pathway reaction is:8 ADP + 3 G P + 5 NAD + 6 NADP + 5 PI P + 8 H + 2 H O + 5 NADH + 6 NADPH + 5 PYR ( − .
44 kJ mol − ) The pathway reaction P corresponds to the NADPH and ATP synthesis discussed in comment4 of Garrett and Grisham (2017, § 22.6d). The negative reaction free energy indicates that thereaction proceeds in the forward direction. As discussed by Gawthrop and Crampin (2016), there are two related but distinct concepts ofmodularity: computational modularity where physical correctness is retained and behaviouralmodularity where module behaviour (such as ultra-sensitivity) is retained. It is the former thatis discussed in this section. As discussed by Gawthrop (2017), modular bond graphs providea way of decomposing complex biomolecular systems into manageable parts (Gawthrop et al.,2015; Gawthrop and Crampin, 2016). In particular, this paper combines the modularity con-cepts of Neal et al. (2016) with the bond graph approach to give a more flexible approach tomodularity. The basic idea (Gawthrop, 2017) is simple: modules are self-contained and have noexplicit ports; but any species, as represented by a Ce component has the potential to becomea port. Thus if two modules share the same species, the corresponding Ce component in eachmodule is replaced by a port with the same name, and the species is explicitly represented as a Ce component on a higher level. Moreover, each module can be individually tested by replacingthe relevant Ce components by chemostats.The algorithm is: 13. Within each module, each Ce component corresponding to a common species is exposed – replaced by a port component. Note that the algorithm of § 2 ensures that each Ce isattached to a junction.2. For each common species, create a Ce component connected to a component.3. Connect all module ports associate with each species to the junction associated with thespecies; all instances of Ce components corresponding to each species are thus unified . [B] [B] Figure 3: Modularity. Modules M1 and M2 correspond to Figures 1(a) & 1(b) respectively.The common species B is exposed as a port in each module and connected to the new Ce : B component via a junction.For example, let modules M1 and M2 correspond to Figures 1(a) & 1(b) respectively. In Figure3, the common species B is exposed as a port in each module and connected to the new Ce : B component via a junction. The composite system contains the two reactions:A r r D + E (4.2)Choosing the set of chemostats to be { A , C , D , E } the corresponding pathway stoichiometricmatrix N p is N p = − − (4.3)where the species are { A , C , D , E , B } and the reactions { r , r } . The pathway reaction P is then:A + 2 C P As in § 3.2, the e.coli
Core Model (Orth et al., 2010a; Palsson, 2015) is used. In particular,reactions corresponding to four modules (Glycolysis, TCA cycle, Electron Transport Chain and14TPase) were extracted as detailed in Appendix B. For simplicity, reaction PDH (convertingPYR to ACCOA) and reaction NADTRHD (converting NADP/NADPHn to NAD/NADH) wereincluded in the TCA cycle module.These modules can be analysed individually. For example the TCA cycle module can beanalysed using the set of chemostats: { PYR , CO , ADP , ATP , H O , NAD , NADH , PI , H , Q , Q H } The two pathways are1. FRD7 + SUCDI2. PDH + CS + ACONTA + ACONTB + ICDHYR + AKGDH - SUCOAS - FRD7 + FUM +MDH + NADTRHDThese two pathways correspond to the two pathway reactions: P ADP + 2 H O + 4 NAD + PI + PYR + Q P ATP + 3 CO + 2 H + 4 NADH + Q H p The first is a type III reaction and the second a type I reaction which utilises the free energy ofPYR to generate two NADH, one NADHP, one ATP and one Q H whilst releasing two CO andtwo H.The overall metabolic system comprises the four modules (Glycolysis, TCA cycle, ElectronTransport Chain and ATPase) connected together. Using the approach of § 4, the modules areinterconnected by declaring the set of species that the modules have in common: { PYR , ATP , ADP , PI , H , H E , NAD , NADH , H O , Q , Q H } These species are unified as described in § 4. To analyse the composite system, the set ofchemostats was chosen as: { GLCD E , CO , O , ADP , ATP , H O , PI , H } . The three pathways are1. PFK + FBP2. FRD7 + SUCDI3. 2 GLCPTS + 2 PGI + 2 PFK + 2 FBA + 2 TPI + 4 GAPD - 4 PGK - 4 PGM + 4 ENO+ 2 PYK + 4 PDH + 4 CS + 4 ACONTA + 4 ACONTB + 4 ICDHYR + 4 AKGDH - 4SUCOAS - 4 FRD7 + 4 FUM + 4 MDH + 4 NADTRHD + 20 NADH16 + 12 CYTBD +27 ATPS4R 15hese three pathways correspond to the three pathway reactions:ATP + H O P ADP + PI + H P E + 12 O + 35 ADP + 35 PI + 35 H P
12 CO + 35 ATP + 47 H OAs in § 3.1, pathway reaction P1 corresponds to a type II pathway, pathway reaction P2 to atype III pathway and pathway reaction P3 to a type I pathway. Pathway 3 corresponds to themetabolic generation of ATP using the free energy of GLCD E . The ratio of ATP to GLCD E is17.5; this is the value quoted by Palsson (2015, § 19.2). The standard FBA approach is to create open systems from closed systems by adding “exchangereactions ” to species which connect to the outside world – for example: ATP (cid:11) . In con-trast, the bond graph approach would declare ATP to be a chemostat. Chemostats provide a moreflexible approach as they can be created without changing system structure and are used in thesequel.FBA (Orth et al., 2010b) uses the linear equation (3.3) within a constrained linear optimisa-tion to compute pathway flows. EBA adds two sorts of nonlinear constraint arising from ther-modynamics. This section shows that the bond graph approach automatically includes the EBAconstraint equations by considering Inequality (2.6) and Equation (3.2). In particular:1. Inequality (2.6) corresponds to Equation 8 of Beard et al. (2002). This inequality can bere-expressed as: Φ i = r i ( φ ) V i (5.1)where r i ( φ ) > (5.2) r i corresponds to the “flux resistances” on p.83 of Beard et al. (2002)].2. If K is the right null matrix of N , it follows from Equation (3.2) that K T Φ = 0 (5.3)This corresponds to Equation 7 of Beard et al. (2002). Note that K defines the pathwaysof the closed system system (with no chemostats).Moreover, the pathways of the open system as defined by K cd can be considered by defining R = diag r i and using Equation (3.3): K T RK cd v = 0 (5.4)Equation (5.4) and inequality (5.2) constrain the pathway flows v ; this is illustrated in the fol-lowing examples drawn from Beard et al. (2002).16 .1 Example: Parallel reactions (a) Example: Parallel reac-tion. C:C 0 Re:r3 0C:A Re:r1Re:r2 0 C:B (b) Example: three-reaction cycle.
Figure 4: Bond graphs coresponding to examples from Beard et al. (2002) ( junctions are notshown for clarity)). (a) (Beard et al., 2002, Fig. 2), (b) (Beard et al., 2002, Fig. 3)Beard et al. (2002, Fig. 2) motivate EBA using the example of two resistors in parallel. Figure4(a) shows the bond graph of the analogous reaction system: the species A and B are joined bytwo reactions: A r B (5.5)A r B (5.6)The stoichiometric matrix is: N = (cid:18) − −
11 1 (cid:19) (5.7)and the null space matrix K is K = (cid:18) − (cid:19) (5.8)corresponding to the pathway: − r + r .Setting A and B as chemostats: N cd = (cid:18) (cid:19) (5.9) K cd = (cid:18) (cid:19) (5.10)Equation (5.4) then becomes: − r v + r v = 0 (5.11)As r i > , it follows that v and v must either be zero or have the same sign.17 .2 Example: three-reaction cycle Beard et al. (2002, Fig. 3) give the example of a three-reaction cycle. Figure 4(b) shows thecorresponding bond graph. The species A, B and C are joined by three reactions:A r B (5.12)B r C (5.13)C r A (5.14)The stoichiometric matrix is: N = − − − (5.15)and the null space matrix K is K = (5.16)corresponding to the pathway: r + r + r .Setting A and B as chemostats: N cd = − (5.17) K cd = (5.18)Equation (5.4) then becomes: r v + r v + r v = r v + ( r + r ) v = 0 (5.19)As r i > , it follows that v and v must either be zero or have the opposite sign.Alternatively, setting A, B and C as chemostats: N cd = (5.20) K cd = (5.21)18quation (5.4) then becomes: r v + r v + r v = 0 (5.22)As r i > , there are three possibilities: all flows are zero; one of the three pathway flows musthave one sign and the other two flows the opposite sign; or one flow is zero and the other twohave opposite signs.
1. It has been shown that the bond graph of a biomolecular system can be derived fromthe stoichiometric matrix. Thus the plethora of existing stoichiometric models can beautomatically endowed with a number of features including(a) thermodynamic compliance(b) modularity(c) explicit energy flows allowing exploration of, for example, efficiency (Gawthrop andCrampin, 2018)(d) generation of reduced-order models using pathway analysis (Gawthrop and Crampin,2017; Gawthrop et al., 2019).(e) energy compliant connections to other physical domains including models of chemo-electric transduction (Gawthrop et al., 2017; Gawthrop, 2017), membrane transporters(Pan et al., 2019), cardiac action potential (Pan et al., 2018a), chemomechanical trans-duction and photosynthesis.2. The key equations of the EBA approach of Beard et al. (2002) have been shown to beimplicit in the system bond graph.3. Via the modular approach of § 4, the Re components of § 2, representing mass-action ki-netics, can be replaced by thermodynamically compliant models of more complex kineticsCornish-Bowden (2013) driven by enzymes and inhibitors including feedback inhibition,allosteric modulation and cooperativity.4. This approach provides a basis for thermodynamically compliant whole-cell models. I would like to thank the Melbourne School of Engineering for its support via a ProfessorialFellowship, and Edmund Crampin and Michael Pan for help, advice and encouragement.19 eferences
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Glycolysis & Pentose Phosphate Pathways: Reactions
GLCD E + PEP GLCPTS G P + PYR ( − .
81 kJ mol − ) G P PGI F P ( − .
60 kJ mol − ) ATP + F P PFK
ADP + FDP + H ( − .
71 kJ mol − ) FDP
FBA
DHAP + G P ( − .
98 kJ mol − ) DHAP
TPI G P ( − .
79 kJ mol − ) G P + NAD + PI
GAPD DPG + H + NADH ( − .
32 kJ mol − ) PG + ATP
PGK DPG + ADP ( − .
42 kJ mol − ) PG PGM PG ( − .
17 kJ mol − ) PG ENO H O + PEP ( − .
75 kJ mol − ) ADP + H + PEP
PYK
ATP + PYR ( − .
09 kJ mol − ) G P + NADP G PDH R PGL + H + NADPH ( − .
96 kJ mol − ) PGL + H O PGL PGC + H ( − .
96 kJ mol − ) PGC + NADP
GND CO + NADPH + RU PD ( − .
08 kJ mol − ) RU PD RPI R P ( − .
00 kJ mol − ) E P + XU PD TKT F P + G P ( − .
61 kJ mol − ) G P + S P TALA E P + F P ( − .
43 kJ mol − ) R P + XU PD TKT G P + S P ( − .
40 kJ mol − ) RU PD RPE XU PD ( − .
08 kJ mol − ) Modular representation of Metabolism: Reactions
B.1 Glycolysis
GLCD E + PEP GLCPTS G P + PYR (B.1)G P PGI F P (B.2)ATP + F P PFK
ADP + FDP + H (B.3)FDP + H O FBP F P + PI (B.4)FDP
FBA
DHAP + G P (B.5)DHAP
TPI G P (B.6)G P + NAD + PI
GAPD DPG + H + NADH (B.7) PG + ATP
PGK DPG + ADP (B.8) PG PGM PG (B.9) PG ENO H O + PEP (B.10)ADP + H + PEP
PYK
ATP + PYR (B.11)
B.2 TCA cycle
COA + NAD + PYR
PDH
ACCOA + CO + NADH (B.12)ACCOA + H O + OAA CS CIT + COA + H (B.13)CIT
ACONTA
ACONC + H O (B.14)ACONC + H O ACONTB
ICIT (B.15)ICIT + NADP
ICDHYR
AKG + CO + NADPH (B.16)AKG + COA + NAD AKGDH CO + NADH + SUCCOA (B.17)ATP + COA + SUCC SUCOAS
ADP + PI + SUCCOA (B.18)FUM + Q H FRD Q + SUCC (B.19)Q + SUCC SUCDI
FUM + Q H (B.20)FUM + H O FUM
MALL (B.21)27ALL + NAD
MDH
H + NADH + OAA (B.22)NAD + NADPH
NADTRHD
NADH + NADP (B.23)
B.3 Electron Transport Chain NADH E + NAD + Q H (B.24)4 H + O + 2 Q H CYTBD O + 4 H E + 2 Q (B.25) B.4 ATPase
ADP + 4 H E + PI ATPS R ATP + 3 H + H2