Systematic quantiﬁcation of complex metabolic ﬂux...

Systematic quantification of complex metabolic flux networks usingstable isotopes and mass spectrometry

Maria I. Klapa*, Juan-Carlos Aon† and Gregory Stephanopoulos

Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

Metabolic fluxes provide a detailed metric of the cellularmetabolic phenotype. Fluxes are estimated indirectly fromavailable measurements and various methods have beendeveloped for this purpose. Of particular interest are meth-ods making use of stable isotopic tracers as they enable theestimation of fluxes at a high resolution. In this paper, wepresent data validating the use of mass spectrometry (MS)for the quantification of complex metabolic flux networks.In the context of the lysine biosynthesis flux network ofCorynebacterium glutamicum (ATCC 21799) under glucoselimitation in continuous culture, operating at 0.1Æh)1 afterthe introduction of 50% [1-13C]glucose, we deploy a bio-reaction network analysis methodology for flux determin-ation from mass isotopomer measurements of biomasshydrolysates, while thoroughly addressing the issues ofmeasurement accuracy, flux observability and data recon-ciliation. The analysis enabled the resolution of the involvedanaplerotic activity of the microorganism using only one

labeled substrate, the determination of the range of most ofthe exchange fluxes and the validation of the flux estimatesthrough satisfaction of redundancies. Specifically, we deter-mined that phosphoenolpyruvate carboxykinase and syn-thase do not carry flux at these experimental conditions andidentified a high futile cycle between oxaloacetate andpyruvate, indicating a highly active in vivo oxaloacetatedecarboxylase. Both results validated previous in vitroactivity measurements. The flux estimates obtained passedthe v2 statistical test. This is a very important result consid-ering that prior flux analyses of extensive metabolic net-works from isotopic measurements have failed criteria ofstatistical consistency.

Keywords:Corynebacteriumglutamicum; data reconciliation;GC-MS; metabolic flux determination; observabilityanalysis.

Defining flux as the rate at which material is processedthrough a metabolic pathway in a conversion process [1],the fluxes of a metabolic bioreaction network emerge as

fundamental metric of the cellular metabolic phenotype inthe absence of in vivo kinetic information [1–3]. In thiscontext, it becomes obvious why accurate and complete fluxmaps are essential in bioreaction network analysis, meta-bolic engineering, diagnosis of medical problems and drugdevelopment [1]. In light of the inability to measuremetabolic fluxes directly, various methods have beendeveloped for their estimation from available measure-ments, based on the fact that mass is conserved in ametabolic network. Among these, the methods that use onlyextracellular metabolite net excretion rate measurements arelimited to the estimation of net fluxes [4–6]. However,methods that make use of stable isotopic tracers, andmeasure the fate of the isotopic label in various metabolitepools, can enhance the resolution of a metabolic fluxnetwork in two ways: by increasing the number of estimablefluxes and by improving the accuracy of flux estimatesthrough measurement redundancy [4,7,8]. In this paper, weuse the stable isotope of carbon (13C) and ion-trap MSof biomass hydrolysates [9] for flux quantification. If 13C isused as tracer, MS can, in principle, measure the fractions ofa metabolite pool that are labeled at the same number ofcarbon atoms. These are the 13C mass isotopomer fractionsof the metabolite and provide a measure of the tracerdistribution in this metabolite pool. MS combined with theseparation ability of GC has been used for many years tomeasure the mass isotopomer distribution of intracellularmetabolites in cell lysates for flux quantification in thecontext of disease diagnosis (e.g. [10–13]). Wittmann and

Correspondence to G. Stephanopoulos, Bayer Professor of Chemical

Engineering and Biotechnology, Department of Chemical Engine-

ering, MIT, Room 56-469, Cambridge, MA 02139, USA.

Fax: +1 617 253 3122, Tel.: +1 617 253 4583,

E-mail: [email protected]

Abbreviations: 1,3-BPG, 1,3-bis-phosphoglycerate; 2-PG, 2-phospho-

glycerate; aKG, a-ketoglutarate; CER, carbon dioxide evolution rate;

DHAP, dihydroxyacetone phosphate; E4P, erythrose 4-phosphate;

FRU1,6bisP, fructose-1,6-bis-phosphate; FRU6P, fructose 6-phos-

phate; FUM, fumarate; G3P, 3-phosphoglycerate; G6P, glucose

6-phosphate; GAMS, General Algebraic Modelling System; GAP,

glyceraldehyde-3-phosphate; H4D, tetrahydrodipicolinate; ISOCIT,

isocitrate; LysEXTRA, lysine excreted extracellularly; LysINTRA, lysine

produced intracellularly; MAL, malate; meso-DAP, meso-diamino-

pimelate; OAA, oxaloacetate; OUR, oxygen uptake rate; P5P, pentose

5-phosphate; PEP, phosphoenolpyruvate; PPP, pentose phosphate

pathway; PYR, pyruvate; RQ, respiratory quotient; SED7P, sedo-

heptulose 7-phosphate; SUC, succinate; SUCCoA, succinyl coenzyme

A; SVD, singular value decomposition analysis; TBDMS, tributyl

dimethyl silyl.

*Present address: Department of Chemical Engineering, University of

Maryland, College Park, MD 20742, USA.

�Present address: GlaxoSmithKline, King of Prussia, PA, USA.

(Received 16 April 2003, revised 17 June 2003, accepted 26 June 2003)

Eur. J. Biochem. 270, 3525–3542 (2003) � FEBS 2003 doi:10.1046/j.1432-1033.2003.03732.x

Heizle (2001) [14] used MALDI-TOF-MS to measure themass isotopomer distribution of extracellular metabolitesfor the determination of the Corynebacterium glutamicummetabolic flux network. Using GC-quadrupole MS, Chris-tensen and Nielsen [15,16] reported the analysis of thePenicillium chrysogenum flux network from the massisotopomer fractions of biomass hydrolysates. Variousother networks were analyzed in subsequent studies usingthe same method [17–19].

In the present paper we expand on the idea of Christensenand Nielsen [15] describing the quantification of the lysinebiosynthesis flux network of C. glutamicum ATCC 21799under glucose limitation in continuous culture from massisotopomer measurements of biomass hydrolysates after theintroduction of 50% [1-13C] glucose. In the context of thismodel system, we thoroughly discuss all issues concerningthe use of stable isotopes, MS and bioreaction networkanalysis for flux quantification of complex metabolicnetworks. In this sense, we provide for the first time acomplete picture of the methodology. Specifically weaddress: (a) the validity of flux estimates from biomasshydrolysate measurements in the context of metabolic andisotopic steady-state only; (b) the accuracy of the MSmeasurements and which of them can be considered reliableto be used for flux determination (the latter question wasalso raised by [20]); (c) flux observability from the availablemeasurements; and (d) measurement redundancy andstatistical consistency analysis.

Apart from presenting a valid methodology for fluxdetermination, the second objective of this work was toapply it in the analysis of the C. glutamicum physiology.C. glutamicum is of special industrial interest primarily forlysine production from inexpensive carbon sources [21,22].While this is the main reason for which C. glutamicummetabolism has been under study for the last 40 years invarious groups [14,23–42], the C. glutamicum flux networkalso constitutes a good model system to illustrate issuesconcerning the application of stable isotope techniques. Itincludes an involved set of anaplerotic reactions and twoparallel pathways in the lysine biosynthesis route. Both ofthese groups of reactions have been shown to play animportant role in lysine biosynthesis [38,43], but theindependent quantification of their activity in vivo requiresthe use of isotopic tracers [5,35,38]. The extent to whichthe use of MS measurements of biomass hydrolysatesafter the introduction of the 13C tracer through theglucose substrate enables the accurate estimation of thesefluxes was explored in this work. Moreover, because ion-trap MS was used, the reported experimental data andflux analysis results provide material for comparisonbetween ion-trap and quadrupole MS in the context offlux quantification.

Finally, we need to underline that the flux analysismethodology presented here in the context of a particularmicroorganism is generic and it could be used for themetabolic reconstruction of any biological system withminor changes to adjust to its specifics. Additionally, whilethe methodology is validated in the context of metabolicand isotopic steady state, it is not per se limited to steady-state systems. Its application, however, to transient biolo-gical systems needs to be investigated further and validatedin the presence of a series of controls to guarantee correct

flux estimation from the isotopic tracer measurements ofbiomass hydrolysates.

Materials and methods

The aspartate kinase enzyme of C. glutamicum ATCC21799 is insensitive to feedback inhibition from threonineand lysine [5]. An excess of threonine, methionine andleucine was added in the preculture and reactor feed mediato inhibit their synthesis and direct the entire carbon fluxthrough aspartate kinase towards lysine production. Cul-tures for chemostat inoculation started from a seed culturein a 250-mL shake flask containing 50 mL of definedmedium. The seed culture was inoculated from a loop ofstock culture grown for 24 h on a Petri dish with complexagar medium. The seed culture medium was modifiedLuria–Bertani broth, containing: 5 gÆL)1 glucose, 5 gÆL)1

yeast extract, 10 gÆL)1 tryptone, 5 gÆL)1 NaCl [31].The shake flask was incubated overnight at 30 �C withagitation at 300 r.p.m. The preculture and chemostat feedmedium consisted of (per liter distilled water): 5 g glucose,50 mg CaCl2, 400 mg MgSO4Æ7H2O, 25 mg FeSO4Æ7H2O,0.1 g NaCl, 10 mL 100 · mineral salts solution, 3 gK2HPO4, 1 g KH2PO4, 1 g threonine, 0.3 g methionine,1 g leucine, 1 mg biotin, 1 mg thiamineÆHCl, 10 mg panto-thenic acid, 5 g (NH4)2SO4 and 0.1 lL antifoam. The100 · mineral salts solution consisted of (per liter distilledwater): 200 mg FeCl3Æ6H2O, 200 mg MnSO4ÆH2O, 50 mgZnSO4Æ7H2O, 20 mg CuCl2Æ2H2O, 20 mg Na2B4O7Æ10H2O,10 mg (NH4)6Mo7O24Æ4H2O (pH was adjusted to 2.0 byaddition of HCl to avoid precipitation). Preliminary meas-urements from shake flask cultures (data not shown) hadindicated that cells grown at 5 gÆL)1 glucose were underglucose limitation. Five hundred milliliters of the preculturewere incubated at 30 �C with agitation at 300 r.p.m. Whenthe attenuance (D) measurement indicated exponentialgrowth, the microbial broth was transferred into a 1-Lchemostat (Applicon Inc., the Netherlands). A D of 1.0corresponded to 0.265 gÆL)1 dry cell weight. Continuousfeed was initiated at dilution rate of 0.1Æh)1 using aperistaltic pump. Temperature and pH were kept at 30 �Cand 7.0, respectively, the latter with external addition of 2 M

NaOH. CO2-free compressed air (CO2 concentration<1 p.p.m.) was provided at 1 LÆmin)1, in an effort toeliminate input of 13C from sources other than glucose. Thecomposition of the air out of the gas cylinder was measuredfor 20 h prior to the experiment using a Perkin-Elmer MGA1600 mass spectrometer. The average concentration ofoxygen, nitrogen and carbon dioxide over this period oftime was considered the inlet air composition in theestimation of oxygen uptake (OUR) and carbon dioxideevolution (CER) rates [31,32]. Five milliliters samples werewithdrawn from the reactor every 10 h (residence time).Each sample was used partly for immediate measurement ofD and the rest was processed as described in the nextparagraph for subsequent analysis. The concentration ofoxygen, carbon dioxide and nitrogen in the outlet air streamwere measured online using the mass spectrometer describedabove. Outlet air composition provided an additional (tothe D) measurement, whose change over time was used tomonitor online the state of the culture. After six residencetimes, i.e. 60 h, and while the online measurements were

3526 M. I. Klapa et al. (Eur. J. Biochem. 270) � FEBS 2003

indicating that the culture was at metabolic steady state, thereactor was switched to labeled feed. In this, 50% of glucosewas 99.9% labeled at carbon 1 (Cambridge IsotopeLaboratories Inc.), everything else remaining the same asin the unlabeled feed. Five-milliliter samples were with-drawn every half-residence time (5 h) up to six residencetimes (60 h); by then, the culture was expected to havereached isotopic steady state.

All samples were kept in ice and (almost immediatelyafter sampling) were centrifuged for 5 min at 5040 g and2–4 �C; the rotor of the centrifuge had been precooled to)20 �C. The supernatant was separated from the pellet aftercentrifugation. The pellet was then washed once with 50%(v/v) methanol/water quenching solution precooled to)20 �C and centrifuged again for 5 min at 5040 g and2–4 �C (in a rotor precooled to)20 �C). The pellet was thendried under a flow of nitrogen; of note, the pellet was kept inice and the duration of drying was the shortest possible. Thedried pellets were stored at )20 �C for subsequent MSanalysis. The MS analysis protocol followed is described indetail in [44]. The supernatant was analyzed to determinethe concentration of glucose, trehalose, organic acids,amino acids and ammonia in the chemostat medium. Theconcentration of amino acids was measured by HPLC.Specifically, all amino acids were analyzed as ortho-phthaldialdehyde (OPA) derivatives using a Hewlett-Pack-ard reverse phase Amino Quant column on a series 1050HPLC system. The solvents used were acetonitrile, 0.1 M

sodium acetate pH 5.02 and water in a gradient mode at40 �C and a flow rate of 0.45 mLÆmin)1, monitoring UVabsorbance at 338 nm. The Boehringer Mannheim enzy-matic kits #716251, #139084, #1112732 and #148261 wereused for the measurement of the glucose (and trehalose),lactate, ammonia, and acetate concentrations, respectively.Specifically for the determination of trehalose concentra-tion, trehalose was initially broken down into glucose usingthe enzyme trehalase (Sigma catalog #T8778). The Sigmakit #726 was used for the determination of the pyruvateconcentration.

Flux analysis

Metabolic flux quantification is simultaneously a parameterestimation and a data reconciliation problem. Specifically,metabolic flux quantification refers to the estimation of theunknown net and exchange fluxes of a metabolic network(�parameters�) from available macroscopic data, based onmetabolite and isotopomer balances, the latter relevant inthe case of labeled substrate use [4,5]. The exchange flux of abiochemical reaction is a measure of the extent of itsreversibility [45]. The metabolite- and isotopomer balancesare formulated based on a stoichiometric model for theintracellular metabolic reactions and describe the conserva-tion of mass and isotopic label in a metabolic network.Clearly then, the first requirement for a successful fluxestimation is for the available measurements to containadequate information about the unknown fluxes. However,measurements are not, in general, expected to strictly satisfythe conservation balances, due to random experimentalerrors and process variability. Therefore, flux estimationproblems have to be defined as data reconciliation problems(i.e. weighted least-squares constrained minimization

problems), where the measured variables are optimallyadjusted, so that their adjusted values satisfy the metabo-lite- and isotopomer balance constraints [46]. Occasionallythough, some measurements may contain gross biases. Inthese cases, including this data in flux estimation will distortthe adjustments of all the measured variables, leading toerroneous metabolic flux estimates. These measurementsshould be isolated and discarded. Hence, the secondrequirement for the success of flux estimation is thereliability of the available experimental data. It becomesobvious then, that addressing the issues of flux observabilityand clever experimental design, along with data consistencyand identification of gross errors through satisfaction ofredundancies, constitutes a major part of flux quantificationanalysis. These issues are sequentially discussed in this paperin the context of the analysis of the C. glutamicum lysinebiosynthesis flux network using extracellular metabolite netexcretion rate- and mass isotopomer measurements (forfurther details see [7]).

Specifically, the metabolic flux quantification problemfrom extracellular metabolite net excretion rate- and massisotopomer measurements can be divided into two sub-problems, which can then be processed sequentially: (a)metabolite balancing analysis, which is the linear regressionof the extracellular metabolite net excretion rate measure-ments based on the metabolite balance constraints. Frommetabolite balancing analysis, only the fluxes of theindependent linear pathways of a network can be deter-mined. Consequently, all exchange fluxes and the net fluxesof the reactions involved in parallel competing pathwaysare unobservable [4–7,32,45,47–49]; (b) mass isotopomerdistribution analysis, which is the nonlinear regression ofthe mass isotopomer measurements based on (i) the13C- (positional) isotopomer balance constraints, (ii) thebalances relating the 13C- mass isotopomer measurementswith the 13C- positional isotopomer fractions of thecorresponding metabolite pools, (iii) the equations relatingthe net and exchange fluxes to the forward and reversefluxes of the network reactions, and (iv) the equationsdescribing the linear dependency between the net reactionfluxes in the groups of parallel competing pathways. If anamino acid is not part of the considered network, but itsmass isotopomer distribution is measured (e.g. phenyl-alanine), then balances (ii) contain the equations that relatethe measured mass isotopomer distribution of the aminoacid with the positional isotopomer fractions of networkmetabolites (e.g. erythrose-4-phosphate and phosphoenol-pyruvate, for the case of phenylalanine). Due to derivati-zation prior to GC, the raw MS measurements must be�corrected� for the natural abundance of the derivatizingagent constituents [50] to obtain the 13C-mass isotopomerfractions of the �bare� amino acid fragments. This correc-tion can be processed separately, and the correctedmeasurements can then be used in the objective functionof the regression problem [50]. Equivalently, the originalMS measurements can be included in the objectivefunction, in which case the correction equations have tobe considered as the last set of constraints in this part ofthe analysis. The latter approach was followed in thepresent study. The net fluxes, which have already beenestimated in metabolite balancing analysis, are includedhere as constants.

� FEBS 2003 Systematic flux analysis using stable isotopes and MS (Eur. J. Biochem. 270) 3527

Biochemistry: stoichiometric model

The analysis of the C. glutamicum lysine biosynthesis fluxnetwork under glucose limitation was based on the stoichio-metric model shown in the Appendix (for further details see[5,29,31]).

Results

Extracellular metabolite net excretion ratemeasurements

Figure 1A shows the time profiles of OUR and CERthroughout the continuous culture, from which the timeprofile of the respiratory quotient (RQ ¼ CER/OUR) isgenerated (Fig. 1B). Considering constant OUR and CERas an indication of metabolic steady state, it is observed thatthe cells reached steady state after approximately threeresidence times (30 h) of continuous feed and remainedat this state for almost 100 h (10 residence times). Theintroduction of the labeled feed after 60 h of continuousfeed did not disturb the physiological state of the cells. Thisis also validated by the concentration profiles of all themetabolites present in the chemostat medium, shown inFig. 2. In continuous culture, a constant concentration of ametabolite in the medium implies constant metabolite netexcretion rate [51]. The small decrease observed in lysineconcentration is expected in overproducers of lysine [40]. Inaddition, the glucose profile indicates that the cells wereindeed under glucose limitation. This guarantees that theentire amount of the isotopic tracer provided to the cellsthrough the glucose feed was assimilated by the culture. Thecells were using the carbon source primarily to grow(� 90%) and produce lysine (� 10%). Of the other aminoacids or organic acids, only valine was detected in trace

quantities in the medium. Threonine, methionine andleucine remained in excess throughout the continuousculture, supporting the assumption that the cells did notproduce any homoserine (or threonine and methionine).

The net excretion rates (in mMÆh)1) of the extracellularmetabolites, averaged over all steady-state samples, and thestandard deviations assigned to them, are shown in Table 1.The elemental composition and ash content of biomass wereconsidered to be C3.97H6.46O1.94N0.845 and 3.02%, respect-ively [32]. Trehalose, acetate, lactate and alanine wereincluded in the set of measured net excretion rates, even

Fig. 1. (A) The time profile of the oxygen uptake rate (OUR) and

carbon dioxide evolution rate (CER) and (B) the profile of the respiratory

quotient, throughout the continuous culture.

Fig. 2. The time profiles of the concentration of glucose, biomass, lysine,

ammonia, threonine, valine, methionine, leucine and pyruvate in the

chemostat medium throughout the continuous culture.

Table 1. The extracellular metabolite net secretion rates at metabolic

steady-state, estimated from the data shown in Figs 1 and 2. Columns 2

and 3 show the SD assigned to each of the rates as a fraction of the

measured value or in absolute terms, respectively.

Extracellular metabolite net

secretion rates (mMÆh)1)

SD

(%)

SD

(mMÆh)1)

Acetate 0 ± 0.02

Ala 0 ± 0.02

Biomass 1.99 4 ± 0.08

CER 6.42 10 ± 0.64

Glc )2.66 2 ± 0.05

lactate 0 ± 0.02

Lys 0.19 13 ± 0.025

Ammonia )2.54 15 ± 0.38

OUR )7.04 10 ± 0.70

PYR 7.7E)4 1 ± 7.7E)6

Trehalose 0 ± 0.02

Val 0.09 15 ± 0.01


though they were not detected in the medium. As explainedin greater detail in [31], there is a slight probability that thesemetabolites, which are known products of C. glutamicumunder some experimental conditions, might have beenproduced, but either accumulated intracellularly or excretedextracellularly at concentrations lower than the limits of thedetection methods. To account for these uncertainties, therates of these four metabolites were assigned a standarddeviation equal to 10% of the lysine excretion rate (i.e.0.02 mMÆh)1), lysine being the amino acid detected at thehighest concentration in the medium. This is smaller thanthe error considered by [31,32], i.e. 20% of the lysineproduction rate at the exponential phase of the batchculture, but the intracellular accumulation of these metabo-lites, if any, is expected to be low at the conditions of theexperiment [52]. The coefficient of variation assigned to therates of pyruvate, glucose and biomass reflects the accuracyof the detection equipment or kit. The standard deviationassigned to the net excretion rates of lysine and valineaccounted for their variation among the steady-state sam-ples. While the decrease in lysine concentration can beexplained from the physiologyof the strain [40], the observedfluctuations in valine concentration should be attributed tothe fact that the concentration of valine was at the limits ofthe detection method (HPLC). The high standard deviationsassigned to CER, OUR and the net consumption rate ofammonia (i.e. 10%, 10% and 15% of the rate value,respectively) reflect the high degree of uncertainty associatedwith these measurements. Specifically for ammonia, Vallino(1991) [31] speculated that the high (NH4)2SO4 concentra-tion in the medium throughout the continuous cultureincreases the difficulty of accurately determining the extentof ammonia assimilation from the cells. The measured CERand OUR values are based on a constant inlet airflow rate(1 LÆmin)1) and composition. Because the air was not pulledout of the air cylinder using a peristaltic pump and its flowrate was controlled manually, observed fluctuations were inthe range of ±0.2ÆL min)1 around the set value. Thestandard deviations assigned to CER and OUR account forthese errors in the airflow rate measurement.

MS measurements

Fig. 3 shows the time profiles of the (M + 0) and (M + 1)mass isotopomer fractions of selected tributyl dimethyl silyl(TBDMS)-amino acid fragments. M depicts the molecularweight of a fragment, i.e. all its atoms are in their naturallymost abundant isotopic form. Similar profiles wereobserved for the rest of the measured fragments. It becomesapparent that the cells reached isotopic steady-state 40 h(i.e. four residence times) after the initiation of the labeledfeed. Therefore, the MS measurements along with theextracellular metabolite net excretion rate measurementsestablish that the culture was at metabolic and isotopicsteady state for the last 30 h of the experiment.

The steady-state values of all MS measurements areshown in Table 2 along with the standard deviation associ-ated with each measurement. The steady-state values wereestimated as the average over the measurements of duplicatesamples and three injections per sample at the fourth, fifthand sixth residence times after the initiation of the labeledfeed.Thismeans that eachmeasurement is a combined result

of 18 GC-MS runs and its standard deviation reflects thevariance of its value among the 18 runs. This high degree ofredundancy enabled us to detect erroneous measurementsdue to saturation phenomena in the ion-trap (see [7,44]),while it obviously increases significantlyour confidence in thevalidity of the experimental data. If necessary, the standarddeviation also accounts for any systematic differencebetween the measured and the real MS values of an aminoacid fragment, as detected during the calibration of the entireMS measurement acquisition process with amino acidsamples of known labeling (for further details see [7,44]).All values depicted were also corrected for the presence of(M–n)+ peaks, as explained in [44]. Fragments of theTBDMS-derivatives of methionine and threonine were alsomeasured, but are not shown in Table 2, because they werenotused influxquantification,aswill be explained later in thetext.Most of themeasurements are associatedwith the lowerpart of the network [below phosphoenolpyruvate (PEP)],while the upper part of the network (glycolysis and pentosephosphate pathway) is �monitored� only from phenylalanineand glycine measurements.

Due to the selected substrate labeling, the most abundantmass isotopomers of each fragment are the three lightest.From Table 2, it can be observed that the error associatedwith these isotopomers is usually <7% of the MS value,while the coefficient of variation of the most abundant(M + 0) fraction can be as low as 0.3% (e.g. for alaninefragments). On the other hand, there is a large coefficient ofvariation (50–250%) associated with the heavier massisotopomer fractions. Under the experimental conditionsdescribed, these fractions are usually smaller than 3%.Calibration experiments had shown that the degree ofreliability and reproducibility of such measurements is verylow [44].

Flux determination: metabolite balancing analysis

The considered lysine biosynthesis network of C. glutami-cum (see Appendix) consists of 45 net fluxes and 46metabolites. Of the 47 reactions in the stoichiometricmodel, PEP carboxylase (reaction 23) and PEP

Fig. 3. Time profiles of the M + 0 and M + 1 mass isotopomer

fractions of selected TBDMS-amino acid fragments. M denotes the

molecular weight of a fragment, i.e. all its atoms are in their naturally

most abundant isotopic form. The number after the name of an amino

acid in the legend refers to the weight of the depicted fragment ion of

the TBDMS-derivative of the amino acid.


Table 2. The steady-state mass isotopomer fractions of the measured TBDMS-amino acid fragments and their estimated values, optimally adjusted

to satisfy the constraints of the flux quantification problem. The part of the amino acid carbon skeleton included in each fragment is depicted in

the first column of the table under the molecular weight of the fragment. The standard deviation associated with each measurement is shown in

the fourth column of the table; the number in parenthesis depicts the standard deviation as a percentage of the measured value (coefficient of

variation). The sixth column of the table shows the difference of the estimated from the measured values divided by the standard deviation of

the measurement. The last column of the table shows the square of the relative difference for each mass isotopomer fraction. The sum of the

elements in that column is equal to the total error of the flux analysis and it is compared with the v2 (0.9,53), 53 being the number of

redundant measurements. The last two columns show the values of the relative differences and their squares, respectively, only for the

measurements considered in the flux quantification analysis.

Fragment

Mass

isotopomer

Measured

fraction (%) SD (%)

Estimated

fraction (%)

Relative

difference

Relative

difference2

Ala260

[1–3]

M + 0 60.43 ± 0.20 (0.33) 59.34 5.45 29.70

M + 1 26.81 ± 0.54 (2.0) 28.24 )2.65 7.01

M + 2 9.73 ± 0.26 (2.7) 9.59 0.54 0.29

M + 3 2.55 ± 0.25 (9.8) 2.36

M + 4 0.47 ± 0.27 (57) 0.41

Ala232

[2–3]

M + 0 63.00 ± 0.21 (0.33) 63.36 )1.71 2.94

M + 1 25.93 ± 0.12 (0.46) 26.09 )1.33 1.78

M + 2 8.69 ± 0.18 (2.1) 8.30 2.16 4.65

M + 3 2.06 ± 0.12 (5.8) 1.90

M + 4 0.32 ± 0.01 (3) 0.29

Gly246

[1–2]

M + 0 74.99 ± 0.83 (1.1) 74.48 0.61 0.38

M + 1 16.57 ± 0.81 (4.9) 17.17 )0.74 0.55

M + 2 7.09 ± 0.39 (5.5) 7.04 0.13 0.02

M + 3 1.32 ± 0.59 (45) 1.07

M + 4 0.02 ± 0.04 (2E+2) 0.18

Gly218

[2]

M + 0 74.97 ± 1.63 (2.17) 76.75 )1.09 1.19

M + 1 16.18 ± 0.75 (4.6) 15.54 0.85 0.73

M + 2 6.94 ± 0.51 (7.3) 6.65 0.57 0.32

M + 3 1.44 ± 0.26 (18) 0.88

M + 4 0.47 ± 0.31 (66) 0.15

Val260

[2–5]

M + 0 50.70 ± 0.70 (1.4) 51.94 )1.77 3.14

M + 1 32.72 ± 0.55 (1.7) 32.63 0.16 0.03

M + 2 12.24 ± 0.22 (1.8) 11.58 3.00 9.00

M + 3 3.50 ± 0.25 (7.1) 3.13 1.48 2.19

M + 4 0.73 ± 0.13 (18) 0.60

M + 5 0.11 ± 0.07 (6E+1) 0.09

Val288

[1–5]

M + 0 51.39 ± 0.65 (1.3) 48.64 4.23 17.90

M + 1 32.68 ± 1.19 (3.64) 33.65 )0.82 0.66

M + 2 12.16 ± 0.87 (7.2) 13.03 )1.00 1.00

M + 3 3.31 ± 0.50 (15) 3.71 )0.80 0.64

M + 4 0.46 ± 0.29 (63) 0.78

Val186

[2–5]

M + 0 55.96 ± 0.53 (0.95) 57.71 )3.30 10.90

M + 1 30.69 ± 0.64 (2.1) 32.04 )2.11 4.45

M + 2 9.66 ± 0.49 (5.1) 8.39 2.59 6.72

M + 3 2.90 ± 0.39 (13) 1.63

M + 4 0.59 ± 0.27 (45) 0.20

M + 5 0.20 ± 0.17 (85) 0.02

Val302

[1–2]

M + 0 64.04 ± 0.22 (0.34) 64.50 )2.09 4.37

M + 1 24.71 ± 0.20 (0.81) 24.40 1.55 2.40

M + 2 9.00 ± 0.65 (7.2) 8.74 0.40 0.16

M + 3 1.99 ± 0.38 (19) 1.93

M + 4 0.14 ± 0.25 (1.8E+2) 0.34

Glu432

[1–5]

M + 0 40.83 ± 0.30 (0.73) 40.81 0.07 0.00

M + 1 36.99 ± 4.29 (11.6) 34.13 0.67 0.44

M + 2 16.77 ± 0.14 (0.83) 16.73 0.29 0.08

M + 3 4.22 ± 2.65 (62.8) 6.06

M + 4 0.95 ± 1.34 (1.4E+2) 1.69


Table 2. (Continued).

Fragment

Mass

isotopomer

Measured

fraction (%) SD (%)

Estimated

fraction (%)

Relative

difference

Relative

difference2

Glu272

[2–5]

M + 0 51.24 ± 1.21 (2.36) 51.80 )0.46 0.21

M + 1 31.71 ± 1.41 (4.45) 32.53 )0.58 0.34

M + 2 12.69 ± 0.41 (3.2) 11.69 2.44 5.95

M + 3 3.58 ± 0.40 (11) 3.21

Asp418

[1–4]

M + 0 47.18 ± 0.78 (1.7) 46.13 1.35 1.81

M + 1 32.60 ± 0.68 (2.1) 32.24 0.53 0.28

M + 2 14.44 ± 1.00 (6.89) 14.83 )0.39 0.15

M + 3 4.69 ± 0.31 (6.6) 5.02 )1.06 1.13

M + 4 0.94 ± 0.38 (4.0E+1) 1.32

M + 5 0.15 ± 0.12 (8.0E+1) 0.28

Asp390

[2–4]

M + 0 50.35 ± 1.65 (3.28) 49.93 0.25 0.06

M + 1 32.55 ± 1.41 (4.33) 30.98 1.11 1.24

M + 2 14.52 ± 1.14 (7.85) 13.51 0.89 0.78

M + 3 2.46 ± 1.88 (76.4) 4.30

M + 4 0.12 ± 0.23 (1.9E+2) 1.06

Asp316

[2–4]

M + 0 56.65 ± 2.02 (3.57) 55.45 0.59 0.35

M + 1 32.24 ± 1.20 (3.72) 30.33 1.59 2.53

M + 2 8.72 ± 1.77 (20.3) 10.76 )1.16 1.34

M + 3 2.38 ± 0.84 (35) 2.83

Lys431

[1–6]

M + 0 40.44 ± 3.03 (7.48) 37.81 0.87 0.76

M + 1 34.83 ± 0.95 (2.7) 34.78 0.05 0.00

M + 2 18.29 ± 2.88 (15.7) 18.04 0.09 0.01

M + 3 5.43 ± 1.22 (22.5) 6.79

M + 4 0.97 ± 0.95 (98) 1.97

Lys272

[2–6]

M + 0 46.83 ± 1.98 (4.23) 47.52 )0.35 0.12

M + 1 32.97 ± 1.53 (4.64) 34.38 )0.92 0.85

M + 2 14.53 ± 0.85 (5.8) 13.42 1.31 1.71

M + 3 4.80 ± 0.53 (11) 3.80

M + 4 0.84 ± 0.49 (58) 0.80

Phe336

[1–9]

M + 0 44.96 + 2.92 (6.49) 43.20 0.60 0.36

M + 1 36.02 ± 2.50 (6.94) 34.86 0.46 0.22

M + 2 14.75 ± 2.78 (18.9) 15.50 )0.27 0.07

M + 3 3.58 ± 1.09 (30.4) 4.97

M + 4 0.06 ± 0.13 (2E+2) 1.21

Phe308

[2–9]

M + 0 44.11 ± 1.26 (2.86) 44.30 )0.15 0.02

M + 1 34.80 ± 1.50 (4.31) 34.98 )0.12 0.02

M + 2 15.56 ± 1.00 (6.43) 14.86 0.70 0.49

M + 3 4.59 ± 0.19 (4.1) 4.57 0.11

M + 4 0.94 ± 0.56 (59) 1.06

Phe234

[2–9]

M + 0 50.34 ± 2.17 (4.31) 49.23 0.51 0.26

M + 1 35.56 ± 1.83 (5.15) 35.27 0.16 0.03

M + 2 11.96 ± 0.50 (42) 12.12 )0.32 0.10

M + 3 2.07 ± 1.74 (84.3) 2.85

M + 4 0.09 + 0.17 (2E+2) 0.48

Phe302

[1–2]

M + 0 71.93 ± 1.28 (1.78) 71.24 0.54 0.29

M + 1 19.89 ± 1.20 (6.03) 19.64 0.21 0.04

M + 2 7.09 ± 0.74 (1.0E+1) 7.52 )0.58 0.34

M + 3 0.78 ± 0.49 (63) 1.34

M + 4 0.04 ± 0.08 (2E+2) 0.23

Consistency index (value of least squares) 135.53 > 66.55


carboxykinase (reaction 24) are considered the oppositedirections of a single biochemical reaction (the ATP balanceis not included in the model). Similarly for pyruvatecarboxylase (reaction 25) and oxaloacetate decarboxylase(reaction 26). In metabolite balancing, the stoichiometricmatrix coincides with the sensitivity or derivative matrixthat connects the vector of the unknown net fluxes to thevector of the extracellular metabolite net excretion ratemeasurements. In the case of the considered network, therank of the stoichiometric matrix is 43. This indicates thepresence of two groups of parallel competing pathways (i.e.two groups of unobservable net fluxes) in the considered netflux network. Singular value decomposition analysis (SVD)[7,31,53,54] of the reduced low-echelon form of the stoichio-metric matrix enabled the identification of the net fluxes ineach group (i.e. the nonzero elements of the two vectors inthe null space [53] of the reduced row-echelon form of thestoichiometric matrix) and the determination of the equa-tions describing their linear dependency (equivalently thiscan be accomplished by identifying the cycles of flow in thenet flux network as described in [49]): (all numbers belowrefer to the corresponding reactions in the Appendix)

Group 1: the net fluxes of reaction 10 and combinedreactions 23–24 and 25–26.

Group 2: the net fluxes of reactions 38, 39, 40, 41, 42, 18 and28.

Both groups include two parallel pathways, competingfor PEP in the case of group 1 (anaplerotic pathways) andtetrahydrodipicolinate (H4D) in the case of group 2 (lysinebiosynthesis). If at least one net flux from each group or thenet flux ratio at PEP or H4D, respectively, were known,then all net fluxes in the respective group would beestimable. Since such information is unavailable, the 10net fluxes in groups 1 and 2 remain unobservable at thisstage of the analysis.

The number of redundant measurements, estimated fromthe difference between the number of measurements and therank of the stoichiometric matrix, is three. Redundantmeasurements are essential for data reconciliation. Datareconciliation analysis (see [55–57] for data reconciliation inlinear balance systems in general and [31,58] for datareconciliation analysis in metabolite balance systems) indi-cated that the extracellular net excretion rates of ammoniaand carbon dioxide were suspect of containing gross errors.When these measurements were excluded, the total error ofthe analysis (consistency index) was almost equal to 0 [7].The net fluxes as estimated after excluding these erroneousmeasurements from the data are shown in Fig. 4, normal-ized with respect to the uptake rate of glucose; the latter isconsidered to be 100. The estimated net fluxes were consid-ered constant in the rest of the analysis, while the net fluxesof the 10 reactions in the singular groups 1 and 2 wereexpressed as a function of the net flux ratio at PEPandH4D,respectively, based on the SVD analysis described earlier.

Flux determination: mass isotopomer distributionanalysis

In this part of the flux analysis, the independent unknownsare the net flux ratios at PEP and H4D nodes and the

exchange fluxes of all reversible reactions in the network.Apart from reactions 3, 11, 15–19, 27, 29–33, 36–44 and thebiomass equation which was decomposed in its constituentsfrom the beginning, the rest of the network reactions wereconsidered potentially reversible, setting the number ofunknown exchange fluxes to 19.

Observability analysis (1). In mass isotopomer distribu-tion analysis, the relationship between the measurements(mass isotopomer fractions) and the unknown fluxes isnonlinear due to the format of the positional isotopomerbalances. In this case, the numerical representation of thesensitivity matrix that connects the measurement vector tothe unknown flux vector and represents the mapping of thefluxes into the available measurements depends not only onthe structure and connectivity of the network, but also onthe substrate labeling and the actual value of the unknownfluxes. It is through the analysis of this matrix that thenumber and the identity of the unobservable fluxes, andconsequently the number of redundant measurements usedin data reconciliation analysis can be determined [46,55–57,59]. Structural observability analysis [7,55–57,59] takesinto consideration only the structural and not the numericalrepresentation of the sensitivity matrix. It can identify onlythe unknown fluxes that cannot be estimated fromthe available measurements due to the connectivity of theconsidered metabolic network as this is mapped in thestructure of the sensitivity matrix. Structural observabilityanalysis has only negative value, i.e. a structurally unob-servable flux is also numerically unobservable (i.e. it isunobservable independently of the substrate labeling usedand the value of the unknown fluxes), but the opposite doesnot necessarily hold true. It cannot identify numerical

Fig. 4. The estimated net flux distribution.


singularities neither differentiate between substrate labelingsif they do not clearly change the connectivity of the network.However, one important aspect of structural observabilityanalysis is that by studying the connectivity of potentialmeasurements to the unknown fluxes, it is possible todetermine which additional data could, in principle, increasethe resolution of the flux network in the absence ofnumerical singularities (further details about structuralobservability analysis of complex metabolic networks fromisotopic tracer data can be found in [7]). Fig. 5 shows anexample of structural observability analysis in the context ofa linear pathway of two reversible reactions.

In the present study the structurally unobservable fluxesare: (a) the exchange fluxes of fructose-6-phosphate aldo-lase (reaction 4) and triose-phosphate isomerase (reaction5) – Based on the structure of these two reactions, for theirexchange fluxes to be estimable, appropriate information

about the isotopic tracer distribution of fructose 1,6-bisphosphate (FRU1,6bisP) and dihydroxyacetone phos-phate (DHAP), respectively, should be available [7](Fig. 5). With the existing measurements the reactions 3,4 and 5 are actually observed as one irreversible reactionproducing two molecules of glyceraldehydes-3-phosphate(GAP) from one molecule of fructose-6-phosphate(FRU6P) (see Figs 5 and 6); (b) the exchange fluxes ofGAP dehydrogenase (no. 6) and phosphoglycerate kinase(no. 7) – These exchange fluxes would have been estimableonly if appropriate information about the isotopic tracerdistribution of GAP and 1,3-bis-phosphoglycerate(1,3BPG) had been provided (Fig. 5). With the existingmeasurements the pools of GAP, 1,3BPG and 3-phospho-glycerate (G3P) are observed as one pool depicted in Fig. 6as GAP/G3P. Information about the isotopic tracerdistribution of GAP/G3P pool is provided from the massisotopomer measurements of glycine; (c) the exchangefluxes of phosphoglycerate mutase (no. 8) and 2-phospho-glycerate enolase (no. 9) cannot be determined independ-ently – Since information about the isotopic tracerdistribution of GAP/G3P and PEP (from phenylalanine)pools, but not for this of 2-phosphoglycerate (2-PG), isavailable, the two reactions are observed as one reversiblereaction between the GAP/G3P and PEP pools; (d) theexchange flux of glutamate synthase reaction (no. 28) –Because no information about the isotopic tracer distribu-tion of alpha-ketoglutarate (aKG) is available, the aKG

Fig. 5. Structural observability analysis of a linear pathway comprising

two reversible reactions. It is assumed that the net flux through the

linear pathway and the isotopic tracer distribution of metabolite C are

known. (A) If the isotopic tracer distribution of neither A or B is

measurable, then the exchange fluxes of the two reactions are not

observable and the pools A and B cannot be considered independently

of pool C. (B) If only the isotopic tracer distribution of metabolite B is

measurable, then the pools of A and B are observed as one, i.e. they

have to be grouped. (C) If only the isotopic distribution of metabolite

A is measurable, then the B metabolite pool is not observable and the

two reversible reactions are conceived as one consuming A to produce

C. The exchange flux of this reaction is, in principle, estimable.

Fig. 6. The structurally observable C. glutamicum flux network, based

on the available mass isotopomer measurements (the zero acetate, lactate

and trehalose production rates are known from metabolite balancing

analysis). The metabolite pools whose mass isotopomer distribution is

reflected in the mass isotopomer measurements of the biomass

hydrolysates are depicted within a gray box.


and glutamate pools are observed as one (depicted byaKG/Glu in Fig. 6); (e) the exchange flux of aspartateamino transferase reaction (no. 34) – Because no informa-tion about the isotopic tracer distribution of oxaloacetate isavailable, the pools of aspartate (Asp) and oxaloacetate(OAA) are observed as one pool; (f) the exchange fluxes offumarase (no. 21) and malate dehydrogenase (no. 22)reactions – Because information about the isotopic tracerdistribution of neither fumarate (FUM) nor malate (MAL),respectively, is available, the pools of OAA, MAL, FUMare observed as one (along with Asp as discussed in theprevious paragraph) (see Fig. 6); (g) the exchange flux ofaspartate kinase reaction (no. 35) – Independently of thisexchange flux value, the pools of aspartate and aspartic-semialdehyde will always have the same isotopic tracerdistribution. This holds true because aspartic semialdehyde�receives� the isotopic tracer only from aspartate, while itsdownstream pathway towards lysine is irreversible.

Thus, 10 out of the 19 initially unknown exchangefluxes are not observable from the available measurementsas mandated from the structure of the network. Fig. 6shows the metabolic flux network of C. glutamicum that isstructurally observable from the available MS measure-ments. At this point, flux quantification (i.e. weightednonlinear regression of the mass isotopomer measure-ments) can be performed with all the structurally observ-able fluxes as unknowns. Any numerical singularities, dueto the values of the measurements (based on the chosensubstrate labeling) and the error associated with them,that render a structurally observable flux numericallyunobservable, can be determined after flux quantification,when the flux confidence intervals are estimated. Theconfidence interval of a numerically unobservable flux willbe equal or exceed the feasible range of values for thisflux. In the next paragraphs, we describe the quantifica-tion of the 9 exchange and 2 net fluxes from 61 (seeexplanation later) MS measurements.

Validation of assumptions and measurement accuracy inthe context of the C. glutamicum intracellularbiochemistry (2). There are three topics to discuss: (a)culture does not produce homoserine (or threonine andmethionine) – The C. glutamicum lysine biosynthesis net-work considered in flux quantification (see Appendix) doesnot include the reactions for homoserine biosynthesis anddownstream reactions for threonine and methionineproduction (see Fig. 4). Even though the ATCC 21799strain can produce homoserine, it was assumed that it didnot, because threonine and methionine were provided inexcess in the chemostat feed. The mass isotopomermeasurements of threonine and methionine validated thisassumption. Neither the mass isotopomer distribution ofthreonine nor that of methionine indicated the presence ofisotopic tracer in these pools at levels higher than naturalabundance (data not shown). If the cells had beensynthesizing any homoserine, then threonine andmethionine would have been isotopically enriched fromthe labeling of glucose; (b) validation of mass isotopomermeasurements through satisfaction of redundancies –Redundant measurements can be used to validatemeasurement accuracy. In the considered network, suchan example is provided by the measured mass isotopomer

fractions of Asp, Ala and Glu derivatives. As discussed inthe observability section, Asp and OAA are seen as a singlepool, the same holding for the pools of aKG and Glu.According to the assumed stoichiometry of the first threereactions of the TCA cycle (no. 16, 17 and 18), the last threecarbon atoms of OAA/Asp become the first three carbonatoms of aKG/Glu (see Fig. 7), while the carbon atoms ofacetyl-CoA (AcCoA) become the last two carbon atoms ofaKG/Glu. The carbon atoms of AcCoA originate from thelast two carbon atoms of pyruvate, the mass isotopomerdistribution of which is reflected in this of alanine.Therefore, the mass isotopomer distribution of Glu can beestimated from the mass isotopomer distribution offragment [2-4] of OAA (or Asp) and fragment [2-3] ofpyruvate (PYR) (or Ala) based on the followingrelationships:

ðM þ 0ÞGlu½1-5� ¼ ðM þ 0ÞOAA ½2-4� � ðM þ 0ÞPYR ½2-3�


þ ðM þ 1ÞOAA ½2-4� � ðM þ 0ÞPYR ½2-3�




ðM þ 3ÞGlu½1-5� ¼ ðM þ 1ÞOAA½2-4� � ðM þ 2ÞPYR ½2-3�

þ ðM þ 2ÞOAA½2-4� � ðM þ 1ÞPYR ½2-3�



þ ðM þ 3ÞOAA½2-4� � ðM þ 1ÞPYR ½2-3�


ð1Þ

If the measured mass isotopomer distributions of Glu,fragment [2-4] of Asp and fragment 2-3 of Ala do notcontain any gross errors, then the estimated (from Eqn 1)and measured mass isotopomer distribution of glutamateshould be statistically identical. As there are 11 unknownfluxes and 61 measurements, this kind of redundancy isexpected in other parts of the network as well, thusenhancing the accuracy of flux estimates; (c) flux distribu-tion around the PEP and PYR nodes – Figure 8A shows thestoichiometry of the pathways responsible for the labeltransfer to Gly and Val. When glucose (substrate) is labeledonly at carbon 1, then, due to the stoichiometry of carbontransfer through the pentose phosphate and glycolysispathways, most of the isotopic tracer of glucose is expectedto be transferred to the third carbon atom of the GAP/G3Ppool. Assuming that this is indeed the case and the first twocarbon atoms of GAP/G3P are at natural abundance,Fig. 8B illustrates the fate of the isotopic tracer throughoutthe depicted metabolic network, if all the involved reactionswere irreversible. All four carbon atoms of oxaloacetate areexpected to be labeled due to the label scrambling throughthe TCA cycle. In this scenario, the first two carbon atomsof the GAP/G3P pool, and consequently Gly, these of PEP,and thereby Phe, and these of PYR, and thereby Val, areexpected to be at natural abundance. Fig. 8C, on the other


hand, shows what the fate of the isotopic tracer would be, ifall the reactions of the depicted network were reversible. Inthis case, the isotopic tracer of the substrate is expected toalso �reach� the first two carbon atoms of the GAP/G3P,Gly, PEP, Phe, PYR and Val pools. Fig. 9A shows the

measured steady-state 13C mass isotopomer distribution ofGly, fragment [1-2] of Phe and fragment [1-2] of Val. Thefirst column of the histogram represents the theoretical massisotopomer distribution of a two-carbon atom molecule atnatural abundance. It is clear that both Gly and fragment[1-2] of Phe are practically unlabeled, while fragment [1-2] ofVal is labeled. The metabolic scenario that allows this tohappen involves irreversibility between the pool of PEP andthose of OAA and PYR. In other words, prelimary analysisof the measurements provides evidence that PEP carboxy-kinase (no. 24) and PEP synthase (the reverse direction ofno. 10) do not carry flux under the conditions of theexperiment. OAA decarboxylase (no 26), on the other hand,should carry flux to allow the isotopic tracer to reach thefirst two carbon atoms of pyruvate and thereby valine (seeFig. 9B). These results are in agreement with previousphysiological studies (see Discussion).

Flux quantification using MS measurements (3). Massisotopomer distribution analysis was performed using only61 measurements out of the set depicted in Table 2. Due tothe low reliability and reproducibility of mass isotopomer

Fig. 7. The measured mass isotopomer distribution of glutamate (cal-

culated from the TBDMS-Glu fragment 432) compared with its esti-

mated value, as it was calculated from the measured fragments [2-4] of

OAA (Asp) ) from TBDMS-Asp fragment 390 – and [2-3] of PYR

(Ala) ) from TBDMS-Ala fragment 232 ) based on Eq. (1). The

relationship between the carbon atoms of Glu and these of OAA [2-4]

and Ala [2-3] are shown in the upper right part of the figure. In the

calculations only the mass isotopomer fractions of the TBDMS-

derivatives considered in flux quantification (as shown in Table 2) were

used.

Fig. 8. Labeling scrambling through the pathways connecting Gly to

Val. (A) The stoichiometry of the pathways connecting Gly to Val.

The color code illustrates the stoichiometry of carbon transfer among

the metabolites of these pathways. (B) As Glc (substrate) is labeled at

carbon 1, almost the entire amount of isotopic tracer is expected to

reach the third carbon of the GAP/G3P pool (if a carbon is labeled, it

has an asterisk next to it). If all the reactions of the depicted network

were irreversible, then Gly and fragments [1-2] of Phe and Val are

expected to be (almost) at natural abundance. All four carbon atoms of

OAA are labeled because of the labeling scrambling through the TCA

cycle. (C) If all the involved reactions were reversible, then glycine and

fragments [1-2] of Phe and Val are expected to be (at a low level)

labeled.

Fig. 9. Preliminary data analysis provides evidence that PEP carboxy-

lase (no. 24) and PEP synthase (the reverse direction of no 10) do not

carry flux under the conditions of the experiment. (A) The measured

steady-state mass isotopomer distribution of Gly and fragments [1-2]

of Phe and Val (�bare� carbon skeleton). Compared with the mass

isotopomer distribution of a 2-carbon atom molecule at natural

abundance, Gly and fragment [1-2] of Phe are unlabeled, while frag-

ment [1-2] of Val is labeled. (B) The metabolic scenario that is con-

sistent with the experimental data (see Fig. 8 for definition of the

metabolites and the color code).


fractions smaller than 3% [7,44], only mass isotopomerfractions greater than 3% and associated error smaller than20% of the measured values were deemed reliable sensors ofthe in vivo fluxes. In addition, the one set of non-continuously differentiable constraints [45], i.e. theequations defining the exchange fluxes as a function of theforward and reverse fluxes of the reversible biochemicalreactions, was transformed into continuously differentiableafter using information about the direction of thecorresponding net fluxes from metabolite balancinganalysis. Only the direction of the net flux of thecombined PEP carboxylase and carboxykinase, and PYRcarboxylase and OAA decarboxylase reactions remainedunknown after metabolite balancing analysis. It wasassumed that both net fluxes follow the direction towardsOAA, which had been determined as the direction of theirsum (nongluconeogenic conditions). All exchange fluxeswere assigned an upper bound (additional constraint) toincrease the stability of the convergence process. Potentialinstability was also reported by Schmidt et al. (1999) [60](see also [8,45]). Defining

mexchj ½0; 1� ¼

mexchj

mexchj þ 1

½45�

where mexchj the j-th unknown exchange flux of the

network, each mexchj [0,1] was upper bounded by 0.95.

All constraints of the problem were generated by aWINDOWS-based software written in Object Pascal in theDEPLHI 2 environment (�Borland International Inc. 1996,www.borland.com). The software comprised a relational

database of all biochemical reactions in Escherichia coliand C. glutamicum, allowing the automatic choice of thenetwork structure by the user. The constraints were thenintroduced into the General Algebraic ModelingSystem (GAMS) environment (�GAMS DevelopmentCorporation, 1998, www.gams.com), a high-level mode-ling system for mathematical programming problems, andthe weighted least-squares mass isotopomer analysisproblem with nonlinear constraints was solved usingCONOPT (documentation about the CONOPT solvercan be found in www.gams.com).

Multiple initial guesses were used; the system, however,converged to the same solution for all the numericallyobservable fluxes (see footnote d, Table 3). The net fluxratios at PEP and H4D and the exchange flux of 1-trans-ketolase reaction (no 12) in the pentose phosphate pathwaywere numerically unobservable. Table 3 shows the 90%marginal confidence intervals of the eight observableexchange fluxes. The confidence intervals of the exchangefluxes were first estimated in the mexch [0,1] space, and thentransformed back to the mexch space, as described in [45]. Forthis set of flux estimates, the least-squares sum (i.e. totalerror) was 135.53. Table 2 shows the fitted values of themass isotopomer fractions relatively to the measured ones.Since the v2-value for (61–8) ¼ 53 redundant measurementsand 90% confidence is equal to 66.55, the considered set ofmeasurements does not initially pass the v2 statistical test.Although the difference between the total error of thisanalysis and the v2-value is much smaller than othersreported in the literature both for NMR and MS data (see[20]), it is important to investigate the sources of such an

Table 3. Estimated values of the two unknown net flux ratios (around PEP and 2-amino-6-ketopimelate) and the nine exchange fluxes. The last column

of the table shows the marginal 90% confidence intervals for each of the flux estimates. All values have been normalized with respect to the uptake

rate of glucose.

Flux name

Value (normalized with

respect to glucose rate)

Marginal 90%

confidence interval

Net flux of PEP carboxylasea Not observable

Exchange flux of PEP carboxylasea 0 [0,0]

Net flux of pyruvate carboxylaseb Not observable

Exchange flux of pyruvate carboxylaseb 20.3 · total anaplerotic net flux [13,43] · total anaplerotic net flux

Exchange flux of pyruvate kinasec 0 [0,0]

Net flux ratio between the four-step and the

one-step pathway of lysine biosynthesis

Not observable

Exchange flux of glucose-6-phosphate isomerase reaction 0 [0, 0.6 · 73.6]

Exchange flux of 1-transketolase reaction Not observable

Exchange flux of 2-transketolase reaction 281 · 4.3 [48.7 · 4.3,1]

Exchange flux of transaldolase reaction in PPP 9.5 · 6.9 [3.3 · 6,9,1]

Exchange flux of reaction GAP/G3P fi PEP 0d [0,0]

Exchange flux of reaction SUC fi FUM 15 · 72.3 [7.9 · 72.3, 235 · 72.3]

a To formulate the mass balances, PEP carboxylase and PEP carboxykinase were considered as the two opposite directions of one reaction

under the name PEP carboxylase. The fact that the exchange flux of this reaction was found equal to 0 means that PEP carboxykinase does

not carry flux in vivo. b To formulate the mass balances, pyruvate carboxylase and oxaloacetate decarboxylase were considered as the two

opposite directions of one reaction under the name pyruvate carboxylase. c To formulate the mass balances, pyruvate kinase and PEP

synthase were considered as the two opposite directions of one reaction under the name pyruvate kinase. The fact that the exchange flux of this

reaction was found equal to 0 means that PEP synthase does not carry flux in vivo. d At a second identified local minimum, the exchange flux of

that reaction is estimated larger than 20 times the net flux of the reaction (the rest of the flux estimates and the value of the objective function

remain the same). This is true, because in this case the two scenarios: (a) irreversible reaction and (b) reaction at equilibrium, are practically

identical. Because of the irreversibility between PEP and the lower part of the network, both cases indicate that the pools of GAP, G3P and

PEP (and all the other intermediates) have the same labeling and are practically observed as one metabolite pool.


inconsistency and propose ways to decrease it. FromTable 2, it can be observed that the measurements contri-buting most to the total error are the fractions (M + 0) and(M + 1) of fragment 260 of Ala, fraction (M + 0) offragment 288 of Val and all the measurements associatedwith fragment 186 of Val. All six measurements areredundant and can be eliminated with no loss of informa-tion about the flux network (i.e. the elimination will notaffect the degree of observability of the flux network). Dueto the irreversibility at PEP (see Discussion) and thestructure of the lower part of the network, this eliminationcould, in principle, affect only the exchange flux of thecombined reaction 25–26. Using, however, the smallermeasurement set, the problem converged to the samesolution as earlier. The 90% confidence intervals of theexchange fluxes were minimally affected from the elimin-ation. The total error of the analysis, however, decreased to58.84, relatively to a v2-value for 55–8 ¼ 47 redundantmeasurements and 90% confidence of 59.8. This is anindication that the six mass isotopomer fractions are themain sources of error in the measurement set. Moreover, theflux estimates presented are now based on a consistentmeasurement set. This is a noteworthy result consideringthat prior flux analyses of entire networks from isotopicmeasurements have failed criteria of statistical consistency[20]. We believe that in the case of prior flux analyses basedon MS measurements, this was due primarily to the fact thateven the smaller least reliable mass isotopomer fractionswere considered in the flux analysis.

Discussion

In the work described, we examined the extent to whichmass isotopomer measurements of biomass hydrolysates,after the introduction of only one labeled substrate([1-13C]glucose), can elucidate the lysine biosynthesis fluxnetwork of C. glutamicum under glucose-limited growthin continuous culture (D ¼ 0.1Æh)1). Probing the massisotopomer distribution of biomass hydrolysates is benefi-cial, because it allows the measurement of the isotopic tracerdistribution of central carbon metabolism intermediatesthat are not excreted extracellularly and would have beenchallenging to be measured otherwise. Observability andbioreaction network analysis enabled the identification ofthe intracellular interactions that can indeed be determinedfrom the available measurements. Very few assumptionsabout the reaction reversibility were initially made. Only thereactions that participate in product synthesis pathways orare known to have high negative free energy under thepresent experimental conditions were considered irreversible(see also notes in Appendix). Any subsequent metabolitepool lumping was mandated by observability analysis onlyand initial assumptions were validated through dataredundancies. For example, isocitrate synthase, which inour network analysis was considered irreversible, has beenreported to be reversible in vivo [61]. However, structuralobservability analysis indicates that the experimental setupof Des Rosiers et al. [61] allowed the determination of thisexchange flux, while, in our case, such estimation was notpossible. When the isotopic tracer distribution of OAA andGlu are measurable, the exchange flux of isocitrate dehy-drogenase is observable only if proper information about

the isotopic tracer distribution of isocitrate is also available.We lacked any such information, while Des Rosiers et al.(1994) [61] obtained it indirectly from the isotopic tracerdistribution of AcCoA produced from isocitrate.

An important result of our analysis concerning C. glu-tamicum physiology was the determination of the exchangeflux of PEP carboxykinase and pyruvate synthase. Thesereactions are part of the structurally entangled set ofC. glutamicum anaplerotic reactions, which makes thedetermination of their flux a challenging task. Accordingto our measurements, PEP carboxykinase and pyruvatesynthase do not carry flux under the described experimentalconditions. This is in agreement with previous (in vitro)physiological studies that indicated no PEP carboxykinaseactivity under nongluconeogenic conditions [30], while, inmost studies, pyruvate synthase is assumed to be inactive inC. glutamicum [42]. This result contradicts, however, theflux estimates of Petersen et al. (2001) [42], who reported arelatively high in vivo PEP carboxykinase activity even whenglucose was the main carbon source. While it is always riskyto compare two different strains, it is speculated that thehigh PEP carboxykinase flux in [42] is the result of lactateuse ) even at small quantities ) as a second labeling source.Lactate might have triggered gluconeogenesis, hence thehigh PEP carboxykinase activity.

The very high futile cycle identified between PYR andOAA pools has double significance: (a) it indicates a highlyactive OAA decarboxylase enzyme. This validates for thefirst time previous physiological studies which identified ahigh in vitroOAA decarboxylase activity in glucose culturedcell-free extracts [31], when PEP carboxykinase was com-pletely inactive. In their analysis, Marx et al. (1996) [40,41]reported a futile cycle between OAA and the combinedPEP/PYR pool, while Petersen et al. (2001) [42] reported arelatively high activity of the PEP carboxykinase reactionand a lower activity of OAA decarboxylase in vivo. Theseresults support the presence of futile cycles among the OAApool and the pools of PEP or PYR. However, the very highin vivo activity of OAA decarboxylase observed in thepresent study has not been reported elsewhere in bacterialstudies. Only in liver cells, pyruvate recycling at rates up tothree times that of the TCA cycle has been reported [62,63],when a mixture of substrates is used; this is in agreementwith our results. Interestingly though, because OAA andMAL are considered as one pool based on the availableaspartate and glutamate measurements, OAA decarboxy-lase activity cannot be distinguished from malic enzymeactivity. Therefore, the high flux from OAA/MAL pool topyruvate could be due to malic enzyme. This enzyme,however, was deemed inactive, when glucose is used assubstrate [31]; (b) the flux of pyruvate carboxylase is at least13 times larger than this of PEP carboxylase. Assuming thatthe net fluxes of the combined (23–24) and (25–26) reactionsfollow the same direction as their normalized sum (i.e. 23.5),then the normalized flux of PEP carboxylase (reaction 23)can vary between 0 and 23.5, since PEP carboxykinase (24)is inactive. On the other hand, the exchange flux of thecombined (25–26) reaction is estimated to be at least 13times the total anaplerotic net flux (i.e. 23.5), implying thatthe flux of pyruvate carboxylase varies between 13 · 23.5and (43 · 23.5 + 23.5). This is a very important physiolo-gical result in agreement with previous studies that indicated


pyruvate carboxylase to be the main anaplerotic route ofC. glutamicum under nongluconeogenic conditions. Speci-fically, Park et al. [38] estimated the flux through pyruvatecarboxylase to be equal to 90% of the total anapleroticactivity. The analysis of Petersen et al. [42] showed also thepyruvate carboxylase activity to be higher than this of PEPcarboxylase. The present analysis connects the very highin vivo activity of pyruvate carboxylase to the function of thefutile cycle between OAA and pyruvate.

Comparing our analysis to that of Marx et al. [40,41],who used NMR measurements of biomass hydrolysatesand [1-13C] labeling to elucidate the flux network ofC. glutamicum grown only in glucose, MS measurementsproved superior in elucidating the anaplerotic flux distri-bution of this microorganism. The zero flux of PEPcarboxykinase and PEP synthase observed in our analysiswas based on the difference in the mass isotopomerdistribution of fragment [1-2] between Gly, Phe and Val. Inthe data of Marx et al. (1996) [40,41], the label enrichmentof the first carbon atoms of Ala and Phe were notdetectable (see also [42]), while the label enrichment of thesecond carbon atoms of these two metabolites and Glywere statistically identical, due to relatively large associatederrors. Additionally, the first two carbon atoms of PEP andPYR were indistinguishable, necessitating the lumping ofthe two pools and the two reactions, PEP and pyruvatecarboxylase, as well as PEP carboxykinase and OAAdecarboxylase, into one. Additionally, in MS, but not inNMR, the unlabeled fraction of a metabolite or fragmentcan be measured. This is advantageous, because, if the�labeled� mass isotopomer fractions are small and prone tomeasurement errors, the unlabeled fraction is larger andmore accurate, validating the less accurate �labeled� meas-urements (redundancy).

On the other hand, MS measurements proved inferior toNMR measurements in elucidating the relative activity ofthe two parallel pathways in lysine biosynthesis. Whilethis flux ratio is structurally observable from the massisotopomer distribution of lysine, fragments [1-4] and [2-4]of OAA and [1-3] and [2-3] of PYR [7], the obtainedmeasurements of these mass isotopomer fractions are notadequate to allow determination of the unknown flux ratio.Using these MS measurements, estimation of the flux splitratio is based only on the difference in the label enrichmentof the first lysine carbon atom depending on the pathwayfollowed. Structurally, the enrichment of the first lysinecarbon atom enables the discrimination between the twolysine biosynthesis pathways, because it originates from thefirst carbon atom either of OAA or PYR depending on thefollowed pathway. However, both these carbon atoms areexpected to be almost unlabeled when [1-13C] glucose isused, thus practically limiting the numerical discriminatorypower of this measurement. Previous extensive studies[35,36] showed that, when [1-13C] glucose is used, it isprimarily the difference in the label enrichment of carbons 3and 5 and to a lesser extent of 2 and 6 of lysine that enablesthe estimation of the flux ratio at H4D using NMRmeasurements.

Finally, the irreversibility between the PEP and the PYRand OAA pools renders the upper (up to PEP) and lowerpart of the network practically independent. Therefore, fluxanalysis (including observability and data reconciliation

analysis) of the upper and lower parts could be performedseparately. This means that flux unobservability or meas-urement inaccuracies in one part of the network should notaffect the flux estimates in the other part.

Acknowledgements

We acknowledge with gratitude support by the National Science

Foundation, Grant No. BES – 9985421 and the DuPont-MIT Alliance

program.

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Supplementary material

This material describes the exact definition of the weightedleast-squares constrained minimization problem, which wasused in this work to quantify the intracellular net and ex-changefluxesfromextracellularmetabolitenetexcretionrate-and mass isotopomer measurements and is available from:http://www.blackwellpublishing.com/products/journals/suppmat/EJB/EJB3732/ejb3732sm.htm

Appendix

The considered lysine biosynthesis reaction network inC. glutamicum, when glucose is used as substrate (for themain metabolic pathways, see [64]; for lysine biosynthesisand respiration pathways of C. glutamicum, see [31]).

PEP-Glc Phosphotransferase system

1. Glc + PEP fi G6P + PYR (irreversible)

Glycolysis

2. G6P « FRU6P (reversible)3. FRU6P + ATP fi FRU1,6bisP (irreversible)4. FRU1,6bisP « DHAP +GAP (reversible)5. DHAP « GAP (reversible)6. GAP + NAD+ « 1,3-BPG + NADH (reversible)7. 1,3-BPG + ADP « G3P +ATP (reversible)8. G3P « 2-PG (reversible)9. 2-PG « PEP +H2O (reversible)

10. PEP +ADP « ATP +PYR (reversible)

Pentose Phosphate Pathway (PPP)

Oxidative11. G6P + 2NADP+ + H2O fi

P5P + CO2 + 2NADPH (irreversible)Non-Oxidative12. 2 P5P « SED7P + GAP (reversible)13. SED7P + GAP « E4P + FRU6P (reversible)14. P5P + E4P « GAP + FRU6P (reversible)

Pyruvate Dehydrogenase Reaction

15. PYR + CoA + NAD+ fi AcCoA + CO2

+ NADH (irreversible)

Tricarboxylic Acid Cycle

16. AcCoA + OAA + H2O fi ISOCIT+ CoA (irreversible)

17. ISOCIT +NADP+ fi aKG + CO2

+ NADPH (irreversible)18. aKG + CoA + NAD+ fi SUCCoA + CO2

+ NADH (irreversible)19. SUCCoA + ADP fi SUC + CoA

+ ATP (irreversible)20. SUC + FAD + H2O « FUM

+ FADH2 (reversible)21. FUM + H2O « MAL (reversible)22. MAL + NAD+ « OAA + NADH (reversible)

Anaplerotic Pathways

23. PEP + CO2 fi OAA (irreversible)24. OAA + ATP fi PEP + CO2 + ADP (irreversible)25. PYR + CO2 + ATP fi OAA + ADP (irreversible)26. OAA fi PYR + CO2 (irreversible)

Trehalose synthesis

27. 2 G6P + ATP fi TREHALOSE+ ADP (irreversible)

Glutamate and Glutamine Synthesis

28. aKG + NH3 + NADPH « Glu +H2O + NADP+ (reversible)

29. Glu + NH3 + ATP fi Gln + ADP (irreversible)


Alanine, Valine and Lactate synthesis

30. PYR + NADH fi LAC + NAD+ (irreversible)31. PYR + Glu fi Ala + aKG (irreversible)32. 2 PYR + NADPH + Glu fi Val + CO2 +

H2O + NADP+ + aKG (irreversible)

Acetate Synthesis

33. AcCoA + ADP fi Ac + CoA + ATP (irreversible)

Aspartate Synthesis

34. OAA + Glu « Asp + aKG (reversible)

Lysine synthesis

35. Asp + ATP + NADPH « aspartic semialdehyde +ADP + NADP+ (reversible)

36. aspartic semialdehyde + PYR + NADPH fiH4D + NADP + 2H2O (irreversible)

37. H4D + H2O fi 2-amino-6-ketopimelate(irreversible)

Four-step pathway38. 2-amino-6-ketopimelate + SUCCoA fi N-succinyl-

2-amino-6-ketopimelate + CoA (irreversible)39. N-succinyl-2-amino-6-ketopimelate + Glu fi N-suc-

cinyl-2,6-diaminopimelate + aKG (irreversible)40. N-succinyl-2,6-diaminopimelate + H2O fi LL-2,6-

diaminopimelate + SUC (irreversible)41. LL-2,6-diaminopimelate fi meso-DAP

(irreversible)One-step pathway42. 2-amino-6-ketopimelate + NH3 + NADPH fi

meso-DAP + NADP+ + H2O (irreversible)

43. meso-DAP fi LysINTRA + CO2 (irreversible)

Lysine Transport

44. LysINTRA fi LysEXTRA (irreversible)

Biomass synthesis

45. 0.021 G6P + 0.007 FRU6P + 0.09 RIB5P + 0.036E4P + 0.013 GAP + 0.15 G3P + 0.052 PEP +0.03 PYR + 0.332 AcCoA + 0.08 Asp + 0.033LysINTRA + 0.446 Glu + 0.025 Gln + 0.054 Ala +0.04 Val + 3.82 ATP + 0.476 NADPH + 0.312NAD+ fi biomass + 3.82 ADP + 0.364 aKG +0.476 NADP+ + 0.312 NADH + 0.143 CO2

(irreversible)

Oxidative Phosphorylation: P/O = 2

46. 2 NADH + O2 + 4 ADP fi 2 H2O + 4 ATP+ 2NAD+

47. 2 FADH2 + O2 + 2 ADP fi 2 H2O+ 2 ATP + 2 FAD

Notes

1. For the reversible reactions, the default direction isconsidered from left to right in the depicted equations. If thenet fluxof these reactions is estimated tobenegative, then thenet direction of the reaction flux is the reverse of the default.The net fluxes of the irreversible reactions are nonnegative.

2. The ATP balance was not included in flux quantifi-cation.

3. Each direction of reaction (10) is catalyzed by adifferent enzyme: the default direction by pyruvate kinase;while the reverse direction by PEP synthase.

4. Ribulose 5-phosphate, ribose 5-phosphate and xylu-lose 5-phosphate are considered at equilibrium in one pool,named pentose 5-phosphate (P5P) [65]. This grouping isimposed by observability limitations. The exchange fluxes ofthe intermediate reactions can be differentiated only whenappropriate information about the isotopic distribution ofall three metabolites is available.

5. The irreversible citrate synthase and the (potentially)reversible aconitase reactions [64] have been combined intoone irreversible reaction. Even if they were consideredindependently, observability analysis indicates that theyhave the same net flux and the exchange flux of aconitasecannot be determined, since isotopic tracer measurements ofcis-aconitate (i.e. the reactant of the reaction) are notavailable. The pools of cis-aconitate and isocitrate arelumped then into one pool.

6. There have been indications that icocitrate dehydrog-enase is reversible in vivo [61]. However in the present case,as it is explained in the text (see Discussion), the extent of itsreversibility cannot be determined, since the isotopic tracerdistribution of isocitrate is not measured.

7. Reaction 19 is indeed reversible in vivo producing asymmetric molecule (SUC) in the forward direction, whilethere is an one-to-one correspondence between the carbonatoms of SUC and SUCCoA in the reverse direction (nosymmetric molecule is produced). Reaction 20 is alsoreversible producing a symmetric molecule (SUC) in thereverse direction (from FUM), while FUM is not symmetricand there is an one-to-one correspondence between thecarbon atoms of SUC and FUM in the default direction.However, to simplify the estimation of the exchange fluxbetween SUC and OAA in the TCA cycle, since succinyl-CoA synthetase reaction (no 19) is also coupled with theirreversible reactions (38, 39, 40) in lysine biosynthesispathway, SUCCoA synthetase is considered irreversiblewith one-to-one correspondence between the carbon atomsof SUCCoA and SUC. Additionally, SUC dehydro-genase (no 20) is considered reversible forming symmetricmolecules in both directions. The latter model is equivalentto the in vivo situation with respect to carbon transferstoichiometry. Therefore, it does not distort the results offlux analysis.

8. In flux quantification, reactions 23 and 24 areconsidered as the two directions of one reversible reaction.However, reaction 23 is catalyzed by enzyme PEP carboxy-lase, while reaction 24 is catalyzed by enzyme PEPcarboxykinase.


9. Reaction 25 is catalyzed by pyruvate carboxylase. Influx quantification, reactions 25 and 26 are considered theopposite directions of one reversible reaction. Reaction 26,however, is catalyzed by a different enzyme, OAA decarb-oxylase.

10. All pathways synthesizing extracellularly excretedproducts from central carbon metabolism intermediates areconsidered irreversible.

11. The two (potentially) reversible in vivo reactions:aspartate kinase and aspartic, b-semialdehyde dehydro-genase (see [31]) have been combined into one reversiblereaction. The exchange fluxes of the two reactions cannot bedistinguished, because the isotopic distribution of aspartylphosphate (the intermediate of the two reactions) is notaccessible.

12. The same biomass equation as in [31] was considered.


Systematic quantiﬁcation of complex metabolic ﬂux...

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