ARTICLE IN PRESS

Metabolic Engineering 10 (2008) 178– 186

Contents lists available at ScienceDirect

Metabolic Engineering

1096-71

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ymben

Isotopic non-stationary 13C gluconate tracer method for accuratedetermination of the pentose phosphate pathway split-ratio inPenicillium chrysogenum

Zheng Zhao �, Karel Kuijvenhoven, Cor Ras, Walter M. van Gulik, Joseph J. Heijnen,Peter J.T. Verheijen, Wouter A. van Winden

Department of Biotechnology, Delft University of Technology, Julianalaan 67, 2628BC Delft, The Netherlands

a r t i c l e i n f o

Article history:

Received 5 November 2007

Received in revised form

16 April 2008

Accepted 17 April 2008Available online 4 May 2008

Keywords:

Isotopic non-stationary 13C flux analysis

Metabolic flux ratio analysis

Pentose phosphate pathway

Statistical analysis

Metabolic engineering

76/$ - see front matter & 2008 Elsevier Inc. A

016/j.ymben.2008.04.003

esponding author.

ail address: Z.Zhao@tudelft.nl (Z. Zhao).

a b s t r a c t

Current 13C labeling experiments for metabolic flux analysis (MFA) are mostly limited by either the

requirement of isotopic steady state or the extremely high computational effort due to the size and

complexity of large metabolic networks. The presented novel approach circumvents these limitations by

applying the isotopic non-stationary approach to a local metabolic network. The procedure is

demonstrated in a study of the pentose phosphate pathway (PPP) split-ratio of Penicillium chrysogenum

in a penicillin-G producing chemostat-culture grown aerobically at a dilution rate of 0:06 h�1 on

glucose, using a tracer amount of uniformly labeled [U-13C6] gluconate. The rate of labeling inflow can

be controlled by using different cell densities and/or different fractions of the labeled tracer in the feed.

Due to the simplicity of the local metabolic network structure around the 6-phosphogluconate (6pg)

node, only three metabolites need to be measured for the pool size and isotopomer distribution.

Furthermore, the mathematical modeling of isotopomer distributions for the flux estimation has been

reduced from large scale differential equations to algebraic equations. Under the studied cultivation

condition, the estimated split-ratio ð41:2� 0:6%Þ using the novel approach, shows statistically no

difference with the split-ratio obtained from the originally proposed isotopic stationary gluconate

tracing method.

& 2008 Elsevier Inc. All rights reserved.

1. Introduction

Metabolic flux analysis (MFA) has become an important toolfor the complementation of transcriptomic and proteomic data inthe last decades (Stephanopoulos, 1999; Szyperski, 1998; Daunerand Sauer, 2000; van Winden et al., 2005). In order to resolve thebewildering complexity of the metabolic network, isotopic tracerexperiments are required because conventional metabolic fluxbalancing (MFB) analysis is unable to interrogate the occurringparallel and bidirectional structure in the metabolic pathways(Sauer, 2006).

13C is the most frequently used isotopic tracer and it is usuallyintroduced by specifically labeled substrate under metabolicsteady state. The labeling patterns of all the metabolites withinthe network therefore contain rich information for intracellularflux. The detection of these labeling patterns has been developedfrom the analysis of proteinogenic amino acids using nuclearmagnetic resonance (NMR) (Szyperski, 1998; Sauer et al., 1999;

ll rights reserved.

van Winden et al., 2001) and/or gas chromatography–massspectrometry (GC–MS) (dos Santos et al., 2003; Antoniewiczet al., 2007a) to the analysis of intracellular metabolites (Langeet al., 2001) using liquid chromatography–tandem quadrupolemass spectrometry (LC–MS/MS) (van Winden et al., 2005; Luoet al., 2007). Nevertheless, most of the 13C labeling experiments todate still require both metabolic and isotopic steady state, whichis recognized as a major limitation for the further application(Sauer, 2006; Wiechert et al., 2007).

Antoniewicz et al. (2007b) and Noh et al. (2007) proved in fed-batch cultures that sampling the isotopic transients could yieldimportant additional in vivo flux information. However, thededicated technology for ultrashort time scale sampling and theextremely high computational power required by the optimiza-tion of the large scale ordinary differential equations (Noh et al.,2007) remains a challenge for the implementation of the method.

In the current contribution, we applied the isotopic non-stationary 13C labeling method using labeled gluconate tracer to aglucose limited chemostat culture of Penicillium chrysogenum at adilution rate of 0:06 h�1. The purpose of using this labelingmethod is to be able to determine the flux-ratio of the oxidativebranch of pentose phosphate pathway (PPP) to the glucose uptake

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Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186 179

(PPP split-ratio) with a minimized analysis and computationaleffort and maximal accuracy.

Recently, Kleijn et al. (2006) have developed a method tomeasure the PPP split-ratio based on the isotopic steady statelabeling pattern of metabolites around the 6-phosphogluconate(6pg) node (from here on this method is referred to as isotopicstationary gluconate tracer method). This method allows theextraction of labeling information (flux ratio in parallel routes)without modeling large metabolic networks. Applying thismethod to the isotopic dynamic phase will minimize the requiredexperiment duration as well as the analytical and computationalefforts associated with the isotopic transients, and thus broadenthe applicability and makes it easier to implement. The presentedisotopic non-stationary gluconate tracer method will be validatedby comparing the measured split-ratio result with the resultsobtained from both the isotopic stationary gluconate tracermethod and the classical MFB method.

Fig. 1. Local metabolic pathways of gluconate uptake.

2. Theory

2.1. Mass isotopomer dynamics under metabolic steady state

In a labeling experiment in a chemostat, when the feedmedium is switched from natural to 13C-labeled substrate undermetabolic steady state, the replacement of extracellular residualsubstrate in the broth follows first-order dynamics, which is alsotrue for each individual mass isotopomer. The mass balances forthe different mass isotopomers are expressed as

dCSfS

dt¼

Fin

VCin

S finS �

Fout

VCSfS � rSfS, (1)

where CS is the concentration of residual substrate (includingboth labeled and natural substrate) in the broth. fS is the massisotopomer distribution vector (MDV) of labeled substrate in thebroth. For a six-carbon compound (e.g. glucose, gluconate, etc.), itholds

fS ¼ ½f mþ0 f mþ1 � � � f mþ6�Tsubstrate

f mþi ði ¼ 0;1; . . . ;6Þ is the fraction of the corresponding massisotopomer. fin

S is the MDV of substrate in the feed medium. rS isthe volumetric substrate consumption rate. Fin and Fout are theflow rate of feed medium inflow and broth outflow, and V is thevolume of broth. At metabolic steady state, CS is constant.Therefore Eq. (1) can be rewritten as

CSdfS

dt¼

Fin

VCin

S finS �

Fout

VCSfS � rSfS. (2)

In the steady state, the substrate balance is

Fout

VCS þ rS ¼

Fin

VCin

S . (3)

Combining Eqs. (2) and (3) and dividing by CS yields:

dfS

dt¼ðfin

S � fSÞ

tS, (4)

where tS is the time constant for the turnover of the extracellularsubstrate pool and is given by

tS ¼CSV

FinCinS

. (5)

Integration of Eq. (4) gives Eq. (6) for the step response of themass isotopomer distribution of the residual substrate:

fS � finS ¼ ðf

0S � fin

S Þ exp �t

tS

� �. (6)

This equation describes the change in extracellular gluconate(glnec) mass isotopomers upon a switch of the feed from 12C to13C gluconate.

2.2. Gluconate tracer method

The original gluconate tracer method was designed by Kleijnet al. (2006) for accurate determination of the PPP split-ratio (theflux entering the oxidative branch of the PPP from glucose-6-phosphate (g6p) normalized by the total uptake rate of glucose,i.e. v2=v1 in Fig. 1). In this method, uniformly labeled gluconate isthe only labeling source and the gluconate concentration in thefeed medium is designed to be 20 times smaller than the glucose(in order to act as a tracer). Because the residual concentration ofgluconate is nearly the same as glucose (Kleijn et al., 2006), theturnover time of the residual gluconate pool is much largercompared with the experiments where glucose is the labelingsource (see Eq. (5)). The labeling of glnec becomes a bottleneck inthis case. Because CS is independent of Cin

S , the time constant ofthe labeling system can be manipulated by changing thegluconate concentration in the feed medium ðCin

S Þ under a definedcultivation condition. Maintaining the tracer ratio (1:19) meansthat also the glucose concentration is changed, leading to differentbiomass concentrations.

Fig. 1 shows the local metabolic reactions of a cell simulta-neously consuming glucose and gluconate. The reaction of acidphosphatase (in Fig. 1, from 6pg to glnic) (Shimada et al., 1977) isadded to this local network based on early observation thatintracellular gluconate (glnic) has less labeling compared with thelabeled gluconate in the feed medium (Kleijn et al., 2006). Atmetabolic steady state, it holds for 6pg and glnic pools that

v2 þ v4f ¼ v4b þ v5

and

v3 þ v4b ¼ v4f . (7)

v2; v3, and v4b need to be known to calculate v4f and v5 fromEq. (7). This can be achieved from the labeling experiments.The labeling dynamics of the intracellular compounds can be

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Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186180

described by the general mass isotopomer balance equation:

_F ¼

�v3

CglnecI Z Z

v3

CglnicI �

v4f

CglnicI

v4b

CglnicI

Zv4f

C6pgI �

ðv4b þ v5Þ

C6pgI

0BBBBBBBB@

1CCCCCCCCA� F

þ

v3

CglnecI Z

Z Z

Zv2

C6pgI

0BBBBB@

1CCCCCA � Fin, (8)

where

F ¼

fglnec

fglnic

f6pg

0B@

1CA; Fin ¼

fingln

fg6p

0@

1A.

I is a 6 by 6 identity matrix and Z is a 6 by 6 zero matrix. Note thatin these mass balances, only the mass fractions caused by thenumber of the carbon isotopes are considered. It should also benoted that the naturally occurring stable isotopes of hydrogen andoxygen will also influence the mass isotopomer distributions (vanWinden et al., 2002; Wahl et al., 2004). This has to be correctedwhen comparing the simulation data and the measurements (aninversed method of van Winden et al., 2002).

At the isotopic steady state, _F is 0. Under this condition, Eqs. (7)and (8) give the isotopic stationary solution for the fluxes v2 andv4b:

v2 ¼ v3

f1glnec � f16pg

f16pg � f1g6p

(9)

and

v4b ¼ v3

f1glnec � f1glnic

f1glnic � f16pg

. (10)

Eqs. (9) and (10) are consistent with the metabolic flux ratioanalysis theory introduced by Szyperski et al. (1999).

During the isotopic dynamic period after the start of thelabeled feed, fluxes v2 and v4b together with the extracellulargluconate concentration ðCglnec) can be estimated using the leastsquare approach

½v2; v4b; Cglnec� ¼ arg minv2 ;v4b ;Cglnec

SSðv2; v4b;Cglnec;uÞ, (11)

where SS is the sum of squared residues between the simulatedand measured time series of mass fractions of 6pg (f6pg) andintracellular gluconate (fglnic). Eq. (8) is used for the simulation. uis a vector containing all the model inputs of Eq. (8), including themeasured mass distribution of g6p during the isotopic dynamicperiod ðfg6pÞ, the mass distribution of gluconate in the feedmedium (fin

gln), the intracellular concentration of 6pg ðC6pgÞ andgluconate ðCglnicÞ, and as well as the gluconate uptake rate ðv3Þ

obtained from the glnec mass balances. A summary of all termsused in the parameter estimation is listed in Table 1.

Table 1Summary of terms used in parameter estimation

Flux Concentration Mass isotopomer

distribution

Measured input for Eq. (8) v3 C6pg, Cglnic fg6p, fingln

Simulated by Eq. (8) – – f6pg, fglnic

Estimated by Eq. (11) v2, v4b Cglnec –

The PPP split-ratio (sr) is obtained by normalizing theoxidative PPP flux v2 with respect to the glucose uptake rate v1.The obtained PPP split-ratio is corrected to take into account theeffect of the tracer feed (Kleijn et al., 2006), i.e. the extra carbonfeed supply from the tracer gluconate and the difference on theNADPH production between gluconate and glucose is corrected toobtain the actual split-ratio.

2.3. Sensitivity analysis

The accuracy of the isotopic stationary gluconate tracermethod can be evaluated as described by Kleijn et al. (2006),where the F-distribution is used to calculate the 95% confidenceinterval for the estimated split-ratio.

The accuracy of the dynamic gluconate tracer method issubsequently evaluated in two steps. The first step is to re-fit theproblem stated by Eq. (11) at a fixed value of v2 such that the SS

reaches SS95%res (see Eq. (12), cf. Appendix B).

SS95%res ¼ SSðbÞ þ

SSðbÞn� p

� F0:05ð1;n� pÞ. (12)

Using the same approach as the isotopic stationary gluconatetracer method, the upper and lower bounds of the net confidenceinterval of the split-ratio can then be determined. When the 95%confidence interval of the fitted parameter is symmetrical, the netstandard error of the fitted parameter, in this case the split-ratio,can be approximated by

ssr;net ¼ ðsr95%upperbond � sr95%lowerbondÞ=3:92, (13)

The factor 3.92 was introduced based on normal distribution. Forasymmetrical confidence intervals, the skewness of the distribu-tion needs to be considered when converting the confidenceinterval into the standard error.

The gross confidence interval of the estimated split-ratio has toalso incorporate the measurement errors of all model inputs inEq. (8). This is achieved in the second step to re-fit Eq. (11) at fixedvalues of each parameter to obtain their sensitivities ðDsr=Dui).The gross variance of split-ratio can then be approximated by

s2sr;gross � s2

sr;net þXn

i¼1

Dsr

Dui

� �2

s2ui

, (14)

where ui � u.

3. Materials and methods

3.1. Strain

The strain used was a high-yielding industrial P. chrysogenum

strain (code name DS17690) which was kindly donated by DSMAnti-Infectives (Delft, The Netherlands).

3.2. Cultivation conditions

P. chrysogenum was cultivated in an aerobic glucose limitedchemostat system at dilution rate 0:06 h�1. The experiments werecarried out in a 1 L (working volume 0.6 L) bioreactor (Applikon,Schiedam, The Netherlands) equipped with a six-bladed Rusthonturbine stirrer ðD ¼ 45 mmÞ. The stirrer speed was 800 rpm andthe aeration rate was 0.2 L/min (0.33 vvm). These conditions weresufficient to maintain the dissolved oxygen tension always above50%. Temperature was controlled at 25 1C, the pH at 6.5 and thehead space overpressure at 0.1 bar. Effluent was intermittentlyremoved by an overflow system, using a peristaltic pump whichwas operated at fixed time intervals (7 s per 15 min) to assure that

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1 P-value p5% is considered statistically rejectable.

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186 181

biomass concentration in the fermentor and the effluent are thesame. Basildons antifoam was fed to the reactor at a fixed feedingrate of 0.06 mL/min intermittently (1 min in every 6 h). It wasassumed that the metabolic steady state was reached after feedingthe unlabeled medium for a period of approximately fiveresidence times. This was verified by measuring if CX andqpenicillin had reached a constant level. This medium was thenreplaced by 13C gluconate tracer labeled medium until the end ofthe experiment.

3.3. Medium

A chemically defined chemostat medium was used, which wasderived from a previously described chemostat medium for P.

chrysogenum (Kleijn et al., 2007; Nasution et al., 2006). Thecomposition of the unlabeled medium was 15.68 g/L glucose�H2O,0.74 g/L glucono-d-lactone, 0.8 g/L KH2PO4, 5 g/L (NH4)2SO4, 0.5 g/LMgSO4 �H2O and 2 mL/L trace element solution. The trace elementsolution contained 75 g/L Na2–EDTA � 2H2O, 2.5 g/L CuSO4 � 5H2O,10 g/L ZnSO4 � 7H2O, 10 g/L MnSO4 �H2O, 20 g/L FeSO4 � 7H2O and2.5 g/L CaCl2 � 2H2O. The labeled medium was chemically identicalto the unlabeled medium except that the glucono-d-lactone wasreplaced by the same molar amount of [U-13C]glucono-d-lactone(Omicron, South Bend, IN, USA).

The concentration of the side-chain precursor phenylaceticacid (PAA) in medium was 0.675 g/L. The PAA concentration waschosen such that the steady state concentration in the chemostatwas approximately 3 mM.

3.4. Medium preparation

The appropriate amount of PAA was dissolved in demineralizedwater and the pH is adjusted to 5.4 by KOH pellets. This PAAsolution is autoclaved at 121 1C for 40 min. The remaining mediumcomponents were filter-sterilized using an Acropak20 filter (PALL,East-Hills, NY, USA) and added to the PAA solution. After thepreparation the minimal medium was stirred for at least 12 h on amagnetic stirrer to ensure that all the added glucono-d-lactonewas hydrolyzed to gluconate.

3.5. Biomass dry weight determination

To determine the biomass dry weight, a broth sample waswithdrawn from the bioreactor. Triplicate 5 mL broth samples aretaken daily during the fermentation. Samples were filtered overpre-weighed glass fiber filters (PALL, East-Hills, NY, USA) and driedat 70 1C for 24 h and subsequently weighed.

3.6. Rapid sampling, quenching and metabolite extraction

Extracellular samples were taken from the bioreactor byrapidly sampling of 2 mL broth into a syringe containing pre-cooled stainless steel beads (�18 1C) followed by immediatefiltration as described by Mashego et al. (2003). After switching tothe medium containing uniformly labeled tracer gluconate, a timeseries of samples for intracellular metabolite analysis are taken ina designed interval. In total 28� 1 mL samples are taken at 14time points (two samples per time point) during the 5 h after themedium switch. The first eight sample points are taken atdesigned interval (see Fig. 2) based on the a priori calculatedlabeling wash-in dynamics. At the operating dilution rate, thesampled volume (2 mL) is smaller than the produced effluentvolume (3.5 mL) at each sample interval. This minimizes thesampling disturbance in the metabolic steady state. Eachintracellular metabolite sample was obtained by rapidly with-

drawing 1 mL broth, followed by direct injection of the sample in5 mL of a 60% (v/v) methanol/water mixture (�40 1C) forinstantaneous quenching of the cell metabolism as described byLange et al. (2001). At each time point, the two intracellularmetabolite samples were taken within 30 s. These two sampleswere then pooled and extracted as described by Kleijn et al. (2006)in order to obtain higher metabolite concentration in the sample.

3.7. Analytical procedures

The intracellular concentrations of 6pg and gluconate weremeasured using isotope dilution mass spectrometry (IDMS)method as described by Wu et al. (2005). The mass isotopomerdistributions of g6p, 6pg and the glnic were measured by LC–MSas described by van Winden et al. (2005). Triplicate LC–MSanalysis were made for each sample to ensure the reproducibility.

The concentrations of glucose in the medium and the brothwere determined enzymatically (Enzytec, Scil Diagnostics, Viern-heim, Germany).

The extracellular concentrations of PAA and penicillin-G wereanalyzed by isocratic HPLC method using a Zorbaxs column at30 1C. The mobile phase consisted of 28% acetonitrile (v/v) with5 mM KH2PO4 and 6 mM H3PO4.

3.8. Calculation methods

The mass fractions and the errors of the measured massfractions were calculated from the measured peak areas of eachmass isotopomer as described in the appendix. The masscorrection for naturally occurring stable isotopes of hydrogenand oxygen is applied to calculate the mass fractions of glenic, g6pand 6pg in order to get the MDV for the carbon isotopes (aninversed method of van Winden et al., 2002). A flow diagram isgiven in Fig. 3 for an overview of the procedures. All calculationswere performed in Matlab R2006b (The Mathworks Inc, Natick,MA, USA).

3.9. Elemental balancing of fermentation data

The measured primary fermentation data were reconciledbased on element balances using a variance weighed optimizationprocedure (van der Heijden et al., 1994). The balanced primarydata were used further on to calculate the balanced biomassspecific conversion rates. A w2-test was used to evaluate thestatistical acceptance of the balanced results.

4. Results and discussion

4.1. Uptake and production rates

P. chrysogenum was cultivated in carbon limited chemostats ona defined medium containing glucose and gluconate as mixedcarbon source as described in the materials and methods section.The macroscopic data obtained from the chemostat are presentedin Table 2, together with biomass specific conversion rates, carbonbalance and degree of reduction balance. The balanced data werecalculated based on the data reconciliation procedure usingelemental conservation constrains according to van der Heijdenet al. (1994). A good consistency was found between the measuredmacroscopic rates and the balanced rates (P-value1 91%). Thecarbon and degree of reduction balances for the non-reconcileddata closed with minor error (less than 3.2%). It was therefore

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0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Time [hour]

G6P

mas

s fra

ctio

n M+0M+1M+2M+3M+4M+5M+6

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Time [hour]

Gln

ic m

ass

fract

ion

M+ 0M+ 1M+ 2M+ 3M+ 4M+ 5M+ 6

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Time [hour]

6PG

mas

s fra

ctio

n

M+ 0M+ 1M+ 2M+ 3M+ 4M+ 5M+ 6

Fig. 2. mass distribution of g6p, 6pg and glnic during the labeling experiment, error bar is the calculated 95% confidence interval.

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186182

considered that these data did not contain gross measurementerrors.

4.2. Measured dynamics of mass distribution

Increasing the total concentration of the mixed carbon sourcein the feed vessel will result in a higher steady state biomassconcentration in the chemostat if carbon is the growth limitingnutrient. The effect of such an increase on the wash-in dynamicsof the labeled gluconate has been independently verified. Theresults clearly support the calculated decrease in the turnovertime of the glnec pool at a higher cell density, and thus a highervolumetric gluconate consumption rate (data not shown). Thisjustifies the use of gluconate tracer method at a higher cell densitythan it was proposed by Kleijn et al. (2006). The mass distribu-tions of g6p, 6pg and glnic were measured in a time series afterswitching to the labeled medium. The results are shown in Fig. 2.It is clear from this figure that all relevant metabolites reachedisotopic steady state within 1 h of labeling. All mass fractionsfollow a simple first-order wash-in kinetics. From the obtainedresults, it was decided to use the dynamic data obtained duringthe first hour for the isotopic non-stationary gluconate tracermethod, and the data after the fourth hour for the isotopicstationary gluconate tracer method. The error bars are the

standard deviations calculated by triplicate analysis of thesamples using the method for calculating mass fraction errorsdescribed in the appendix.

4.3. Isotopic stationary and non-stationary gluconate tracer method

The original data used for the isotopic stationary gluconatetracer method are listed in Table 3. After removing the effect ofstable isotopes of hydrogen and oxygen, the pure mass isotopomerfractions due to carbon isotopes are obtained. The results arelisted in Table 4 and are used with Eq. (10) to calculate the split-ratio. The thus calculated split-ratio, after the correction for thegluconate uptake, was found to be 40.6% for the appliedcultivation conditions.

The data used for the isotopic non-stationary gluconate tracermethod are plotted in Fig. 2. The MDVs were corrected for otherstable isotopes than carbon as described in Materials and methodssection. From Eq. (11), fluxes v2 and v4b (see Fig. 1) are estimatedto be 8.40 and 0.201 mmol/Cmol/h, respectively. The glnecconcentration is estimated to be 0.0261 mmol/L. After normalizingv2 to glucose uptake rate and correcting for the gluconate uptake,the split-ratio is found to be 41.2%. The simulated first hour timecourse of the mass distribution of 6pg and glnic correspondedwell with the measured data (Fig. 4). The results obtained with

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Fig. 3. Flow diagram for isotopic non-stationary gluconate tracer method.

Table 2Measured and balanced steady-state biomass specific rates, biomass and

penicillin-G concentrations, carbon balance and degree of reduction balance

Biomass specific conversion rate

ðmmol CmolX�1 h�1Þ

Measured Balanced

Glucose consumption �20:8� 1:8 �20:3� 1:7

Gluconate consumption �1:014� 0:090 �1:001� 0:086

PAA consumption �0:370� 0:086 �0:36� 0:084

O2 consumption �46:4� 4:3 �45:9� 4:1

CO2 consumption 50:0� 4:5 50:2� 4:4

Biomass production 61:0� 4:9 61:1� 4:9

Pen-G production 0:339� 0:039 0:337� 0:039

Byproduct 14:2� 1:8 14:3� 1:8

Biomass concentration (g/L) 5:41� 0:11 5:43� 0:11

CpenG (mmol/L) 1:085� 0:084 1:084� 0:084

Carbon balance 97.8%

Degree of reduction balance 98.3%

Table 3Measured mass isotopomer distribution for isotopic stationary gluconate tracer

method

g6p 6pg Gluconatea

Average (%) Stdev (%) Average (%) Stdev (%) Average (%) Stdev (%)

M þ 0 86.5 1.3 77.07 0.54 0.0 0.0

M þ 1 8.9 1.0 7.89 0.27 0.0 0.0

M þ 2 2.81 0.33 3.00 0.28 0.0 0.0

M þ 3 1.23 0.15 1.11 0.28 0.0 0.0

M þ 4 0.539 0.065 0.38 0.28 0.2 0.0

M þ 5 0.065 0.014 0.84 0.28 5.9 0.1

M þ 6 0.006 0.015 9.71 0.27 93.9 0.1

a Gluconate in the labeled feed. Data taken from Kleijn et al. (2006).

Table 4Calculated mass isotopomer distribution due to only carbon labeling

g6p 6pg Gluconatea

Average (%) Stdev (%) Average (%) Stdev (%) Average (%) Stdev (%)

M þ 0 88.5 1.3 78.86 0.54 0.0 0.0

M þ 1 8.7 1.0 7.66 0.27 0.0 0.0

M þ 2 1.19 0.33 1.41 0.28 0.0 0.0

M þ 3 1.08 0.15 0.96 0.28 0.0 0.0

M þ 4 0.509 0.065 0.34 0.28 0.2 0.0

M þ 5 0.041 0.014 0.84 0.28 5.9 0.1

M þ 6 0.001 0.015 9.92 0.27 93.9 0.1

a Gluconate in the labeled feed. Data taken from Kleijn et al. (2006).

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186 183

the two methods are compared with traditional MFB result39:3� 0:7%2 (van Gulik et al., 2000, with the NADPH balance). Itcan be concluded that the split-ratios obtained by these threemethods are very comparable. This also confirms the NADPHmetabolic balances in classical MFB analysis.

4.4. Time constant analysis

The three metabolite pools involved in the dynamics oflabeling inflow in this experiment (glnec, glnic and 6pg) haveturnover times that are very different. For simplicity theseturnover times were calculated as the ratio between the poolsize and the total incoming flux to that pool. The results arecompared in Fig. 5. It is clear that the slowest step is the labelingof the glnec pool.

The time constant of glnic and 6pg are far less than the timeconstant of the tracer gluconate in the extracellular pool, whichmakes it possible to simplify the model by assuming pseudoisotopic steady state (see Eq. (8)), i.e. assuming _Fglnic and _F6pg

equal 0. As a result, both MDV of the time dependent 6pg andglnic can be expressed as a function of time varying fglnecðtÞ andfg6pðtÞ (Eq. (15)):

v5f6pgðtÞ ¼ ðv3fglnecðtÞ þ v2fg6pðtÞÞ (15)

and

v4f fglnicðtÞ ¼1

v5ðv3ðv4b þ v5ÞfglnecðtÞ þ v2v4bfg6pðtÞÞ, (16)

where fglnecðtÞ is expressed by Eq. (5). When v3, fg6pðtÞ and f6pgðtÞ

are measured, fluxes v2 and v5 can be calculated using Eqs. (15)and (7). An interesting observation from Eq. (15) is that the

2 The notation � implies standard error.

ARTICLE IN PRESS

Fig. 4. Simulated (solid line) and measured (circle) mass fractions.

glnec glnic 6pg0

1

2

3

4

5

6

7

Tim

e co

nsta

nt [m

in]

Fig. 5. Time constant comparison of different pools.

0.9 0.95 1 1.05 1.1−0.010

−0.027

−0.013

0

0.013

0.027

0.040

Relative value of the model inputs [−]

Rel

ativ

e ch

ange

in e

stim

ated

spl

it−ra

tio [−

]

Cglnic

C6pg

v3

Fig. 6. Sensitivity of the estimated split-ratio (sr) for the individual model inputs.

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186184

dynamics of the MDV of 6pg becomes independent of theexchange rate between the glnic pool and 6pg pool and is only afunction of the gluconate uptake rate ðv3Þ, and the MDVs of glnecand intracellular g6p. This would reduce the number of com-pounds to be analyzed to only 6pg and g6p. This simplificationalso resulted in a much shorter calculation time. Essentially Eqs.(15) and (16) are the same as Eqs. (9) and (10) used in the isotopicstationary gluconate tracer method, except that the glnec massisotopomer fractions are not constants but replaced by a first-order wash-in response as a function of time. It was found thatsplit-ratio obtained using this algebraic equation was notstatistically different from the result obtained from Eq. (11) (datanot shown).

4.5. Sensitivity analysis

It was calculated that the 95% confidence interval estimated forthe isotopic stationary gluconate tracer method was (37.8% and44.0%), i.e. the split-ratio was 40:6� 1:6%. For the isotopic non-stationary gluconate tracer method, the 95% confidence intervalestimated by Eq. (12) was (40.1% and 42.3%). Because of the

symmetry, the net standard error of the estimated split-ratio wascalculated by Eq. (13) and the result was 0.6%.

Fig. 6 shows the sensitivity of the fitted split-ratio (y-axis) onthe relative percentage changes (x-axis) in the model inputs of Eq.(11) ðC6pg;Cglnic and v3Þ. The sensitivity of the mass fractions ofg6p was also investigated (data not shown). Among all the modelinputs, it was only found that 1% of the gluconate uptake rateerror will change the estimated split-ratio by 0.4%. It is clear thatthe fitting result was only strongly dependent on the gluconateuptake rate, while relatively independent on the glnic and 6pgconcentrations or the mass fractions of g6p. This can be explainedby the large difference in the time constants of the differentcompounds (see Eq. (8)). Furthermore, it is shown that the error islinear with the gluconate uptake rate. As a result, Eq. (14) can besimplified into

s2sr;gross ¼ s2

sr;net þDsr

Dv3

� �2

s2v3

. (17)

From Eq. (17) it was calculated that, the split-ratio was41:2� 4:8%. It is clear from this result that only 12% of the errorin the estimated split-ratio comes from the net error of the

ARTICLE IN PRESS

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186 185

measured mass fractions. The error in the substrate uptake ratesincreased the standard error from 41:2� 0:6 to 41:2� 4:8%.

5. Conclusion

The measured split-ratio, using both isotopic stationary andnon-stationary gluconate tracer method in a carbon limitedchemostat culture of P. chrysogenum grown at a dilution rate of0:06 h�1, were 40:6� 1:6 and 41:2� 0:6%, respectively. Theseresults were very close to the 39:3� 0:7% obtained by the methodof van Gulik et al. (2000) using metabolic flux balancing andassumptions on the NADPH balance at similar cultivationconditions. This justifies the use of the isotopic non-stationarygluconate tracer method as an alternative to the originalgluconate tracer method, which was based on isotopic steadystate data.

Increasing the steady state biomass concentration in chemo-stat cultures fed with trace amounts of labeled gluconateeffectively shortened the time needed to reach isotopic steadystate. With the presented isotopic non-stationary gluconate tracermethod, it is possible to reduce the time needed for labelingfurther down to approximately 0.06 residence time (1 h). This ismuch shorter than the originally proposed three residence times(60 h) (Kleijn et al., 2006). It is also shown that the rate of labelinginflow and as well as the sampling frequency can be designedusing Eq. (6).

With the presented gluconate tracer method, the complexity ofthe modeling of the dynamic labeling inflow has been reducedfrom a set of differential equations to a system of algebraicequations. This makes it much easier to be used compared withsimulation of isotope distribution of large metabolic networks.Moreover, the number of metabolite needs to be analyzed aremuch less due to the compact size of the studied metabolicnetwork. Consequently there are a significant reduction of theanalytical measurement and computational work.

Finally, it is shown in the sensitivity analysis that the accuracyof the chemostat macroscopic rates are of great importance forobtaining accurate results in the labeling experiments.

Acknowledgments

This project is financially supported by the NetherlandsMinistry of Economic Affairs and the B-Basic partner organiza-tions (www.b-basic.nl) through B-Basic, a public-private NWO–-ACTS (Advanced Chemical Technologies for Sustainability)programme.

Appendix A. Calculation of measured mass fractions and theirerrors from the measured peak areas

In order to obtain the accuracy of this measurement, usuallyreplicate measurements are made (e.g. n replicates). It has beengenerally accepted that the mass fractions can be simplycalculated by dividing the peak area of one mass isotopomer bythe total peak area of all mass isotopomers

f k ¼1

n

Xn

i¼1

Ai;k

Ai;Sðk ¼ 1; . . . ;mÞ, (18)

where f k is the average mass fraction of the kth isotopomer of acompound, which has m different masses, and has been analyzedn times, Ai;k is the peak area of the ith replicate analysis of the kthmass isotopomer, Ai;S is

Pmj¼1Ai;j. Subsequently, the standard

deviation of the measured mass fractions are calculated using the

fractions obtained from these replicates in a general approach inthe following equation:

sf k¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xn

i¼1

ðf i;k � f kÞ2

vuut . (19)

The error of the measured mass fraction ðsf kÞ is usually assumed

to be constant for one compound in a series of measurements(Noh et al., 2007). However, sf k

can be calculated more accuratelybased on the individual measurement error ðsAi

Þ of theircorresponding peak area ðAiÞ, using an error propagationapproach:

s2f k¼ rf T

� s2A � rf

¼qf k

qA1

!2

s2A1þ

qf k

qA2

!2

s2A2þ � � � þ

qf k

qAm

!2

s2Am

, (20)

where

qf k

qAk

¼1

AS�

Ak

A2S

and

qf k

qAl

¼ �Ak

A2S

ðlakÞ. (21)

Meanwhile, the most suitable mass fractions can be estimated bya parameter estimation approach:

½f; AR� ¼ arg minf;AR

SSðf; A; s2Ai;jÞ (22)

and

SSðf; AR; s2Ai;jÞ ¼

Xn

i¼1

Xmj¼1

ðAi;j � Ai;S � f jÞ2

s2Ai;j

, (23)

where f ¼ ½f 1; . . . ; f m�T , AR ¼ ½A1;S; . . . ; An;S�

T , f j is the estimatedmass isotopomer fraction of the jth mass. As;i is the estimatedtotal peak area of all mass isotopomers in the ith replicate. sAi;j

isthe standard deviation of the jth peak area of the ith replicate. Forthe case of g6p, there are seven different masses due to the carbonisotopomers, i.e. f ¼ ½f 1; . . . ; f 7�

T . If the sample is analyzed threetimes, then AR ¼ ½A1;S; . . . ; A3;S�

T . Therefore, we have 7� 3 ¼ 21peak area data and 7þ 3 ¼ 10 parameters, which means 11degrees of freedom. Experience with our mass spectrometry datashows that the error can usually be described by an error model

containing a constant absolute error sa and a constant relativeerror sr for one compound:

sAi;j¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

a þ ðsr � Ai;jÞ2

q. (24)

The parameters of the error model can be estimated by themaximum likelihood method using a large data set containing w

different samples (each have m mass and n replicates). Assumingthe error follows a normal distribution, the logarithm of thelikelihood of occurrence of all samples can be calculated by thefollowing equation:

lnðLÞ ¼Xw

s¼1

�SSsðf; AR; s2

Ai;jÞ

2

0@

1A

þXw

s¼1

ð� lnðsAi;j ;sÞÞ þw ln1ffiffiffiffiffiffi2pp

� �, (25)

where w is the total number of samples, SSs is the varianceweighed total sum of squares of the sth sample ð1pspwÞ. sAi;j ;s isthe estimated error of the jth isotopomer in the ith replicate of the

ARTICLE IN PRESS

Table 5Statistics of the measured peak area data

Absolute error Relative error No. of data Median Max

g6p 7.36% 336 502.6 87241.3

6pg 6.46 0 336 41.1 2863.0

Gluconate 14.4% 720 6.2 16661.9

Z. Zhao et al. / Metabolic Engineering 10 (2008) 178–186186

sth sample, which is used in Eq. (23) as the variance of the peakarea.

Repeated analysis of blank samples indicate a constantabsolute error of the peak area sa of 6.46. From the completemeasured mass isotopomer data set in a time series, the constantrelative error of each compound can be identified by maximizingthe logarithm of the maximum likelihood estimation function:

sr ¼ arg maxsr ;f;AS

L. (26)

The result is shown in Table 5. The possible reason for theconstant absolute error for both compounds can be the instabilityof the instrument.

The estimated errors of mass fractions by the error modelmethod are consistent with the general approach (data notshown). It can be concluded that the error model approach givesstable and fair estimations of the errors in the estimated massfractions. It is therefore used in our calculations instead of thegeneral approach.

Appendix B. List of symbols

Scalars3: A, the measured peak area; b, set of all estimatedparameters; C, concentration; f, mass isotopomer fraction; L,likelihood of occurrence; m, total number of mass isotopomers; n,total number of replicate measurements; p, total number ofparameters to be estimated; F, volumetric flow rate; r, volumetricconsumption rate; s, standard deviation; s2

A, variance of themeasured peak area; s2

f , variance of the measured mass fraction;SS, sum of squared residues; t, time constant; V, working volume;v, metabolic fluxes; w, total amount of samples.

Subscripts: S, all isotopomers; i, index of replicate measure-ments; j, index of mass isotopomers; k, number of massisotopomers; l, number of mass isotopomers, to be distinguishedwith k; S, residual substrate in the broth; s, index of samplenumber; sr, split-ratio.

Superscripts: 0, initial status;1, steady state; in, feed mediuminflow; out, effluent.

Matrix: I, 6 by 6 identify matrix; Z, 6 by 6 zero matrix.

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