A quantitative comparison ofCalvin–Benson cycle modelsAnne Arnold1,2 and Zoran Nikoloski1,2

1 Institute of Biochemistry and Biology, University of Potsdam, 14476 Potsdam, Germany2 Systems Biology and Mathematical Modeling Group, Max-Planck Institute of Molecular Plant Physiology, 14476 Potsdam,

Germany

Review

The Calvin–Benson cycle (CBC) provides the precursorsfor biomass synthesis necessary for plant growth. Thedynamic behavior and yield of the CBC depend on theenvironmental conditions and regulation of the cellularstate. Accurate quantitative models hold the promise ofidentifying the key determinants of the tightly regulatedCBC function and their effects on the responses in futureclimates. We provide an integrative analysis of the larg-est compendium of existing models for photosyntheticprocesses. Based on the proposed ranking, our frame-work facilitates the discovery of best-performing modelswith regard to metabolomics data and of candidates formetabolic engineering.

Modeling as a step towards understandingMathematical modeling of integrated processes lends itselfas a useful tool for in silico probing of biological systems.The existing modeling paradigms hold the promise totackle one of the greatest challenges in plant physiology,improving the understanding of the Calvin–Benson cycle(CBC1), its limiting steps, and the relation to plant growth[1]. Mathematical modeling allows for placing this impor-tant metabolic pathway in the context of its cellular milieu(e.g. surrounding pathways) and the entire carbon cycle.Moreover, it provides the means for predicting systemicbehavior on various levels of the system, rendering itvaluable in planning laborious experiments aimed at con-firming posited hypotheses.

Here we present a comprehensive critical review of theexisting models of the CBC. By assembling and implement-ing a compendium of 15 models, our aim is to identify thosemodel candidates that provide quantitatively accuratepredictions for the levels of CBC intermediates and showbiologically plausible dynamics. These candidates, in turn,can be used in metabolic engineering [2] and in the designof synthetic metabolic pathways for improved carbon fixa-tion, growth and yield [3].

Overview of model componentsThe challenge of modeling involves selection of the relevantbiochemical reactions. In the case of the CBC, the mostgeneral description includes the three stages: carboxyla-tion, reduction and regeneration, which can further bedivided into more specific sub-processes down to the levelof single reactions. One of the key reactions is the initial

Corresponding author: Nikoloski, Z. (nikoloski@mpimp-golm.mpg.de)1 See abbreviation table (Supplementary Material Table S30).

676 1360-1385/$ – see front matter � 2011 Elsevier Ltd. All rights reserved. d

step, whereby CO2 enters the CBC. It is termed the ribu-lose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO;EC 4.1.1.39) reaction, although the corresponding enzymename is misleading [4]. In fact, the enzyme initially reactswith ribulose-1,5-bisphosphate (RuBP), resulting in theenediol–enzyme complex which can then capture CO2

(carboxylation) or O2 (oxygenation) [5,6]. Due to its bio-chemical importance, this step has been included in almostall modeling attempts either at reaction level or its repre-sentation as substrate–enzyme and product–enzyme steps[7]. The other key reactions of the CBC, such as thosecatalyzed by fructose-1,6-bisphosphatase (FBPase; EC3.1.3.11), sedoheptulose-1,7-bisphosphatase (SBPase; EC3.1.3.37) and ribulose-5-phosphate (Ru5P) kinase (EC2.7.1.19), also appear in almost every model at the reactionlevel [2,7–12].

In the case of the end-product processes, the key reactionsinvolve ADP-glucose pyrophosphorylase (AGPase, EC2.7.7.27) for starch synthesis, UTP-glucose-1-phosphate uri-dylyltransferase (EC 2.7.7.9) for sucrose synthesis andRuBisCO (oxygenase function) for photorespiration. More-over, the steps branching from the CBC and competing forthe branch-point intermediates have to be carefully consid-ered. For instance, the triose phosphates (TP) are the branchintermediates for sucrose synthesis, and the export of TP outof the chloroplast is the corresponding branching point.There is a strict counter-exchange of TP with inorganicphosphate (Pi) via the triose-phosphate translocator (TPT)which does not follow established kinetics, thus requiringcareful selection of modeling strategies [13].

The regulatory processes, activation and inhibition,enable the adaptation to different conditions and representyet another aspect of the modeled pathways. Light activa-tion, as the best characterized effect of an external regula-tor, affects several enzymes of CBC and of the end-productprocesses, which undergo a 2 to 40-fold increase in activityafter the onset of light [14–16].

A common example of an external metabolic regulator ofthe CBC is the CO2–O2 competition for RuBisCO. In both(i.e. carboxylation and oxygenation reactions), each gas-eous substrate acts as an inhibitor of the reaction involvingthe other, i.e. CO2 and O2 influence the CBC/photorespi-ration ratio (Supplementary Material 1.3.2) [17].

Internal metabolic regulators can be distinguished bythe direction of their effect in a reaction chain. If a reactiondoes not take place, its substrate(s) will accumulate and itsproduct(s) will be depleted. The concentration of substrates

oi:10.1016/j.tplants.2011.09.004 Trends in Plant Science, December 2011, Vol. 16, No. 12

Review Trends in Plant Science December 2011, Vol. 16, No. 12

or products could then inhibit the downstream reactions,termed forward regulation. However, in backward regula-tion, metabolites may activate or inhibit previous reactionsby affecting the activity of the corresponding enzymes or byproducing the necessary cofactor.

Forward regulation for photosynthetic processes is ex-perimentally proven for AGPase. The CBC intermediate 3-phosphoglycerate (PGA) increases the activity of AGPaseby 9 to 80-fold and enforces starch production [18,19]. Bycontrast, Pi inhibits this reaction and it was shown that thePGA/Pi ratio, in fact, exerts control over CO2 incorporationinto starch [20,21].

A typical backward regulation within the CBC is theproduct inhibition (also termed product competition) ofFBPase, whereby increased concentration of fructose-6-phosphate (F6P) inhibits the enzyme activity of the previ-ous reaction [22]. More complex cases of regulatory pro-cesses exist, such as substrate competition for a CBCenzyme. Transketolase (EC 2.2.1.1) catalyzes two revers-ible reactions, the transformation of F6P or sedoheptulose-7-phosphate with glyceraldehyde-3-phosphate (GAP) toerythrose-4-phosphate or ribose-5-phosphate (R5P) withxylulose-5-phosphate (X5P), respectively (SupplementaryMaterial Equation S32). It may be expected that thedifferent substrates inhibit the competing reaction, al-though only the inhibition of F6P and R5P has beenexperimentally established. However, the unproven inhi-bition by the remaining substrates has often been used inmodeling of the CBC [2,7,9], although it may lead todoubtful conclusions. Moreover, this inhibitory effect hasbeen proven only for Leishmania mexicana [23], an obli-gate intracellular protozoan parasite. The integration ofsuch regulatory processes, not experimentally proven forhigher plants, or even for the model plant Arabidopsis

Table 1. Classification of the 15 CBC modelsa

Model Classificat ion

Level Bounda ry K

Farquhar et al.

Leaf

Medlyn et al.

Schultz

Sharkey et al.

Damou r and Urban

Fridlyand and Scheibe Compartment

CB

C

Zhu et al. (20 09)

Gier sch et al.

Cell

Hahn ++b

Poolman et al. +

Petterss on and Ryde -Pettersso n +

Woodrow and Mott +

Laisk et al. (1989 ) ++

Laisk et al. (2006 ) ++

Zhu et al. (20 07) +++

RuB

isC

O

Abbreviations: a, activation; c, competitive inhibition; m, mixed inhibition; n, non-comaDetails and regulatory processes are included in Supplementary Material Tables S3 ab+ CBC & starch; ++ CBC & starch & sucrose; +++ CBC & starch & sucrose & photorescRed, mass-action; green, equilibrium-approximation; dark blue, Michaelis–Menten; p

thaliana (Arabidopsis), could result in more implausiblepredictions. This exemplifies the fact that regulatory pro-cesses are the most challenging part of modeling the CBCand, therefore, have to be examined with special care.

Classification of CBC modelsBy thoroughly reviewing the literature spanning the pastthree decades, we assembled the largest existing compen-dium of models for the CBC. The compendium consists of15 models, including the initial modeling attempts cover-ing various contexts of photosynthesis-related processesand some of their widely cited extensions [2,7–12,24–31](Supplementary Material Table S1). We provide a detailedclassification of the models included in the compendiumbased on (i) model boundaries, i.e. the coverage of CBCtogether with the end-product pathways; (ii) levels ofcellular organization, i.e. leaf, cell or compartment; (iii)complexity of kinetics [32], translating the model structureinto mathematical equations for analyzing spatiotemporalproperties; and (iv) included regulatory processes, specify-ing the regulators, their types and the resulting formaliza-tion [33].

The considered models differ in their boundaries due tothe aspects of photosynthesis they cover. Five of the modelsfocus on the RuBisCO reaction and merge the remainingsteps of the CBC (Table 1, column 3). There are threemodels describing the CBC in detail but omit relatedprocesses. A group of three models investigate the process-es taking place in the chloroplast, namely the CBC andstarch synthesis, and the remaining four models addition-ally integrate the different end-product pathways. Themodel of Laisk et al. (2006) [7] originally includes thephotosystems and electron transport chain; however, inthe model comparison, we use a reduced version capturing

Refs ine ticsc Regulatio n

c [25]

c [28]

c [29]

c [30]

c [24]

[8]

[31]

p [26]

[27]

a c m n p [11]

a c m n p [10]

c p [12]

a c [9]

a x [7]

a c m n p [2]

petitive inhibition; p, competitive product; x, unidentified inhibition.

nd S4.

piration.

urple, Michaelis–Menten-like; black, special-functions kinetics.

677

Review Trends in Plant Science December 2011, Vol. 16, No. 12

the CBC and the end-product pathways (SupplementaryMaterial 1.2).

The model boundaries partly affect the levels of cellularorganization. Models focusing exclusively on the CBC andthe processes in the chloroplast were modeled at compart-ment level (Table 1, column 2), whereas models includingsucrose synthesis (and photorespiration) span the cell level.As a result, the cell-level models include details concerningcompartmentation and the related transport steps. Theremaining models describe photosynthesis at the most com-plex biological level, i.e. the leaf. This level necessitatesconsideration of diffusion and the difference of atmosphericand intercellular partial pressure of gases. However, thesedetails were merely included in the RuBisCO-focusinggroup, consisting of the smallest models.

A critical part of kinetic modeling is the translation ofthe model structure into mathematical equations, requir-ing specification of the reaction kinetics. The most commonapproaches for modeling reaction kinetics include mass-action and Michaelis–Menten kinetics (Table 1, column 4;Supplementary Material 1.3.1). In the compendium, othertypes of kinetics in the form of Michaelis–Menten-like andspecial functions are also considered. The special functionscomprise the kinetics proposed by Giersch, specificallytailored to TPT [13], and all remaining kinetics withinthe compendium which do not fall into the common theo-retical frameworks. Regardless of the type, the kineticfunctions describe the velocities of the modeled reactions,and determine the temporal changes of the metaboliteconcentrations which can be mathematically describedby differential equations.

0

10

20

30

210 PGA

0

0.1

0.2

0.3

0.4

0.5

Farq

uhar

[25]

Med

lyn

[28]

Sch

ultz

[29]

Sha

rkey

[30]

Dam

our

[24]

∗∗Fr

idly

and

[8]

Zhu

’09

[31]

Gie

rsch

[26]

Hah

n[2

7]P

ool

man

[11]

∗Pet

ters

son

[10]

∗Wo

odr

ow[1

2]∗∗

Lais

k’8

9[9

]La

isk

’06

[7]

Zhu

’07

[2]

Ara

bido

psis

[43]

0.7 Ru5P

Met

abol

iteco

ncen

trat

ion

[mM

]

Figure 1. Quantitative steady-state concentrations of PGA, TP, Ru5P and RuBP for each

(gray). For Zhu et al. (2009) [31], only GAP is available instead of TP ( ). Experimentally

extrapolated concentration of Ru5P using the assignment of Zhu et al. (2007) [2]. The val

could not be evaluated (and are excluded for this reason) and the remaining are simul

678

Nevertheless, there exist simplifications of the kinetics,such as the equilibrium approximation, which are notsuitable for time-resolved description. For reactions mod-eled by this approximation such as TP isomerase (EC5.3.1.1), one assumes rapid settling into equilibrium. Thisassumption may be advantageous as it allows reducing thesystem size. The metabolites involved in reactions, whichare very fast compared to the adjacent reactions, aremerged into a metabolic pool and their transient behavioris determined by the dependence to this pool (Supplemen-tary Material Equation S20) [2,7,9]. Such simplificationsare common and reliable for small metabolic pools includ-ing two or three different intermediates. However, equilib-rium approximation of many reactions forming one poolmay restrict or even disable any temporal analysis of theremaining system, as the differential equations can beevaluated solely at steady state [10,12], without any addi-tional assumption (Supplementary Material 1.3.1). As aresult, equilibrium approximation has not been used tomodel the key reactions discussed above.

The employed kinetics can be regarded as the determi-nants of the regulatory processes. For instance, the mass-action kinetics and the equilibrium approximation do notallow the inclusion of any regulation term. For the CBCmodels, Michaelis–Menten, Michaelis–Menten-like as wellas special-functions kinetics can integrate regulatory pro-cesses and reinforce their usage for modeling the keyreactions. In the compendium, there are several combina-tions of regulators and regulated reactions: altogether, 19reactions are regulated by 19 regulators (SupplementaryMaterial 1.3.2). We further distinguished six different

0

1

2

3

8 TP

0

1

2

3

Farq

uhar

[25]

Med

lyn

[28]

Sch

ultz

[29]

Sha

rkey

[30]

Dam

our

[24]

∗∗Fr

idly

and

[8]

Zhu

’09

[31]

Gie

rsch

[26]

Hah

n[2

7]P

ool

man

[11]

∗Pet

ters

son

[10]

∗Wo

odr

ow[1

2]∗∗

Lais

k’8

9[9

]La

isk

’06

[7]

Zhu

’07

[2]

Ara

bido

psis

[43]

RuBP

TRENDS in Plant Science

model within the compendium (green) and the corresponding experimental data

, Ru5P could be measured only together with X5P ( ). The brighter gray shows the

ues of the models marked with * are analytically calculated, models marked with **

ated.

Table 2. Results and model rankings of the 15 investigated models

ModelResults Ranking

RefsStabilityb Robustness RSS [mM2]c Preliminary Similarity Final

Farquhar et al. 0.41557 11.0248 1 – 1 [25]

Medlyn et al. 0.59599 14.9133 4 – 4 [28]

Schultz 0.66280 994.0520 9 – 10 [29]

Sharkey et al. 0.03036 4532 11 – 11 [30]

Damour and Urban 1 55.5920 5 – 6 [24]

Fridlyand and Scheibe – – 14 – 14 [8]

Zhu et al. (2009) a 0.06418 25.4759 8 [27] 9 [31]

Giersch et al.a 0.02934 11.8853 3 [7] 3 [26]

Hahna 0 14.3666 5 [31] 7 [27]

Poolman et al.a 0.01873 0.7460 1 [2] 2 [11]

Pettersson and Ryde-Pettersson – 0.2811 12 – 12 [10]

Woodrow and Mott – 4.2994 13 – 13 [12]

Laisk et al. (1989) – – 14 – 14 [9]

Laisk et al. (2006) a 0.01500 14.4639 5 [26] 5 [7]

Zhu et al. (2007) a 0.00042 14.5347 9 [11] 8 [2]

aCandidates for metabolic engineering.

b stable; unstable; no steady state at all.

cRSS, averaged residual sum of squares.

Review Trends in Plant Science December 2011, Vol. 16, No. 12

(sub)types of regulation: one activation and five inhibition(i.e. competitive, mixed, non-competitive, competitiveproduct and unidentified) (Table 1, column 5; Supplemen-tary Material Tables S3 and S4). This resulted in 29different combinations of regulators and regulation types,and up to four different terms for a single regulated reac-tion.

Model analysis and comparisonTo capture biologically realistic scenarios, a model of theCBC should be stable and robust to small parameterperturbations. Moreover, it should reflect the experimentaldata. These three criteria form the basis for the ranking ofthe models in the compendium. Furthermore, modelswhose dynamics are more similar to those determinedas well-performing will be considered more reliable. Tothis end, we carried out the following analyses with respectto (i) sensitivity, (ii) stability, (iii) robustness and (iv)residual sum of squares (RSS) at the resulting steadystates.

The implementation of the models in the compendiumhas proven a challenging step, especially the case of repro-ducing the published results. To facilitate the usage of thecompendium, we implemented the included models inSystems Biology Markup Language (SBML) suitable foranalysis tools, such as: SBtoolbox2 and COPASI (Supple-mentary Material 1.4, Table S5). Nevertheless, furtherchallenges arose: the models of Pettersson and Ryde-Pet-tersson [10], Poolman et al. [11] and Zhu et al. (2007) [2] usethe same incorrect velocity for the AGPase reaction due toa unit inconsistency first appearing in Pettersson andRyde-Pettersson [10] (Supplementary Material 1.2). Weresolved this issue in accordance with the authors’ sugges-tion. A similar inconsistency was corrected for the reactiondescribing the transformation of PGA to sink in the modelof Zhu et al. (2009) [31]. In addition, for some models not allkinetic parameters are given. Here, we resolved this issue

by using values obtained from prior model versions[9,34,35], alternative models [2,7,9,10,25,28] and litera-ture [36–42] (Supplementary Material Table S2). Never-theless, even after this step, the model of Fridlyand andScheibe [8] has missing parameter values. As a result, thismodel was excluded from further analyses.

As the temporal analysis requires model output in theform of time series, the models based on the steady-stateassumption cannot be evaluated in the time domain; there-fore, the models of Pettersson and Ryde-Pettersson [10]and Woodrow and Mott [12] were excluded from temporalanalyses and were not implemented in SBML. Further-more, for the model of Laisk et al. (1989) [9], imaginaryconcentrations for ADP arise (Supplementary MaterialEquation S11) at physiologically plausible values forADP-glucose and adenylphosphate (Supplementary Mate-rial Equations S13 and S14). Therefore, this model wasalso been excluded from any further analysis.

To enable a fair comparison, the initial conditions for allmodels must be the same. To this end, we chose the innerand outer metabolite concentrations from Zhu et al. (2007)[2] as a reference data set of initial values, due to itsextensive coverage and literature support. This includesthe boundary conditions for the CBC – namely, CO2, O2,the energy equivalents provided by the electron transportchain and the Pi pool. Whether the boundary conditions areinner metabolites, outer metabolites, or even not integrat-ed, varies from model to model. However, if integrated, theinitial values of the boundary conditions are almost thesame within the compendium (Supplementary MaterialTable S6). Only the model of Hahn [27] uses very differentinitial values due to its special metabolites, such as thia-mine pyrophosphate glycoaldehyde. Based on these values,we determined the steady state by using the resulting timeseries or calculated it by using the steady-state assump-tion. Startlingly, very different steady-state solutions wereobtained from the thirteen suitable models, e.g. for PGA

679

Zhu ’07 [2]Laisk ’06 [7]

Poolman [11 ]Hah n [27 ]

Gier sch [26 ]Zhu ’09 [31 ]

Zhu

’09

[31]

Gie

rsch

[26]

Hah

n[2

7]

Po

olm

an[1

1]

Lais

k’0

6[7

]

Zhu

’07

[2]

τ of elasti city coeffic ients for s4r3·

-1

-0.5

0

0.5

1

Sco

re

Zhu ’07 [2]Laisk ’06 [7]

Poolman [11 ]Hah n [27 ]

Gier sch [26 ]Zhu ’09 [31 ]

Zhu

’09

[31]

Gie

rsch

[26]

Hah

n[2

7]

Po

olm

an[1

1]

Lais

k’0

6[7

]

Zhu

’07

[2]

τ of flux control coefficients for s4r3

-1

-0.5

0

0.5

1

Sco

re

TRENDS in Plant Science

Figure 2. Kendall t0s for pairwise comparison of category s4r3 (Supplementary Material Tables S10 and S11) where: red indicates very similar, black neutral and turquoise

very different dynamic behavior (legend right-hand side).

Review Trends in Plant Science December 2011, Vol. 16, No. 12

and TP within 0.147–215.042 and 0.113–8.503 mM(Figure 1, Supplementary Material Table S7), respectively.The set of obtained steady-state solutions for the metabo-lite concentrations is the starting point for the analyses.

Stability and robustness analyses

Metabolic systems have evolved to provide stable androbust operation in a well-defined physiological range,reflecting the effects of external and internal perturba-tions, respectively. To check which of the models areendowed with these properties, we carried out a stabilityanalysis (Supplementary Material 2.3) to investigate theeffects of external perturbations on the system. All of the11 models suitable for this analysis were stable at the givensteady state (Table 2, column 2; Supplementary MaterialTables S22–S27).

To investigate the robustness, namely, the effects ofchanging values for the kinetic parameters, as internalperturbations, we tested whether the system exhibitssmall deviations from the given steady state (Supplemen-tary Material 2.4). This evaluation was repeated 105 timesand the relative frequency of robust instances wasrecorded (Table 2, column 3; Supplementary MaterialAlgorithm S1). Among the six most robust models arethe five of the RuBisCO-focusing group [24,25,28–30],and the model of Zhu et al. (2009) [31]. These modelsare also the six smallest within the compendium, leadingto the claim that robustness decreases with increasingmodel complexity. The order of the remaining modelssupports this claim: Giersch et al. [26] (ranked seventh)covers the CBC only (but is larger than that of Zhu et al.2009 [31]); whereas Poolman et al. [11] (ranked eighth),Laisk et al. (2006) [7] (ninth) and Zhu et al. (2007) [2](tenth) comprise each one additional end-product process.The model of Hahn [27] is the only which does not followthe rule. The underlying mass-action kinetics does notpermit inclusion of regulatory processes which are neededto stabilize thermodynamically highly irreversible fluxes,in particular, the carbon flux through the CBC down to theend-product processes.

Residual sum of squares

The biological relevance of the models is reinforced by thecompliance of the computational predictions to experimen-tal data. We calculated the RSS between the given steady-state concentrations and metabolomics data of Arabidopsis

680

[43] to investigate the physiological plausibility of thepredictions of each model. The data include one of themost recent measurements capturing almost all CBCmetabolites. Moreover, the experimental data wereobtained under non-limiting CO2, O2 and light conditionsin accordance with the assumptions of most of the models.As mentioned above, the model of Hahn [27] is the onlyoutlier (Supplementary Material Table S5). We note thatCBC metabolites are compartmentalized, but only partialexperimental support is available for them [44,45]. To usethese data, we converted them by using the subcellularvolumes [46] together with assumptions, such as GAP is5% of dihydroxyacetone phosphate [47] (SupplementaryMaterial 2.5, Table S28). We would like to stress that notime series is required for these calculations and, conse-quently, the RSS can be calculated for 13 models in thiscompendium. Because of the different model boundaries,the RSS involves different numbers of metabolites for eachmodel. For model comparison, we employed the averagedRSS, RSS (Table 2, column 4; Supplementary Material 2.5).The models of Pettersson and Ryde-Pettersson [10], Pool-man et al. [11] and Woodrow and Mott [12] best describethe Arabidopsis data. The high ranking of this group ofmodels is comprehensible taking into account that the twolast models are extensions of the first and, therefore, aresimilar in structure, size and model boundaries. However,the model of Farquhar et al. [25] has the fourth highcompliance to the data, whereas the other RuBisCO-focus-ing models are ranked between ninth and thirteenth. Thisshows that similar model structure only does not implysimilar compliance to data. The relationship between lowrank and high model complexity observed in the case ofrobustness analysis is not found for the case of RSS.

Sensitivity analysis

Models describing the same pathway should show similardynamic behavior in the vicinity of their steady states. Atpresent, a comparison of dynamic behavior across themodels is hampered by the lack of time-resolved experi-mental data for the concentration of metabolites appearingin more than one compartment. Therefore, to determinethe similarity for two models, we carried out a sensitivityanalysis comprising metabolic control analysis (MCA) foreach model [48] and the Kendall rank correlation of theoutcomes for each pair of models [49]. The concept of MCAallows investigation of the effects of small parameter

Review Trends in Plant Science December 2011, Vol. 16, No. 12

perturbations on a steady state. We focused on theelasticity and flux control coefficients which describe thecontribution of perturbations in metabolite concentrationson the velocity of a single reaction and changes in enzymeactivity on all velocities, respectively. To enable the com-parison of the different models, six new matrices werecreated from each original MCA matrix by reduction andmerging of entries, corresponding to the model boundaries(Supplementary Material Tables S8 and S9). The similari-ties between the models could then be investigated for themost general form of the CBC, more detailed versions of theCBC with five steps, as well as for the entire CBC withstarch and/or sucrose synthesis.

In addition, for any pair of models, the Kendall rankcorrelation t was used to quantify the similarity of theMCA results (Supplementary Material 2.2.2, EquationS37). Larger t implies more similar behavior betweenthe compared models (Figure 2, Supplementary MaterialTables S10–S20, Figures S1 and S2). We note that t cannotbe calculated in the case of only one substrate and tworeactions, since it requires at least six arguments. There-fore, five of the remaining 11 models, focusing on theRuBisCO reaction, are excluded from this analysis.

For the models of Zhu et al. (2009) [31], Giersch et al.[26], Hahn [27], Poolman et al. [11], Laisk et al. (2006) [7]and Zhu et al. (2007) [2], the Kendall correlation of theelasticities, tE, yields a value greater than zero in each ofthe six cases for each possible pair of models. This providesa similarity between all models with respect to local effectson reaction velocities. However, regarding the globaleffects captured by the Kendall correlation of the fluxcontrol coefficients, tF, the models differ considerably,which is demonstrated by tF 2 [� 1, 1]. Large variationsdo not only appear for different pairwise combinations, butalso for the same combination among the different cases.For a given model, the average of the six cases is used toidentify the most similar candidate from the compendium.This is done for the elasticity (t̄E) and the flux controlcoefficients (t̄F), as well as for their combination, t̄av (Sup-plementary Material Table S21). The mutually most simi-lar pairs of models are: Zhu et al. (2009) [31] and Hahn [27],Giersch et al. [26] and Laisk et al. (2006) [7], and Poolmanet al. [11] and Zhu et al. (2007) [2]. These similarities, t̄av,were integrated into the final ranking (SupplementaryMaterial Table S29, column 11), so that models showingsimilar dynamic behavior to well-performing ones becamehigher ranked. Obviously, worse-performing models andthose similar to them are then ranked lower.

RankingThe applied analyses favor different models as best-performing (Supplementary Material Table S29). Todetermine the best-performing models, we worked out ascore combining the ranking of each criterion with respectto their relevance. In our opinion, for a biologically reliableCBC model, (i) the stability is of paramount importance,since small perturbations of metabolite concentrationsshould not lead to the system’s break-down; (ii) the com-pliance of the model predictions to experimental data isthe second crucial property, because regardless of howstable a model is, if the steady state is physiologically

unreliable the model is useless for biological predictions;and (iii) the model robustness is consequently the thirdbasic principle. To demonstrate their relevance, the crite-ria are weighted by decreasing factors, here chosen as:four, two and one, respectively (Supplementary MaterialEquation S43).

The resulting preliminary score for each model (Table 2,column 5; Supplementary Material Table S29) is furthercombined with the one of the most similar model via linearcombination. For the models applicable for sensitivityanalysis, the similarity value (t̄av) is used as the weightingfactor (Supplementary Material Equation S44). So, thegreater the similarity of the mutually most similar models,the more the preliminary ranking of the most similarmodel influence the final score of a model (Table 2, column7; Supplementary Material Table S29). For the modelsunsuitable for sensitivity analysis, the weighting factoris set to zero. Therefore, their final rank is affected only bythe preliminary score.

The overall best-performing model of this ranking isthat of Farquhar et al. [25]. Although the model and itsextensions are criticized [4], when restricted to the rankingcriteria it performs best to investigate carbon fixationprocesses. Due to the focus on the RuBisCO reaction andthe merging of the remaining steps of the CBC, i.e. thesmall model size, the RuBisCO-focusing models [24,25,28–

30] (ranked first, fourth, sixth, tenth and eleventh) provideinsufficient information for metabolic engineering. More-over, the last four models [8–10,12] (with respect to theranking) are inappropriate for such applications becausethey cannot be evaluated in the time domain.

By dividing the models into these two groups, best-performing regarding carbon fixation or metabolic engi-neering, the RuBisCO-focusing [24,25,28–30] as well as thelast four models [8–10,12] should be excluded from themetabolic engineering application category. Thus, themodel of Poolman et al. [11], ranked second, is the best-performing model suitable for metabolic engineering appli-cations. The compliance to the Arabidopsis data is veryhigh, promising reliable predictions. Furthermore, themodels of Giersch et al. [26] and Laisk et al. (2006) [7]are promising candidates for metabolic engineering appli-cations. The model of Giersch et al. [26] is more robust thanthat of Poolman et al. [11], whereas the model of Laisk et al.(2006) [7] provides the connection in the original modelboundaries to include photosystems and the electron trans-port chain. Interestingly, the next model of the ranking,that of Hahn [27] (ranked seventh), is one of the simplest,as it is based on mass-action kinetics and, consequently,does not consider regulatory processes. However, thesemodeling assumptions undermine the importance of met-abolic regulation, and may lead to inaccurate predictions,especially under changing environmental scenarios [50].

Modeling challenges aheadThis newly assembled compendium of CBC models allowsfor some important questions directly related to biotech-nology applications to be readdressed. To this end, ourmodel classification can serve as an easy reference ofCBC models, components and relations between them,based on different modeling aspects. Moreover, the proposed

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analysis framework can be employed to address the optimi-zation of the net carbon fixation, identification of missingregulators, or the maximization of biomass production.Furthermore, it can facilitate analysis of the CBC spatio-temporal dynamics of which rigorous treatments are scarce[32]. As challenges of metabolic engineering, these issuesare largely unexplored in plant research. Our findings fur-ther reinforce the potential of the framework in developingmodels of better performance for other metabolic processes(e.g. cancer development).

As modeling is an iterative process, the compendiumcan only point out the best-performing models if theranking criteria together with the compendium itselfare regularly updated. To this end, the latest models,as well as the new insights into the underlying biology ofthe photosynthetic processes, can be incorporated. Forinstance, at present there are no time-resolved experi-mental data that clearly include the metabolite concen-trations in the different compartments to enable thecomparison of predictions regarding the dynamic behav-ior of the investigated process. In this context, methodscan also be incorporated to push forth various aspectsof the biologically relevant theoretical analyses. Forinstance, the integration of the Variational Bayes mea-sure can then take into account the model complexitywithin the analysis of compliance to data which will bepart of our future studies.

AcknowledgmentsA.A. and Z.N. are supported by the GoFORSYS project funded by theGerman Federal Ministry of Education and Research, Grant No. 0313924.The authors would like to thank Alisdair Fernie and Mark Stitt forcritically reviewing drafts of the manuscript.

Appendix A. Supplementary dataSupplementary data associated with this article can befound, in the online version, at doi:10.1016/j.tplants.2011.09.004.

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