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Asian Journal of Control, Vol. 13, No. 5, pp. 1 14, September 2011Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/asjc.359

A PARAMETER CONDITION FOR RULING OUT MULTIPLE

EQUILIBRIA OF THE PHOTOSYNTHETIC CARBON METABOLISM

Hong-Bo Lei, Xin Wang, Ruiqi Wang, Xin-Guang Zhu, Luonan Chen, and Ji-Feng Zhang

ABSTRACT

In this paper, we propose a reduced molecular network for the photosyn-thetic carbonmetabolism, which can describe the following key characteristics:Calvin cycle, utilization of photosynthate, and photorespiration. Taking theconcentrations of the nine major metabolites as variables, we represent thereduced network by deriving a nonlinear differential-algebraic system with48 parameters, and theoretically study the multi-equilibrium property in thephotosynthetic carbon metabolism. Specifically, we equivalently transform theoriginal 9-dimensional system into an independent 2-dimensional subsystemwith ten parameters, and show that the original system has no more than onephysiologically feasible equilibrium when the ten parameters of the subsystemstay in a certain field around the nominal value of each parameter, no matterwhat values the other 38 parameters in the original model are taken. Such atheoretical result not only provides profound insights for qualitatively under-standing of the dynamic features of the photosynthetic carbon metabolism,but also can be used to make an accurate judgement on a correct strategy forimproving the photosynthesis in plants.

Key Words: Metabolic network, multi-equilibrium property, photosynthesis,photosynthetic carbon metabolism.

I. INTRODUCTION

The grain yield of crops has doubled during thepast century, but it is still unable to meet the growing

Manuscript received June 25, 2010; revised September 28,2010; accepted November 15, 2010.Hong-Bo Lei, Xin Wang and Ji-Feng Zhang are with

the Key Laboratory of Systems and Control, Academy ofMathematics and Systems Science, Chinese Academy ofSciences, Beijing 100190, China (e-mail: [email protected];[email protected]; [email protected]).Hong-Bo Lei and Ji-Feng Zhang were supported by the

National Natural Science Foundation of China (under grant60821091).Ruiqi Wang is with the Institute of Systems Biology,

Shanghai University, Shanghai 200444, China (e-mail:[email protected]). He was supported by the NationalNatural Science Foundation of China (Grant No. 10832006 andYouth Research Grant No. 10701052).

demand [1, 2]. Even worse, some studies suggest thatthere is not much probability of getting any furtherincrease in grain yield by the traditional breedingapproaches [3–5]. Instead, scientists have found thatimproving photosynthesis is an effective way to furtherdramatically increase crop yield [4–6]. There are twoapproaches to increase crop yield by improving photo-synthesis: one is to increase the total photosynthesis,such as increasing the leaf area and extending the dailyduration of photosynthesis; and the other is to improve

Xin-Guang Zhu is with the CAS-MPG Partner Institutefor Computational Biology, Shanghai Institutes for BiologicalSciences, Chinese Academy of Sciences, Shanghai 200031,China (e-mail: [email protected]).Luonan Chen is with the Key Laboratory of Systems

Biology, Shanghai Institutes for Biological Sciences, ChineseAcademy of Sciences, Shanghai 200031, China (e-mail:[email protected]).

q 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Asian Journal of Control, Vol. 13, No. 5, pp. 1 14, September 2011

the rate of photosynthesis per unit leaf area (i.e. the rateof CO2 assimilation) [1, 4, 6]. It has also been shownthat increasing leaf photosynthesis rate will boost yieldpotential when other factors are held constant [4, 6].

Photosynthesis is a complex system that includesa large number of biophysical and biochemical reac-tions, such as absorption of light energy, conversionof light energy to chemical energy, and some otherbiochemical reactions involved in the photosyntheticcarbon metabolism [7, 8]. The carbon in crop yield ismainly from the CO2 fixed during the photosyntheticcarbon metabolism. Much attention has been rivetedon photosynthetic carbon metabolism since it is closelyrelated to increasing crop yield.

From a systems viewpoint, the photosyntheticcarbon metabolism can be viewed as a molecularnetwork which has many important dynamic charac-teristics, such as the oscillation driven by variation ofexternal conditions, the sensitivity to each enzyme,stability, and multi-equilibrium property (i.e. whetheror not the network can admit multiple equilibria).In particular, the multi-equilibrium property in thephotosynthetic carbon metabolism is intimately asso-ciated with increasing crop yield for the followingreason. If the photosynthetic carbon metabolism hastwo or more equilibria, one of them will correspond tothe higher or highest photosynthesis rate, which clearlycan be used to increase the grain yields by driving thesystem into this equilibrium; otherwise, the only thingone can do is to improve the photosynthesis at theexisting equilibrium. Thus, it is crucial to accuratelyjudge whether or not this molecular network is ableto admit multiple equilibria so that a correct strategycan be adopted. Since there is no current biologicalexperiment available to answer this question [9–12],one has to resort to the systems modeling approachand control theory to gain profound insights on it.Actually, control theory contributes a lot in systemsbiology [13–21]. Cheng et al. proposed a control routharray method to analyze biomolecular networks [13].Sontag et al. used monotone theory to study biologicalsystems [14]. Wellstead et al. provided a review of therole of control and system theory in systems biology[16]. Wang et al. modeled and analyzed biologicaloscillations in molecular networks [17]. Chesi proposeda recurive algorithm to compute equilibrium point ofgenetic regulatory network [18] and analyzed theirglobal asymptotic stability [19].

Up to now, some reasonable and effective modelshave been proposed for the photosynthetic carbonmetabolism to study its multi-equilibrium property[10, 22–27]. Pettersson and Ryde-Pettersson [23]proposed a model for the Calvin cycle (a key part of

the photosynthetic carbon metabolism) and found thatthere are two equilibria when the cytosolic phosphateconcentration does not exceed 1.9 mM, but one ofthem is not stable. Poolman et al. [24, 28] showedthat the Calvin cycle has two different equilibria inplant leaves at different ages. Zhu et al. [12] proposeda simple model of the Calvin cycle which has twokey ingredients of the Calvin cycle: Calvin cycleand utilization of photosynthate. For a group of fixedparameter values, Zhu et al. found that the model hasmultiple equilibria by numerical computation, but onlyone is physiologically feasible.

In the previous works, the model parameterswere obtained from different experiments with variousconditions. In fact, the parameter values are alwaysdifferent for the photosynthetic carbon metabolismin different mesophyll cells, not to mention differentleaves and different plants. Therefore, rather than fixedvalues, it is more biologically reasonable to let themodel parameters vary in an appropriate neighborhoodaround their experimental values when investigatingthe multi-equilibrium property in the photosyntheticcarbon metabolism, and the results obtained in such away will have a good suitability for a wide variety ofplant species or conditions. However, it is a difficulttask to derive such a theoretical result due to thecomplicated nonlinearity of the model.

In this paper, we develop a reduced molecularnetwork for the photosynthetic carbon metabolism,which describes the following key characteristics:Calvin cycle, utilization of photosynthate, and photores-piration. While we investigate the multi-equilibriumproperty in the photosynthetic carbon metabolism,nine major metabolites are considered. We propose anonlinear differential-algebraic model with nine vari-ables and 48 parameters. We first explore the effectof the photorespiration pathway and then study themulti-equilibrium property of the model. Specifically,we equivalently transform the model into an indepen-dent 2-dimensional subsystem with ten parameters,and show that the equilibria of the original systemcan be determined by the 2-dimensional subsystemuniquely. Then, we prove that when the ten parametersin the 2-dimensional subsystem stay in an appro-priate neighborhood around their nominal values (i.e.the experimental values), the original 9-dimensionalsystem has no more than one equilibrium, no matterwhat values the other 38 parameters take. Such a resultcan help us to make an accurate judgement on a correctstrategy for improving the photosynthesis in plants.

This paper is organized as follows. In Section II, anintroduction of the photosynthetic carbon metabolismis given, and a nonlinear differential-algebraic model


H.-B. Lei et al.: A Parameter Condition for Ruling Out Multiple Equilibria

323331

23

22

21

7 6

122

123

5

7

12

8

9

3

4

1010

1010

5

101a

124

121

101b

1121 111

PGA GCAGCEA

RuBP PGCA

GCEA

HPR

SER

GCA

GOA

GLY

F6P

FBP

GAPSBP

DHAP

3111 111 Ru5P

Xu5PRi5P

DPGA

S7P

E4P

7

7

131

2

G6P

G1P

Starch

1138

PGADHAP GAP

Fig. 1. Complete photosynthetic carbon metabolic network.

is derived. In Section III, a parameter condition isobtained to ensure that the model has no more thanone equilibrium, and a certain parameter field meetingsuch a condition is given by numeric computation. InSection IV, several general remarks and future topicsare given to conclude this paper.

II. MODEL OF THE PHOTOSYNTHETICCARBON METABOLISM

2.1 Photosynthetic carbon metabolism

The photosynthetic carbon metabolism contains alarge number of metabolites and biochemical reactionswith several major modules: Calvin cycle, photores-piration pathway, starch synthesis, triose-P export andsucrose synthesis, which have been widely studied andmathematically described in detail in [22]. Taking eachmetabolite as a node and each reaction as an edge,we obtain the photosynthetic carbon metabolic networkshown in Fig. 1. The symbols in the boxes are metabo-lites. An arrow indicates the direction of a reaction. Thenumber on each arrow represents the reaction number.The symbol “· · ·” represents a series of reactions thatutilize the triose phosphate PGA, GAP and DHAP. Thedouble dotted line represents the chloroplast membrane.Reactions above the line occur in the chloroplast stromaand these below occur in the cytosol.

The abbreviations used for each metabolite in thispaper are as follows. RuBP, Ribulose 1,5-bisphosphate;PGA, 3-Phosphoglycerate; DPGA, 1,3-bisphosphogly-cerate; GAP, Glyceraldehyde 3-phosphate; Ru5P,Ribulose 5-phosphate; PGCA, 3-Phosphoglycollate;GCA, Glycollate; GCEA, Glycerate; DHAP,Dihydroxyacetone-phosphate; E4P, Erythrose 4-phosphate; SBP, Sedoheptulose 1,7-phosphate; S7P,Sedoheptulose 7-phosphate; FBP, Fructose 1,6-phosphate; F6P, Fructose 6-phosphate; G6P, Glucose6-phosphate; G1P, Glucose 1-phosphate; Ri5P, Ribose5-phosphate; Xu5P, Xylulose 5-phosphate; GOA,Glyoxylate; GLY, Glycine; SER, Serine; HPR,Hydroxypyruvate; Rubisco, Ribulose1,5-bisphosphateCarboxylase/Oxygenase.

Our primary interest is whether or not the photo-synthetic carbon metabolism can admit multiple equi-libria when the model parameters vary in a certain field.Such a theoretical result not only can be used to under-stand the qualitative dynamics of the photosyntheticcarbon metabolism but also may lead a correct deci-sion on the strategy for improving the photosynthesisrate on plants. Although the complete network shownin Fig. 1 provides relatively detailed information onthe photosynthetic carbon metabolism, it is difficult totheoretically analyze its asymptotical behaviors evenfor fixed parameter values due to the nonlinearity ofsuch a complicated system. Hence, we next convert the



complete network shown in Fig. 1 to a reduced onebased on some biological principles, which is tractablefor theoretical analysis.

GAP is a 3-carbon compound and Ru5P is a5-carbon compound. The yield of 3 Ru5P moleculeswill consume 5 GAP molecules. Hence, from such anobservation, we simply take reaction GAP→0.6Ru5Pto equivalently represent the complicated conversionof GAP into Ru5P. Another part of GAP is convertedinto starch in the chloroplast stroma by the pathwayGAP→FBP→F6P→G6P→G1P→Starch, and werepresent this utilization of GAP by GAP→Sink.Part of the triose phosphate PGA, GAP and DHAPare translocated into the cytosol for different cellularfunctions, such as sucrose synthesis. We representthis utilization of the triose phosphate GAP, PGA andDHAP by PGA→Sink and GAP→Sink. The transfor-mation of GCA to GCEA occurs in the cytosol. GCAis first translocated from stroma to cytosol, and thengoes through a series of reactions to become GCEA.GCEA is finally translocated back to stroma. Wereduce this process as GCA→GCEA. Then we derivea reduced metabolic network of the photosyntheticcarbon metabolism, which is shown in Fig. 2. Sinkrepresents the utilization of the photosynthate PGAand GAP. The symbol �i on each arrow represents therate of each reaction. The subscript of �i represents thereaction number. The cycle RuBP→PGA→DPGA→GAP→Ru5P→RuBP represents the Calvin cycle, andthe pathway RuBP→PGCA→GCA→GCEA→PGArepresents the photorespiration.

The reactions in Fig. 2 are

RN Reaction1 RuBP+CO2−→ 2PGA2 PGA+ ATP−→DPGA + ADP3 DPGA+NADPH+H+−→GAP+Pi+NADP4 GAP−→ 0.6Ru5P5 PGA−→ Sink6 GAP−→ Sink

13 Ru5P + ATP−→ RuBP + ADP111 RuBP+O2−→PGA + PGCA112 PGCA+H2O−→ GCA + Pi7 GCA −→ GCEA

113 GCEA + ATP −→ PGA + ADP

where RN represents the reaction number. Clearly,such a reduced metabolic network not only simplifiesthe model but also represents the key processes ofthe photosynthetic carbon metabolism: Calvin cycle,utilization of photosynthate, and photorespiration.Moreover, GAP and Ru5P can be viewed as input andoutput in the conversion of GAP to Ru5P, respectively.Such a process can be taken as a functional module,

111v13v

112v4v

GCAGCEA 7v

1v

Ru5P RuBP PGCA

GAP DPGA PGA3v 113v2v

5v6vSink

Fig. 2. Reduced photosynthetic carbon metabolic network.

and then reduced as GAP→0.6Ru5P . Similarly,PGA→ Sink and GAP→ Sink. From the view offunction, the reduced network (Fig. 2) is equivalent tothe complete one (Fig. 1).

2.2 Rate equation of each reaction

Generally, a metabolic network can be modeledby ordinary differential equations (ODE) or stochasticdifferential equations (SDE). Different modelsmay leadto different results. Lipshtat et al. studied the stochasticeffects on bistability of genetic switch systems [29].Since a reasonable and effective ODE model has beenproposed and improved [10, 22–27], we model thephotosynthetic carbon metabolic systems in a deter-ministic approach based on the existed works. We nowderive some appropriate expressions to describe the ratefor each reaction in Fig. 2 in a mathematical manner.We use the symbols of the metabolites to representtheir own concentrations.

Badger and Lorimer [30] found that some interme-diates of the Calvin cycle, such as PGA, SBP and FBP,can also bind to the Rubisco active sites and compet-itively inhibit RuBP carboxylation. To model such aninhibition, Badger and Lorimer [30], Pettersson andRyde-Pettersson [23] took the reaction rate v1 of RuBPcarboxylation as

v1 = Vmax1RuBP/(RuBP+KM13�),

� = 1+ PGA

KI11+ FBP

KI12+ SBP

KI13

+ Pi

KI14+ N ADPH

KI15,

(1)

where Vmax1 represents the maximal velocity of theenzymatic reaction, KM13 is the Michaelis-Mentenconstant for RuBP, KI11, KI12, KI13, KI14 and KI15are respective constants for PGA, FBP, SBP, Pi andNADPH inhibition of RuBP binding to Rubisco activesites. Moreover, the concentration of Rubisco activesites in the chloroplast stroma can be as high as that



of the substrate RuBP [26, 31–33]. Thus, Farquhar andCaemmerer [10, 25] represented the reaction rate ofRuBP carboxylation approximately as

v1=Vmax1CO2min

(1,

RuBP

Et

)

CO2+KM11

(1+ O2

KM12

) ,

where min(·, ·) is the function which returns the lowestvalue in its elements, Et is the total concentration ofRubisco, KM11 and KM12 are respective Michaelis-Menten constants for CO2 and O2. Based on thoseprevious works, Zhu et al. [22] gave

v1= RuBP

RuBP+KM13�

Vmax1CO2min

(1,

RuBP

Et

)

CO2+KM11

(1+ O2

KM12

) , (2)

where � is given in (1).Since FBP and SBP do not exist in the reduced

network (Fig. 2), the inhibition of these two metabo-lites can be equivalently represented with their upstreammetabolite GAP by choosing an appropriate inhibitionparameter KI16 from mathematical viewpoint. Morespecifically, the term FBP

KI12+ SBP

KI13in the denominator of

v1 in (2) can be replaced by GAPKI16

. Hence, we take

v1 = RuBP

RuBP+�

Vmax1CO2min

(1,

RuBP

Et

)

CO2+KM11

(1+ O2

KM12

) ,

� = KM13

(1+ PGA

KI11+ GAP

KI16+ Pi

KI14+NADPH

KI15

).

(3)

RuBP oxygenation, i.e. reaction 111 (see Fig. 2), is alsocatalyzed by Rubisco. Thus, we take

v111= RuBP

RuBP+�

Vmax111O2min

(1,

RuBP

Et

)

O2+KM12

(1+ CO2

KM11

) . (4)

The reactions 4, 5, 6 and 7 (see Fig. 2) are allsimplifications of a series of biochemical reactions. Weassume that all these reactions obey Michaelis-Mentenkinetics, and the corresponding reaction rates are

v4 = Vmax4GAP

GAP+KM4(5)

v5 = Vmax5PGA

PGA+KM5(6)

v6 = Vmax6GAP

GAP+KM6(7)

v7 = Vmax7GCA

GCA+KM7. (8)

Note that the reverse reaction of the reaction 2(see Fig. 2) is not considered since it is a very weakprocess. The rate equations of the reactions 3, 13, 112and 113 (see Fig. 2) are assumed to be consistent withthose developed in [22]. The mathematical expressionsare

v2 = Vmax2PGA×ATP

(PGA+KM21)(ATP+KM22)(9)

v3 = Vmax3DPGA×NADPH

(DPGA+KM31)(NADPH+KM32)(10)

v112 = Vmax112PGCA

PGCA+KM112

(1+ GCA

KI1121

)(1+ Pi

KI1122

) (11)

v13 =Vmax13

(Ru5P×ATP− ADP×RuBP

KE13

)(Ru5P+KM131

(1+ GAP

KI131+ RuBP

KI132+ Pi

KI133

))ATP

(1+ ADP

KI134

)+KM132

(1+ ADP

KI135

) (12)

v113 =Vmax113

(GCEA×ATP− PGA×ADP

KE113

)(ATP+KM1131

(1+ PGA

KI113

))GCEA+KM1132

. (13)



2.3 Model of the reduced metabolic network

With the above preparation, we take the concen-trations of the orthophosphate Pi in the stroma andthe eight metabolites RuBP, PGA, DPGA, GAP, Ru5P,PGCA, GCA and GCEA in Fig. 2 as the variables, andtake the concentrations of ATP, ADP, NADPH, CO2and O2 as the parameters. Then, the rate of change ofeach metabolite concentration is given by the differ-ence between the rates of the reactions that generate themetabolites and the rates of the reactions that consumethe metabolites:

dRuBP/dt = v13−v1−v111 (14a)

dPGA/dt = 2v1+v111+v113−v2−v5 (14b)

dDPGA/dt = v2−v3 (14c)

dGAP/dt = v3−v4−v6 (14d)

dRu5P/dt = 0.6v4−v13 (14e)

dPGCA/dt = v111−v112 (14f)

dGCA/dt = v112−v7 (14g)

dGCEA/dt = v7−v113, (14h)

where the rates vi are given in (3)–(13). The export ofphotosynthate PGA, GAP and DHAP from the chloro-plast to the cytosol is mediated by the phosphate translo-cator of chloroplast membrane, and is associated witha counter-import of orthophosphate from the cytosolto the chloroplast. Therefore, the total concentration ofphosphate (CP) in stroma remains constant [22, 23, 34].We write the conserved quantity of phosphate approxi-mately as

CP = Pi+PGA+2DPGA+ATP+PGCA

+2RuBP+Ru5P+GAP. (15)

Thus, the differential Equations (14) and the algebraicEquation (15) form a coupled nonlinear differential-algebraic system that represents a reduced model of thephotosynthetic carbon metabolism with nine variablesand 48 parameters.

III. THEORETICAL ANALYSIS

3.1 Effect of the photorespiration

For a dynamic system

dX

dt= f (X), (16)

where X is a vector-valued function of t and f (·) is aknown vector-valued function with appropriate dimen-sion, the equilibrium is defined as the solution of thesystem of equations obtained by setting the right-handside of (16) to zero, i.e. the solution of f (X)=0.

The difference between the reduced photosyn-thetic carbon metabolic network (Fig. 2) and that in [12]is that our reduced network includes the photorespira-tion pathway RuBP→PGCA→GCA→GCEA→PGA, which represents a key biological process in thephotosynthetic carbon metabolism. Hence, we will firstinvestigate the effect of the photorespiration pathwayon the photosynthetic carbon metabolism. We take theorthophosphate Pi as a parameter and consider themodel (14) as the original model here. Without thephotorespiration pathway, the model (14) becomes

dRuBP/dt = v13−v1 (17a)

dPGA/dt = 2v1−v2−v5 (17b)

dDPGA/dt = v2−v3 (17c)

dGAP/dt = v3−v4−v6 (17d)

dRu5P/dt = 0.6v4−v13. (17e)

We find that under a mild condition on the reaction rates,the metabolites PGA, DPGA and GAP have the sameequilibria regardless of the photorespiration pathway.This property is summarized in the following proposi-tion, which is proven in Appendix 5.1.

Proposition 1. Let {RuBP=RuBP0,PGA=PGA0,

DPGA=DPGA0,GAP=GAP0, Ru5P= Ru5P0} be anequilibrium of the system (17), where RuBP0, PGA0,DPGA0, GAP0 and Ru5P0 are fixed positive numbers.Assume that the rate equations v2,v3,v4,v5 and v6 donot depend on the variables RuBP, Ru5P, PGCA, GCAand GCEA. Then, if the system (14) has an equilibrium,it must have the form {RuBP=RuBP1, PGA=PGA0,DPGA=DPGA0, GAP=GAP0, Ru5P= Ru5P1,PGCA=PGCA0, GCA=GCA0, GCEA=GCEA0},where RuBP1, Ru5P1, PGCA0, GCA0 and GCEA0 aresome positive numbers.

Remark 1. In Proposition 3.1, there is no requirementon the detailed form of the reaction rate vi . It requiresonly that the same vi in system (17) and (14) has thesame expression.

Generally, the biochemical reactions 2, 3, 4, 5 and6 (see Fig. 2) are not affected by the metabolites RuBP,Ru5P, PGCA, GCA and GCEA [22]. Hence, the condi-tion on the reaction rates in Proposition 1 is always heldfor the photosynthetic carbon metabolism. Since there



is no requirement on detailed expressions of reactionrates v2,v3,v4,v5 and v6, Proposition 1 is suitable fora wide variety of models of the photosynthetic carbonmetabolism.

3.2 A parameter condition for ruling out multipleequilibria

For the model composed by the differential equa-tions (14) and the algebraic equation (15), it is stilldifficult to analyze its multi-equilibrium property whenall the 48 parameters vary in certain intervals. Thus,we need to equivalently transform this system into asimplified one.

By setting the right-hand side of (14) to zero andwith an equivalent transformation, we get the followingalgebraic equations,

0.6v4−v1−v111 = 0 (18a)

1.2v4−v2−v5 = 0 (18b)

v2−v3 = 0 (18c)

v2−v4−v6 = 0 (18d)

0.6v4−v13 = 0 (18e)

v111−v7 = 0 (18f)

v112−v7 = 0 (18g)

v7−v113 = 0. (18h)

With the rate Equations (3)–(13), (18b) and (18d) forman independent subsystem,

1.2Vmax4GAP

GAP+KM4− Vmax5PGA

PGA+KM5

− Vmax2PGA×ATP

(PGA+KM21)(ATP+KM22)=0 (19a)

Vmax2PGA×ATP

(PGA+KM21)(ATP+KM22)

− Vmax4GAP

GAP+KM4− Vmax6GAP

GAP+KM6=0, (19b)

which contains just two variables (i.e. PGA and GAP)and 10 parameters (i.e. Vmax2, Vmax4, Vmax5, Vmax6,KM21, KM22, KM4, KM5, KM6 and ATP ).

Lemma 1. If the subsystem (19) has only one positivesolution, then the original system (18) has no more thanone positive solution.

The proof of this lemma is given in Appendix 5.2.Based on Lemma 1, we only need to discuss the

subsystem (19). Let x= 1PGA and y= 1

GAP . Then, from(19) we have

1.2Vmax4

1+KM4y− Vmax2ATP

(1+KM21x)(ATP+KM22)

− Vmax5

1+KM5x=0 (20a)

Vmax2ATP

(1+KM21x)(ATP+KM22)− Vmax4

1+KM4y

− Vmax6

1+KM6y=0. (20b)

This transformation only loses the zero root {PGA=0,GAP=0} of (19), which has no meaning. Thus, theroots of the systems (19) and (20) are a one-to-onecorrespondence, and have the same signs.

Eliminating y in (20), we obtain a fractional equa-tion with one variable x that can be written as a poly-nomial equation of degree 3,

ax3+bx2+cx+d=0, (21)

where a, b, c and d are all polynomials of (KM ,Vmax,ATP), KM=(KM21,KM22,KM4,KM5,KM6) andVmax=(Vmax2,Vmax4,Vmax5,Vmax6).

We take the values of the parameters (KM ,Vmax,

ATP) used in [12, 22, 23] as the nominal values inour work, see Table I. For such nominal values(KM0,Vmax0,ATP0), the corresponding roots of (21) are

x10=−1.385, x20=−1.134, x30=18.521. (22)

That is, (21) has only one positive solution x30=18.5209. The corresponding solution of y is y30=0.562646. Thus, {PGA=0.054,GAP=1.777} is theonly positive solution to the subsystem (19), which falls

Table I. Nominal values of the parameters Vmax,KM ,ATP.

Parameter Value Reference

Vmax2 10.3 [22, 26, 36]Vmax4 1.5 [22, 37, 38]Vmax5 0.3 [22, 37, 38]Vmax6 0.7 [22, 37, 38]KM21 0.240 [22, 39]KM22 0.390 [22, 39]KM4 0.84 [12]KM5 0.75 [12]KM6 5.0 [12]ATP 0.68 [22, 26]



within the physiologically relevant range (0.0001−5mM) [35].

Next, we discuss the multi-equilibrium prop-erty of the original system when the 10 parameters(KM ,Vmax,ATP) vary in some field around theirnominal values. Before deriving the main result, wegive the following lemma about polynomial equationof degree 3, whose proof is given in Appendix 5.3.

Lemma 2. Suppose that the coefficients a, b, c and dof the polynomial equation

ax3+bx2+cx+d=0 (23)

are continuous real functions of P= (p1, p2, · · ·, pn)∈Rn (n is a positive integer), i.e. a=a(P), b=b(P),c=c(P) and d=d(P). Let xi0= xi0(a,b,c,d)=xi0(P0), i =1,2,3, be the roots of (23) with respectto the parameter P0= (p10, p20, · · ·, pn0) and x10<0,x20<0, x30>0. Assume that �⊂Rn is connected andP0∈�. If ad(ad−bc) has the same sign for all P∈�,then the positive root x3= x3(P) will keep its sign andthe other two roots will stay in the left open half planewhen P varies in �.

Remark 2. The two roots of (23) that stay in theleft half plane could be a pair of conjugate complexnumbers, two distinct negative numbers, or two repeatednegative numbers.

Theorem 1. Let �⊂R10+ ={(z1, · · ·, z10) : zi ∈R+ =(0,∞), i=1, · · ·,10} be connected and contain thenominal values (KM0,Vmax0,ATP0) listed in Table I,and a,b,c and d be the coefficients of (21). Then, (18)has no more than one equilibrium if ad(ad−bc) doesnot change its sign whenever (KM ,Vmax,ATP)∈�.

Proof. The first two roots in (22) are negative and thelast one is positive. Noticing that ad(ad−bc) has thesame sign for all (KM ,Vmax,ATP)∈�, we can claimthat (21) has one and only one positive root for each(KM ,Vmax,ATP)∈� by Lemma 2. This implies that thesubsystem (20) has no more than one positive root. ByLemma 1, the original system (18) has no more thanone equilibrium for any (KM ,Vmax,ATP)∈�. �

Remark 3. To get the result in Theorem 3.1, it onlyneeds some condition on the subsystem parameter(KM ,Vmax,ATP), which implies the other 38 parame-ters can vary arbitrarily.

3.3 Numeric results

Based on Theorem 1, we can find a certain param-eter field in which the original system (18) has no morethan one equilibrium by some numeric computation.

Let�={(KM ,Vmax,ATP):LKMi ≤KMi ≤UKMi ,

i =21,22,4,5,6, LVmax j ≤Vmax j≤UVmax j , j=2,4,5,6, LAT P≤ATP≤U AT P}, where LKMi and UKMi ,LVmax j and UVmax j , LATP and UATP are some posi-tive numbers. In other words, � is a neighborhood ofthe nominal values (KM0,Vmax0,ATP0).

For the nominal values (KM0,Vmax0,ATP0)listed in Table I, the coefficients of (21) satisfya0d0−b0c0<0,a0d0<0. Thus, we can find some �satisfying ad(ad−bc)>0 for all (KM ,Vmax,ATP)∈�.Actually, if the minimal value of the function

f (KM ,Vmax,ATP)=ad(ad−bc),

is positive on �, then such an � will meet the require-ment. It is difficult to obtain the largest �. But for agiven �, it is relatively easy to verify whether or notthe condition is satisfied. Varying 25% , 20% and 20%around the nominal values for Vmax’s, KM ’s and ATP,respectively, we can get the following field,

Vmax2∈[7.725,12.875], KM21∈[0.192,0.288],Vmax4∈[1.125,1.875], KM22∈[0.312,0.468],Vmax5∈[0.225,0.375], KM4∈[0.672,1.008],Vmax6∈[0.525,0.875], KM5∈[0.6,0.9],ATP∈[0.544,0.816], KM6∈[4.0,6.0].

(24)

Using Mathematica, we find that the function f (KM ,

Vmax,ATP) takes its minimal value 7.17058×106 in �at Vmax2=7.75853, Vmax4=1.13436, Vmax5=0.373942,Vmax6=0.534386, KM21=0.192427, KM22=0.312,KM4=1.00787, KM5=0.602485, KM6=4.00263 andATP=0.555017. Therefore, based on Theorem 1, weclaim that the model composed by the differentialEquations (14) and the algebraic Equation (15) has nomore than one equilibrium, when the 10 parametersVmax2, Vmax4, Vmax5, Vmax6, KM21, KM22, KM4, KM5,KM6 and ATP stay in the field (24) and the other 38parameters are arbitrary.

IV. CONCLUSION AND FUTURE WORK

In this paper, we first proposed a reduced molec-ular network for the photosynthetic carbon metabolism,which describes the key characteristics of the photosyn-thetic carbon metabolism: Calvin cycle, utilization ofphotosynthate, and photorespiration. Then a nonlineardifferential-algebraic model is derived to representthe reduced network. By investigating the effect ofthe photorespiration pathway on the multi-equilibriumproperty in the photosynthetic carbon metabolism, we



found that under a mild condition on the reaction ratesv2, v3, v4, v5 and v6, the metabolites PGA, DPGA andGAP have robust dynamic behavior and are indepen-dent of the dynamics of the photorespiration pathway.Moreover, we studied the multi-equilibrium propertyof the network allowing the parameters to vary in anappropriate domain. Although there are 48 parametersin our model, we proved that if the 10 parameters in thesubsystem stay in a certain field, no matter what valuesthe other 38 parameters take, there exists no more thanone equilibrium in the original system. Such a resultnot only provides profound insights for qualitativelyunderstanding dynamic features of the photosyntheticcarbon metabolism but also can be adopted as a quan-titative criteria to find a correct strategy to improve thephotosynthesis in plants.

From the view of function, the reduced networkis equivalent to the entire one. This paper only gives aparameter condition for ruling out multiple equilibria ofthe reduced network. The parameter condition for thepresence of multiple equilibria and the stability of eachequilibrium are still worth investigating.

V. APPENDIX

5.1 Proof of Proposition 1

Proof. By setting the right-hand side of the ordinarydifferential Equations (17) and (14) to zero and withsome equivalent transformations, respectively, we getthe following algebraic equations,

v13−v1 = 0 (A1a)

1.2v4−v2−v5 = 0 (A1b)

v2−v3 = 0 (A1c)

v3−v4−v6 = 0 (A1d)

0.6v4−v13 = 0, (A1e)

v13−v1−v111 = 0 (A2a)

1.2v4−v2−v5 = 0 (A2b)

v2−v3 = 0 (A2c)

v3−v4−v6 = 0 (A2d)

0.6v4−v13 = 0 (A2e)

v111−v112 = 0 (A2f)

v112−v7 = 0 (A2g)

v7−v113 = 0. (A2h)

Since v2, v3, v4, v5 and v6 do not depend on {RuBP,Ru5P, PGCA, GCA, GCEA}, (A1b)–(A1d) and (A2b)–(A2d) just include the variables PGA, DPGA and GAP,and form two independent subsystems of the systems(A.1) and (A.2), respectively. It is obvious that thesubsystems (A1b)–(A1d) and (A2b)–(A2d) are thesame. That is, the two systems (A.1) and (A.2) have thesame independent subsystem. Thus, the values of PGA,DPGA and GAP in the solutions of systems (A1a) and(A2a) are the same. �

5.2 Proof of Lemma 1

Proof. Since the subsystem (19) (i.e. (18b) and (18d))contains only two variables PGA andGAP, we can solvePGA and GAP first. Then, by the following procedure,we can uniquely solve the other seven variables DPGA,GCEA,GCA, Pi , PGCA, RuBP andRu5P, whichmeansGAP and PGA can determine the other seven metabo-lites uniquely. Thus, if the subsystem (19) has only onepositive solution, then the original system (18) has nomore than one positive solution.

Procedure for solving DPGA, GCEA, GCA, Pi ,PGCA, RuBP and Ru5P.

Step 1: Obtaining the value of each reaction rate.After having obtained the values of {PGA,GAP}

by the subsystem (19), we can get the values of v2, v4,v5 and v6 accordingly. By (18c) and (18e), it is obviousthat

v3 = v2, (B1)

v13 = 0.6v4. (B2)

Denote

WC = Vmax1CO2

CO2+KM11

(1+ O2

KM12

) ,

WO = Vmax111O2

O2+KM12

(1+ CO2

KM11

) ,

W = WC +WO .

Then

v1 = WC

W(v1+v111),

v111 = WO

W(v1+v111).

Let

�= 0.6

Wv4.



Then, by (18a) we have (18f), (18g) and (18h),

v1 = WC ·� (B3)

v111 = WO ·� (B4)

v7 = WO ·� (B5)

v112 = WO ·� (B6)

v113 = WO ·�. (B7)

Step 2: Solving DPGA, GCA and GCEA.With the rate equations (8), (10) and (13), we can

obtain unique DPGA , GCA and GCEA by (B1), (B5)and (B7), respectively.

Step 3: Solving PGCA, RuBP and Ru5P forfixed Pi .

For fixed Pi , GCA can determine PGCA(Pi)uniquely by (B6).

For given PGA and GAP,

�(Pi) = KM13

(1+ PGA

KI11+ GAP

KI16

+ Pi

KI14+ N ADPH

KI15

)

is only a function of Pi . Combining (3) and (B3), wehave

RuBPmin

(1,

RuBP

Et

)

RuBP+�(Pi)=�. (B8)

If RuBP≥Et , then (B8) becomes a linear equation

RuBP

RuBP+�(Pi)=�. (B9)

If RuBP<Et , then (B8) becomes a quadratic equation

RuBPRuBP

Et

RuBP+�(Pi)=�, (B10)

or equivalently,

RuBP2−�EtRuBP−�(Pi)�Et =0. (B11)

Noting that Et , �(Pi) and � are all positive, we have−�(Pi)·� ·Et<0. Therefore, one root of the quadraticequation (B11) is positive, and the other is negative.

For fixed Pi , denote the root of (B9) byRuBP1(Pi), and the positive root of (B11) byRuBP2(Pi). Then we can show that RuBP1(Pi) andRuBP2(Pi) are not positive roots of the original equa-tion (B8) simultaneously, since otherwise, there would

be that RuBP1(Pi) and RuBP2(Pi) were both positiveroots of the original equation (B8) simultaneously. Thisresults in 0<RuBP2(Pi)<Et ≤ RuBP1(Pi),

� =RuBP2(Pi)

RuBP2(Pi)

Et

RuBP2(Pi)+�(Pi)<

RuBP2(Pi)

RuBP2(Pi)+�(Pi)

<RuBP1(Pi)

RuBP1(Pi)+�(Pi)=�,

which is obviously a contradiction. Therefore, we canobtain an unique positive RuBP(Pi) for fixed Pi .Thus, RuBP(Pi) and GAP can determine an uniqueRu5P(Pi) for fixed Pi by (B2).

Step 4: Solving Pi , PGCA, RuBP and Ru5P.We will first show that PGCA(Pi), RuBP(Pi)

and Ru5P(Pi) obtained for fixed Pi in Step 3 are allstrictly increasing functions of Pi .

Define function f (·, ·) as

f (RuBP, Pi)=RuBPmin

(1,

RuBP

Et

)

RuBP+�(Pi).

Then f (RuBP, Pi) is strictly increasing in RuBP anddecreasing in Pi . Let Pi2>Pi1>0 be any two fixedvalues of Pi , and RuBP(Pi1) and RuBP(Pi2) be thecorresponding roots of (B8). That is, f (RuBP(Pi1),Pi1)=� and f (RuBP(Pi2), Pi2)=�. Noticing themonotonicity of f (·, ·), we have

f (RuBP(Pi1), Pi2)< f (RuBP(Pi1), Pi1)

= f (RuBP(Pi2), Pi2),

which implies

RuBP(Pi1)<RuBP(Pi2).

Thus, RuBP(Pi) is strictly increasing in Pi . Similarly,we can show that PGCA(Pi) and Ru5P(Pi) are alsostrictly increasing in Pi .

Now, we will solve Pi by (15). Define functiong(·) as

g(Pi) = Pi+PGA+2DPGA+ATP+GAP

+PGCA(Pi)+2RuBP(Pi)+Ru5P(Pi),

where PGCA(Pi), RuBP(Pi) and Ru5P(Pi) areobtained in Step 2. Then (15) becomes

g(Pi)=CP . (B12)

By the above argument, g(Pi) is strictly increasing inPi . Thus, if the solution of (B12) exists for a givenparameter CP , then it must be unique. Suppose that Pi



is the unique solution of (B12). Then we can obtainPGCA=PGCA(Pi), RuBP=RuBP(Pi) and Ru5P=Ru5P(Pi) uniquely. �

5.3 Proof of Lemma 2

Proof. We will first show that “ad(ad−bc) has thesame sign for all P∈�” is equivalent to “ − d

a>0 andbc>ad ( or bc<ad) for all P∈� ”. Note that bc>ad (orbc<ad) for all P∈� means that ad−bc keeps its signfor all P∈�, and − d

a>0 is equivalent to ad<0. Thus,− d

a>0 and bc>ad (or bc<ad) for all P ∈� implies thatad(ad−bc) keeps its sign for all P∈�. Conversely,assume that ad(ad−bc) keeps its sign for all P∈�.First, − d(P0)

a(P0)= x10x20x30>0 implies a(P0)d(P0)<0. If

there would exist a P2∈� such that a(P2)d(P2)>0,then there would be a P3∈� such that a(P3)d(P3)=0by the connectivity of � and the continuity of a(P) andd(P). Then, a(P3)d(P3)(a(P3)d(P3)−b(P3)c(P3))=0. This contradicts the condition that ad(ad−bc) keepsits sign for all P∈�. Thus, ad<0 for all P in �, whichimplies − d

a>0. Similarly, we can show that bc>ad (orbc<ad) for all P∈�.

Next, we will show that the roots xi = xi(P) (i =1,2,3) cannot be on the imaginary axis for all P∈�.Noting that x1x2x3=− d

a and the assumption− da>0 for

all P∈�, we have xi �=0 (i =1,2,3). Assume, to arriveat a contradiction, that there existed P1∈� such thatthe corresponding roots x11(P1), x21(P1) were a pair ofconjugate imaginary roots. Then (23) would have theform

a1(x+ p)(x2+q)=0, q>0,

or equivalently,

a1x3+a1 px

2+a1qx+a1 pq=0, q>0.

That is, b1=a1 p, c1=a1q and d1=a1 pq, whichimplies b1c1=a1d1. This contradicts the assumptionthat bc>ad (or bc<ad) for all P∈�. Hence, theremust be no root of (23) on the imaginary axis forany parameter P∈�. This means the root set of (23)corresponding to the parameter P∈� is divided intotwo parts by the imaginary axis.

Since the roots of a polynomial equation of degree3 depend on the parameters continuously, the roots xi =xi(P) (i =1,2,3) cannot cross the imaginary axis whenthe parameter P varies in � continuously. Noticing theconnectivity of�, we can claim that the roots xi = xi(P)

(i=1,2,3) will stay in its original field, either the lefthalf plane or the right half plane, no matter how theparameter P varies in �.

Finally, we will show that the positive root x3=x3(P) will stay on the positive real axis when P variesin�. Otherwise, x3= x3(P)would become into a pair ofconjugate complex roots x31 and x32 since they cannotbe two positive roots, and the other two roots (i.e. x1and x2) would merge into an negative number, say x12.Thus, the product of the three roots x12x31x32 wouldbe negative, that is − d

a = x12x31x32<0. This contradictsthe condition − d

a>0 for all P∈�. �

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Hong-Bo Lei received his B.S.degree in Statistics from theUniversity of Science and Tech-nology of China, Hefei, China, in2006.Now he is pursuing his Ph.D.degree at the Key Laboratoryof Systems and Control, ChineseAcademy of Sciences, Beijing,China. His current research inter-ests are sytem modeling and

control, metabolic control analysis and systems biology.

Xin Wang received his B.S.degree in Information and ScienceComputing from ShandongUniversity, Shandong, China, in2006. He is currently workingtoward his Ph.D. degree at theKey Laboratory of Systems andControl, Chinese Academy ofSciences, Beijing, China. Hisresearch interests are systems

biology, subspace identification, and identification ofgenetic networks.

Ruiqi Wang received his M.S.degree in Mathematics fromYunnan University, Kunming,China, in 1999, and his Ph. D.degree in Mathematics from theAcademy of Mathematics andSystems Science, CAS, Beijing,China, in 2002. Since 2007, hehas been with the faculty of theShanghai University, Shanghai,

China, where he is currently an Associate Professor atthe Institute of Systems Biology. His fields of interestare systems biology and nonlinear dynamics.

Xin-Guang Zhu , is the Ph.D.Junior Independent Group Leader,CAS-MPG Partner Institute ofComputational Biology. Hismajor research interests: includephotosynthesis, plant systemsbiology, options to engineer higherphotosynthetic energy conver-sion efficiency, synthetic biology.(received his Ph.D. degree from

the University of Illinois at Urbana Champaign in2004).

Luonan Chen received his M.E.and Ph.D. degrees in the ElectricalEngineering, from Tohoku Univer-sity, Sendai, Japan, in 1988 and1991, respectively. From 1997, hewas an Associate Professor of theOsaka Sangyo University, Osaka,Japan, and then a full Professor.Since 2010, he has been the Excu-tive Director at Key Laboratory

of Systems Biology, Shanghai Institutes for Biolog-ical Sciences, Chinese Academy of Sciences. He wasthe Founding Director of Institute of Systems Biology,Shanghai University, and from 2010 he has also beena Visiting Professor in Institute of Industrial Science,The University of Tokyo, Japan. He serves as Chairof Technical Committee of Systems Biology at IEEESMC Society, and President of Computational SystemsBiology of ORS China. His fields of interest are systemsbiology, bioinformatics and nonlinear dynamics. Heserves as editor or editorial board member for manysystems biology related journals, e.g. BMC SystemsBiology, IEEE/ACM Trans. on Computational Biologyand Bioinformatics, IET Systems Biology, Mathemat-ical Biosciences, Neural Processing Letters, Frontier inSystems Physiology, International Journal of Systemsand Synthetic Biology, and Journal of Systems Scienceand Complexity. In the past five years, he published over100 journal papers and two monographs on the area ofsystems biology.

Ji-Feng Zhang received his B.S.degree in Mathematics from Shan-dong University in 1985, andhis Ph.D. degree from the Insti-tute of Systems Science (ISS),Chinese Academy of Sciences(CAS) in 1991. Since 1985 he hasbeen with ISS, CAS, where he isnow a Professor of Academy of



Mathematics and Systems Science, the Vice-Directorof the ISS. His current research interests include systemmodeling and identification, adaptive control, stochasticsystems, and multi-agent systems. He received theDistinguished Young Scholar Fund from NationalNatural Science Foundation of China in 1997, the FirstPrize of the Young Scientist Award of CAS in 1995,the Outstanding Advisor Award of CAS in 2007, 2008

and 2009, respectively, and served as Managing Editorof the Journal of Systems Science and Complexity;Deputy Editor-in-Chief of the following three journals:Acta Automatica Sinica, Journal of Systems Scienceand Mathematical Sciences, Control Theory and Appli-cations; and Associate Editor of several other journals,including the IEEE Transactions on Automatic Control,SIAM Journal on Control and Optimization, etc.


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