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Research ArticleReceived: 10 June 2008 Revised: 18 October 2008 Accepted: 28 October 2008 Published online in Wiley Interscience: 30 December 2008

(www.interscience.wiley.com) DOI 10.1002/jctb.2097

A model for L-methionine productiondescribing oxygen–productivity relationshipAmalendu P. Ranjan,† Rajib Nayak† and James Gomes∗

Abstract

BACKGROUND: The state-time profile of cell mass, substrate and methionine concentrations of a methionine synthesis processshows strongly nonlinear features. A mathematical representation of this process was developed that conformed to systemsanalysis required for monitoring and controlling methionine production. The specific growth rate was defined by an exponentialterm to describe the lag phase in growth, extended before the onset of methionine production and substrate inhibition observedfor this process. A switching function was used to describe the relation between methionine synthesis and dissolved oxygenconcentration. In addition, the product formation kinetics of this model described the reutilization of methionine feedbackregulation whenever the residual substrate concentration dropped below a critical value.

RESULTS: The parameters for the model were determined from experimental data using a nonlinear regression technique. Acomplete nonlinear systems analysis of the model proved that using this model, the system was controllable and observable.The model prediction of methionine production in controlled and uncontrolled environments was satisfactory. Six statisticalmeasures were employed to validate model prediction and its adequacy was shown through simulations.

CONCLUSIONS: The proposed model for methionine production possesses the correct system architecture for application inprocess control. It predicts satisfactorily the relationship between methionine synthesis and dissolved oxygen, and time profilesof state variables.c© 2008 Society of Chemical Industry

Keywords: methionine production; Corynebacterium lilium; metabolic switch; parameter sensitivity; statistical analysis; nonlinear systemsanalysis; observability; controllability

NOTATIONA Cross-sectional area of the reactor (m2)C Nonlinear controllability matrixcL Dissolved oxygen concentration (g L−1)c∗

L Saturation value of dissolved oxygen concentration (gL−1)

d Willmott’s index of agreementD Dilution rate (h−1)Di Diameter of impeller (m)Ei Experimental dataE′ Average deviation of experimental dataf, g Vector fieldFA Air flow rate (L h−1)Fs Glucose feed rate (L h−1)h Vector output functionsKE Exponential equivalent of the Monod constant (g L−1)KI Glucose inhibition constant (g L−1)Km Monod constant (g L−1)kLa Mass transfer coefficient (h−1)Lr

f h rth-order Lie derivative of function h with respect to fm Total number of inputsms Maintenance coefficient (g g−1 h−1)n Power of Hill equation (constant)O Nonlinear observability matrixp Product concentration (g L−1)Pg Gassed power (W)Pi Model predicted data

P′ Average deviation of predicted datar Rate of consumption or production (g L−1 h−1)R2 Correlation coefficients Glucose concentration (g L−1)sc Critical glucose concentration(g L−1)sF Glucose concentration in the feed (g L−1)t Time (h)u Vector of manipulated inputs, [D, kLa]V Volume of reactor (L)X Cell mass concentration (g L−1)x Vector of state variable, [x, s, p, cL]y Vector of measured outputs, [s, cL]Yp/o Product yield coefficient based on oxygen (g g−1)Yp/s Product yield coefficient based on glucose (g g−1)Yx/o Cell mass yield coefficient based on oxygen (g g−1)Yx/s Cell mass yield coefficient based on glucose (g g−1)Greek letters

∗ Correspondence to: James Gomes, Department of Biochemical Engineering andBiotechnology, Indian Institute of Technology, Hauz Khas, New Delhi – 110016,India. E-mail: gomes@dbeb.iitd.ac.in

† The authors, Amalendu P. Ranjan and Rajib Nayak have equal contribution.

Department of Biochemical Engineering and Biotechnology, Indian Institute ofTechnology, Hauz Khas, New Delhi – 110016, India

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α Non-growth associated product synthesis coefficient(g−1L h−1)

β Growth associated product synthesis coefficient (g g−1)χ Constant in Hill equationδ1 Growth associated formation rate for all products and

by-products (g g−1)δ2 Non-growth associated formation rate for all products and

by-products (h−1)φ (1/Yx/o + δ1/Yp/o) (g g−1)γ 1/Yx/s + δ1/Yp/s (g g−1)η (δ2/Yp/s + ms) (g g−1 h−1)µ Specific growth rate (h−1)µm Maximum specific growth rate (h−1)ψ

(δ2/Yp/o

)(g g−1 h−1)

INTRODUCTIONMethionine is an essential amino acid. In humans, dietarydeficiency of methionine may interrupt normal growth and causeseveral ailments.1,2 Congenital methionine metabolism defectscause serious illnesses such as homocystinuria.3,4 L-methionine isexclusively used in the preparation of infusion fluids, electrolytebalancing, parenteral nutrition and pharmaceuticals adjuvant.Methionine is now ranked within the top 800 drugs in humanmedicine.1,2,5 – 7 In its D,L form, methionine is used extensively as asupplement in the animal feed industry. The annual requirementof methionine worldwide is 350 000 tons. Most of the industrialdemand for methionine is met by D,L-methionine producedchemically by Strecker’s synthesis. However, this chemical routeuses hazardous raw material such as acrolein. Since humansare allergic to D,L-methionine, there is a niche market forL-methionine produced by submerged fermentation processes.Submerged fermentation processes have their own limitations.The first requirement is developing a high producing strain. Awild type strain normally does not synthesize excess intra-cellularmethionine because the metabolic pathway for its synthesis isstrictly feedback regulated.8,9 A mutant strain is required tooverproduce methionine to eliminate feedback inhibition andrepression and dissipative carbon flux into branch pathways.10,11

Once this strain is available, the process needs to be optimizedand process operation in terms of scheduling and process controlneeds to be clearly defined.

Process modeling is an important step in the development ofa successful bioreactor operation. A model that can accuratelydescribe the dynamic relationship between state variables may beused to simulate and understand process behaviour under unusualconditions and enable the development of advanced controlstrategies. The level of detail required in a model depends onthe problem that needs to be addressed. For most process controlwork the unstructured mechanistic model is still preferred becauseit is simple, practical and adequate. However, the question whetherthe attributes of the unstructured model fulfill the requirementsof accuracy and system properties are often overlooked.

Mechanistic models for bioprocesses, in general, are meant todescribe the macroscopic observations of nutrients, cell mass andproduct. These models usually have four or five state variables andtheir time variation is represented by coupled-nonlinear ordinarydifferential equations. The time profiles exhibited by these statevariables are a result of cellular metabolism that encompassesthousands of complex reactions occurring at different time scales.Consequently, these profiles show strongly nonlinear features.This is particularly true for methionine production. At the cellular

level, the metabolic pathway for methionine synthesis is strictlyregulated genetically and enzymatically. There are two branchesin the pathway, the first leading to synthesis of lysine and thesecond to threonine and iso-leucine, which are the three otheraspartate family amino acids. Since methionine is the universalmethyl donor and universal N-terminal amino acid in proteinsynthesis, it is utilized within the cell in hundreds of other reactions.Consequently, in the case of methionine production by microbialsubmerged fermentation, several characteristics are evident fromthe time profiles of state measurements.

The main characteristics are extended delay period before theonset of methionine synthesis, existence of a maximum in themethionine concentration profile, a critical residual substrateconcentration below which methionine is re-utilized and amaximum specific growth rate and methionine production rate at40% dissolved oxygen (DO) concentration.8 – 13 In this paper, thedevelopment of an unstructured model for methionine productionby a mutant strain of Corynebacterium lilium (M 128) is presented.Since the growth and production of methionine are inter-dependent, the period of lag in growth and extended delay periodbefore the onset of methionine production is described by usingan exponential kinetic term for growth rate. The dependence of thecell growth rate and methionine production rate on DO is describedapplying a metabolic switching function to the exponential term.Five key parameters of the model are identified by nonlinearregression using the Levenberg–Marquardt algorithm while theothers were calculated from experimental data. The model fitwas validated by parametric sensitivity and statistical analyses.An analysis of nonlinear controllability and observability of themodel is carried out to demonstrate that the model may be usedeffectively for developing control strategies.

FEATURES OF STATE-TIME PROFILESExperimental data is taken from the published work of Gomeset al.10,13 for modeling the process. A series of continuously stirredtank reactor (CSTR) experiments were carried out to determinethe relationship between DO and D on methionine and cellmass productivity. Results from some of these CSTR experimentsare presented in Fig. 1. From the collective results of all CSTRexperiments it was established that methionine productivity washighest when the DO was 40% of the saturation value and Dwas 0.16 h−1.13 Subsequently, batch reactor experiments wereperformed. The average of five batch reactor experiments ofmethionine production by a mutant strain of Corynebacteriumlilium (M 128) is shown in Fig 2. In all these experiments, DO wascontrolled at 40% of the saturation value. An examination of thetime profiles clearly shows a lag phase of about 2 h in cell mass andan extended delay period of 10 h before the onset of methionineproduction. Methionine is produced at a maximum rate between30 and 45 h. At 48 h the methionine time profile passes througha maximum after which it begins to decrease. The low substrateconcentration at this point appears to trigger the re-utilizationof methionine to conserve energy. The data also shows thatmethionine production begins during the mid to late exponentialphase of cell growth and continues even during the stationaryphase. The rate of methionine production increases gradually andreaches a constant value during the stationary phase. During theinitial stage of methionine production, the increase depends bothon the rate of cell growth and the cell mass concentration, whileduring the later phase it depends on the cell mass concentrationsince cells reach a stationary phase and the energy metabolism

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0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320

Time (h)

Met

hio

nin

e C

on

cen

trat

ion

(m

g L

-1)

0

5

10

15

20

25

Res

idu

al G

luco

se a

nd

Cel

lmas

sC

on

cen

trat

ion

(g L

-1)

DO = 30%

D = 0.17

DO = 10%D = 0.25

DO = 50%

D = 0.25

DO = 30%D = 0.17

DO = 58.2D = 0.17

DO = 30D = 0.17

Figure 1. Data for CSTR runs carried out at different dilution rate and dissolved oxygen concentration (� Methionine, � Residual glucose and ♦ Cell massconcentration)13.

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Time (h)

Res

idu

al G

luco

se a

nd

Cel

lm

ass

Co

nce

ntr

atio

n (g

L-1

)

0

0.5

1

1.5

2

2.5

Met

hio

nin

e C

on

cen

trat

ion

(g L

-1)

Figure 2. The average time profiles of residual glucose, cell mass andmethionine concentrations in a batch reactor (� Methionine, � Residualglucose and ♦ Cell mass concentration); each data point is an average offive experiments10.

is directed towards maintenance and methionine production.Hence, methionine production is partly growth associated andpartly non-growth associated.

Extensive kinetic studies and metabolic control analysisperformed by Shimizu’s group14 – 16 on lysine show that the ex-ternal lysine time profile exhibits a similar delay and trend. Anexamination of the data reported by other researchers on valine,17

glutamic acid18 and lysine19,20 also shows a delay of about 12 hbefore the onset of amino acid production. Ensari and Lim20 hadalso studied the effect of the dilution rate D on lysine production.Their reported data shows that lysine production goes through amaximum near a dilution rate of 0.18 h−1.

For methionine production, flux analysis was carried out to studythe energy requirements, theoretical maximum for methionineproduction and the optimal RQ for the process.9 These flux analyseswere performed at various instances of time using experimentaldata. From the time snapshots of the fluxes, it was possible toframe a rough idea about the influence of oxygen on methionineproduction. The results obtained from this analysis gives sufficientevidence to indicate that an intermediate DO level of 30–50%would be suitable for most amino acid production processes.

DEVELOPMENT OF KINETIC MODELSpecific growth rate – exponential kinetic structureMonod kinetics was developed to describe exponential growthobserved in microbial cultures.21,22 However there exist manyvariations of the Monod kinetics that describe extended lag phasesoften observed in submerged fermentation data, and particularlyin secondary metabolite production. Gomes and Menawat23 usedan exponential variation of the Monod equation to describe thecell mass growth in Spectinomycin production. It was shown thatthe Monod kinetics is in fact a special case of the exponentialkinetics and can be derived from it by making some assumptions.The exponential kinetic term adopted for describing the growthkinetics of the methionine process can be written as

µ(s) = µm exp(−KE/s) exp(−s/KI) (1)

where KE is the exponential equivalent of the Monod constantand KI is the substrate inhibition constant. The maximum specificgrowth rate is µm and µ is the specific growth rate at any instant oftime. The exponential structure has also appeared in the descrip-tion of amino acid production by other researchers. Yao et al.19

expressed the parameters of their model in terms of an exponen-tial dependence on the dissolved oxygen tension. Ratkov et al.17

defined the dependence of specific growth rate as an exponentialof a polynomial, where the polynomial is a quadratic function ofsubstrate and dissolved oxygen concentrations. The exponentialstructure proposed here enables some degree of flexibility anda quicker response to changing process dynamics. The logarithmof Equation (1) separates out the parameters KE and KI andthereby enables higher sensitivity during the estimation of theseparameters compared with the hyperbolic form of the Monod andHaldane-Monod structure.14 In addition, the derivative of the ex-ponential structure is zero at s = 0 and is a correct representationof the process in the absence of substrate in the culture medium;whereas for the Monod kinetics this value equals µm/Km.

Oxygen dependence – metabolic switching functionExperimental results suggest that DO concentration significantlyaffects growth and methionine productivity. The contour plotsobtained from experimental data showed that at around 40% DOthe productivity of methionine is highest,10,13 and is indicative of

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different metabolic pathways that are active at different levels ofoxygen. The flux analysis data shows that at DO levels lower than40%, certain other organic acids are produced.9 At DO levels higherthan 40% the TCA cycle becomes more active and energy is wastedin futile cycles. This metabolic switching is best represented bythe Hill function for biological systems. Since the cell growth andmethionine production are both affected by oxygen uptake, aswitching function based on the DO is defined and coupled withthe exponential kinetic term to describe the observed trends. Thefunction is given by

f (cL) = (cL)n

(χ )n + (cL)n (2)

where cL is concentration of DO in g L−1:

cL =( cL

0.008

)and χ =

(0.0032

0.008

)

After an examination of errors in model prediction from severalsimulation experiments, the value of n = 6 was selected for thefunction. Figure 3 shows the variation of f (cL) with respect to cL.Higher values of n would give sharper changes in the slope butnot describe the data well at these values. The change in the slopeof f (cL), passes through a maximum at 40% DO as defined by χ ,the constant term in the function. When this function is coupledwith the exponential kinetics used to describe the specific growthrate, it affects the cell growth and methionine productivity in afashion similar to the behavior observed in the CSTR experiments.The final representation of the specific growth rate is given below.

µ(s, cL) =(

µmc6

L

χ6 + c6L

)exp

(− KE

s− s

KI

)

= µm exp

(− KE

s− s

KI

)(3)

Cooperative processes are commonly observed in biologicalprocesses and a cascade of these processes results in a sigmoidalcurve that can be described by the Hill function (Equation (2)).The Hill coefficient n represents cooperativity; n > 1 indicatespositive cooperativity and n < 1 indicates negative cooperativity.The integer value gives an indication of the number of bindingsites. When it represents the number of ligand molecules requiredfor activation or deactivation of any given site, values of n > 4

0

0.2

0.4

0.6

0.8

1

0 0.0016 0.0032 0.0048 0.0064 0.008

Dissolved Oxygen Concentration (g L-1)

Var

iati

on

in f

(cL)

n = 2n = 4n = 6n = 8

Figure 3. Variation in f (cL) with changes in the DO concentration fordifferent values of Hill coefficient.

may not be biologically realistic. However, when there is a cascadeof enzyme reactions, n can assume larger values. For example,n = 12 has been reported for analyzing the dynamics of a G0 cellcycle model of pluripotential stem cells to understand the originof the long period oscillations of blood cell levels observed inperiodic chronic myelogenous leukemia.24 Other reported valuesof the Hill coefficient are 4.6,5.9,25 10, 25,26 5,27 and ≥4.28

For the model proposed here, the Hill function was employed todescribe the effect of DO concentration on growth and methionineproduction. Experimental results showed that the production ofmethionine sharply increased from 25% DO to 50% DO and had amaximum at around 40% DO13. Consequently, fixing the inflexionpoint at 40% DO, the value of n = 6 was determined using aniterative simulation-based procedure.

Methionine re-utilization – critical substrate concentrationMethionine production is partly growth and partly non-growthassociated as observed in most amino acid synthesis processes.Amino acid and organic acid synthesis by microorganisms is usuallydescribed by the Leudeking–Piret model.29 However, in the case ofmethionine production (Fig. 2), the concentration passes througha maximum at 48 h. After this point in submerged cultivation, itis observed that methionine concentration decreases. This kindof phenomena is observed in other microorganisms producingprimary or secondary metabolite that has a non-growth associatedcomponent. This behavior is related to the low concentration of theresidual carbon source and is usually in the range 2–3 g L−1.23 Todescribe this behavior the Leudeking–Piret model is modified byusing the residual substrate concentration at the point where themethionine concentration begins to fall. The rate of methionineformation (rP) for the modified Leudeking–Piret model can bewritten as

rP =[α

(s − sc)

spx + βµ(s, cL)x

](4)

where p is methionine concentration, sc is the critical substrateconcentration, α is non-growth associated constant and β is

growth associated constant. The term(

s − scs

)is used to account

for the critical substrate concentration effect. The complete first

term{α

(s − sc)s px

}also shows a dependence on methionine

concentration which decreases as fermentation proceeds. This isin tune with observations that the rate of methionine productiondecreases gradually and that the amount reutilized depends onthe amount of methionine present in the fermentation medium.

Carbon source and oxygen utilization – uptake kineticsThe substrate, usually glucose, is assimilated for the cell masssynthesis, for metabolite production and for the maintenance ofthe microorganisms. The maintenance for mutant microorganismsis usually much higher than that for their wild-type parents.Therefore, it is essential to incorporate a maintenance term toexpress the proportion of the carbon sources that is diverted. Thesubstrate uptake kinetics is described by taking into account therelative proportions that are utilized for cell mass and methionineproduction, and for maintenance. Here we present the finalequation for the rate of substrate utilization (rS) as

rS = −[γµ(s, cl)x + ηx] (5)

where

γ =(

1

Yx/s+ δ1

Yp/s

)and η =

(δ2

Yp/s+ ms

)

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δ1 is growth associated formation rate for all products andby-products. δ2 is non-growth associated formation rate for allproducts and by-products. Yx/s and Yp/s are cell mass yield basedon substrate and product yield based on substrate respectively.

Oxygen is the terminal electron acceptor in aerobic processesand contributes to growth and methionine synthesis by fulfillingthe energy requirements. Its supply in the liquid medium is limitedby the solubility constraint. When air is bubbled at atmosphericpressure, an oxygen concentration of about 8 mg L−1 is themaximum limit. The dissolved oxygen concentrations thereforecan become limiting in many processes where the demand is high.For this process, the oxygen uptake kinetics is described simply by

rO2 = −[φµ(s, cL) + ψ ]x (6)

where φ = (1/Yx/o + δ1/Yp/o) and ψ = (δ2/Yp/o). Yx/o and Yp/o arecell mass yield coefficient and product yield coefficient based onoxygen respectively.

RESULTS AND DISCUSSIONSIdentification of model parametersCompiling the information presented from Equations (1)–(6),the model for methionine production in submerged fed batchcultivation may be written as

xspcL

=

µ(s, cL) x−γ µ(s, cL) x − η x

β µ(s, cL) x + α(s − sc)px/s−φ µ(s, cL)x − ψ x

+

−x 0(sF − s) 0

−p 0−cL (c∗

L − cL)

[

DkLa

](7)

The concentration of glucose in the feed stream is denoted by sF

and the saturation DO concentration is denoted by cL∗. There are

two inputs for this fed-batch process, the flow rate of the glucosefeed stream FS imbedded in the dilution rate D = FS/V , and airflowrate FA that is imbedded in the oxygen mass transfer coefficientkLa,30 given by

kLa = 2.6 × 10−2( pg

V

)0.4(

FA

A

)0.5

(8)

where pg is the power consumption of the aerated (gassed) reactor,V is the volume of the medium, A is the cross-sectional area of thereactor and FA is the air flow rate.

The parameter values Yx/s, Yp/s, Yx/o and Yp/o are directly calcu-lated from the experimental results.10 The remaining parametersof the model were estimated using nonlinear regression basedon the Levenberg–Marquardt algorithm and the values obtainedfor the parameters is given in Table 1. The batch fermentationdata (Fig. 2) suggests that when the value of residual substrateconcentration is approximately 3.25 g L−1 the production rate ofmethionine started to decrease and this is taken as the value forthe critical substrate concentration sc in the model.

Model evaluationA comprehensive statistical analysis was performed using experi-mental and predicted data to fit the model. Six different statistical

indices, namely, the correlation coefficient, slope of linear regres-sion, root mean square error, relative error, index of agreement,and ratio of means were employed. The definitions and otherrelevant details of these statistical indices are given in AppendixA. The results of this analysis are presented Table 2. The model fitbetween the predicted and experimental values are presented inFigs 4(a)–(c) and 5. The model prediction shows good correlationwith the actual experimental data. It is observed that the RMSEvalue 0.558 for substrate concentration is the highest becausethe magnitude of substrate concentration is the largest amongthe variables. Further, data points for substrate concentrationsbetween 20 g L−1 and 30 g L−1 that are displaced from the corre-lation line appear to contribute to this error. It was observed thatthe linear regression results of three state variables determined forthe variables x, s and p, had R2 values above 0.94 and regressionslope m nearly equal to 1. This represents good agreement be-tween experimental data and model prediction. The relative error(RE), which is defined as the normalized RMSE with respect to themean of the experimental data, and the Willmott Index of agree-ment (d) may be used in combination to validate the model.31 Thevalues of RE computed based on model prediction were 0.028,0.020 and 0.015 for the variables x, s and p, respectively; the valuesfor d for the same variables were obtained as 0.989, 0.984 and0.998. Literature suggests32 that when these indices are used incombination, d ≥ 0.95 and RE ≤ 0.10 indicates ‘very good’ predic-tion capability of a model. In addition the largest magnitude of theratio of means (Rm) for the three state variables is 0.158. Althoughthe negative values obtained for the three variables indicate thatthere is a small bias in the model prediction, the model will explainthe average trend reasonably well. Finally, standardized residualanalysis was performed to identify outlier data points. The plotsof the standardized residuals against their corresponding fittedvalues are presented in Fig. 6(a)–(c). Since 95% of the standardizedresidual values are in the range between −2 and +2 and 99% ofthe standardized residual values are in the range −3 to +3, it wasconcluded that none of the data points predicted using the modelwere outliers.33

Considering the results of this comprehensive statisticalvalidation, the overall prediction of state-time profiles (Fig. 5)obtained from the model for experimetnal data where the initialcell mass concentration was 1.25 g L−1 (compared with 1.14 g L−1

for the average initial condition), we may consider the modelto be satisfactory for process development and control systemdesign. The model proposed here describes the variations inthe external process variables. The models of Yao et al.19 andEnsari and Lim20 are unstructured and in that sense are similarto the model developed here. The model for lysine productionby Yang et al.15 was built at a more fundamental level becauseits purpose was different. It describes amino acid synthesis at theenzymatic level by including the kinetics of enzymes occurringin the metabolic pathway. Such a description permits a betterunderstanding of the contribution of each enzyme node tointracellular lysine synthesis and the corresponding influenceon the external lysine concentration. All these models servedwell the purpose for which they were developed. The currentlyproposed model was specially developed for methionine processdevelopment and control applications and fitted using actualexperimental data.

Parameter sensitivity analysisParameter sensitivity analyses for the five parameters µm, KE , KI,α and β were carried out by perturbing one parameter at a time

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Table 1. Operating variable and parameter values for methionine production model

Model parameter Description Units Value

x Cell mass concentration g L−1 1.14#

s Glucose concentration g L−1 50#

p Methionine concentration g L−1 0.0001#

A Cross-sectional area m2 0.02836

c∗L Saturation value of dissolved oxygen concentration g L−1 0.008

D Dilution rate h−1 calculated

FA Air flow rate L min−1 calculated

Fs Glucose feed rate g L−1 h−1 calculated

KE Exponential equivalent of the Monod constant g L−1 27.99

KI Glucose inhibition constant g L−1 400.26

kLa Mass transfer coefficient h−1 calculated

ms Maintenance Coefficient g g−1 h−1 0.0001

sc Critical glucose concentration g L−1 3.25

sF Glucose feed concentration g L−1 calculated

V Volume of reactor m3 calculated

Yp/o Product yield coefficient based on oxygen g g−1 3.26

Yp/s Product yield based on glucose g g−1 0.105

Yx/o Cell mass yield coefficient based on oxygen g g−1 2.33

Yx/s Cell mass yield based on glucose g g−1 0.38

α Non-growth associated product synthesis coefficient g−1L h−1 0.0139

β Growth associated product synthesis coefficient g g−1 0.0039

δ1 Growth associated formation rate for all products and by-products g g−1 1.49 × 10−4

δ2 Non-growth associated formation rate for all products and by-products. h−1 5.5 × 10−4

φ (1/Yx/o + δ1/Yp/o) g g−1 0.4292

γ 1/Yx/s + δ1/Yp/s g g−1 2.633

η δ2/Yp/s + ms g g−1 h−1 0.0052

µm Maximum specific growth rate h−1 0.7

ψ (δ2/Yp/o) g g−1 h−1 0.00025

# Initial value

Table 2. Comparison of model-computed and experimental datawith different statistical indicators

State variables

Statistical indices

Substrateconcentration

(s)

Cell massconcentration

(x)

Methionineconcentration

(p)

Correlation coefficient(R2)

0.963 0.943 0.995

Slope (m) 1.027 0.981 0.991

Root mean squareerror (RMSE)

0.558 0.254 0.009

Relative error (RE) 0.028 0.020 0.015

Index of agreement (d) 0.989 0.984 0.998

Ratio of means (Rm) −0.109 −0.158 −0.018

and studying its effect on the time profile of the state variables.34

The sensitivity analysis was carried out in two categories. Inthe first, each parameter was perturbed ±10% while the otherparameters were held constant at their nominal values. In thesecond, all five parameters were simultaneously subjected toperturbations of +10% and −10% (Table 3). The results showthat methionine concentration is most sensitive to parametricuncertainty. Variations in the exponential parameter KE elicits

the largest change (20.41%) in methionine concentration fora +10% change in this parameter. This fulfills the purpose ofintroducing this structure because the corresponding Km thatappears in the Monod kinetic structure µ(s) = µms/(Km + s),is a relatively insensitive parameter23 and often ill-conditionsparameter identification.35 Similarly, the methionine response issensitive to perturbation in α and β because these are linked to KE

through the growth-rate and non growth-rate associated terms.Among the parameters the first three, µm, KE and KI , may

be placed in one group, and α and β in another. In thefirst group, substrate and methionine concentrations exhibitinverse response to perturbations in µm, and KI, while cell massconcentration exhibits a direct response. Observations are thereverse for KE . The parameters µm, and KI act positively ongrowth and hence negatively on methionine concentrations.This seems to indicate that if much of the available energy(from substrate) is utilized for growth, less is available formethionine synthesis. In the second group, the state variablesexhibit similar behavior to parametric variations α and β . Theresponse is inverse for substrate concentration and direct forcell mass and methionine concentrations. This is consistentwith the fact that substrate is utilized for growth and productformation, as shown in the equation describing product dynamics.This observation is further supported by the results obtainedfor cases in which all the parameters were simultaneouslyperturbed by +10% or −10%. A positive perturbation elicits

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y = 0.9808x

0

3

6

9

12

15

18

0 3 6 9 12 15 18

Experimental Cell Mass Concentration (g L-1)

Pre

dic

ted

Cel

lmas

sC

on

cen

trat

ion

(g L

-1)

y = 1.027x

0

10

20

30

40

50

0 10 20 30 40 50

Experimental Substrate Concentration (g L-1)

Pre

dic

ted

Su

bst

rate

Co

nce

ntr

atio

n (g

L-1

)

y = 0.9906x

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5

Experimental Methionine Concentration (g L-1)

Pre

dic

ted

Met

hio

nin

eC

on

cen

trat

ion

(g L

-1)

(a)

(b)

(c)

Figure 4. Correlation between experimental data and model prediction(a) cell mass (b) substrate (c) methionine concentration for the batchprocess.

an inverse response in substrate concentration and a directresponse in cell mass and methionine concentrations and viceversa.

Nonlinear controllability and observability analysisThe model for methionine production by fed batch cultivation asdescribed by Equation (7) may be rewritten in the general form

x = f(x) + g(x)u (9)

y = h(x)

where x ∈ �n is the vector of state variables, u ∈ �m is the vector ofmanipulated inputs and y ∈ �p is the vector of measured outputs.Here the outputs y are the measurements that can be made from

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Time (h)

Res

idu

al G

luco

se a

nd

Cel

lmas

s C

on

cen

trat

ion

(g L

-1)

0

0.5

1

1.5

2

2.5

Met

hio

nin

e C

on

cen

trat

ion

(g L

-1)

Figure 5. Comparison of experimental cell mass, substrate and methionineconcentration with model prediction for a batch experiment having adifferent initial cell mass concentration (Experimental data: � Methionine,� Residual glucose and ♦ Cell mass; model prediction is shown withcontinuous lines).

-2

-1

0

1

2

3

0 12 18

Cellmass Concentration (g L-1)

Sta

nd

ard

ized

Res

idu

al

-3

-2

-1

0

1

2

0 10 20 30 40 50

Residual Glucose Concentration (g L-1)

Sta

nd

ard

ized

Res

idu

al

-3

-2

-1

0

1

2

3

0 0.5 1 1.5 2 2.5

Methionine Concentration (g L-1)

Sta

nd

ard

ized

Res

idu

al

6

(a)

(b)

(c)

Figure 6. Plot between standardized residuals and corresponding statevariable (a) cell mass (b) residual glucose (c) methionine concentration.

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Table 3. Parameter sensitivity analysis

Perturbation ofPercentage change

in concentrationparameters

Parameter (%) Substrate Cell mass Methionine

µm +10 −7.09 +2.62 −16.46

−10 +8.39 −3.19 +20.33

KE +10 +2.44 −0.97 +20.41

−10 −2.52 +0.92 −15.53

KI +10 −0.67 +0.21 −1.34

−10 +0.73 −0.31 +2.04

A +10 −0.05 +0.08 +19.89

−10 +0.15 −0.09 −13.78

B +10 −0.08 +0.06 +11.63

−10 +0.06 −0.07 −8.23

µm, KE , KI , α and β +10 −8.77 +3.83 +40.04

µm, KE , KI , α and β −10 +6.64 −4.49 −21.05

among all the state variables. We have included this in the generalrepresentation because the state of the system can only be knownthrough the measurements. Very often not all the fermentationstates can be measured because sensors are not available. Hence,although the model may be satisfactory for prediction it cannotbe used for online control implementations The functions f and gare smooth vector fields, and represent the kinetics and transportterms. For this process, the state vector is x = [x s p cL]T andthe input vector is u = [D kLa]T . In actual implementation, theinputs to the process are the substrate feed rate FS and the airflowrate FA. The input FS is determined from D and FA is determinedfrom Equation (8). In the analysis, we assume different cases ofmeasurement options and check the system. For example, if itis assumed that one can measure substrate concentration anddissolved oxygen concentration online, the output is y = [s cL]T .The output scalar functions constituting h(x) are simply the valuesof glucose and DO in the appropriate units of measurement.Hence, for this example, the system is square because there aretwo measurements and two control inputs, that is, m = p = 2. Inthe analysis we consider single measurements because it gives anidea about the information content in each of the measurements.This also enables one to choose the measurements that will givethe maximum benefit.

A systems analysis can mean many things depending on thecontext and application. In this case, we wish to show that themodel developed has the necessary structure to make the processreachable (controllable) and reconstructible (observable) with thegiven inputs and outputs. This is an important first step in theanalysis of nonlinear systems and is, in some sense, a mathematicalvalidation of the structural correctness of the model. The detailsof the nonlinear controllability and observability analysis arepresented in Appendix–B.

Proposition 1The system (7) is controllable with any input ui if the nonlinear

controllability matrix defined by C = �gi ad1f gi ad2

f gi ad3f gi�

is of full rankProposition 2The system (7) is observable with any measurement

yi = hi(x), if the observability matrix defined by O =∇[L0

f hi L1f hi L2

f hi L3f hi]T is of full rank

Where Lfh = ∇h · f and Lifh = (∇Li−1

fh) · f is the directionalderivative called the Lie derivative; ad1

f g = [f, g] = ∇g · f − ∇f · gand adi

fg = �f, adi−1f g� defines the successive Lie brackets.

Analysis shows that the system is controllable with eachinput, D or kLa, alone. Therefore, we have the freedom of usingeither or both inputs in designing control strategies. Normally,for nonlinear systems, it would be possible to construct aninput–output linearizing controller that may be implementedonline. A decoupled input–output linearizing nonlinear controllerhas been presented elsewhere.36 The observability analysis, on theother hand, shows that if individual states were measured, only themethionine concentration p contains enough information to makethe system observable. All other state variables, when measuredindividually, fail to provide observability of the system. Sincemethionine must be measured by off-line methods, this featureof system (7) will be disadvantageous in system reconstructionand prediction. However, this will not restrict the application ofthe system equation (7) in controller design. For example, theinput–output linearization method is not restrictive and may beeasily applied for a suitable controller design.

To ensure non-singularity of the controllability and observabilitymatrices, the constraints arising from expansion of the matricesmust be satisfied. However, as the system evolves, the valuesof the states may result in a violation of these constraints.Therefore, in applications based on this analysis, for examplein controller design, it will be necessary to carry out a full stabilityanalysis to guarantee performance. Similarly, in observer design,the observation function h(x) must be chosen in such a waythat within the acceptable boundary of system operation, fullreconstructibility of the state space is permitted.

Model prediction under controlled and uncontrolledconditionsPrediction for different DO setpoints

The model equations were used with the three state variables x,s and p, when DO was controlled at different levels (25%, 30%,35%, 40%, 50%, 70% and 100%) and the results were comparedwith the trends in experimental values.13 It is expected that themodel should predict the methionine production when the DOis controlled at 40% and decrease as the DO set point changesto either lower or higher values. The results of these simulationsare presented in Fig. 7(c). The corresponding time profiles of thecell mass and substrate concentration is presented in Fig. 7(a) and7(b).

When DO concentration is controlled at 40% the productionof methionine passes through a maximum, as observed in theexperimental data (Fig. 2). In simulations where DO is controlledat values lower or higher than 40%, the maximum value ofmethionine produced is lower. Further, at higher DO, the timeprofiles of methionine concentration exhibit a maximum butwith lower peak values. At DO set points lower than 40%,the methionine production begins later and the maximum isnot observed during the experiment. At higher DO set pointexperiments, the substrate consumption (Fig. 7(b)) is faster andthis results in a major portion of carbon source being diverted tocell growth. At lower DO set points, the rate of substrate uptakeis slower and consequently methionine production is delayedbecause the cell mass concentration is insufficient to supporthigher rates of methionine synthesis.

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0

5

10

15

20

25

0 10 20 30 40 50 60

Time (h)

Cel

lmas

s C

on

cen

trat

ion

(g L

-1) X (25% DO) X (30% DO) X (35% DO)

X (40% DO) X (50% DO) X (70% DO)X (100% DO)

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Time (h)

Res

idu

al G

luco

se C

on

cen

trat

ion

(g L

-1)

S (25% DO) S (30% DO) S (35% DO)S (40% DO) S (50% DO) S (70% DO)S (100% DO)

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50 60

Time (h)

Met

hio

nin

e C

on

cen

trat

ion

(g L

-1)

P (25% DO)

P (30% DO)P (35% DO)

P (40% DO)P (50% DO)

P (70% DO)P (100% DO)

(a)

(b)

(c)

Figure 7. Effect of changes in DO concentration on the time profiles of statevariables (a) cell mass concentration (b) residual glucose concentration(c) methionine concentration.

Prediction in the absence of DO controlThe prediction of the model was examined in fermentationexperiments in which DO was not controlled. The initial conditionsof the simulation were the same as in the actual experiments.The initial DO concentration was taken to be 100% of thesaturation level. The process was supplied with a constantairflow rate of 6 L min−1. The time profiles of x, s and p arepresented in Fig. 8(a) and the DO time profile in Fig. 8(b). In theregion of active growth, the oxygen demand is highest. Sinceairflow rate is supplied at a constant rate, DO concentrationdecreases between 10 and 24 h of fermentation. Once the activegrowth phase is over, DO concentration increases and reachessaturation at about 45 h. The oxygen demand for methionine

0

10

20

30

40

50

0 10 20 30 40 50 60

Time (h)

Res

idu

al G

luco

se a

nd

Cel

lmas

s co

nce

ntr

atio

n (g

L-1

)

0

0.4

0.8

1.2

1.6

2

Met

hio

nin

e C

on

cen

trat

ion

(g L

-1)

Substrate

Cell mass

Methioninie

0

0.0016

0.0032

0.0048

0.0064

0.008

0 10 20 30 40 50 60

Time (h)D

isso

lved

Oxy

gen

on

cen

trat

ion

(g L

-1)

(a)

(b)

Figure 8. (a) Prediction of time profiles of cell mass, residual glucose andmethionine concentration in the absence of DO control (b) Prediction oftime profile of DO in the absence of control.

production is low; this result is similar to the results obtainedfrom the metabolic flux analysis.9 The optimal RQ value formethionine synthesis is about 0.75 compared with 1.17 forlysine and other amino acids.37 Methionine requires seven ATPswhen synthesized from glucose. Hence, one of the reasons forthis strict regulation of the methionine pathway, is to preventthe wastage of energy used for methionine synthesis.38 Theproduction of methionine is less when DO is not controlledbecause the cells preferentially utilize oxygen for cell growth. In anuncontrolled environment, as DO decreases, the cells synthesizeonly as much methionine as is necessary for cellular growthand maintenance. Methionine must be synthesized by the cellsbecause N-formyl-methionyl-tRNA is the universal initiator ofall protein synthesis. In addition, it acts as a methyl donor innumerous reactions. Consequently, even in adverse environmentsmethionine must be synthesized by the cell but it does so onlyto fulfill cellular requirements because, energetically, it requiresthe highest number of ATPs compared with all other aminoacids. One may also visualize this as the time average of themethionine synthesis over a range of DO concentrations that arenot optimal. Initially, the available oxygen is around 100% andreturns to above 80% around 36 h. Previous simulations showthat the methionine produced under these circumstances is lower(Fig. 7(c)).

A prediction similar to that shown in Fig. 5 has also beenreported by Hua et al.16 for micro-aerobic conditions, where thedecrease in DO is not as sharp. They obtained a maximum cellmass concentration of 12–13 g L−1, which was about 80% of thatobtained for aerobic conditions. They observed that the glucoseconsumption was low in micro- aerobic conditions, with significantaccumulation of lactic acid (39 g L−1 for DO of 1%). Their resultsshow that TCA cycle intermediates influence the excretion of lysine

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into the medium. A similar study was beyond the scope of thiswork. However, the flux analysis carried out seems to indicate asimilar behavior at DO levels well below 40%.9 The cumulativeeffect of oxygen variation is reflected by a 20% reduction inmethionine concentration.

CONCLUSIONSThe model, presented in this work provides a good mathematicalrelationship between cell mass production, substrate consump-tion and product formation in submerged fed-batch cultivation ofCorynebacterium lilium to produce methionine. The effect of DO isaccurately incorporated in the model by employing a metabolicswitching function. Results show that this model can describemethionine and cell mass variation under different controlled DOand uncontrolled DO scenarios. Satisfactory conformity betweenmodel prediction of state variables and experimental data has beendemonstrated. A system analysis was performed which shows thatthe process can be controlled with either of the inputs. However,the system is observable only with the measurement of methio-nine concentration. While a combination of the other variables,although not individually, will also make the process observable.Therefore, the developed model may be used in process controlapplications.

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et al, S-Adenosyl-l-methionine: effects on brain bioenergtic statusand transverse relaxation time in healthy subjects. Biol Psychiatry54:833–839 (2003).

2 Uversky VN, Yamin G, Souillac PO, Goers J, Glaser CB and Fink AL,Methionine oxidation inhibits fibrillation of human α-synucleinin vitro. FEBS Lett 517:239–244 (2002).

3 Guttormsen AB, Ueland PM, Kruger WD, Kim CE, Ose L, Folling Iet al, Disposition of homocysteine in subjects heterozygous forhomocystinuria due to cystathionine beta-synthase deficiency:relationship between genotype and phenotype. Am J Med Genet100:204–213 (2001).

4 McKinley MC, Strain JJ, McPartlin J, Scott JM and McNulty H, Plasmahomocysteine is not subject to seasonal variation. Clin Chem47:1430–1436 (2001).

5 Bedell LS and Hulbert MK, Mosby’s Complete Drug Reference 7th edn.Mosby-Year Book Inc., St Louis, USA (1997).

6 Kroger H, Dietrich A, Ohde M, Lange R, Ehrlich W and Kurpisz M,Protection from acetaminophen-induced liver damage by actionof low doses of the poly(ADP-ribose) polymerase-inhibitornicotinamide and the antioxidant n-acetylcysteine or the aminoacid L-methionine. Gen Pharmacol 28:257–263 (1997).

7 Townsend DM, Tew KD and Tapiero H, Sulfur containing amino acidsand human disease. Biomed Pharmacother 58:47–55 (2004).

8 Kumar D and Gomes J, Methionine production by fermentation.Biotechnol Adv 23:41–61 (2005).

9 Kumar D, Subramanian K, Bisaria VK, Sreekrishnan TS and Gomes J,Effect of cysteine on methionine production by a regulatory mutantof Corynebacterium lilium. Biores Technol 96:287–294 (2005).

10 Kumar D, Garg S, Bisaria VK, Sreekrishnan TS and Gomes J, Productionof methionine by a multi-analogues resistant mutant ofCorynebacterium lilium. Process Biochem 38:1165–1171 (2003).

11 Sharma S, Strain improvement and reactor studies for the productionof methionine by Corynebacterium lilium. PhD thesis, IndianInstitute of Technology, Delhi, India (2003).

12 Gomes J and Kumar D, Production of L-methionine by submergedfermentation: a review. Enzyme Microbial Technol 37:3–18 (2005).

13 Sharma S and Gomes J, Effect of dissolved oxygen on continuousproduction of methionine. Eng Life Sci 1:69–73 (2001).

14 Hua Q, Yang C and Shimizu K, Metabolic control analysis for lysinesynthesis using Corynebacterium glutamicum and experimentalverification. J Biosci Bioeng 90:184–192 (2000).

15 Yang C, Hua Q and Shimizu K, Development of a kinetic modelfor L-lysine biosynthesis of Corynebacterium glutamicum and itsapplication to metabolic control analysis. JBiosciBioeng 88:393–403(1999).

16 Hua Q, Fu PC, Yang C and Shimizu K, Micro aerobic lysinefermentations and metabolic flux analysis. Biochem Eng J 2:89–100(1998).

17 Ratkov A, Georgiev T, Ivanova V, Kristeva J and Ratkov B, Comparativestudies of fed-batch fermentation process for production of L-lysineand L-valine based on mathematical models. Proceedings of 25thInternational Conference on Information Technology Interfaces ITI2003, Cavtat, Croatia, pp. 513–518 (2003).

18 Zhang XW, Sun T, Sun ZY, Liu X and Gu DX, Time-dependent kineticmodels for glutamic acid fermentation. Enzyme Microbial Technol22:205–209 (1998).

19 Yao HM, Tian YC, Tade MO and Ang HM, Variations and modellingof oxygen demand in amino acid production. Chem Eng Process40:401–409 (2001).

20 Ensari S and Lim HC, Kinetics of l-lysine fermentation: a continuousculture model incorporate oxygen uptake rate. Appl MicrobiolBiotechnol 62:35–40 (2003).

21 Monod J, The growth of bacterial cultures. Ann Rev Microbiol3:371–394 (1949).

22 Levenspiel O, The Monod equation: a revisit and a generalizationto product inhibition situations. Biotechnol Bioeng 22:1671–1687(1980).

23 Gomes J and Menawat AS, Fed-batch bioproduction ofspectinomycine. Adv Biochem Eng Biotechnol 59:1–45 (1998).

24 Pujo-Menjouet L, Bernard S and Mackey MC, Long period oscillationsin a G0 model of hematopoietic stem cells. SIAM J Appl DynamSystems 4:312–322 (2005).

25 Zielinski S, Sartoris FJ and Portner HO, Temperature effect onHemocyanin oxygen binding in an Antarctic cephalopod. Biol Bull200:67–76 (2001).

26 Mustacich RV and Weber G, Ligand-promoted transfer of proteinsbetween phases: Spontaneous and electrically helped. Proc NatlAcad Sci USA 75:779–783 (1978).

27 Ferrell JE, Tripping the switch fantastic: how a protein kinase cascadecan convert graded inputs into switch like outputs. Trends BiochemSci 21:460–466 (1996).

28 Bernard S, Cajavec B, Pujo-Menjouet L and Herzel MCM, Modelingtranscriptional feedback loops: the role of Gro/TLE1 in Hes1oscillations. Proc Roy Soc 364:1155–1170 (2004).

29 Luedeking EL and Piret AJ, A kinetic study of the lactic acidfermentation. Batch process at controlled pH. Biochem MicrobiolTechnol Eng 1:393–412 (1959).

30 Van’t Riet K, Review of measuring methods and results in nonviscousgas-liquid mass transferred in stirred vessels. Ind Eng Chem ProcessDes Dev 18:357–364 (1979).

31 Willmott CJ, Some comments on the evaluation of modelperformance. Bulletin Am Meteorol Soc 63:1309–1313 (1982).

32 Stockle CO, Kjelgaard J and Bellocchi G, Evaluation of estimatedweather data for calculating Penman-Monteith reference cropevapotranspiration. Irrig Sci 23:39–46 (2004).

33 Dielman TE, Applied Regression Analysis for Business and Economics.Duxbury Press (2000).

34 Tang Y-J and Zhong J-J, Modeling the kinetics of cell growth andganoderic acid production in liquid static cultures of the medicinalmushroom Ganoderma lucidum, Biochem Eng J, 21:259–254(2004).

35 Gomes J, Roychoudhury PK and Menawat AS, Effect of inputs onparameter estimation and prediction of unmeasured states. InProceedings of the IIChE Golden Jubilee Congress, Vol 1, pp. 239–248(1997).

36 Ranjan AP and Gomes J, Simultaneous oxygen and glucoseregulation in fed-batch methionine production using decoupledinput output linearizing control. J Process Control (2008).DOI:10.1016/j.jprocont.2008.07.008.

37 Vallino JL and Stephanopoulos G, Metabolic characterization of a L-lysine-producing strain by continuous culture. Biotechnol Bioeng41:633–646.

38 Kromer JO, Wittmann C, Schroder H and Heinzle E, Metabolic pathwayanalysis for rational design of L-methionine production byEscherichia coli and Corynebacterium glutamicum. Metabol Eng8:353–369 (2006).

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APPENDIX AStatistical indicesThe following statistical indices are used to assess the performanceof the mathematical model developed in this study.

(i) Root mean square error (RMSE) calculated as

RMSE =

√√√√√√n∑

i−1

(Pi − Ei)2

n(Al)

where Ei and Pi are experimental and model predicted values.n is the number of data points analyzed.

(ii) Relative error (RE) calculated as

RE = RMSE

E(A2)

where E is the mean of the experimental data.(iii) The Willmott index of agreement (d)31 calculated as

d = 1 −

n∑i=1

(Pi − Ei)2

n∑i=1

{∣∣(Pi − E)| + |(Ei − E)∣∣}2

(A3)

(iv) The slope (m) and the correlation coefficient (R2) of the linearregression between experimental (X variable) and modelpredicted (Y variable).

(v) Ratio of means (Rm)37 calculated as

Rm =(

E′ − P′

P′

)(A4)

where E′ and P′ are the average deviation of experimental andmodel predicted data. It represents a measure of the relativedifferences between experimental and predicted averages.

To interpret the performance of model prediction the followingcriteria were used:32

∇f.g =

µ(s, cL) µs(s, cL)x 0 µc(s, cL)x−γµ(s, cL) − η −γµs(s, cL)x 0 −γµc(s, cL)x

α(s − sc)psv + βµ(s) αpxsc

vs2 + βµs(s, cL)x α(s − sc)xsv βµc(s, cL)x

−ϕµ(s, cL) − φ −ϕµ2(s, cL)x 0 ϕµc(s, cL)x

0 0sF 00 00 c∗

L

= sF c∗L

µs(s, cL)x µc(s, cL)x−γµs(s, cL)x −γµc(s, cL)

αpxsvs2 + βµs(s, cL)x βµc(s, cL)x

−ϕµs(s, cL)x −ϕµc(s, cL)x

= [ψ1 ψ2] (B5)

Combination of values Model performanced ≥ 0.95 and RE ≤ 0.10 Very goodd ≥ 0.95 and 0.15 ≥ RE > 0.10 Goodd ≥ 0.95 and 0.20 ≥ RE > 0.15 Acceptabled ≥ 0.95 and 0.25 ≥ RE > 0.20 MarginalAll other combination of d and RE Poor

In addition, all combination with, R2 < 0.85 and 0.9 > m orm > 1.1 were considered poor. Low absolute values of Rm

(Rm < 0.3) specify that the model can estimate the overall mean

value of the experimental data with acceptable accuracy. Thesign of Rm explains the model tendency towards overestimation(Rm > 0) or underestimation (Rm < 0) of the average experimentaldata.

APPENDIX BNonlinear controllability analysisThe nonlinear model for methionine production is first rewrittenin mass units to simplify the algebra. Although the same notationhas been used for the variables, in the analysis that follows, thevariables refer to the states expressed in mass units g h−1. Thesystem (7) is now represented as (B1); note that the transportmatrix g is simpler.

xspcl

=

µ(s, cL)x−γµ(s, cL)x − ηx

βµ(s, cL)x + α((s − sc)/s)px−ϕµ(s, cL)x − φx

+

0 0sF 00 00 c∗

L

[

FkLa

](B1)

The nonlinear system (B1) is controllable with the inputs F and kLa(which contains the airflow rate) if the controllability matrix C is offull rank for each of the inputs. The controllability matrix is definedas

C = �g ad1f g ad2

f g ad3f g� (B2)

where the successive Lie brackets are defined by

ad0f g = g

ad1f g = [f, g] = ∇g · f − ∇f · g

..

.

adkf g = [f, adk−1

f g] (B3)

Evaluating the first-order Lie bracket we get

adfg = [f, g] = ∇g.f − ∇f.g = 0 − ∇f.g (B4)

Evaluating the second-order Lie bracket we get

∇[f, g] · f =

ψ11x ψ11s 0 ψ11c | ψ21x ψ21s 0 ψ21c

ψ12x ψ12s 0 ψ12c | ψ22x ψ22s 0 ψ22c

ψ13x ψ13s ψ13p ψ13c | ψ23x ψ23s 0 ψ23c

ψ14x ψ14s 0 ψ14c | ψ24x ψ24s 0 ψ24c

f1

f2

f3

f4

(B6)

∇f · [f, g] =

f1x f1s 0 f1c

f2x f2s 0 f2c

f3x f3s f3p f3c

f4x f4s 0 f4c

ψ11 ψ11

ψ12 ψ22

ψ13 ψ23

ψ14 ψ24

(B7)

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Hence

ad2f g = �f, ad1

f g� = ∇[f, g] · f − ∇f · [f, g] = [ξ1 ξ2] (B8)

where the expansions of the matrix can be obtained fromEquations (B6) and (B7). Similarly, the third-order Lie bracketcan evaluated as follows

ad3f g = ∇ad2

f g · f − ∇f · ad2f g = [ζ1 ζ2] (B9)

where the expansions of ζ1 and ζ2 will have similar forms is thosegiven in Equations (B6) and (B7). Finally, the controllability matrixis obtained as

C =

0 ψ11 ξ11 ζ11 | 0 ψ21 ξ21 ζ21

sF ψ12 ξ12 ζ12 | 0 ψ22 ξ22 ζ22

0 ψ13 ξ13 ζ13 | 0 ψ23 ξ23 ζ23

0 ψ14 ξ14 ζ14 | c∗L ψ24 ξ24 ζ24

(B10)

The rank condition for each of the inputs is fulfilled and thereforethe system is controllable with either of the inputs. When both theinputs are used together there is interaction and the control designshould incorporate a decoupling scheme. The individual terms ofthe matrix can be expanded and from the two determinants thatcan be constructed, the constraints on the controllable space canbe determined.

Nonlinear observability analysisThe observability analysis presented here has been carried outconsidering that each state variable can be measured in turn.The definition of the nonlinear observability stated as the rankcondition for this system is

O = ∇

L0

f hL1

f hL2

f hL3

f h

(B11)

If the rank of the observability matrixO is full rank then the systemis observable with that measurement.

Methionine concentration p as observation (h(x) = p)Compute L0

f h

L0f h = [0 0 1 0]

xspcL

= p (B12)

Compute L1f h

L1f h = ∇h · f = [0 0 1 0]

f1

f2

f3

f4

= f3(x, s, p, cL) (B13)

Compute L2f h

L2f h = ∇(L1

f h) · f =[

∂f3

∂x

∂f3

∂s

∂f3

∂p

∂f3

∂cl

]

f1

f2

f3

f4

= ψ1(x, s, p, cL) (B14)

Compute L3f h

L3f h = ∇(L2

f h) · f =[

∂ψ1

∂x

∂ψ1

∂s

∂ψ1

∂p

∂ψ1

∂cL

]

f1

f2

f3

f4

= ψ2(x, s, p, cL) (B15)

Construct the observability rank matrix

O =

0 0 1 0∂f3∂x

∂f3∂s

∂f3∂p

∂f3∂cL

∂ψ1∂x

∂ψ1∂s

∂ψ1∂p

∂ψ1∂cL

∂ψ2∂x

∂ψ2∂s

∂ψ2∂p

∂ψ2∂cL

(B16)

The determinant needs to be evaluated to establish that it is non-singular. In general, the functions are complex and do not leadto zero elements. In biochemical systems, the determinant usuallygives the constraints for the observability condition.

Dissolved oxygen concentration cL as observation (h(x) = cL)Compute L0

f h

L0f h = [0 0 0 1]

xspcL

= cL (B17)

Compute L1f h

L1f h = ∇h · f = [0 0 0 1]

f1

f2

f3

f4

= f4(x, s, cL) (B18)

Compute L2f h

L1f h = ∇(L2

f h) · f =[

∂f4

∂x

∂f4

∂s0

∂f4

∂cL

]

f1

f2

f3

f4

= ψ1(x, s, cL) (B19)

Compute L3f h

L3f h = ∇(L2

f h) · f =[

∂ψ1

∂x

∂ψ1

∂s0

∂ψ1

∂cL

]

f1

f2

f3

f4

= ψ2(x, s, cL) (B20)

Construct the observability rank matrix

O =

0 0 0 1∂f4∂x

∂f4∂s 0 ∂f4

∂cL∂ψ1∂x

∂ψ1∂s 0 ∂ψ1

∂cL∂ψ2∂x

∂ψ2∂s 0 ∂ψ2

∂cL

(B21)

In this case it is evident that that rank of the matrix is at most 3 andhence the system is not fully observable with cL as a measurement.

J Chem Technol Biotechnol 2009; 84: 662–674 c© 2008 Society of Chemical Industry www.interscience.wiley.com/jctb

67

4

www.soci.org AP Ranjan, R Nayak, J Gomes

Substrate concentration s as observation (h(x) = s)Compute L0

f h

L0f h = [0 1 0 0]

xspcL

= s (B22)

Compute L1f h

L1f h = ∇h · f = [0 1 0 0]

f1

f2

f3

f4

= f2(x, s, cL) (B23)

Compute L2f h

L2f h = ∇(L1

f h) · f =[

∂f2

∂x

∂f2

∂s0

∂f2

∂cL

]

f1

f2

f3

f4

= ψ1(x, s, cL) (B24)

Compute L3f h

L3f h = ∇(L2

f h) · f =[

∂ψ1

∂x

∂ψ1

∂s0

∂ψ1

∂cL

]

f1

f2

f3

f4

= ψ2(x, s, cL) (B25)

Construct the observability rank matrix

O =

0 1 0 0∂f2∂x

∂f2∂s 0 ∂f2

∂cL∂ψ1∂x

∂ψ1∂s 0 ∂ψ1

∂cL∂ψ2∂x

∂ψ2∂s 0 ∂ψ2

∂cL

(B26)

In this case again the rank of the matrix O is at most 3 and so thesystem is not observable with s as measurement.

Cell mass concentration x as observation (h(x) = x)Compute L0

f h

L0f h = [1 0 0 0]

xspcL

= x (B27)

Compute L1f h

L1f h = ∇h · f = [1 0 0 0]

f1

f2

f3

f4

= f1(x, s, cL) (B28)

Compute L2f h

L2f h = ∇(L1

f h) · f =[

∂f1

∂x

∂f1

∂s0

∂f1

∂cL

]

f1

f2

f3

f4

= ψ1(x, s, cL) (B29)

Compute L3f h

L3f h = ∇(L2

f h) · f =[

∂ψ1

∂x

∂ψ1

∂s0

∂ψ1

∂cL

]

f1

f2

f3

f4

= ψ2(x, s, cL) (B30)

Construct the observability rank matrix

O =

1 0 0 0∂f1∂x

∂f1∂s 0 ∂f1

∂cL∂ψ1∂x

∂ψ1∂s 0 ∂ψ1

∂cL∂ψ2∂x

∂ψ2∂s 0 ∂ψ2

∂cL

(B31)

In this case also the rank of the matrix O is at most 3 and so thesystem is not observable with x as measurement.

www.interscience.wiley.com/jctb c© 2008 Society of Chemical Industry J Chem Technol Biotechnol 2009; 84: 662–674