error propagation from prime variables into specific rates and metabolic fluxes for mammalian cells...

Error Propagation from Prime Variables into Specific Rates and Metabolic Fluxes

for Mammalian Cells in Perfusion Culture

Chetan T. GoudarCell Culture Development, Global Biologics Development, Bayer HealthCare, 800 DwightWay, Berkeley, CA 94710

Richard BienerUniversity of Applied Sciences Esslingen, Department of Natural Sciences, Esslingen, Germany

Konstantin B. KonstantinovGenzyme Corporation, 45 New York Ave, Framingham, MA 01701

James M. PiretMichael Smith Laboratories and Dept. of Chemical and Biological Engineering, University of British Columbia,Vancouver, BC V6T 1Z3, Canada

DOI 10.1021/bp.155Published online June 23, 2009 in Wiley InterScience (www.interscience.wiley.com).

Error propagation from prime variables into specific rates and metabolic fluxes was quan-tified for high-concentration CHO cell perfusion cultivation. Prime variable errors were firstdetermined from repeated measurements and ranged from 4.8 to 12.2%. Errors in nutrientuptake and metabolite/product formation rates for 5–15% error in prime variables rangedfrom 8–22%. The specific growth rate, however, was characterized by higher uncertainty as15% errors in the bioreactor and harvest cell concentration resulted in 37.8% error. Meta-bolic fluxes were estimated for 12 experimental conditions, each of 10 day duration, during120-day perfusion cultivation and were used to determine error propagation from specificrates into metabolic fluxes. Errors of the greater metabolic fluxes (those related to glycoly-sis, lactate production, TCA cycle and oxidative phosphorylation) were similar in magnitudeto those of the related greater specific rates (glucose, lactate, oxygen and CO2 rates) andwere insensitive to errors of the lesser specific rates (amino acid catabolism and biosynthesisrates). Errors of the lesser metabolic fluxes (those related to amino acid metabolism), how-ever, were extremely sensitive to errors of the greater specific rates to the extent that theywere no longer representative of cellular metabolism and were much less affected by errorsin the lesser specific rates. We show that the relationship between specific rate and meta-bolic flux error could be accurately described by normalized sensitivity coefficients, whichwere readily calculated once metabolic fluxes were estimated. Their ease of calculation,along with their ability to accurately describe the specific rate-metabolic flux error relation-ship, makes them a necessary component of metabolic flux analysis. VVC 2009 American Insti-tute of Chemical Engineers Biotechnol. Prog., 25: 986–998, 2009Keywords: CHO cells, error propagation, metabolic flux analysis, normalized sensitivitycoefficients, perfusion culture

Introduction

Metabolic fluxes are considered a fundamental determinantof cell physiology,1 and metabolic flux analysis has beenincreasingly used to characterize the physiology and metabo-lism of mammalian cell cultures.2–7 Flux data provide aquantitative description of cellular response to changingenvironmental conditions, such as those encountered duringbioprocess development, and are hence useful for bioprocessoptimization.

The first step in metabolic flux estimation is the construc-tion of a bioreaction network that describes the conversion

of substrates to metabolites and biomass. These bioreactionnetworks are typically simplified to enable flux estimationfrom available experimental data. For mammalian cells,these include the main reactions of central carbon and aminoacid metabolism.4,5,8 The unknown fluxes in the bioreactionnetwork are subsequently estimated either using metabolitebalancing2,3,5,7,9–13 or isotope tracer techniques.8,14–19 In themetabolite balancing approach, fluxes are estimated byapplying mass balances around the intracellular metabolitesusing the measured extracellular rates as input data. The ana-lytical and computational techniques associated with themetabolite balancing approach are relatively simple1 and canbe readily applied to most experimental systems. Thisapproach, however, cannot determine fluxes in cyclic andbidirectional reactions. Additional shortcomings and

Correspondence concerning this article should be addressed to C. T.Goudar at [email protected].

986 VVC 2009 American Institute of Chemical Engineers

approaches to overcome them have been discussed indetail.9,20 Despite these limitations, metabolite balancingremains the method of choice for a majority of process de-velopment experiments and for all pilot and manufacturing-scale studies given the expense of the isotope traceralternative.

Information on the error associated with metabolic fluxesobtained by the metabolite balancing approach is critical tomeaningfully interpret changes in cellular metabolism. Ascell specific rates including growth, nutrient consumptionand metabolite production comprise the input data for fluxestimation, flux values can be strongly influenced by specificrate errors. Cell specific rates, however, are not experimen-tally measured but are calculated from measured prime vari-ables such as cell, nutrient, metabolite, and productconcentrations. Information on prime variable error is thusnecessary to characterize their influence on specific rate errorand ultimately on flux values.

The need to have specific rate data with no gross measure-ment error has been long recognized and a framework hasbeen proposed to check for the presence of gross errors.13,21–24

However, error propagation from prime variables to metabolicfluxes has not been reported. This study is aimed at systemati-cally characterizing error propagation from prime variables tometabolic fluxes for mammalian cells in perfusion culture.Prime variable errors were first determined and their propaga-tion into specific rates and metabolic fluxes was quantifiedusing a combination of experimental data and Monte-Carloanalysis.

Materials and Methods

Cell line, medium, and cell culture system

Chinese hamster ovary (CHO) cells were cultivated in per-fusion mode25 with glucose and glutamine as the main carbonand energy sources in a proprietary medium formulation. Thebioreactor was inoculated at 0.92 � 106 cells/mL and cellswere accumulated until the bioreactor reached 20 � 106

cells/mL at which point the cell concentration was main-tained constant by controlling the bleed stream from the bio-reactor. Experiments were conducted in a 15 L bioreactor(Applikon, Foster City, CA) with a 10 L working volume.Under standard operating conditions, the temperature wasmaintained at 36.5�C and the agitation at 40 rpm. The dis-solved oxygen (DO) concentration was maintained at 50% airsaturation by sparging a mixture of oxygen and nitrogenthrough 0.5 lm spargers. The bioreactor pH was maintainedat 6.8 by the addition of 0.3 M NaOH which helped neutral-ize the lactic acid produced by the cells. Only at the begin-ning of perfusion cultivation was CO2 addition necessary tomaintain bioreactor pH. Temperature, DO and pH were var-ied during the course of the cultivation resulting in a total of12 experimental conditions, each of 10 day duration to iden-tify valid operating ranges for these variables. All prime vari-able data from the last 4 days of each experimental conditionwere considered representative (variation \15%) and usedfor specific rate and metabolic flux calculations.

Analytical methods

Samples from the bioreactor were analyzed daily for cellconcentration and viability using the Cedex system (Innova-tis, Bielefeld, Germany). The samples were subsequentlycentrifuged (Beckman Coulter, Fullerton, CA), and the su-

pernatant was analyzed for nutrient and metabolite concen-trations. Glucose, lactate, glutamine, and glutamateconcentrations were determined using a YSI Model 2700analyzer (Yellow Springs Instruments, Yellow Springs, OH)whereas ammonium was measured using an Ektachem DT60analyzer (Eastman Kodak, Rochester, NY). The pH and DOwere measured online using retractable electrodes (Mettler-Toledo Inc., Columbus, OH) and their measurement accuracywas verified through off-line analysis in a Stat Profile 9blood gas analyzer (Nova Biomedical, Waltham, MA). Thesame instrument also measured the dissolved CO2 concentra-tion. On-line measurements of cell concentration were madewith a retractable optical density probe (Aquasant Messtech-nik, Bubendorf, Switzerland) that was calibrated with cellconcentrations estimated using the Cedex system. Oxygenand carbon-dioxide concentrations in the exit gas were deter-mined using a MGA-1200 Mass Spectrometer (AppliedInstrument Technologies, Pomona, CA).

Prime variables and specific rates

Errors in prime variable (cell concentration, product, glu-cose, glutamine, lactate, ammonium, and oxygen) measure-ments were estimated by analyzing multiple bioreactorsamples with replicate numbers determined by power analy-sis. A significance level of 0.05 was assumed and the detect-able difference was set equal to the assumed experimentalerror. The sample size was determined at a power value of0.95.

Specific rate expressions were derived from mass balanceequations for all prime variables of interest (Table 1). Errorin specific rates calculated from these equations were deter-mined using the Gaussian approach26 retaining only the first-order term in the Taylor series expansion

Df x1; x2;…::xnð Þ ffi dfdx1

��Dx1 þ

dfdx2

��Dx2 þ …::

dfdxn

��Dxn

(1)

Table 1. Expressions for Growth Rate, Specific Productivity and

Specific Uptake/Production Rates of Key Nutrients and Metabolites

in a Perfusion System

Parameter Expression

Specific growth ratel0 ¼ Fd

Vþ Fh

V

� �XH

V

XBV

þ 1

XBV

dXBV

dt

Specific productivityqp ¼ 1

XBV

FmP

Vþ dP

dt

� �

Specific glucoseconsumption rate qG ¼ 1

XBV

FmðGm � GÞV

� dG

dt

� �

Specific glutamineconsumptionrate

qGln ¼ 1

XBV

FmðGlnm � GlnÞV

� dGln

dt� kGlnGln

� �

Specific lactateproduction rate qL ¼ 1

XBV

FmL

Vþ dL

dt

� �

Specific ammoniumproduction rate qA ¼ 1

XBV

FmA

Vþ dA

dtþ kGlnGln

� �

Specific oxygenuptake rate qO2

¼ 1

XBV

FgðO2in� O2out

ÞV

8>>:9>>;

Biotechnol. Prog., 2009, Vol. 25, No. 4 987

where Df(x1,x2,…xn) is the error in the function f, x1,x2,...xnare the true values of the prime variables and Dx1,Dx2,…Dxnthe measurement errors.

Recognizing the truncation associated limitation of theGaussian approach at high prime variable errors, a Monte-Carlo approach was also used for specific rate error estima-tion. Normally distributed noise with mean ¼ 0 and desiredstandard deviation was introduced in the prime variables andspecific rates were computed. As most specific rates werefunctions of multiple prime variables, errors in each primevariable were changed one at a time to calculate the corre-sponding specific rate errors. This allowed comprehensivespecific rate error characterization in a multidimensional gridover the desired range of prime variable errors. For eachprime variable error value, 10,000 normally distributed ran-dom error values were generated and 10,000 specific ratescalculated. Thus the specific rate data reported from Monte-Carlo analysis are an average of 10,000 estimates. This pro-cedure was repeated when all associated prime variableswere in error.

Metabolic fluxes

A biochemical network previously developed for CHOcells7 was used in this study. This includes the major reac-tions of central carbon metabolism along with reactions foramino acid metabolism and has been previously described indetail.5,7 The stoichiometric matrix for this reaction networkwas of full rank and had a low condition number (69) indi-cating that flux estimates were not overly sensitive to spe-cific rate variations. Metabolic fluxes were estimated usingweighted least squares1

x ¼ ATw�1A� ��1

ATw�1r (2)

where x is the flux vector, A the stoichiometric matrix, r therate vector, and w the variance-covariance matrix of r. Thebioreaction network was characterized by two degrees offreedom and the two redundant measurements were used totest the consistency of experimental data and the assumedbiochemistry. The consistency index, h, was calculated foreach of the 12 experimental conditions according to methodspreviously described13,21,24 and compared with v2 ¼ 5.99(95% confidence level for 2 degrees of freedom).

The sensitivity matrix, S, of the metabolic network wasdetermined as1

S ¼ ATA� ��1

AT (3)

with the individual elements of S defined as

si;j ¼@xi@rj

8>>:9>>; (4)

where si,j is the sensitivity of the ith flux with respect to thejth rate. The normalized sensitivity coefficient matrix wasdetermined by multiplying the right hand side of Eq. 4 witha ratio of the specific rate and metabolic flux

sNi;j ¼

@xi@rj

8>>:9>>; rj

xi(5)

where SNi;j is the normalized sensitivity coefficient (NSC) for

flux xi with respect to rate rj.

To characterize error propagation from specific rates intometabolic fluxes, an initial metabolic flux vector wasassumed and the corresponding specific rate vector wasdetermined as r ¼ Ax. Subsequently, error was introducedin r using normally distributed noise with zero mean andstandard deviation corresponding to the desired error level.Initially, error was separately added to each element in r

(10,000 points at each error magnitude) and the resultingflux vector was computed. The flux data were averaged andcompared with the error-free flux vector. The differencebetween these flux values was caused by the specific rateerror and helped quantify error propagation from the specificrates into the metabolic fluxes. For a more realistic represen-tation of experimental conditions, this procedure wasrepeated with all elements of the specific rate vector weresimultaneously in error.

Results and Discussion

Perfusion cultivation

DO, temperature and pH set points were varied during thecourse of the cultivation resulting in a total of 12 experimen-tal phases, each of 10 day duration. Time courses of viablecell concentration and cell viability are shown in Figure 1.While the target cell concentration throughout the cultivationwas 20 � 106 cells/mL, cell concentrations for T ¼ 30.5�C(condition H) and pH ¼ 6.6 (condition K) were substantiallylower because of reduced growth rates. Cell viability wasgreater than 90% throughout the experiment. Specific ratesincluding growth, nutrient consumption and metabolite/pro-duct formation were calculated using the Table 1 equationswith the variables defined in the nomenclature section.

The average specific glucose consumption and lactate pro-duction rates are shown in Figure 2. Changes to DO (conditionsB and D) had minimal effect on glucose consumption and lac-tate production while temperature (conditions F–H) and pHreduction (condition K) slowed down glucose metabolism. Glu-cose consumption and lactate production increased at highertemperature (condition J) and pH ¼ 7 (condition L).

Figure 1. Viable cell concentration (*) and viability (&) timeprofiles over the 12 conditions examined in thisstudy.

Under standard conditions, DO ¼ 50%, T ¼ 36.5�C, pH ¼ 6.8,and the target cell concentration was 20 � 106 cells/mL for allconditions.

988 Biotechnol. Prog., 2009, Vol. 25, No. 4

Metabolic fluxes were computed using the average specificrates as inputs from the later portion of each of the 12 exper-imental conditions and are shown in Figure 3 for experimen-tal phase E (standard bioreactor conditions). The fluxesthrough glycolysis, the TCA cycle and oxidative phosphoryl-ation were one to three orders of magnitude higher thanthose for amino acid biosynthesis and catabolism as weresome fluxes for biomass synthesis. Similar observations onrelative flux magnitudes were made for the 11 other experi-mental conditions (not shown). The actual flux values, how-ever, did change between different experimental phases,especially when temperature and pH were varied.

Prime variable error

Errors in prime variable measurements were determinedby analyzing multiple samples and the results are shown inTable 2. Glucose, lactate and glutamine measurements haderrors close to 5% of the measured value, among the lowest.The highest errors were at 12.2 and 10.4%, respectively, forammonium and oxygen. Errors in the bioreactor volume andthe harvest, cell discard and gas flow rates were assumed tobe 5% based on manufacturer specifications.

Specific rate error

Mass balances around the bioreactor and cell retentiondevice were used to obtain expressions for growth rate,

specific productivity and specific uptake/consumption ratesfor nutrients and metabolites (Table 1). As perfusion systemsare typically operated at constant cell concentration and per-fusion rates, the prime variables would ideally be time invar-iant. However, imperfect cell concentration control and shiftsin cellular metabolism require retention of the accumulationterms in the mass balance expressions as their contributionsto the specific rates can typically range from 5 to 15%.

Gaussian error estimation versus Monte Carlo analysis

Specific glucose consumption rate, qG, was used to com-pare the Gaussian and Monte-Carlo approaches for specificrate error estimation. The specific glucose consumption rateis a function of five prime variables, V,Fh,Gm,G, and XB

V (Ta-ble 1) and is thus affected by error in all of them. For sim-plicity, however, the bioreactor volume, V, the harvest flowrate, Fh, and the glucose concentration in the medium, Gm,were assumed to be error-free for this comparison. The errorin bioreactor glucose concentration was varied from 0 to10% whereas that in bioreactor viable cell concentration,was varied from 0 to 20%. For each pair of G and XB

V errors,the corresponding error in qG was calculated using both theGaussian and Monte-Carlo approaches and representativedata at 10% error in glucose concentration is shown in Fig-ure 4. For XB

V error \8%, both the Gaussian and Monte-Carlo approaches resulted in similar qG errors whereas theGaussian approach under predicted qG error at XB

V error[8% for all G errors (Error estimates from the Monte-Carlomethod are representative because no assumptions andapproximations are made). As XB

V errors of 8.9% (Table 2)and higher are commonly observed in practice, the Gaussianapproach as defined in Eq. 1 has limited utility.

This limitation of the Gaussian approach is due the lack ofhigher order terms in Eq. 1. Inclusion of the second-orderterm considerably increased the complexity of the Gaussianerror expression (not shown) with only a minor improvementin error prediction (Figure 4). For example, 10% error in Gand 20% error in XB

V resulted in 26.9% qG error by the MonteCarlo method whereaas the corresponding Gaussian errorestimates were 22.4 and 23.1%, respectively, using the first-and second-order terms. Although addition of third andhigher-order terms can further increase the accuracy of theGaussian approach, the resulting expressions are quite com-plex. The Monte-Carlo approach with its ability to accuratelyestimate error over any desired range without derivative com-putation is superior to the Gaussian approach and has beenused to obtain the data presented in subsequent sections.

Error in specific growth rate

The apparent specific growth rate, l, is a function of fiveprime variables (Table 1) and using values from ConditionE, the dominant contributor is the cell bleed stream followedby cell loss in the harvest

l ¼ 4:9

10þ 100

10

0:21

20

� �þ 1

20

0

1

� �(6)

0:60 ¼ 0:49 þ 0:11 þ 0:0 (7)

The bleed stream term makes up 82% of the growth ratewhile the remaining 18% is from the harvest stream term.The dXB

V has been set to zero to reflect an ideal steady-state

Figure 2. Average specific glucose consumption and lactateproduction rates (mean 6 standard deviation) forthe 12 experimental conditions in this study.

More information on conditions A–L is in Figure 1.


with perfect cell concentration control. It is common toobserve �10% variation in cell density that can be morebecause of sampling and instrument error than a true changein cell density. Including this variation in the above expres-sion will misrepresent contributions of the cell bleed andharvest streams to growth rate. It is the dXB

V/dt term, how-ever, that largely affects growth rate error as will be shownbelow.

Figure 5 shows the error in l as a function of errors in the5 prime variables that make up the specific growth rateexpression. The impacts of 0–10% error in V, Fd, and Fh areshown in Figure 5a (XB

V and XHV were assumed error free)

where the l errors increased monotonically with those in V,

Figure 3. Flux map for experimental condition E using the network of Follstad et al.5

Reaction numbers (1–33) and flux values (in parenthesis as pmol/cell-d) are also shown.

Table 2. Error in Prime Variable Measurements*

Prime Variable Error (%)

Bioreactor viable cell concentration (XBV) 8.9

Harvest viable cell concentration (XHV) 7.9

Product concentration (P) 8.0Glucose concentration (G) 4.9Glutamine concentration (Gln) 5.1Lactate concentration (L) 4.8Ammonoum Concentration (A) 12.2Oxygen concentration (O2) 10.4

*Errors in pump flow rates and bioreactor volume were conserva-tively assumed to be 5% based on manufacturer specifications.

Figure 4. Comparison of Gaussian and Monte-Carlo qG errorestimates at 10% glucose error and 0–20% XB

Verror.

Both the first and second order Gaussian qG errror estimateswere lower than the Monte-Carlo error at higher XB

V errors.


Fd, and Fh. Errors in V had the highest impact on l and theaverage l/V error ratio was 1.03 (standard deviation of 0.01)suggesting a one-to-one relationship. The l/Fd and l/Fh errorratios were 0.83 (SD ¼ 3.3 � 10�3) and 0.17 (SD ¼ 6.4 �10�4), respectively, indicating lower sensitivity of l to Fd,and Fh errors. This difference in error sensitivity is consist-ent with the relative prime variable contributions to the lvalue. The fermentor volume, V, is in both the terms thatcontribute to l in Eq. 6 resulting in the one-to-one error de-pendence. The discard rate is present only in the first termthat contributes 82% to the l value, consistent with the l/Fd

error ratio of 0.83. Errors in the harvest flow rate have theleast impact because Fh is present only in the second termwith a 18% contribution to l, consistent with the l/ Fh errorratio of 0.17.

Impacts on l errors from errors in bioreactor and harvestcell concentrations are shown in Figure 5b (V, Fd, and Fh

were assumed to be error free). Cell concentration estimatesare more prone to error as manual cell counting techniquescontinue to be widespread. Although this has been alleviatedwith the advent of reliable automated cell concentration esti-mators, viable cell concentration in the harvest stream, XH

V,is especially susceptible to experimental error as there arerelatively few cells. However, given the minor contributionof the harvest stream term to the growth rate, a 20% error inXH

V results only in 3.3% error in the corresponding l esti-mate (Figure 5b). Errors in XB

V however have a dramaticeffect on the error in l with the third term in Eq. 6 largelyresponsible for the strong influence of XB

V error on l. Thiswas primarily due to the error associated with derivativeestimation that is typically done by finite forward differencesusing XB

V values from two consecutive days. More accuratederivative estimation approaches should thus be used to min-imize the error in l.

Figure 5c shows the calculated error in l when all the fiveprime variables are in error, reflective of experimental condi-tions. For a 5% error in V, Fd, and Fh and a 10% error inXB

V and XHV (representative conditions in this study), the cor-

responding l error was 24.4%, emphasizing the need foraccurate cell concentration determination and subsequent de-rivative estimation.

Error in specific uptake and production rates

The Monte-Carlo approach was used to estimate error innutrient consumption and metabolite production rates from

the Table 1 expressions. With the exception of oxygen, thesespecific rates were functions of V, Fm, XB

V and the corre-sponding nutrient/metabolite concentration while the oxygenuptake rate expression had Fg in place of Fh. A 5% errorwas assumed for V, Fm and FO2 while XB

V and the nutrient/metabolite concentration were evaluated over a 2–20% error

Table 3. Consistency Index Values for the 12 Experimental

Conditions Examined in this Study

Experimental Condition Bioreactor Set Point h*

A Standard† 5.93B DO ¼ 20% 2.54C Standard 4.64D DO ¼ 100% 3.08E Standard 1.59F T ¼ 34.5�C 0.95G T ¼ 32.5�C 0.80H T ¼ 30.5�C 4.28I Standard 0.89J T ¼ 37.5�C 0.26K pH ¼ 6.6 4.84L pH ¼ 7.0 3.56

*No gross measurement errors were present because h \ 5.99 (v2

value at 95% confidence level for 2 degrees of freedom) for all experi-mental conditions.† DO ¼ 50%, T ¼ 36.5�C and pH ¼ 6.8.

Figure 5. Error in l as a function of error in the five associ-ated prime variables.

Panel (a) is for V, Fd, and Fh whereas panel (b) is for XHV and

XBV. Panel (c) is when all prime variables are simultaneously in

error (V, Fd,and Fh at 5%; XHV ¼ 5–20%; XB

V ¼ 0–20%). XBV

error legend for panel c: (l) 0%; (*) 5%; (n) 10%; (&)15%; (~) 20%.


range. For each combination of XBV and nutrient/metabolite

error, 10,000 specific rates were calculated and average errorvalues are shown in Figure 6. For 0% error in XB

V, error inall specific rates increased monotonically with error in thecorresponding prime variable. For instance, the qG error was7.3% at a 2% G error (V and Fm error ¼ 5%, XB

V error ¼0%) and this value increased to 19.5% at a 20% G error.Increases in the XB

V error caused an upward shift in the errorprofile while maintaining the monotonic dependence on thecorresponding prime variable error. There were slight differ-ences in the specific rate errors for their corresponding primevariables and this is due to differences in the specific rateexpressions (Table 1). Error profiles for qP were identical tothose for qL given the identical specific rate expressions.

Of all specific rates evaluated in this study (Table 1), thespecific growth rate was characterized by the highest errorwith 5% errors in V, Fd, and Fh and 10% errors in XH

V andXB

V resulting in a 24.4% error in l (Figure 5). For 5% mea-surement errors in glucose, lactate and glutamine concentra-tions, errors in their respective specific rates at a 10% XB

V

error were in the 12–14% range (Figure 6). The estimatederror in oxygen uptake rate at 10% XB

V and oxygen errorswas 16.1% (Figure 6). Overall, specific rate errors were�10% with 5% errors in prime variables and 20–25% with15% prime variable errors (Figure 6). Thus the specific rateerrors in a perfusion system can be expected to span a 10–25% range depending upon the accuracy of prime variablemeasurements.

Error in metabolic fluxes

Metabolic fluxes were computed for all 12 experimentalconditions and the consistency of the experimental data was

verified by calculating the consistency index (h) values (Table3) using methods described earlier.5,24 The h values for all ex-perimental conditions passed the v2 distribution test with a95% confidence level (h \ 5.99 for 2 degrees of freedom)indicating that the experimental data for all conditions wereconsistent and unlikely to contain gross measurement errors.This observation coupled with the stoichiometirc matrix beingof full rank and having a low condition number (69) clearlyattests to the robustness of the bioreaction network and thequality of the experimental data. Experimental condition E,where the bioreactor was operated under standard conditions(Table 3) was arbitrarily chosen to quantify the effect of spe-cific rate errors on those in the metabolic fluxes.

Lesser metabolic fluxes

The effect of specific rate errors on the lesser metabolicfluxes (amino acid metabolism) is shown in Figure 7. Panelsa–d are for relatively greater specific rates (glycolysis, oxida-tive phosphorylation, and biomass production) whereas e–hare for amino acid metabolism (lesser specific rates). Despiteall 4 metabolic fluxes in Figure 7 being associated with theTCA cycle (Figure 3: threonine, valine and isoleucine arecatabolized to SuCoA, asparagine is produced from oxaloac-etate), they were greatly affected by the glucose uptake rateerror. A 15% error in glucose uptake rate resulted in 36, 11,62, and 33% errors, respectively (Figure 7a–d). The lactateproduction rate had a similar effect resulting in errors of 21,6, 35, and 19%, respectively, (Figure 7a–d) for a 15% lactateproduction rate error. As expected, the Figure 7a–d fluxeswere affected by errors in the oxygen uptake and carbondioxide production rates given their close relation to theTCA cycle (threonine catabolism was less affected because

Figure 6. Errors in qG, qL, qGln, and qO2as functions of error in XB

V and the corresponding prime variable.

XBV error legend: (l) 0%; (*) 5%; (n) 10%; (&) 15%; (~) 20%.


this reaction does not directly involve O2 or CO2). A 15%error in the oxygen uptake rate resulted in respective errorsof 15, 33, 222, and 137% whereas that in the CO2 produc-tion rate caused 6, 29, 208, and 174% errors, respectively, inthe Figure 7a–d fluxes. Thus errors in the greater specificrates very substantially influence the lesser metabolic fluxesto the extent that the values are far from accurately repre-senting cellular metabolism.

With the exception of the Ile ! SucCoA flux, errors inamino acid metabolic rates did not significantly affect themetabolic fluxes. Overall, the maximum flux error was lessthan 2.5% even when the specific rate error was 25% (Figure7e,f,h). As expected, the Ile ! SucCoA flux was influencedby errors in isoleucine catabolism with a 25% error resulting

in a 20% error in the flux (Figure 7g) and this dependencewas true in all instances where the specific rate and fluxwere closely related.

Greater metabolic fluxes

The effect of specific rate errors in the 5–25% range onthe greater metabolic flux errors was examined for experi-mental condition E and data for four fluxes representing gly-colysis, lactate production, the TCA cycle and oxidativephosphorylation are shown in Figure 8. While the influenceof all 35 specific rates in the bioreaction network wereexamined, Figure 8 shows representative results for the fivegreater specific rates (glucose, lactate, oxygen, carbon

Figure 7. Effect of specific rate error on the error in lesser metabolic fluxes (amino acid metabolism).

Panels (a–d) are for errors in the five greater specific rates whereas (e–h) are for errors in lesser specific rates (amino acid metabolism).


dioxide, Bio_NADH; panels a–d) and five lesser specificrates representing amino acid metabolism (serine, glycine,lysine, isoleucine, aspartate; panels e–h). As expected, spe-cific rates that were not closely related to the flux had alower impact on the flux error. For instance, a 15% error inglucose uptake rate caused 0.4, 0.9, and 1.6% errors, respec-tively, in Figures 8b–d whereas the error in Figure 8a was14.1%. Similarly, a 15% error in lactate production ratecaused errors of 0.6, 0.5, and 1.0%, respectively, in Figure8a,c,d whereas that in Figure 8b was 14.8%.

Thus, errors in the greater specific rates had a substantialeffect on the errors in the most closely related fluxes. Forinstance, a 15% error in glucose uptake resulted in a 14%error in the Glc ! GCP flux (Figure 8a) and a similar de-

pendence was seen between the error in the lactate produc-tion rate and the Pyr ! Lac flux (Figure 8b). The error inthe TCA cycle flux, aKG ! SuCoA, was most influencedby error in CO2 production and oxygen uptake (Figure 8c)whereas that for oxidative phosphorylation was primarilyaffected by error in the oxygen uptake rate (Figure 8d).

Errors in the amino acid metabolic rates, however, hadminimal impact on the flux errors even when they wererelated to the flux. For instance, the specific production ratesof serine and glycine (both synthesized from GAP (Figure 3)had a negligible impact on the glycolytic fluxes. A 25%error in serine or glycine production rates resulted in 2.34 �10�4 and 7.02 � 10�4% error, respectively, in the Glc !GCP flux (Figure 8e). Although lysine and isoleucine are

Figure 8. Effect of specific rate error on the error in four greater metabolic fluxes.

Panels (a–d) are for errors in five greater specific rates (glycolysis, oxidative phosphorylation and biomass production) whereas (e–h) are for errorsin lesser specific rates (amino acid metabolism).


catabolized to AcCoA which enters the TCA cycle, their rateerrors had little impact on the TCA cycle flux (aKG ! Suc-CoA). A 25% error in their catabolic rates resulted in respec-tive flux errors of 0.14 and 0.12% (Figure 8g). Aspartate isformed in the TCA cycle from oxaloacetate and a 25% errorin aspartate production rate caused a 3.67 � 10�2% error inthe aKG ! SucCoA flux. Thus, errors from lesser magni-tude specific rates have negligible impact on the error in thegreater metabolic fluxes even when the specific rates andmetabolic fluxes are related.

Overall flux errors in perfusion cultivation

Figures 7 and 8 show flux error data when only one spe-cific rate is in error. In a typical experiment, all specific rateshave error and their combined influences on the flux errorare shown in Figure 9. Specific rate errors in the 5–25%range were examined and when all specific rate errors were15%, the greater flux errors ranged from 12.3% for aKG !SuCoA to 14.7% for Pyr ! Lac (Figure 9a). For the lesserfluxes, when the specific rate errors were 15%, the fluxerrors were between 46.9% (Thr ! SuCoA) and 312.5% (Ile! SuCoA) (Figure 9b). Hence, lesser metabolic fluxes canbe extremely sensitive to specific rate errors making theiraccurate determination difficult even at relatively low primevariable and specific rate errors. This was despite using a ro-bust bioreaction network with a stoichiometric matrix of fullrank and low condition number.

Normalized sensitivity coefficients for analysis of metabolicflux errors

The flux error data in Figures 7 and 8 were obtained frommultiple simulations using the Monte-Carlo method.Although comprehensive, this approach is cumbersome toapply to new metabolic models and a generalized approachto quantify the relationship between specific rate and meta-bolic flux error is desirable. The sensitivity matrix, S, asdefined in Eq. 3 provides a framework for such quantifica-tion. For the metabolic network examined in this study, S isa 33 � 35 matrix where the jth column contains the sensitiv-ities of the 33 fluxes to the jth rate. Figure 10 shows abso-lute values of the minimum and maximum flux sensitivitiesfor each of the 35 specific rates. The minimum sensitivitiesranged from 9 � 10�4 to 2 � 10�3 whereas the maximumvalues were in the 0.32–1.50 range. Low sensitivity coeffi-cients are favorable from an error analysis standpoint as theinfluence of specific rate errors on flux estimates is minimal.Even the maximum sensitivities obtained were quite lowconsistent with the low condition number (69) of the stoichi-ometric matrix, A.

However, sensitivity coefficients as defined in Eq. 3 do

not completely explain the relationship between specific rate

and flux error. For instance, sensitivity coefficients for the

O2 ! 3ATP flux are �1.354 and �0.587 for oxygen uptake

and glucose uptake rates, respectively, a ratio of 2.3. Errors

in the O2 ! 3ATP flux, however, are scaled differently as

25% error in glucose and oxygen uptake rates result in flux

errors of 21.39 and 2.7%, respectively, a ratio of 7.9. This

discrepancy is due to the difference in the magnitudes of the

oxygen and glucose uptake rates (�5.14 and �1.48 pmol/

cell-d, respectively) which is not accounted for in Eq. 3. If

the sensitivity coefficients �1.35 and �0.59 are multiplied

by their respective specific rates of �5.14 and �1.48, the

resulting values are 6.94 and 0.87 with a ratio of 7.9 that is

consistent with the flux error ratio and the results of Monte-

Carlo analysis (Figure 8).

Figure 9. Error in greater (panel a) and lesser (panel b) meta-bolic fluxes when all specific rates in the bioreactionnetwork have errors in the 5–25% range.

The Thr ! SuCoA and Val ! SuCoA error profiles overlap inpanel b.

Figure 10. Absolute values of the maximum and minimumsensitivity coefficients for the metabolic model usedin this study.

For each of the 35 specific rates, there were 33 sensitivitycoefficients corresponding to the 33 fluxes (Figure 3) in thebioreaction network.


A normalization of the Eq. 3 sensitivity coefficients is

thus necessary for the resulting value to be representative of

the error relationship between the specific rate and metabolic

flux pair. This is described by Eq. 5 and a similar approach

is used to define the flux control coefficients in metabolic

control analysis that describe the change in steady-state flux

due to a change in enzyme activity.27 For the O2 ! 3ATP

flux, normalized sensitivity coefficients from Eq. 5 were

0.849 and 0.106 for oxygen uptake and glucose uptake,

respectively. The ratio of these normalized sensitivity coeffi-

cients is 8, similar to the flux error ratio of 7.9 from Monte-

Carlo analysis with a small difference due to round-off

errors. Normalized sensitivity coefficients as defined in Eq. 5

thus provide accurate quantification of the dependence of

metabolic flux error on specific rate error (this was verified

for other flux-specific rate combinations).

NSCs for the greater fluxes are shown for both greater and

lesser specific rates in Figure 11. For the Glc ! G6P flux,

the NSC with respect to glucose uptake rate from Eq. 5 was

0.923 indicating that a 1% error in glucose uptake rate

would result in a 0.923% error in the Glc ! G6P flux (Fig-

ure 11a). The flux to specific rate error ratio from Figure 8a

was 0.940 � 1.7 � 10�3 (average of 5 data points for the

glucose uptake rate) verifying the ability of the NSC to accu-

rately describe the specific rate and flux error relationship.

NSCs for lactate, oxygen, CO2 and biomass from Eq. 5 were

0.037, 0.027, 0.011, and 0.003, respectively (error ratios

from Figure 8b–d were identical), suggestive of their much

lower impact on the Glc ! G6P flux. The highest NSC for

the Pyr ! Lac flux was for lactate (0.99) while both oxygen

and CO2 were characterized by high NSCs for the aKG !SucCoA flux (0.416 and 0.691, respectively). The O2 !3ATP flux was most affected by errors in oxygen uptake rate

and this dependence was characterized by a normalized sen-

sitivity coefficient of 0.849 (Figure 11d). Normalized sensi-

tivity coefficients with respect to amino acid metabolism

were much smaller (Figure 11e–h) reflecting their minimal

impact on the greater flux errors.

Although both Figures 8 and 11 provide very similar in-formation on the specific rate and flux error relationship,Figure 11 data are easier to generate and are a more compactrepresentation of error dependence. The sensitivity matrixcan be readily estimated from the stoichiometric matrixusing Eq. 3 and once metabolic fluxes are calculated usingthe experimentally measured specific rates (Eq. 2), NSCs canbe determined from Eq. 5. Moreover, a single number com-pletely describes the specific rate—flux error relationship.

Variation in normalized sensitivity coefficients

It must be recognized that NSCs and hence specific rate-flux error relationships can change during the course of anexperiment if either the specific rate or metabolic fluxchange. This is not true of the conventional sensitivity coef-ficients (Eq. 3) that depend only upon the stoichiometry ofthe bioreaction network. Figure 12a shows variation in the

Figure 11. Normalized sensitivity coefficients for the greatermetabolic fluxes for both greater (panels a–d) andlesser (panels e–h) specific rates.

Figure 12. NSC variation with respect to glucose uptake rateduring the course of an experiment.

Data from this study are shown in panel a and those fromFollstad et al.5 in panel b.


normalized sensitivity coefficients for the O2 ! 2ATP andSuCoA ! Fum fluxes with respect to glucose uptake ratefor the 12 experimental conditions in this study. For bothfluxes, the lowest values of the normalized sensitivity coeffi-cients (0.218 and 0.139, respectively) were at T ¼ 30.5�C(condition H), where the flux values were the highest andthe rate values were among the lowest. The opposite wastrue at pH ¼ 7 (condition L) where flux values were thelowest and rate values were the highest (NSCs of 0.949 and0.481). Thus during the course of a single experiment, theNSC for the O2 ! 2ATP flux with respect to glucose uptakerate ranged from 0.218 to 0.949, a 4.4-fold variation whereasa 3.4 fold increase was observed for the SuCoA ! Fumflux. The value of the O2 ! 2ATP flux was thus 4.4 timesmore affected by errors in glucose uptake rate at pH ¼ 7than at T ¼ 30.5�C whereas the SuCoA ! Fum flux was 3.4times more affected.

Equation 5 was also used to calculate NSCs for hybridomacell cultivation in chemostat culture reported by Follstad etal5 at different dilution rates (Figure 12b). The dilution ratescorresponding to experimental conditions A–E were 0.04,0.03, 0.02, 0.01, and 0.04 h�1, respectively, and significantchanges in cellular metabolism were observed over thecourse of the experiment. The sensitivity coefficient for thePyr ! AcCoA flux was 0.67 and the normalized sensitivitycoefficient for experimental condition A was 6.57 reflectingthe 10-fold higher value of glucose uptake when comparedwith this flux. For experimental conditions B–D, Pyr !AcCoA flux increased while the glucose uptake ratedecreased resulting in significant reduction in the NSC. Anincrease in the glucose uptake rate for experimental condi-tion E was responsible for the slight increase in the NSC.Thus Pyr ! AcCoA flux was most sensitive to glucoseuptake rate errors in experimental condition A and thisdecreased by 6.5-fold for experimental condition D. Varia-tions in the SuCoA ! Fum flux were primarily due tochanges in the glucose uptake rate because the SuCoA !Fum flux did not change much over the course of the culti-vation while those for the Gln ! Glu flux were due tochanges in both the flux and glucose uptake rate. Thus theflux and specific rate error relationship can change duringthe course of an experiment and NSCs under all experimen-tal conditions must be calculated to rationally interpret themetabolic flux data.

Conclusions

We have characterized error propagation from prime vari-ables into specific rates and subsequently into metabolicfluxes for mammalian cells in high cell concentration perfu-sion culture. Prime variable errors were in the 5–12% rangeresulting in a 8–22% error in specific rates. The effect ofspecific rate error on the flux error was a function of boththe sensitivity of the flux with respect to the specific rateand relative magnitudes of the flux and the specific rate. Thegreater fluxes in the bioreaction network had errors that werecomparable in magnitude to the related greater specific rateerrors and were virtually unaffected by errors in the lesserspecific rates. Greater flux errors ranged from 12–15% for15% error in the greater specific rates. The lesser fluxes,however, were extremely sensitive to errors in the greaterspecific rates making their accurate estimation difficult givenanalytical limitations in prime variable measurements. Often,errors were so large that the flux values grossly misrepre-

sented cellular metabolism. The relationship between specificrate and flux error was accurately described by the normal-ized sensitivity coefficient that could be readily calculatedonce the metabolic fluxes were estimated. We recommendnormalized sensitivity coefficient calculation be an integralpart of metabolic flux analysis as it describes the relationshipbetween flux and specific rate error through a single numericvalue.

Notation

A ¼ ammonium concentration (mM)Fa ¼ discard rate (L/d)Fh ¼ harvest flow rate (L/d)Fg ¼ gas flow rate (L/d)G ¼ bioreactor glucose concentration (mM)

Gm ¼ medium glucose concentration (mM)Gln ¼ bioreactor glutamine concentration (mM)

Glnm ¼ medium glutamine concentration (mM)kGln ¼ first-order rate constant for glutamine degradation (1/

day)L ¼ bioreactor lactate concentration (mM)

O2in¼ oxygen concentration in the inlet stream (mM)

O2out¼ oxygen concentration in the outlet stream (mM)

P ¼ bioreactor product concentration (mg/L)qA ¼ specific ammonium production rate (pmol/cell-d)qG ¼ specific glucose consumption rate (pmol/cell-d)

qGln ¼ specific glutamine consumption rate (pmol/cell-d)qL ¼ specific lactate consumption rate (pmol/cell-d)qO2

¼ specific oxygen uptake rate (pmol/cell-d)qP ¼ specific productivity (pmol/cell-d)V ¼ bioreactor volume (L)

XBV ¼ bioreactor viable cell density (�106 cells/mL)

XHV ¼ harvest viable cell density (�106 cells/mL)t ¼ time (d)l ¼ specific growth rate (1/d)

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Manuscript received July 29, 2008, and revision received July 29, 2008.


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