ISSN 1751-8849 Steady-state and dynamic ﬂux balance

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Published in IET Systems BiologyReceived on 7th February 2008Revised on 27th August 2008doi: 10.1049/iet-syb.2008.0103

ISSN 1751-8849

Steady-state and dynamic flux balance analysisof ethanol production by SaccharomycescerevisiaeJ.L. Hjersted M.A. HensonDepartment of Chemical Engineering, University of Massachusetts, Amherst, MA 01003-9303, USAE-mail: [email protected]

Abstract: Steady-state and dynamic flux balance analysis (DFBA) was used to investigate the effects of metabolicmodel complexity and parameters on ethanol production predictions for wild-type and engineeredSaccharomyces cerevisiae. Three metabolic network models ranging from a single compartment representationof metabolism to a genome-scale reconstruction with seven compartments and detailed charge balancing werestudied. Steady-state analysis showed that the models generated similar wild-type predictions for the biomassand ethanol yields, but for ten engineered strains the seven compartment model produced smaller ethanol yieldenhancements. Simplification of the seven compartment model to two intracellular compartments producedincreased ethanol yields, suggesting that reaction localisation had an impact on mutant phenotype predictions.Further analysis with the seven compartment model demonstrated that steady-state predictions can be sensitiveto intracellular model parameters, with the biomass yield exhibiting high sensitivity to ATP utilisation parametersand the biomass composition. The incorporation of gene expression data through the zeroing of metabolicreactions associated with unexpressed genes was shown to produce negligible changes in steady-statepredictions when the oxygen uptake rate was suitably constrained. Dynamic extensions of the single and sevencompartment models were developed through the addition of glucose and oxygen uptake expressions andtransient extracellular balances. While the dynamic models produced similar predictions of the optimal batchethanol productivity for the wild type, the single compartment model produced significantly different predictionsfor four implementable gene insertions. A combined deletion/overexpression/insertion mutant with improvedethanol productivity capabilities was computationally identified by dynamically screening multiple combinationsof the ten metabolic engineering strategies. The authors concluded that extensive compartmentalisation anddetailed charge balancing can be important for reliably screening metabolic engineering strategies that rely onmodification of the global redox balance and that DFBA offers the potential to identify novel mutants forenhanced metabolite production in batch and fed-batch cultures.

1 IntroductionGenome-scale models of cellular metabolism [1] haveenabled the development of computational frameworks forthe analysis and design of complex metabolic networks.Flux balance analysis (FBA) and linear programming allowunderdetermined stoichiometric models to be resolvedunder the assumption of a cellular objective such as growthrate maximisation [2]. In addition to characterising

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metabolic capabilities of the wild type, FBA has beenextensively used to identify gene deletion and insertionmutants for metabolite overproduction [3–5]. Despite thetrend to develop increasingly detailed metabolicreconstructions, few studies actually focus on therelationship between model complexity and predictivecapability. Moreover, investigators often claim thatintracellular model parameters associated with the biomassenergy requirement, non-growth associated maintenance

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and biomass composition have little effect on modelpredictions without presenting any quantitative results ontheir effects. A well documented weakness of FBA is thedifficulty associated with incorporating cellular regulationinto the metabolic description. Several studies have shownthat model predictions can be substantially changed byincorporating gene expression data through the eliminationof reactions associated with non-expressed genes [6, 7].

FBA assumes time-invariant extracellular conditions andgenerates steady-state predictions consistent with continuousculture. However, large-scale production of metabolicproducts is often achieved in batch and fed-batch culture.Dynamic flux balance models are obtained by combiningstoichiometric equations for intracellular metabolism withdynamic mass balances on key extracellular substrates andproducts assuming fast intracellular dynamics [8–12]. Theintracellular and extracellular descriptions are coupledthrough the cellular growth rate and substrate uptakekinetics. Dynamic flux balance analysis (DFBA) offers thepossibility of formulating substrate uptake kinetics to accountfor known regulatory processes. DFBA has been primarilyused to generate dynamic predictions of substrate, biomassand product concentrations for wild type growth in batchculture. Our previous research has shown the utility of yeastdynamic flux balance models for optimisation of fed-batchoperating strategies [12] and identification of ethanoloverproduction mutants [13]. The latter study was limited tothe screening of single gene deletions and insertions eventhough combined strategies offer the potential for superiorperformance. Although we used different models for cellularand process optimisation in these two studies, the effect ofmodel complexity on predicted wild type and mutantbehaviour has not been systematically evaluated.

The objective of the present study was to investigate theeffects of metabolic model complexity and parameters onethanol production predictions for wild type and engineeredSaccharomyces cerevisiae. We utilised FBA and DFBA tocompare predictions from a small set of metabolic models toquantitatively access the impact of the following factors:

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† Model complexity ranging from a lumped singlecompartment representation of glucose metabolism [14] toa genome-scale reconstruction with seven intracellularcompartments and detailed charge balancing [15].

† Variations of intracellular model parameters representingthe biomass energy requirement, non-growth associatedmaintenance and biomass composition.

† Incorporation of gene regulation data from aerobic batchand anaerobic chemostat experiments [7].

† Strategies for in silio identification of novel ethanoloverproduction mutants by screening combinations of tenpreviously suggested strategies [3].

While this study is limited to yeast–ethanol production,we believe that that the results provide general insight intothe formulation and analysis of metabolic models forcharacterising wild type behaviour and identifyingoverproduction mutants.

2 Model formulation2.1 Intracellular model

Three intracellular S. cerevisiae metabolic network modelswere considered in this study: a small-scale model [14, 16],the first generation iFF708 genome-scale model [17] andthe second generation iND750 genome-scale model [15].We refer to the small-scale model as iGH99 with thecapital letters ‘GH’ denoting the last names of the primaryauthors and the number ‘99’ referring to the number ofintracellular reactions rather than the number of gene-reaction associations as used in the genome-scale models.We also constructed a decompartmentalised version of theiND750 model to assess the effect of reactionlocalisation on model predictions. The development of thisdecompartmentalised model referred to as iJH732 isexplained below. The general characteristics of eachintracellular model are summarised in Table 1. The numberof fluxes reported for each model includes both the number

Table 1 Summary of Saccharomyces cerevisiae metabolic network models

iGH99 iFF708 iJH732 iND750

genes – 708 732 750

intracellularcompartments

1 2 2 7

metabolites 98 711 935 1059

fluxes 129 1176 1152 1264

elementally balanced � � 3 3

charge balanced � � 3 3

reference van Gulik and Heijnen [14] Forster et al. [17] this paper Duarte et al. [15]

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of intracellular reactions plus the number of membraneexchange fluxes (e.g. the iGH99 model has 99 intracellularreaction fluxes and 30 exchange fluxes).

The iGH99 model has only a single intracellularcompartment and accounts for a comparatively small numberof metabolites. Because the iGH99 model does not explicitlyaccount for genes, we manually analysed the reaction setto establish the gene-reaction associations necessary toimplement metabolic engineering strategies. In addition todividing the intracellular reactions between the cytosol andmitochondria and including gene-reaction associations, thefirst generation iFF708 genome-scale model contains a muchmore extensive list of reactions. The primary enhancements inthe second generation iND750 genome-scale model are moreextensive, and the reaction localisation through inclusion ofcytosol, mitochondria, peroxisome, nucleus, endoplasmicreticulum, Golgi apparatus and vacuole, includes detailedcharge balancing, and full elemental mass balancing withrespect to carbon and hydrogen.

Direct comparisons of iFF708 and iND750 modelpredictions were complicated by differences in both reactionlocalisation and charge balancing. In an attempt to isolatethese effects, the iND750 model was decompartmentalised tohave the same intracellular compartments (cytosol andmitochondria) as the iFF708 model to produce a new modeliJH732. Decompartmentalisation was performed by movingall iND750 reactions in the other compartments (golgiapparatus, nucleus, endoplasmic reticulum, vacuole andperoxisome) to the cytosol and by deleting all transportreactions between these compartments and the cytosol. Atransport reaction was considered to be any reaction thatinvolved species from both the cytosol and one of theeliminated compartments. While most of the deletedreactions involved simple transport of a species betweencompartments, a few transport reactions also involvedchemical conversion. One such transport reaction catalysed bydolichylphosphate d-mannosyltransferase was inserted intothe cytosol because the reaction product was essential forproduction of the biomass precursor mannose. Twenty-sevenreactions in the other compartments that were already presentin the cytosol also were discarded. The number of genes listedfor the iJH732 model (Table 1) was determined byeliminating genes associated solely with reactions removedfrom the iND750 network. The total number of metabolitesdecreased because some metabolites in the removedcompartments were already present in the cytosol of theiND750 network.

The linear program (LP) used to resolve theunderdetermined flux balance models was formulated as

maxv

m ¼X

j

wjvj

subject to: Av ¼ 0

vmin � v � vmax

(1)



where A is the stoichiometric matrix for the metabolicnetwork, v is a vector of reaction and exchange fluxes andvmax and vmin are vectors of upper and lower flux bounds,respectively. The cellular growth rate (m) was calculatedfrom the fluxes producing biomass precursors with theweights (w) determined from the amount of each precursornecessary for biomass formation. Each intracellular modelincluded an artificial biomass formation reaction in whichthe reactants are the biomass precursors and the fluxrepresents the cellular growth rate.

While stoichiometric coefficients are fixed by thebiochemical reactions, the intracellular models have severaladjustable parameters associated with cellular energetics andthe biomass composition. These parameters include cellularenergy requirements in the form of growth and non-growthassociated maintenance. The small-scale model iGH99does not distinguish between the two types of maintenanceand has a single lumped parameter in the biomassformation flux. The genome-scale models iFF708, iJH732and iND750 account for non-growth associatedmaintenance with a separate flux where the upper andlower bounds of the flux are fixed at identical values (mmolATP/gdw/h). Lumped maintenance in the small-scalemodel and growth associated maintenance in the genome-scale models were specified by adjusting the stoichiometryof ATP consumption in the biomass formation flux. In theoriginal studies, the energy related parameters for eachmodel were determined for a single metabolic state andthen assumed to remain constant under different conditions.

The biomass composition parameters (w) determine therelative contribution of each precursor to the biomassformation rate (m). Despite large differences in theirmetabolic descriptions, all three models utilise roughly thesame level of detail for the biomass precursors. In theoriginal studies for iGH99 and iFF708, the biomasscomposition parameters were determined for a particularmetabolic state and then assumed to remain constant. Thesecond generation genome-scale iND750 model utilises thesame biomass composition as iFF708. While metabolicmodels are formulated and analysed under the assumptionof constant biomass composition, continuous cultureexperiments with S. cerevisiae have shown that the relativecontributions of proteins and carbohydrates to the biomasscomposition change significantly with varying dilution rate[18]. Known variations in biomass composition at differentculture conditions can be incorporated by manipulating thestoichiometric coefficients of the precursors in the biomassformation rate. The precursor coefficients must becollectively adjusted so that the total biomass is conserved.

2.2 Dynamic model

A limitation of the steady-state flux balance models describedin the previous section is their inability to account fordynamic and non-balanced growth conditions encounteredin batch and fed-batch cell culture. Dynamic extensions of

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steady-state FBA have been developed to address thisshortcoming [8, 9, 12]. In this study, we developed adynamic flux balance model for combined aerobic andanaerobic batch growth of S. cerevisiae. The modelconsisted of intracellular steady-state mass balances anddynamic extracellular mass balances coupled throughkinetic uptake expressions for glucose (vg) and oxygen (vo)

vg ¼ vgm

G

Kg þ G

1

1þ (E=Kie)(2)

vo ¼ vom

O

Ko þ O(3)

where G, E and O are the glucose, ethanol and dissolvedoxygen concentrations, respectively, Kg and Ko aresaturation constants, vgm and vom are maximum uptakerates and Kie is an inhibition constant. The glucose uptakerate follows Michaelis–Menten kinetics with an additionalregulatory term to capture growth rate suppression due tohigh ethanol concentrations [10]. Ethanol uptake wasexcluded from the model because ethanol consumption isoxidative and only experimentally observed when glucose isnearly exhausted [19], conditions not observed in oursimulations.

The dissolved oxygen concentration was treated as anindependent variable under the assumption that it could beregulated by a suitably designed feedback controller. Thissimplification was deemed reasonable because anaerobicconditions were used to promote ethanol production duringlater stages of the batch when high cell densities mightlimit oxygen mass transfer. Consequently, extracellularoxygen balances were omitted and the dissolved oxygenconcentration was simply represented as the percent ofsaturation: DO ¼ O=Osat, where Osat is the saturationconcentration. This relation can be directly substituted into(3) to yield

vo ¼ vom

DO

(Ko=Osat)þDO(4)

The dynamic mass balances on the extracellular environmentwere posed as

dX

dt¼ mX (5)

dG

dt¼ �vgX (6)

dE

dt¼ veX (7)

dY

dt¼ vyX (8)

where X is the biomass concentration and Y is the glycerolconcentration. The growth rate (m), the ethanol exchange

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flux (ve) and the glycerol exchange flux (vy) were resolvedby solution of the intracellular flux balance model.

Nominal parameter values for batch simulations are listedin Table 2. Parameter values for the maximum specific uptakerates (vgm and vom) and the saturation constants (Kg and Ko)were based on experimental estimates. The saturation oxygenconcentration (Osat) was determined using Henry’s law at30 8C and 0.21 atm oxygen. The inhibition constant (Kie)was chosen to give reasonable model predictions, and theinitial biomass, glucose and ethanol concentrations(X0, G0, E0) were selected to be representative ofexperimental batch cultures.

3 Steady-state analysis of ethanolproductionThe three intracellular models were used to analyse predictedsteady-state ethanol production for anaerobic growth onglucose in continuous culture. The optimisation problem(1) was solved using the MATLAB interface to the LPcode MOSEK. A possible problem with FBA is thepresence of multiple optimal solutions, which implies theexistence of an infinite number of different fluxdistributions that produce the same optimal biomassformation rate [20]. Multiple optimal solutions withrespect to the ethanol secretion rate were handled by firstsolving the LP for maximum biomass, and then by fixingthe biomass at this maximum value and resolving the LPfor maximum ethanol secretion. This approach allowedvariability in the ethanol production rate as a result ofmultiple optima to be eliminated by selecting thetheoretical maximum ethanol production with respect tomaximised cell growth.

3.1 Effect of model complexity

A comparison of steady-state FBA predictions from thesmall-scale (iGH99), the first generation genome-scale

Table 2 Nominal model parameter values

Variable Value Reference

vgm 20 mmol/gdw/h [25]

Kg 0.5 g/L [25]

vom 8 mmol/gdw/h [25]

Ko 0.003 mmol/L [25]

Osat 0.30 mmol/L –

Kie 10 g/L –

X0 0.1 g/L –

G0 50 g/L –

E0 0 g/L –



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(iFF708) and the second generation genome-scale (iND750)S. cerevisiae metabolic network models is presented inTable 3. Ethanol, glycerol and biomass yield predictions forthe wild type and ten previously identified mutants [3] areshown for each intracellular model. Each mutant wasdesigned to redirect carbon flux from glycerol to ethanol byfavourable alteration of the intracellular redox balance. Thefirst two mutants listed correspond to the deletion of onegene (D gdh1) coupled with the overexpression of either theglt1 and gln1 genes or the gdh2 gene. The final eightmutants result from single-gene insertions, with the labelfor each insertion (e.g. R00365) corresponding to theassociated reaction entry in the KEGG database(Kyoto Encyclopedia of Genes and Genomes: http://www.genome.jp). Each gene deletion was implementedby removing the associated flux from the intracellularmodel, and each gene insertion was incorporated by addinga new flux to the network. Gene overexpressions wereimplemented by removing wild-type flux constraints fromthe associated reactions. The basis for the wild-type fluxconstraints and their removal for gene overexpressions isoutlined in Bro et al. [3].

Only four mutants could be evaluated with the iGH99model due to the lack of the necessary gene-reactionassociations. We attempted to reproduce previouslypublished results [3] for the iFF708 model using the mostrecent version of the model available at http://www.cmb.dtu.dk/models.aspx. We were not ableto reproduce wild-type or mutant predictions, most likely



due to undocumented differences between the iFF708models used in the two studies. Therefore Table 3contained results taken directly from the original study [3].

The three models produced reasonably close wild-typepredictions, with �10% differences in the biomass andethanol yields. Much larger differences were observed inthe glycerol yields, especially between the iND750 andiGH99 models. Mutant glycerol predictions for theiND750 model differed from values previously reported inHjersted et al. [21] due to the selection of differentalternative optima. In this study, we used the alternativeoptima with maximum glycerol production with respect tooptimal ethanol production and growth. For each mutant,the optimum was found by fixing the ethanol and growthfluxes at their maximum values and resolving the LP formaximum glycerol secretion. The four mutant predictionsobtained with the iGH99 model appeared to be quiteunreliable when compared to the corresponding results forthe iFF708 and iND750 models, demonstrating that theadditional complexity of genome-scale models producedsignificant differences in phenotypic predictions. TheiFF708 model consistently produced larger ethanol yieldincreases for the ten mutants as compared to the iND750model, while no discernable trend was observed for thebiomass yield. Experimental evaluation of the R01058strategy ( gapN insertion) produced a 2.4% ethanol yieldincrease and 3.0% biomass yield increase compared to thewild type [3]. The iND750 model produced closeragreement with this data.

Table 3 Comparison of steady-state FBA predictions for anaerobic growth on glucose

iND750 iFF708a iGH99

Mutant Ye/g inc.,%

Ygly/g dec.,%

Ym/g inc.,%

Ye/g inc.,%

Ygly/g

dec., %Ym/g inc.,

%Ye/g inc.,

%Ygly/g

dec., %Ym/g inc.,

%

D gdh 1 glt1and gln1

3.4 50.6 5.4 5.2 49.3 5.2 – – –

D gdh1 gdh2 3.7 100.0 11.0 4.2 46.8 9.7 – – –

R00365 3.8 100.0 18.0 4.2 46.8 9.7 0.1 6.6 2.3

R01866 3.9 100.0 18.1 9.0 100.0 16.5 – – –

R00112 3.8 100.0 18.0 9.0 100.0 16.5 8.5 68.4 24.7

R00105 6.1 100.0 5.6 10.4 100.0 7.6 0.1 6.6 2.3

R01039 6.1 100.0 5.6 10.4 100.0 6.6 – – –

R00845 3.8 100.0 18.0 9.0 100.0 16.5 – – –

R01063 3.8 100.0 18.0 9.0 100.0 16.5 – – –

R01058 6.1 100.0 5.6 10.4 100.0 7.6 11.2 67.0 11.2

wild-type yield,g/g

0.424 0.060 0.085 0.40 0.09 0.10 0.382 0.126 0.087

inc. indicates increase and dec. indicates decrease compared to the wild typea Predictions taken from Bro et al. [3]

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The highly compartmentalised iND750 model was expectedto be particularly sensitive to charge balancing since hydrogenion balances in each compartment as well as in the entire cellmust be satisfied [15]. In an attempt to evaluate the impact ofcharge balancing, metabolites in the small-scale iGH99model were mapped to species in the genome-scale iND750model and then the iGH99 model was charged balanced.The charge balancing was performed according to theprocedure outlined in Hjersted et al. [21]. The iGH99 modelwas found to have an almost fully satisfied charge balance asonly three reactions required modification by adjustinghydrogen ion stoichiometry to simultaneously satisfy chargeand elemental balances. The incorporation of these balancedreactions into the iGH99 model was found to have no effecton the wild type and mutant predictions in Table 3,presumably because the newly balanced reactions involvedsmall fluxes that negligibly affected the global redox balance.However, the iGH99 model was found to be potentiallysensitive to the insertion of reactions with charge imbalances.The effect of such charge imbalances was evaluated by directincorporation of the eight inserted reactions as listed in theoriginal reference [3], whereas balanced reactions as reportedin Hjersted et al. [21] were used to generate Table 3. TheR01058 and R00365 insertions showed no sensitivity toproper charge balancing, but the R0012 prediction changedslightly and the R00105 insertion was predicted to produceno ethanol increase compared to the wild type.

We constructed the decompartmentalised model iJH732to have limited reaction localisation similar to the iFF708model and detailed charge balancing similar to the

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iND750 model. Table 4 contains a comparison of theethanol and biomass yields obtained with the threemodels. For the wild type, the iJH732 and iND750models produced similar predictions while theiFF708 model produced a smaller ethanol yield and alarger biomass yield. The results suggest that wild-type predictions were relatively insensitive tocompartmentalisation beyond mitochondrial and cytosolicreaction localisation. Ethanol yield predictions for the tengenetic manipulation strategies proved more sensitive tocompartmentalisation. The iJH732 model producedethanol yields closer to the iFF708 model for the twocombined gene deletions and overexpressions, while theiJH732 and iND750 models showed much closeragreement for the eight gene insertions. Less conclusiveresults were obtained for the biomass yield, with no cleartrend between the three models. These results indicatethat ethanol overproduction with insertion strategies basedon modification of the global redox balance may appear tobe more effective when implemented on models with lessdetail for reaction localisation. Our results collectivelysuggest that both reaction localisation and chargebalancing impact the computational evaluation ofmetabolite overproduction strategies that depend onmodification of the redox balance.

3.2 Sensitivity of FBA predictions tointracellular parameters

We utilised the iND750 model to evaluate the sensitivity ofwild-type predictions to adjustable intracellular parameters

Table 4 Effect of compartmentalisation on steady-state FBA predictions foranaerobic growth on glucose

Ye/g increase % Ym/g increase %

iND750a iJH732b iFF708a iND750a iJH732b iFF708a

D gdh1 glt1 and gln1 3.4 5.2 5.2 5.4 5.9 5.2

D gdh1 gdh2 3.7 4.4 4.2 11.0 11.6 9.7

R00365 3.8 4.5 4.2 18.0 16.1 9.7

R01866 3.9 5.0 9.0 18.1 18.8 16.5

R00112 3.8 4.5 9.0 18.0 16.1 16.5

R00105 6.1 6.6 10.4 5.6 6.2 7.6

R01039 6.1 6.6 10.4 5.6 6.2 6.6

R00845 3.8 4.5 9.0 18.0 16.1 16.5

R01063 3.8 4.5 9.0 18.0 16.1 16.5

R01058 6.1 6.6 10.4 5.6 6.2 7.6

wild-type, g/g 0.424 0.425 0.40 0.085 0.088 0.10

aResults repeated from Table 3bDecompartmentalised iND750 model with same number of compartments asiFF708



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for growth and non-growth associated maintenance and thebiomass composition. Sensitivity of the biomass andethanol yield predictions to the growth and non-growthassociated maintenance parameters is shown in Fig. 1. Theglycerol yield was directly proportional to the biomass yieldand is not shown. An increase in either energy requirementproduced a decrease in the biomass yield and acorresponding increase in the ethanol yield. The biomassand ethanol yields were both sensitive to variations in themaintenance parameters, with a 6% difference in biomassyield and a 2% difference in ethanol yield resulting from a10% change in growth associated maintenance. Because themutants listed in Table 3 produced 5–18% increases inbiomass yield and 3–6% increases in ethanol yield, smallerrors in the maintenance parameters have the potential tobias in silico predictions of metabolite overproductionmutants.

The constant biomass composition used in the iND750model was determined from an anaerobic continuousculture experiment at a dilution rate of 0.1 h21 [15].However, aerobic continuous culture experiments have

Figure 1 Sensitivity of iND750 model predictions tointracellular parameters for growth (top) and non-growth(bottom) associated energy requirements

The nominal model parameters are indicated by vertical dottedlines



shown that biomass composition varies as a function of thedilution rate [18]. These experiments were performed atseven dilution rates between 0.022 and 0.211 h21, and thebiomass was partitioned into six groups: proteins,carbohydrates, lipids, RNA, DNA and sulfate. We usedthese data to examine the sensitivity of iND750 modelpredictions to biomass composition variations. Theprecursors for biomass formation in the iND750 modelwere mapped to the six groups as follows: 20 aminoacids to proteins; glucan, glycogen, mannan andtrehalose to carbohydrates; two sterols, tryglyceride and fivephospholipids to lipids; AMP, CMP, GMP and UMP toRNA; dAMP, dCMP, dGMP and dTMP to DNA; andSO4 to sulfate.

Fig. 2a shows the percentage changes in eachcompositional group at the seven dilution rates where thedilution rate 0.107 h21 was defined as the nominalcondition. Fig. 2b shows corresponding yield predictionsfor anaerobic and aerobic growth. Only the biomass yield isshown for aerobic growth since the ethanol and glycerolyields were identically zero. For anaerobic growth, thelargest total percentage change was observed in the glycerolyield (16%) followed by the biomass yield (3.5%) and theethanol yield (0.75%). These yield variations are significantcompared to increases in the biomass yield (5–18%) andethanol yield (3–6%) listed in Table 3 for the ethanoloverproduction mutants. Our results suggest that failure toaccount for biomass composition variations could causesignificant prediction errors at different conditions thanthose used to parameterise the composition equation. Thisconclusion is contrary to the conventional wisdom thatmodel predictions should be relatively insensitive to realisticvariations in the biomass composition [22].

3.3 Incorporation of metabolicconstraints derived from microarray data

The availability of gene expression data from highthroughput microarray technology enables genome-scaleanalysis of metabolic regulation. A recent study [7]presented a framework for integration of gene expressiondata into genome-scale metabolic networks by constrainingreaction fluxes exclusively associated with non-expressedgenes. Statistical analysis of the microarray data was used todetermine which genes were not expressed. Experimentaldata (EXP) was compared with two iFF708 modelformulations to assess the impact of unmodelled regulatoryeffects: FBA with unconstrained oxygen uptake rate (FBA);and FBA with unconstrained oxygen uptake rate and geneexpression constraints (FBAþGE). Fig. 3 shows thatimproved FBA predictions for the exponential phase ofaerobic batch growth were obtained by applying the geneexpression constraints.

To further assess the impact of gene expression constraints,we considered two additional iND750 model formulations inwhich the oxygen uptake rate was subject to a maximum

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Figure 2 Biomass composition

a Change in S. cerevisiae biomass composition at various dilutionrates D [18]b Sensitivity of iND750 model predictions to varying biomasscompositionThe reported changes are relative to the nominal D ¼ 0.107 h21


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constraint: FBA with constrained oxygen uptake rate(FBAþO2); and FBA with constrained oxygen uptake rateand gene expression constraints (FBAþO2þGE). Themaximum oxygen uptake rate was determined from asensitivity analysis, with the value 0.032 mmol O2/mmolglucose providing the best agreement between theFBAþGE and FBAþO2 predictions. Fig. 3 comparesthe biomass, ethanol and glycerol yields obtained fromexperiment and the four model formulations. Despite usingdifferent versions of the S cerevisiae genome-scale metabolicnetwork (FBA and FBAþGE: iFF708; FBAþO2 andFBAþO2þGE: iND750), the results show thatpredictions from the model constrained with geneexpression data (FBAþGE) can be very closely matchedby the model with a fixed maximum oxygen uptake rate(FBAþO2). Inspection of the individual fluxes exclusivelyassociated with absent genes for the FBAþO2 caserevealed that only 4 of the 51 fluxes were non-zero, andthat these non-zero fluxes had a minimal effect onphenotypic behaviour. Further constraining the metabolicnetwork with gene expression data (FBAþO2þGE)produced only small decreases in the growth and ethanolyields and a small increase in the glycerol yield. Thereforethe cellular objective of growth maximisation combinedwith a reasonable upper bound on oxygen uptake effectivelysatisfies the constraints derived from gene expression data.Because measurement of substrate uptake rates is muchsimpler than the collection and analysis of gene expressiondata, we believe that more compelling illustrations areneeded to clearly demonstrate the value of gene expressiondata for constraining metabolic models.

Figure 3 Comparison of biomass, ethanol and glycerolyields for aerobic batch culture from [7] (EXP, FBA,FBA þ GE) to two additional cases (FBA þO2,FBAþO2þ GE)

The labels indicate EXP, unconstrained FBA prediction (FBA), geneexpression constrained FBA prediction (FBAþ GE), oxygen uptakeconstrained FBA prediction (FBAþO2) and oxygen uptake andgene expression constrained FBA prediction (FBAþO2þ GE)


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4 Dynamic analysis of ethanolproduction in batch cultureDFBA provides a more meaningful characterisation ofethanol production in batch culture than is possible withsteady-state FBA. The ethanol productivity P is defined asthe total ethanol production over the batch

P ¼E(tf )

tf

(9)

where E(tf ) denotes the ethanol concentration at the finalbatch time tf. The productivity represents a single measureof dynamic culture performance that implicitly accounts forboth the biomass yield and the ethanol yield. We utiliseDFBA to analyse ethanol productivity for combinedaerobic/anaerobic growth on glucose in batch culture.

4.1 Effect of model complexity

Batch culture simulations with an intermediate switch from apartially aerobic phase (50% DO) to a purely anaerobic phasewere performed for the wild type and each of the ten mutantslisted in Table 3. The aerobic to anaerobic switch wasapproximated as being instantaneous based on theassumption that DO dynamics were much faster than thecellular dynamics. More discussion on implementation ofthe aerobic-anaerobic switch is available in Hjersted andHenson [12]. Fig. 4 shows the sensitivities of the predictedbatch ethanol productivities computed with the iND750

Figure 4 Sensitivity of ethanol productivity in batch cultureto the aerobic to anaerobic switch time for variousmetabolic engineering strategies computed using theiND750 intracellular model

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intracellular model to the aerobic to anaerobic switch time.The final batch time (tf ) was determined individually foreach case as the time when the extracellular glucoseconcentration reached 0.1 g/L. A grouping of the geneinsertion mutants, which is also observed in the steady-state analysis (Table 3), into two distinct classes is evidentwith five of the eight insertions outperforming all otherstrategies. The Dgdh1 glt1 and gln1 mutant underperformsthe wild type except at large switching times. The peakobserved in each productivity curve corresponds to theoptimal switching time for the corresponding cell type. Thefact that this peak occurs at different times indicates thatthe process operating policy and the metabolic engineeringstrategy must be considered simultaneously to achieveoptimal batch performance.

A comparison between the small-scale iGH99 model andthe genome-scale iND750 model was performed to examinethe effect of model complexity on batch productivityoptimisation. The first generation genome-scale iFF708model was omitted from this analysis since we were unableto reproduce steady-state predictions published elsewhere[3]. The final batch time was determined as the time whenglucose was nearly exhausted (G � 0:1 g/L), whichsimplified the optimisation problem by eliminating adecision variable. Therefore the optimisation probleminvolved an objective of ethanol productivity maximisationwith the aerobic-anaerobic switch time as the only decisionvariable. The problem was solved with a direct searchalgorithm using the MATLAB function fminsearch,thereby verifying the optimal points in Fig. 4.

Table 5 shows the aerobic to anaerobic switch time, finalbatch time, and ethanol productivity increase over the wildtype for each mutant implementable with the twointracellular models. Wild-type predictions were roughlysimilar, with the iGH99 model producing an optimalethanol productivity �20% less than that obtained with theiND750 model. With the exception of the D gdh1 glt1 andgln1 mutant, all the metabolic engineering strategiesimplemented in the iND750 model were predicted to yieldethanol productivity increases over the wild type. Thepredicted productivity enhancements were modest due tothe need for an extended aerobic growth phase, where themetabolic engineering strategies were predicted to havelittle or no effect on the biomass and ethanol yields [13].The iGH99 model produced three distinct grouping ofmutants with a much larger range of ethanol productivitiesthan observed with the iND750 model. Two sets ofinsertion mutants (R00105/R01058 and R00365/R00112)that produced identical results with the iND750 modelexhibited vastly different behaviour with the iGH99 model.Conversely, two mutants (R00105/R00365) predicted bythe iGH99 model to have the same performance differedwhen analysed with the iND750 model. These resultsreinforce our previous conclusion that small-scale metabolicmodels are generally not suitable for analysis of metabolicengineering strategies.

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Table 5 Comparison of DFBA predictions for combined aerobic/anerobic growthon glucose

iND750 iGH99

EtOH EtOH

Strategy ts, h tf, h Prod. Inc., % ts, h tf, h Prod. Inc., %

D gdh1 glt1 and gln1 7.16 9.67 22.03 – – –

D gdh1 gdh2 6.71 9.16 2.97 – – –

R00365 6.64 9.18 3.52 7.48 9.56 0.73

R01866 6.64 9.18 3.56 – – –

R00112 6.64 9.18 3.52 6.96 9.52 7.19

R00105 6.75 9.27 3.14 7.48 9.56 0.73

R01039 6.75 9.27 3.14 – – –

R00845 6.64 9.18 3.52 – – –

R01063 6.64 9.18 3.52 – – –

R01058 6.75 9.27 3.14 7.22 9.86 5.19

wild type 6.94 9.13 2.04 g/Lh 7.57 9.62 1.81 g/Lh

4.2 Sensitivity of DFBA predictions tobiomass composition

Our steady-state analysis showed that measurable variationsin the biomass composition could introduce significanterrors in iND750 model predictions. We tested thisconclusion for batch culture simulations by allowing thebiomass composition to vary between the aerobic andanaerobic growth phases. The biomass composition foraerobic growth at D ¼ 0.2 h21 in Fig. 2a was used for theaerobic phase, and the nominal biomass composition foranaerobic growth at D ¼ 0.1 h21 was used for theanaerobic phase (Fig. 5a). A biomass compositioncorresponding to a higher dilution rate was used for aerobicgrowth to be consistent with the higher growth ratesobserved in this phase of the batch. While notquantitatively accurate, these variations allowed the impactof non-uniform cellular composition in dynamic culture tobe evaluated. The switching and final batch times weretaken to be the same as the optimal values determined forthe wild type with the nominal biomass composition(Table 5). Fig. 5b compares results obtained with theuniform, nominal biomass composition to those generatedwhen the biomass composition was instantaneouslychanged at the switching time of 6.94 h. Only smalldifferences between the simulation profiles were observedfor the two cases, implying that DFBA is less sensitive tobiomass composition errors that FBA. Despite visualsimilarities between the simulation profiles, the two casesproduced biomass concentrations that differed by 1.8% atthe final batch time. Consequently, biomass compositionvariations have the potential to impact the accuracy ofDFBA predictions, similar to FBA predictions.

ineering and Technology 2009

ited to: University of Massachusetts Amherst. Downloaded on Jun

4.3 Identification of ethanoloverproduction mutants

Table 5 contains iND750 model predictions of the optimalbatch ethanol productivities achievable with the wild typeand ten mutants. We performed additional steady-state anddynamic analysis with the iND750 model to identify othermutants with superior ethanol production performance.First all possible combinations of the eight gene insertionswere enumerated and dynamically optimised for ethanolproduction in batch culture. The predicted performance ofthe best single insertion (R01866) could not be exceededby any combination of multiple insertions. The dynamicscreening was subsequently expanded to include the eightgene insertions coupled with all possible single-geneknockouts, but once again no strategy exceeding thepredicted performance of the R01866 insertion wasdiscovered.

Finally, the two combined gene deletion/geneoverexpression strategies, Dgdh1 glt1 and gln1 and Dgdh1gdh2, were individually combined with each of the eightgene insertions to create a small library of 16 mutants forscreening. FBA revealed one combination mutant that hadthe potential to outperform the best single insertion inbatch culture based on anaerobic and partially aerobic yields(Table 6). The gene deletion/overexpression strategy (D)produced a large increase in the aerobic ethanol yield at theexpense of the aerobic biomass yield and modestimprovements in anaerobic yields, which caused inferiorperformance to the wild type in batch culture (Fig. 4). Bycontrast the R00365 insertion (I ) produced large increasesin the aerobic ethanol and biomass yields but identical



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Figure 5 Aerobic and anaerobic biomass compositions (top)that were implemented for batch simulation

Batch simulation profiles (bottom) comparing nominal modelpredictions where the biomass composition remained constantat the anaerobic values to a case where the biomasscomposition instantaneously changed from the aerobic values tothe anaerobic values at the switching time 6.94 h

Table 6 Steady-state FBA for three ethanol overproductionmutants

Anaerobic Aerobica

Strategy Yethb Ymb Yeth

b Ymb

D 3.4 5.4 9.5 27.4

I 3.8 18.0 0.0 0.0

Dþ I 3.5 12.0 21.2 1.0

D: deletion of gdh1 andoverexpression of glt1 and gln1.

I: insertion of NAD-dependentglycine dehydrogenase (R00365).

aPartially aerobic: 50% DObYields reported as %-increase overwild-type S. cerevisiae

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anaerobic yields as the wild type, which allowed superiorperformance in batch culture (Fig. 4). The combinedstrategy, Dgdh1 glt1 and gln1 coupled with the R00365insertion (Dþ I ), was predicted to offer one advantagecompared to the single insertion (I ): a slightly increasedaerobic biomass yield. The other three yields for the Dþ Istrategy were inferior to those of the single insertion (I ),rendering steady-state assessment of their relativeperformance in batch culture impossible.

Despite having lower steady-state anaerobic ethanol,anaerobic biomass and aerobic ethanol yields, the combinedstrategy (Dþ I ) slightly outperformed the single insertion(I ) with respect to optimal batch ethanol productivity(Fig. 6). The peak in the productivity sensitivity curve forthe combined strategy corresponds to a 3.63% increase inethanol productivity over the wild type compared to a 3.52%increase for the single insertion (see R00365 in Table 5).This result suggests that FBA can be an inadequatetechnique for identifying metabolite overproduction mutantsin dynamic cell culture. Optimisation methods based onDFBA are better suited for addressing the synergistic effectsof metabolic engineering and process operation that arise inbatch and fed-batch fermentation.

5 DiscussionWe used steady-state and DFBA to investigate the ethanolproduction capabilities of wild type and engineered S.cerevisiae. The limitations of small-scale metabolic networkswas demonstrated by applying steady-state FBA to alumped single compartment representation of glucose

Figure 6 Sensitivity of ethanol productivity in batch cultureto the aerobic to anaerobic switching for the wild type andthree mutants defined in Table 6

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metabolism (iGH99) [14] for evaluation of ten ethanoloverproduction mutants [3]. Only four mutants could beimplemented due to the lack of gene-reaction associations,and the predictions generated for these mutants were/textbfin disagreement with more detailed models. Thesame mutant set was used to demonstrate the value ofgenome-scale metabolic reconstructions with explicit gene-reaction associations and expansive pathway descriptions.Comparisons between two existing genome-scale models(iFF708, iND750) and a decompartmentalised version ofthe iND750 model (iJH732), suggested that reactionlocalisation and detailed charge balancing may beimportant for evaluating metabolic engineering strategiesbased on modification of the global redox balance. Thesecond generation genome-scale iND750 model was usedto evaluate the sensitivity of model predictions to uncertainintracellular parameters that are usually treated as constantsin the FBA framework. Errors of 10% in the energymaintenance parameters and experimentally measuredvariations in the biomass composition [18] were shown topotentially bias in silico predictions for metaboliteoverproduction mutants. Gene expression data collectedfrom aerobic batch culture experiments [7] were used toassess prediction improvements that resulted from furtherconstraining the iND750 model. The FBA objective ofgrowth maximisation combined with a reasonable upperbound on the oxygen uptake rate was shown to effectivelysatisfy the constraints derived from the gene expressiondata, raising questions about the value of such data forconstraining flux model predictions.

We utilised an optimisation method based on DFBA toevaluate achievable ethanol productivities of wild type andoverproduction mutants in batch culture by optimallyselecting the switching time from partially aerobic to purelyanaerobic growth. Parallelling the steady-state results, adynamic flux balance model based on the small scale (iGH99)metabolic network was shown to be inadequate for screeningethanol production capabilities of the ten mutants. Bycontrast, use of the genome-scale iND750 network allowedan explicit ranking of mutant capabilities that could provide atheoretical basis for guiding metabolic engineering activities.A batch simulation in which the biomass composition wasassumed to change instantaneously at the aerobic-anaerobicswitching time suggested that unmodelled biomasscomposition variations could affect the prediction ofphenotypic behaviour in dynamic cell culture. DFBA wasused to screen a mutant library constructed by combining theten ethanol overproduction strategies with all possible geneknockouts for ethanol productivities that exceeded the bestsingle gene insertion mutant. A promising mutant generatedby combining a single gene deletion, a single geneoverexpression and a single gene insertion was shown to havean increased aerobic biomass yield but decreased aerobicbiomass and anaerobic biomass/ethanol yields compared tothe best single gene insertion mutant, demonstrating thelimitations of FBA for screening mutant phenotypes. DFBAshowed that the identified combination mutant produced a



slightly higher optimal ethanol productivity in batch culturethan the best single gene insertion mutant.

Computational strategies are needed to simultaneouslyoptimise cellular design and dynamic operating policies formaximisation of metabolite production in batch and fed-batch cultures. We believe that the combination of FBAlinear programming and DFBA nonlinear programmingrepresents a powerful methodology for addressing thisproblem. However, the brute force approach utilised in thispaper involving the explicit enumeration and evaluation ofall possible cellular designs would be computationallyinfeasible for screening much larger mutant libraries.Therefore extensions of existing mixed-integer linearprogramming methods [4, 5] that account for culturedynamics are needed. Recent work on mixed-integernonlinear programming [23, 24] has potential applicationsto this problem. While DFBA predictions have beencompared to measured metabolite and biomass profiles[8–10], we are not aware of any studies in which theextensibility of wild-type dynamic flux balance models toengineered mutants has been experimentally validated.Therefore our future work will focus on developing moresophisticated DFBA optimisation strategies andexperimental model validation for mutant strains.

6 AcknowledgmentStimulating conversations with Radhakrishnan Mahadevan(University of Toronto) are gratefully acknowledged.

7 References

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[18] LANGE H.C., HEIJNEN J.J.: ‘Statistical reconciliation of theelemental and molecular biomass composition ofSaccharomyces cerevisiae’, Biotechnol. Bioeng., 2001, 75,pp. 334–344

[19] JONES K.D., KOMPALA D.S.: ‘Cybernetic modeling ofthe growth dynamics of Saccharomyces cerevisiae inbatch and continuous cultures’, J. Biotech., 1999, 71,pp. 105–131

[20] MAHADEVAN R., SCHILLING C.H.: ‘Effects of alternate optimaon constraint-based genome-scale metabolic models’,Metab. Eng., 2003, 5, pp. 264–276

[21] HJERSTED J.L., HENSON M.A., MAHADEVAN R.: ‘Genome-scaleanalysis of Saccharomyces cerevisiae metabolism andethanol production in fed-batch culture’, Biotechnol.Bioeng., 2007, 97, pp. 1190–1204

[22] VARMA A., PALSSON B.O.: ‘Metabolic capabilities ofEscherichia coli. II. Optimal growth patterns’, J. Theor.Biol., 1993, 165, pp. 503–522

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ISSN 1751-8849 Steady-state and dynamic ﬂux balance