alberton etal 2013 parameters identifiability

Computers and Chemical Engineering 55 (2013) 181 197

Contents lists available at SciVerse ScienceDirect

Computers and Chemical Engineering

jo u r n al homep age: www.elsev ier .com/ lo

Acceler ceSet by

Kese P.F. i MMara Soa Programa de 941-9b Instituto de Q -900 c Planta Piloto Km 7,

a r t i c

Article history:Received 17 JaReceived in reAccepted 1 ApAvailable onlin

Keywords:Parameters ideBinary searchParameters esScarce experim

ased y, theters iltanevaluaationthe se

1. Introduction

Several amatical modstatisticallyoften not poexperimentsurement ddifcult anscenario of for estimatestimates; tidentiabili

Briey, pa set of pamated usinproceduresparametersother paramstructure oftainties, andMcAuley, &2006).

CorresponE-mail add

Identiability procedures presents two important features(Kravaris, Chu, & Hahn, 2012): (1) interpretability indicates which

0098-1354/$ http://dx.doi.oreas of human knowledge make use of complex mathe-els containing a large number of parameters. However,

consistent estimation of all these parameters is veryssible, since it would demand a prohibitive number ofal data or would require inexistent/unavailable mea-evices. In fact, experiments are usually time consuming,d expensive. A common approach to overcome thisill-posed estimation is selecting a subset of parametersion, while other parameters are set at their initialhese procedures are commonly known as parametersty.arameters identiability procedures seek to determinerameters of the model that can be consistently esti-g the available experimental data. Theoretically, these

should take into account the inuence of the selected on the model predictions and/or the correlation witheters. Such analyses can be performed based on the

the model, the experimental measurements and uncer- the uncertainties of the initial estimates (Thompson,

McLellan, 2009), which are not easy tasks (Yue et al.,

ding author. Tel.: +55 21 25628307; fax: +55 21 25628300.ress: [email protected] (A.R. Secchi).

parameters are important for a model dynamic behavior and theyalso provide insight about the correlation of the effect that differ-ent parameters have on the outputs and (2) simplication reducesthe numbers of decision variables in parameter estimation problemsince non-selected parameters are all xed at in initial estimates.Thus, several parameters identiability approaches have been pro-posed in the literature. A common sense is the use of sensitivityanalysis, as a natural choice, being a valuable tool for verifying theparameters importance. In the last decade, several methodologieswere developed based on local sensitivity analysis (Brun, Reichert,& Kunsch, 2001; Chu & Hahn, 2007; Li, Henson, & Kurtz, 2004;Lund & Foss, 2008; Sandink, McAuley, & McLellan, 2001; Secchi,Cardozo, Almeida Neto, & Finkler, 2006; Sun & Hahn, 2006; Yao,Shaw, Kou, McAuley, & Bacon, 2003) and less usual global sen-sitivity methods (Chu, Huang, & Hahn, 2011). In most of thesemethodologies, parameters are selected from the most estimable tothe least estimable. Nonetheless, this parameters selection is estab-lished by evaluating the parameters one by one, which can be a timeconsuming procedure for a complex mathematical model contain-ing a large number of parameters. In such scenario, an importantquestion to be answered is: what is the smallest number of neces-sary evaluations to be performed to determine a set of identiableparameters?

In this context, this work presents a numerical procedure forparameters selection based on the binary search that could be

see front matter 2013 Elsevier Ltd. All rights reserved.rg/10.1016/j.compchemeng.2013.04.014ating the parameters identiability proset selection

Albertona, Andr Lus Albertonb, Jimena Andra Dledad Dazc, Argimiro R. Secchia,

Engenharia Qumica-COPPE, Universidade Federal do Rio de Janeiro, P.O. Box 68502, 21umica, Universidade Estadual do Rio de Janeiro-UERJ, So Francisco Xavier 524, 20550de Ingeniera Qumica-CONICET, Universidad Nacional del Sur, Camino La Carrindanga

l e i n f o

nuary 2013vised form 23 March 2013ril 2013e 1 May 2013

ntiability

timationental data

a b s t r a c t

In this paper, a numerical procedure bters identiability procedure. Basicallranking the parameters. Since parameestimates of parameters values, simupaper. Two examples were used to ea signicant reduction of the computare similar to those criteria based on literature.cate /compchemeng

dure:

aggioc,

72 Rio de Janeiro, RJ, BrazilRio de Janeiro, RJ, Brazil

Baha Blanca, Argentina

on the binary search is proposed for accelerating the parame- parameters are selected set by set using a given criterion fordentiability procedures are strongly dependent on the initialous parameters re-estimation step has been proposed in thiste the performance of the proposed criterion. In both cases,al time was observed, and the results regard to the model tlection of parameters one by one, as usually presented in the

2013 Elsevier Ltd. All rights reserved.

182 K.P.F. Alberton et al. / Computers and Chemical Engineering 55 (2013) 181 197

Fig. 1

applied usinparameters& Hahn, 202001; SeccBasically, thtion evaluathe selectedto a signicthe usual pthe proposeproblem ofear model cexperimentparametersEscherichia & Daz, 200data.

2. Parame

Based onavailable exsplits the paestimated, ccannot be enon-identiselection prheuristic mmethods coputational approaches

Based onselection arparameterstwo criteriamagnitude and/or corrdure ill-posand consistAlthough uof the resul

Usually,estimable pThus, the qparametersnately, goodcase they cassumed arsical schemFig. 1.

In the clin the set ootherwise,

and the procedure stops. The rst parameter is included in the setof selected parameters, (S), based on its effect on model predic-tion. For the next parameters, a measure of correlation with theparameters already included in (S) is also taken into account for

king on prameatril., 20ex (001)imumal (

2009alueng thased ier tex mcedu

al. (2nownue priteriot thd Hd pron. Ass (2

resuompabil

holdese (1onal

tranand t

st ofonal

(FIM

(V)

ch B, Vmentst co, to aut v

l formted a

iag(Y

ch Ye vec. Classical scheme of parameters identiability procedures.

g any criteria proposed in the literature for ranking the according to their identiability (Brun et al., 2001; Chu07; Li et al., 2004; Lund & Foss, 2008; Sandink et al.,hi et al., 2006; Sun & Hahn, 2006; Yao et al., 2003).e proposed procedure makes the parameters selec-

ting subsets of parameters divided in half according to criterion to order the parameters. This proposal leadsant reduction in the computational cost compared torocedures. Two examples illustrate the performance ofd procedure: (1) a hypothetical case study a simply

parameters identiability with known solution: a lin-ontained large number of parameters and orthogonalal data, and (2) a real case study a complex problem of

identiability: a dynamic model of the microorganismcoli K-12 W3110 metabolic networks (Di Maggio, Ricci,9) contained 131 parameters and scarce experimental

ters identiability

the structure of the mathematical model and on theperimental data, a parameter identiability procedurerameters set in two subsets: the parameters that can bealled identiable parameters, and the parameters thatstimated and are kept in their initial estimates, calledable parameters. According to Kravaris et al. (2012),ocedures can be divided into the following categories:ethods and optimization-based methods. Optimizationnsist on a combinatorial problem requiring high com-efforts, for such reason, are less used than heuristic.

sensitivity analysis, heuristic methods for parametere derived based on the effects that the variations in the

have on the model outputs and so take into account for parameters evaluation (Kravaris et al., 2012): (1)of the effect, and (2) uncorrelated effect. Non-signicantelated effects make the parameters estimation proce-ed, even when experimental data would be sufcientent for identiability of all parameters of the model.sually these criteria are used together, the performanceting estimation problem can often not be guaranteed.

parameters estimation is performed after the set ofarameters be obtained in the identiability procedure.uality of the initial estimates values admitted to the

re-ranselectithe parlarge m(Li et arity indet al., 2(5) optE-optimBaeza,gular v

Beidure bare eascomplity proYao etwell-koff valthis crin whaChu anmethocriterioand Fosimilargreat cidentiHouseVergheorthogholder1996) 2012).

MoorthogMatrix

FIM = in whimatrixexperi

MoEq. (2)of outpseverapresen

BN = d

in whi is th is fundamental for the adequate selection. Unfortu- parameters initial estimates are seldom known; some

an be obtained from the literature or more commonlybitrarily (Kou, McAuley, Hsu, Bacon, & Yao, 2005). Clas-e employed to parameters identiability is presented in

assical scheme, the parameters are included one by onef selected parameters, verifying if this set is estimable;the last included parameter is removed from the set

onal matrixIn the localone paramThus, each respondingvectors, ||bprediction.

Surely, by one arethe remaining parameters. Thus, at each step of theocedure, one should evaluate the correlation betweenters, which requires costly and complex operations withces (Kravaris et al., 2012) such as: (1) orthogonalization04; Lund & Foss, 2008; Yao et al., 2003), (2) collinea-

Brun et al., 2001), (3) relative gain array RGA (Sandink, (4) principal component analysis PCA (Li et al., 2004),

experimental design criteria D-optimal, A-optimal,Chu & Hahn, 2007; Machado, Tapia, Gabriel, Lafuente, &; Weijers & Vanrolleghem, 1997), and (6) Hankel sin-

(Sun & Hahn, 2006).e most popular, the parameters identiability proce-on orthogonalization methods is simple and the resultso interpret, having been widely used for the analysis ofodels (Kravaris et al., 2012). Among them, identiabil-re proposed by Yao et al. (2003) seems to be most used.003) proposed the parameters identiability based on

GramSchmidt orthogonalization method, using a cut-eviously specied as stop criterion of selection, beingn replaced by Thompson et al. (2009) for the point

e parameters estimation problem becomes ill-posed.ahn (2007) also use GramSchmidt orthogonalizationposing a forward selection which maximizes the D-

lso this orthogonalization method is employed by Lund008) that suggest a methodology that produces verylts to those presented by Yao et al. (2003), but presentsutational complexity. Other approach of parametersity problem employing orthogonalization methods usesr transforms as in Hiskens (2001) and Velez-Reyes and995). Both GramSchmidt and Householder transformsization methods produce identical results, but House-sforms is more stable numerically (Golub & van Loan,he GramSchmidt is simpler to interpret (Kravaris et al.,

the parameters identiability procedures based onization methods make use of the Fisher Information), dened as:

1 = BT (VY )1B (1) = Y/, more commonly known as local sensitivityand VY represent, respectively, the parameters andal covariance uncertainty matrices.mmonly, B matrix is normalized (BN), as presented invoid distortions due to the different magnitude ordersariables and parameters values. The literature reportss for normalization, being one of the most commonly

s follows.

E)1(

Y

)diag() (2)

E is the vector of mean experimental output variables,tor of parameters values, and diag(x) denotes the diag-

of the vector x. Fig. 2 illustrates such normalization. sensitivity matrix, B or BN, each column is related toeter and each row is related to one output variable.column i can be considered as a sensitivity vector cor-

to the parameter i, bi . The norm of these sensitivity

i||, can be used to evaluate the parameters effects on the

procedures for selecting parameters evaluating one computationally costly and time consuming. A rst

K.P.F. Alberton et al. / Computers and Chemical Engineering 55 (2013) 181 197 183

b

nP...

1 2

Parameters

1

1

2

1

1

N

Yn

y

y

B

y

Fig. 2. Norma

propositionHuang, Hahof a numbeAccording tincluded inoptimizatioorthogonaliparametersresults is a represents parametersAccording tsequential sheuristic re

3. Binary s

Binary secharacter ormation. Givwhere one necessary tthat can beway to get if the numbone can elimis not presethe numbercharacteriznumber arethe set of nuillustrative ever is the srequired is

In binargiven by:

nEval = log2in which n i

4. Parame

An impoputational

uery

ion. r of lectio

numIn thed pad selset, bed ining

updducee prramee ide

for istabm.rder to explain the proposed procedure, in Fig. 5 it is appliedodel with four parameters, showing all possible combina-or this case. The parameters are sorted according to theirance for the model prediction (as in: Brun et al., 2001; Chu &...nP

b2

b1

Sensitivity vectors of para meter s

Outpu t var iables

1y

nYy...

2y 1 1 2 1

1121

1 2 2 2

2222

1 2

2

nP

nP

nP

nP

PnYnYn

nY nY nP nY

y yy y y

y yy y y

y yy y y

lized local sensitivity matrix BN and sensitivity vectors of parameters.

to overcome this drawback was done recently by Chu,n, and Hahn (2012), based on heuristics for assumptionr of parameters that can be selected at each iteration.o the authors, at each iteration, new parameters are

the set of selected parameters, determined by ann procedure, making use of a metric based on thezation of the sensitivity matrix. Although appealing,

identiability using optimization-based methodscombinatorial problem, where nP !/[nSet ! (nP nSet) !]the total number of combinations for selecting nSet

from a set of nP parameters (Kravaris et al., 2012).o Kravaris et al. (2012), these techniques include theelection procedures, stochastic search techniques, andduction approaches.

earch

arch is an optimized way to determine an element (e.g., number) in a sorted sequence with no additional infor-en a set of sorted numbers, = {1, 2, 3, 4, 5, 6, 7, 8},number is randomly selected. How many questions areo obtain the selected number making only questions

answered as yes or no? The most guaranteed fastthis answer with no any additional information is asker is in the rst half of the set. Thus, given the answer,inate half of the numbers where the selected number

Fig. 3. Q

a solutnumbeone se

TheFig. 4. selectesucceein this includcontainwill benPS reand ththe paters aroccursrange eproble

In oto a mtions fimportnt. The remaining set is divided by half again, asking if is in the rst half of the remaining set. This procedurees a tree search, in which the branches with no selected

discharged from the search, reducing rapidly the size ofmbers to be investigated. In Fig. 3, the query tree of thisexample of binary search is presented. Note that what-elected number in the set , the number of evaluationsequal to 3.y search the number of necessary evaluations nEval is

n (3)

s the number of possible values in the original set.

ters identiability based on the binary search

rtant aspect in search procedures regarding to the com-cost is the numbers of required evaluations to nd

Hahn, 2007Secchi et a1997; Yao ecedure, theare includesquares in parametersbe evaluatesimultaneosuch purpoFIM leads to

The maiposed procenew paramFIM matrixters may chdo not chaneters is petree of the example of set of numbers using the binary search.

In classical parameters identiability procedures theevaluations may become very high, because the one byn of parameters criteria are chosen.erical procedure proposed in this paper is presented in

algorithm, nPSS represents the number of successfullyrameters as identiable, which is updated at each well-ection. The algorithm tries to include more parameterseing nPS the number of parameters that are temporarily

the set of identiable parameters. However, if the setnPS is not identiable, the set will be reduced and nPSated accordingly. In the case of successive fails untils to nPSS, no additional parameters could be selectedocedure stops. The procedure can also stop when allters are identiable ((NS) is empty), or no parame-ntiable. This last case only happens if some problemdentifying even one parameter, indicating either a badlished for the most important parameter or a modeling; Li et al., 2004; Lund & Foss, 2008; Sandink et al., 2001;l., 2006; Sun & Hahn, 2006; Weijers & Vanrolleghem,t al., 2003, and others). According to the proposed pro-

rst half of the most relevant parameters of the rankd in the set of selected parameters, represented by lledFig. 5. When adding parameters to the set of selected, the conditioning of the estimation procedure shouldd in order to verify if such parameters can be estimatedusly, the invertibility of the matrix FIM can be used forse. In this numerical procedure, the singularity of the

the reduction of the set of selected parameters.n difference on the number of evaluations of the pro-dure regard to the classical binary search is that, when aeter is included in the set of selected parameters and the

is invertible, then the sort of the non-selected parame-ange (in the classical binary search, the sorted sequencege). Specially in the case where re-estimation of param-rformed, even parameters that have already tried to


been includhappen sinFIM may chof non-selethe chosen selected. It once a parafrom the se

In mostparametersprocedure. bility proceanalyses arFig. 4. Numerical procedure proposed for parameter identia

ed without success can be now included, which mayce the values of parameters change, and so the matrixange sufciently to become invertible. Thus, the setcted parameters should be re-evaluated according tocriterion when a new set of parameters is successfullymust be emphasized that, in the proposed procedure,meter is successfully selected it will never be removedt of selected parameters.

procedures presented in the literature, the selected re-estimation is only performed after the identiabilityWithout performing simultaneously with the identia-dure the estimation of the selected parameters, all thee done based on the initial estimates of the parameters

values, carrtaneous paidentiabilparametersmain purpoof parametthat usuallysimultaneomodel t tNeverthelewhen parawith identstep shouldbility: set-by-set selection.

ying great uncertainties out to the results. The simul-rameters re-estimation reduces the dependence of theity procedure regarding to the initial estimates of the

values (Secchi et al., 2006). Moreover, since one of theses of the parameters identiability is to select the seters that can be estimable by optimization algorithms

require the inversion of the Hessian matrix, with theus parameters re-estimation, it is possible to improveo experimental data, resulting in a better prediction.ss, the computational time increases signicantlymeters re-estimation is performed simultaneouslyiability procedure even so, the importance of this

not be despised, becoming relevant the development


Fig. 5. Numer f succeset (S) , (NS)

of identiastages.

Howevetion only amay arise destimates oof the initiproblems tothat the reespecially dmay exceedthe model parametersproblems ahave been a

When pFig. 4, it is ing model of selected performed carded at tevaluation selected parselected pa

Global osian matrixparameterswith complical restrictinto numerition methoEberhart, 19ing good i(Alberton e

An impowhat is the (bs) (Eq. (3selected paworst case,the re-estimthat have amaximum n

tion

Eval

+ Ev

+

+ Ev

1 +

the , whcase,

the ical procedure for 4 parameters to be investigated; nPSS , nPS represent the number ois the set of non-selected parameters.

bility procedures with less parameters estimation

r, if one still decides to perform the parameters estima-fter the identiability procedure, numerical problemsue to modeling errors and/or uncertainties in the initialf the parameters. As previously mentioned, the qualityal estimates of the parameters is one of the major

the identiability procedures. There is no guaranteesulting estimation problems will become well-posed,uring the estimation step, where the parameters values

pre-established parameters ranges, and also duringevaluation step, where some combination of model

values may lead to infeasible solution. Although thesere very common in parameters identiability, theylmost completely neglected in the literature.erforming these steps simultaneously, as described inpossible to overcome problems that may occur dur-solving or parameters re-estimation reducing the setparameters. If the step of parameters re-estimation isin each selection, it is possible that the parameters dis-his step may be selected in a posterior step, when anof the correlation between the set of selected and non-

evalua

NbsEval =

NbsEval =

Forabilityworst only atameters is performed using more suitable values of therameters.ptimization procedures that do not make use of the Hes-

could be used to obtain good initial estimates for the values. However, in many cases of interest, one dealex dynamical models, which present signicant numer-ions regarding to the parameters values without fallingcal problems. Therefore, even derivative-free optimiza-ds, such as particle swarm optimization (Kennedy &95) or genetic algorithm (Goldberg, 1989), for obtain-

nitial estimates, may fail in obtaining such valuest al., 2013).rtant question about the identiability procedure is:number of necessary evaluations? For the binary search)) since there is the possibility of re-sorting the non-rameters when new parameters are selected, in the

only one parameter would be included each time, andation could make possible the inclusion of parameters

lso tried to be included without success. Therefore, theumber of re-estimation procedure calls or FIM matrix

re-evaluatioestimation

NcpEval

= Eval

+ Ev

+ Ev

+ .

Fig. 6 precase in procthe proposecase will sesignicant ed selected parameters and the number of parameters in the selected

is given by:

Isup(log2(nP)) uations to include only the rst parameter

Isup(log2(nP 1)) aluations to include only the second parameter

+ Isup(log2(2)) Evaluations to include only the penult parameter

1aluation to include or remove the last parameter

(4)

nP2k=0

Isup(log2(nP k))

conventional procedures (cp) of parameters identi-ere only one parameter is selected each time, in the

where all the remaining parameters are evaluated andlast evaluation one parameter is included, allowing the

n of all remaining parameters again, the number of re-procedure calls or FIM matrix evaluation is given by:

nPuations to include only the rst parameter

(nP 1) aluations to include only the second parameter

(nP 2) aluations to include only the penult parameter

. . + 1Evaluation to include or remove the last parameter

(5)

sents the number of necessary evaluations for the worstedures that try to select parameters one by one and ford procedure. Although one may expect that the worstldom happen, the proposed procedure really leads to alower number of required evaluations. It is especially


Fig. 6. Compaprocedures thaprocedure ( )

important formed togethe comput

Most of tthe proposeeters to theis evaluatednon-selecteters correlasuch as rel1997) and n1997) can btion previou2001). In pa(6)) is moredifferent fro

NSRGA = B

in which B+

denotes theIn the p

lation is evare denitivinvertible. Tset, then ththe FIM anamatrix is eqcorrespondparametersLi et al. (200

Two exaprocedure astrate that cparametersevaluating t

5. Illustrat

In the rtiability wnumerical order to evcedure whea complex dbe selectedproblem thestimation

The proposed numerical procedure was applied using theparameters identiability criterion of Yao et al. (2003) for eval-uation the importance and correlation of the parameters. Such

n was selected due to its easy implementation and lowtatiocan b

algo all he pl codNewameton prch

(3) tepsereple

ment bothre-eslculeter

amp

us co

1

xk

ch dime

orig, whe

indeto thcreased thtermach xpernded

valuted ica 6le y wby ariancring between numbers of necessary evaluations for the worst case int try to select parameters one by one ( ) and the proposed numerical.

when simultaneous parameters re-estimation is per-ther with identiability procedure, reducing drasticallyational cost.he available criteria to sort the rst set of parameters ind procedure are based on the importance of the param-

model prediction. Usually, the parametric correlation between set of selected parameters (S) and set ofd parameters (NS), avoiding the selection of parame-ted with the parameters previously selected. Methodsative gain array RGA (Bristol, 1966; Cao & Rossiter,on-square relative gain array NSRGA (Cao & Rossiter,e used as criteria for parameters correlation evalua-sly to the rst selection (Botelho, 2012; Sandink et al.,rameters identiability problems, the use of NSRGA (Eq.

indicated because the number of parameters is usuallym the number of measured variables of the model.

(B+)T (6)

is the pseudo inverse of the sensitivity matrix and Hadamard (or element-wise) product.roposed numerical procedure, the parameters corre-aluated at each step of selection, where parametersely considered as identiable parameters if the FIM isherefore, if there are correlated parameters in the rstese parameters will be removed from this set duringlysis. Moreover, the sum of each column of the NSRGAuivalent to the projection of the sensitivity vector of theing parameter on the parametric space of the selected, which is the criterion of linear independence used by4) and Secchi et al. (2006).mples to illustrate the application of the proposedre presented in the next section. The results demon-omputational time and cost are lower than addressing

criteriocomputerion

The95, andRAM. TtationaGauss

Parmizatilinesealimits,these stions wfor comexperi

Forof the tion, caparam

5.1. Ex

Let

y =nXk=

in whitor of d

Theeffectsber ofadded and inexpectorder with elated econfou

ThesimulaStatistvariabadded and va identiability problem where the selection is made byhe parameters one by one.

ive examples

st example, a hypothetical problem of parameters iden-ith known solution is used for verifying if the proposedprocedure is able to make an adequate selection. Inaluate the robustness of the proposed numerical pro-n facing real problems, the second example representsynamic model containing 131 parameters that should

for estimation with scarce experimental data; in thise solution is unknown. On both cases, parameters re-is performed.

nal model f1 = 5, the of successivexperimentstructure is

Two casMoreover, strongly de(Chu & Hahgiven for ththe parameized sensitiwhen a re-epresented inal cost. Nonetheless, any parameters identiability cri-e employed in the procedure.rithm presented in Fig. 4 was implemented in FORTRANcase studies were run in an Intel Core 2 Duo with 3 GBarameter estimation was performed using the compu-e Estima&Planeja (Schwaab et al., 2011), based on theton method.ers estimation with GaussNewton method is an opti-rocedure that includes: (1) model evaluation, (2)for stepsize calculation with checking of parametersevaluation of FIM matrix and its inverse. As each of

may fail interrupting the estimation, some modica- introduced in this code to improve the convergencex parameters identiability problems that have scarceal data.

examples, the results are presented with the valuestimated parameters together with the relative devia-

ated as the ratio between the standard deviation of theand its estimated value (/).

le 1 a simple model

nsider the mathematical model presented in Eq. (7).

k +nX1i=1

nXj=1

xi xj i,j (7)

enotes the parameter, x the independent variable vec-nsion nX, and y the dependent variable.inal mathematical model is described only by mainre the number of parameters, nP, is equal to the num-pendent variables, nX = 31. The crossed effects weree original model to create a super-parameterized modele the complexity of the identiability problem. It isat the 31 30 = 930 parameters relating to the second-s become not identiable because they are correlatedother and were not employed to generate the simu-imental data (y). In this case, the crossed effects are

with the main effects.es of the independent variables x for generating the

experimental data were obtained using the software.0 by Taguchis experimental design. The dependentas generated using the output of the original model

random noise with normal distribution, zero meane 2Y = 0.25. The true parameters values of the origi-orm an arithmetic progression with the rst parameterlast parameter 31 = 155, and the common differencee members equal to 5. In this scenario the simulatedal data are orthogonal and the mathematical model

linear.es were investigated: without and with re-estimation.since the parameters identiability procedures arependent on the initial estimates of parameters valuesn, 2007; Omlin & Reichert, 1999), different values weree parameters as initial estimates. For linear models onters this dependence is generated by using the normal-vity matrix and also by the non-selected parametersstimation step is applied. In both cases, the results aren Table 1.


Table 1Results for parameters identiability procedure based on binary search for cases (a) and (b) of the rst example. The superscripts represent the positions of the parameters.

Symbol Exactvalue

Initial estimates(a.1 and b.1)

Initial estimates(a.2 and b.2)

Estimatedvalues (b.1)

Estimatedvalues (b.2)

123456789101112131415161718192021222324252627282930311,1 to 30,31

a Non-selec

5.1.1. Case The main

and computpared to us4 evaluatiowith convebe requiredbinary searworst case evaluations

In order of the param

Case a.1 magnitudequal to o

Case a.2 magnitudequal to o

As expedecreasing of the seleca.2, the paris, [S] = {1complex prmodels) theon the empseriousnessthe initial eidentiabili

The CPU

Case ilar tinitia5 1.2 31 10 2 30 15 3 29 20 4 28 25 5 27 30 6 26 35 7 25 40 8 24 45 9 23 50 10 22 55 11 21 60 12 20 65 13 19 70 14 18 75 15 17 80 16 16 85 17 15 90 18 14 95 19 13

100 20 12 105 21 11 110 22 10 115 23 9 120 24 8 125 25 7 130 26 6 135 27 5 140 28 4 145 29 3 150 30 2 155 31 1.2

0 1 1

ted parameters.

a without the parameters re-estimation step objective of this case is to demonstrate the lower timeational cost of the proposed numerical procedure com-

5.1.2. Sim

ferent

ual procedures described in the literature. In fact, onlyns were required for selection of 31 parameters, whilentional parameters identiability procedures it would

at least 31 evaluations. Note that the worst case of thech would require 8588 evaluations (Eq. (4)), and theof the conventional procedures would require 461,281

(Eq. (5)).to demonstrate the dependence on the initial estimateseters values, two initial estimates were investigated:

values of the parameters 131 in crescent order ofe, with parameters related to the crossed effects keptne.

values of the parameters 131 in decreasing order ofe, with parameters related to the crossed effects keptne.

cted for the case a.1, the parameters were selected inorder according to their values, being obtained the setted parameters [S] = {31, 30, . . ., 1}. For the caseameters were selected according to their values, that, 2, . . ., 31}. It is important to empathize that for moreoblems (e.g., scarce experimental data, and non-linear

results of parameters selection are highly dependentloyed parameters identiability criteria. Despite the

of the example, this problem of the dependence onstimates also occurs in the conventional parametersty procedures.

time spent for the selections is the same in both cases.

Case b.1 magnitudequal to o

Case b.2 magnitudequal to o

Introducters were ssame as obters valuesdeviations estimated pand in casetion were thto get goodence betweis mainly asexact modeues were keeffects wer

In this eset of paramto the same

5.2. Examp

Represeproblems (c159.265(31) 14.1113(1)

152.752(30) 13.8686(2)

144.396(29) 17.6081(3)

145.181(28) 23.0792(4)

138.824(27) 26.5707(5)

134.280(26) 31.0258(6)

129.384(25) 39.2003(7)

120.687(24) 45.6158(8)

123.449(23) 46.3685(9)

115.398(22) 57.3666(10)

108.222(21) 63.0269(11)

104.095(20) 64.8253(12)

103.921(19) 72.3390(13)

88.3366(18) 68.5984(14)

85.0125(17) 76.4169(15)

86.7994(16) 86.7994(16)

76.4169(15) 85.0125(17)

68.5984(14) 88.3366(18)

72.3390(13) 103.921(19)

64.8253(12) 104.095(20)

63.0269(11) 108.222(21)

57.3666(10) 115.398(22)

46.3685(9) 123.449(23)

45.6158(8) 120.687(24)

39.2003(7) 129.384(25)

31.0258(6) 134.280(26)

26.5707(5) 138.824(27)

23.0792(4) 145.181(28)

17.6081(3) 144.396(29)

13.8686(2) 152.752(30)

14.1113(1) 159.265(31)a a

b with the parameters re-estimation stepo the case (a), investigations were performed with dif-l estimates as follows: values of the parameters 131 in crescent order ofe, with parameters related to the crossed effects keptne.

values of the parameters 131 in decreasing order ofe, with parameters related to the crossed effects keptne.

ing the step of parameters re-estimation, 31 parame-elected, where the parameters selection order was thetained without re-estimation. The estimated parame-

are presented in Table 1; for both cases, the relativewere about 1 102, indicating a good accuracy of thearameters. The CPU time spent in case b.1 was 1.30 min

b.2 was 1.84 min. In both cases, the results of estima-e same, as expected, and the re-estimations were able

estimates for the true parameters values. The differ-en re-estimated parameters and true parameters valuessociated with the crossed effects, that are zero for thel and equal to one for the initial estimates, and such val-pt unchanged during the estimation since the crossed

e not included in the set of estimable parameters.xample, the NSRGA was also applied to select the rsteters, in order to avoid correlated parameters, leading

results.

le 2 a complex dynamic model

nting a complex real case of parameters identiabilityomplex dynamic model, large number of parameters,


parametersrelated, andmetabolic nMauch, & Rexample. Adcostly, timetant variabunknown.

Large scphosphate-K-12 W311

The manetworks istions that rintracellularepresent thFig. 7. Escherichia coli central carbon met

with low inuence for model prediction and/or cor- scarce experimental data), the E. coli K-12 W3110etwork model (Chassagnole, Noisommit-Rizzi, Schimd,euss, 2002; Di Maggio et al., 2009) is used in thisditionally, conducting experiments for such system are

consuming and difcult. Consequently, some impor-les are not measured and their initial conditions are

ale metabolic network of the glycolysis, the pentose-pathway and the phosphotransferase system of E. coli0 is presented in Fig. 7.thematical model of E. coli K-12 W3110 metabolic

described by a system of 18 ordinary differential equa-epresent mass balances for extracellular glycolysis andr metabolites (Eqs. (8)(25)), 7 forcing functions thate co-metabolites (Eqs. (A1)(A7)) and 30 kinetic rates

that repres2002). Thisselected.

ModicaDiaz (2010)sion for theRicci (1996Ratushny ephosphate

dCextracellulaglcdt

dCg6pdt

= rPTabolism.

ent the enzymes (Eqs. (A8)(A37)) (Chassagnole et al., dynamic model has 131 candidate parameters to be

tions were introduced by Di Maggio, Diaz Ricci, and for some enzyme kinetics. In this sense, kinetic expres-

activity of phosphofructokinase was taken from Diaz) (Eq. (A5)) and other modications are given byt al. (2006) for modeling the activity of glucose-6-dehydrogenase (Eq. (A6)).

r

= D(Cfeedglc Cextracellularglc ) + fpulse CxrPTS

x(8)

S rPGI rG6PDH rPGM Cg6p (9)


dCf6pdt

= rPGI rPFK + rTKb + rTA 2rMurSynth Cf6p (10)

dCfdpdt

= rPK

dCgapdt

= rAL

dCdhapdt

= rA

dCpgpdt

= rGA

dC3pgdt

= rPG

dC2pgdt

= rPG

dCpepdt

= rEN

dCpyrdt

= rPK

dC6pgdt

= rG6

dCribu5pdt

= r

dCxyl5pdt

= rR

dCsed7pdt

= r

dCrib5pdt

= rR

dCe4pdt

= rTA

dCglpdt

= rPGThe com

for the numThe exp

Tomita, anand for tdihydroxya(e4p), pho1,6-diphospribose-5-ph2-phosphog6-phosphatintracellulainitial condimportant i

Table 2Results for parameters identiability based on the information theory of E. colimetabolic networks.

Symbol Nominal value Symbol Nominal value

conditions5.5406 101 C3gp 1.28516.7016 102 Csed7p 4.0874 1023.7000 105

eters4.0000 101 KGAPDH,pgp 5.0000 1051.0519 103 KG6PDH,nad,ihn 9.0000 1031.2873 101 rmax

Synthesis25.6900 101

2 2.0000 101 rmaxPK 5.6700 1011.3900 101 rmaxPTS 8.21073 104

lase,fdp 7.0000 101 KPTS1 3.0823 103lase,fdp

q

,gap

be presmatiaintyepen

yi +

a and esti

knom forg. Therimbitrae the

10

tude

Case sideetersical llula

20 pablenon-metrrelared iing E. coli metabolic behavior. Other important factor is

to the initial estimates of parameters values that can be the true parameters values, and are used in the evaluation

meters identiability criterion.

of the estimation of initial conditions for non-measured intracellularites using the proposed numerical procedure with parameters re-on.

l Nominal value Estimated value Relative deviation

5.5406 101 5.2742 101 8.9433 1036.7016 102 6.4919 102 9.8674 1013.7000 105 3.0857 103 3.39571.2851 1.7901 3.55164.0874 102 4.0850 102 1.4311 104F rALDO Cfdp (11)

DO + rTIS rGAPDH + rTKa + rTKb rTA + rTrpSynth Cgap(12)

LDO rTIS rG3PDH Cdhap (13)

PDH rPGK Cpgp (14)

K rPGGluMu rSerSynth C3pg (15)

luMu rENO C2pg (16)

O rPK rPTS rPEPCxylase rDAHPS rSynth1 Cpep(17)

+ rPTS rPDH rSynth2 + rMetSynth + rTrpSynth Cpyr(18)

PDH rPGDH C6pg (19)

PGDH rRu5P rR5PI Cribu5p (20)

u5P rTKa rTKb Cxyl5p (21)

TKa rTA Csed7p (22)

5PI rTKa rRPPK Crib5p (23)

rTKb rDAHPS Ce4p (24)

M rGIPAT Cglp (25)

putational code DASSL (Petzold, 1989) was employederical integration of the dynamic model.erimental data were obtained in Hoque, Ushiyama,d Shimizu (2005) for a time horizont of 300 she following measured variables: glucose (glc),cetonephosphate (dhap), erythrose-4-phosphatesphoenolpyruvate (pep), pyruvate (pyr), fructose-hate (fdp), glyceraldehyde-3-phosphate (gap),osphate (rib5p), ribulose-5-phosphate (ribu5p),licerate (2pg), glucose-6-phosphate (g6p), fructose-e (f6p), and 6-phosphogluconate (6pg). The otherr metabolites were not measured and thus, theiritions are unknown. Since these initial conditions arenformation for the mathematical model, their values

InitialCxyl5pCg1pCpgp

ParamKPGI,g6prmaxPGIrmaxG6PDHKPGI,g6pKTIS,eqKPEPCxynPEPCxyKRU5P,ermaxGAPDHKGAPDH

shouldwill bere-estiuncertthe ind

yi = a

whereor even

Theprobleneerinall expand aracterizand b =magni

5.2.1. Con

paramnumerintracewhereidentiber of of paraare comeasudescribrelatedfar fromof para

Table 3Results metabolestimati

Symbo

Cxyl5pCg1pCpgpC3gpCsed7p4.0000 KSynthesis2,pyr 1.00001.4000 rmaxPGDH 5.22076

50.0000 KPGDH,nadp 5.0600 1021.2000 KPGDH,nadph 1.3769 102

estimated together with the parameters. The resultsented for two cases: without and with the parameterson step. As usual, for both cases, the experimental

is admitted proportional to the measured values ofdent variable (yi) in the form:

b (26)

b can be arbitrarily chosen or obtained experimentally,mated.wledge of experimental uncertainty is a signicant

the majority of investigated systems in chemical engi-e appropriate knowledge would require replications forents. It is important information, but often it is neglectedrily chosen due to the hard work that would be to char-

experimental errors. In our case, we admitted a = 1026 (particularly the value of b was chosen based on theof experimental data values).

a without the parameters re-estimation stepring 5 unknown initial conditions and 131 model, 136 parameters were evaluated by the proposedprocedure using the available experimental data forr metabolites. The results are summarized in Table 2,arameters and 5 initial conditions were selected as, being required only 13 evaluations. The high num-selected parameters may indicate that a large numberers present low inuence for model prediction and/orted with the selected parameters, and that the non-ntracellular metabolites store relevant information for

190K.P.F.

Alberton

et al.

/ Com

puters and

Chemical

Engineering 55 (2013) 181 197

Table 4Results of parameters selection using the proposed numerical procedure with parameters re-estimation.

Parameter Nominal value Estimated value Relative deviation Parameter Nominal value Estimated value Relative deviation

KPGI,g6p 4.0000 101 1.0769 102 3.0814 KPGK,adp 1.8500 101 1.8500 101 1.9104 102rmaxPGI 1.0519 103 5.9234 102 1.2569 KGAPDH,nad 2.5200 101 2.5213 101 6.7565 101rmaxG6PDH 1.2873 101 7.2217 2.0269 101 KG6PDH,nad,inh 9.0000 103 8.6536 103 6.1194 101KPGI,f6p2 2.0000 101 1.1109 105 7.6247 105 rmaxRU5P 5.1816 2.2516 2.5393 101KPGK,3pg 5.1700 101 4.5490 101 4.4129 rmaxR5PI 3.7333 1.9924 4.6917KPEPCxylase,fdp 7.0000 101 4.4571 101 3.7857 rmaxALDO 3.0423 101 4.0000 1.7730 103nPEPCxylase,fdp 4.0000 9.2161 2.3282 KENO,2pg 1.0000 101 1.0000 101 1.6760 101rmaxPGDH 5.2208 9.4885 101 1.2338 103 KTIS,dhap 2.8000 2.5225 2.2587 103rmaxPFK 1.9670 1.4669 101 2.5160 KR5PI,eq 4.0000 1.4200 3.5450 101rmaxGAPDH 5.0000 102 9.4187 9.5864 rmaxPDH 4.0104 6.2336 101 7.4603 101KRU5P,eq 1.4000 1.4855 8.9487 103 KPGLUMU,3pg 2.0000 101 2.0000 101 6.9664 102KGAPDH,gap 1.2000 2.2312 102 1.3122 103 HG6PDH,nadh 1.2000 101 5.1769 101 5.1116 103KGAPDH,pgp 5.0000 105 8.3295 105 1.3434 103 KTKa,eq 1.3500 101 1.1685 101 2.3249 104KPGDH,nadp 5.0600 102 1.1772 1.3215 103 KENO,pep 5.2056 101 1.3500 101 3.9731 101KPGDH,nadph 1.3769 102 7.3728 102 4.0426 102 rmaxTIS 2.2000 8.7199 102 1.9057 103KPGDH,atp 2.0800 102 5.4786 101 4.2688 103 nDAHPS,pep 3.6900 101 2.4990 2.6254 101KPGM,eq 1.5800 101 1.3716 101 8.7306 101 KPGLUMU,2pg 3.0000 102 3.6900 101 6.3087 102KALDO,eq 1.4438 102 1.4935 102 2.6863 KG6PDH,nadph 1.4000 2.8818 102 7.5260 103rmaxSynth2

5.6900 101 4.2606 102 2.6446 101 HG6PDH,nadph 8.8572 103 1.4032 8.6816 103rmaxPK 5.6700 101 3.5304 102 6.0122 101 KR,f6p 2.0000 102 8.8801 103 9.6690 103rmaxPTS 1.0000 1.0745 3.5897 103 KMG6PDH,nadp 1.0500 1.9406 102 3.7786 103KPTS1 2.6000 101 2.3386 103 3.6104 103 KTA,eq 1.3900 101 1.0507 1.4314 104KSynthesis2,pyr 4.6800 102 1.8140 101 1.5514 102 KTIS,eq 9.5000 102 1.3896 101 2.9680 102KPK,adp 2.6000 2.5944 101 2.9091 102 rmaxPEPCxylase 8.8000 102 1.0256 102 6.2233 101KPGK,pgp 1.8700 101 5.4125 102 3.5774 KALDO,dhap 9.9000 101 8.8000 102 1.4290 102nDAHP,se4p 3.1000 102 3.4499 3.5791 e 5.6180 101 4.0301 101 9.6704 103KPGLUMU,eq 1.0000 9.4505 101 7.0688 rmaxPGM 5.4491 9.0650 102 5.7009 101KPK,pep 4.3000 102 3.0968 102 1.4155 102 KPGDH,6pg 3.0000 101 3.6871 1.1154 102nPDH 4.0700 9.2200 1.5091 KG6PDH,nadh 3.0000 101 3.0070 101 1.0491 104KPGI,eq 6.7000 101 4.2994 102 7.1124 101 KG6PDH,dtn 7.0000 103 3.0613 101 4.4899 104KPEPCxylase,pep 8.4613 8.0500 6.6521 101 KMG6PDH,g6p 1.8000 103 6.9652 103 2.2991 102KENO,eq 2.0000 101 7.1033 101 2.0301 KPGK,eq 1.0000 8.9599 103 2.2371 103rmaxTA 4.2260 103 3.2541 102 1.4314 104 KTIS,gap 1.5000 103 3.4455 101 3.0763 102KPGI,g6p2 2.5274 102 5.9448 101 4.9639 rmaxG3PDH 5.3000 103 1.0700 102 5.4517 104rmaxPGK 7.3718 2.7500 101 2.3841 103 KG3PDH,dhap 1.8500 101 6.5999 6.1520 104rmaxENO 9.6972 101 2.5000 1.0603 101 rmaxMurSynth 2.5200 101 1.4961 103 1.6909 102rmaxTKa 4.0000 101 5.7700 8.9491 103 KDAHPS,pep 9.0000 103 5.3000 103 1.5787 102rmaxPGLUMU 1.0519 103 7.0600 101 6.8782 101


Fig. 8. Experimpredicted valu

5.2.2. Case Includin

numerical ental and predicted metabolites concentrations as function of the time: ( ) experime after parameter identiability.

b with the parameters re-estimation stepg the step of parameters re-estimation, the proposedprocedure was able to select 75 parameters and 5

unknown infore, 55 adparametersental value, (--) predicted value using initial estimates and ( )

itial conditions, demanding only 19 iterations. There-ditional parameters were selected by introducing the

re-estimation step with only 6 additional evaluations.


It must be emphasized that the selection of the parameters usingthe one by one procedure would require at least 80 iterations.Actually, the save of computational time is much higher, since theusual procedure demands much higher iterations, forasmuch theinclusion of one parameter at each iteration often is not succeed,and several parameters must be tested before one additionalparameter proposed pperformed each time, aThe resultsboth param

The initTable 3, wesmall timesintracellulathe non-meend of the tions. Withpresent a si

Most of present diffto the initiestimates oas observedprediction parametersestimated vinuenced Special attewhich the meters valuedata.

Applyingters in orderesults. Theparameters

6. Conclus

In this wbased on biof parametewere preseIn the rstthe numeriing to theiscarce expecal model oand 5 unktion. In botreduction ocompared tselection.

The use lation did nIn fact, theevaluating tselection.

A discustogether wiand it was for reducinestimates.

Acknowledgments

The authors thank CNPq (Conselho Nacional de Desen-volvimento Cientco e Tecnolgico), CAPES (Coordenac o deAperfeic oamento de Pessoal de Nvel Superior), National ResearchCouncil (CONICET), Universidad Nacional del Sur and ANPCYT, for

ing scholarships and for supporting this work.

dix A.

Tables A1A4.

lature.

olitesGlycoseGlycose-6-phosphateGlycose-6-phosphateFructose-1,6-biphosphateGlyceraldehyde-3-phosphateDihydroxyacetonephosphate1,3-Diphosphoglycerate3-Phosphoglicerate2-PhosphogliceratePhosphoenolpyruvatePyruvate6-Phosphogluconate

Ribulose-5-phosphateXylulose-5-phosphate

Sedoheptulose-7-phosphateXylulose-5-phosphate

Sedoheptulose-7-phosphateRibose-5-phosphateErythrose-4-phosphateGlycose-1-phosphate

tabolitesAdenosintriphosphateAdenosindiphosphateDiphosphopyridindinucleotide, oxizedDiphosphopyridindinucleotide, reducedDiphosphopyridindinucleotide-phosphate, oxized

Diphosphopyridindinucleotide-phosphate, reducedAdenosinmonophosphate

esAldolase

S DAHPS synthaseEnolase

Glycose-1-phosphate adenyltransferase Glycerol-3-phosphate dehydrogenase Glycose-6-phosphate dehydrogenase

Glyceraldehyde 3-phosphate dehydrogenasenth Methionine synthesisnth Mureine synthesis

Pyruvate dehydrogenaseylase PEP carboxylase

Phosphofructokinase 6-Phosphogluconate dehydrogenase

PhosphoglucoisomerasePhosphoglycerate kinasePhosphoglucomutase

u Phosphoglycerate mutasePyruvate kinasePhosphotransferase systemRibose phosphate isomerase

th 1 2

th is denitively included. The CPU time spent with therocedure was only 3.24 min. For comparison, we havethe identication procedure adding one parameternd obtained the results in 2.28 h, with similar results.

are summarized in Tables 3 and 4, respectively, foreters and initial conditions.ial estimates of the initial conditions, presented inre obtained in two steps: (1) integrate the model with atep and null initial concentrations of the non-measuredr metabolites, and (2) take the calculated values forasured intracellular metabolites concentrations at therst timestep as initial estimates for the initial condi-

the parameters re-estimation step, these values do notgnicant inuence.the estimated parameters values presented in Table 4erence of at least one order of magnitude with respectal estimates. This difference demonstrates that initialf the parameters could be signicantly improved,

in Fig. 8, comparing experimental data with modelusing initial estimates and estimated values after the

identiability. It is important to emphasize that thealues of the selected parameters can be signicantly

by the initial estimates of the non-selected parameters.ntion should be given at Fig. 8(b), (c) and (g)(j), forathematical model behavior with re-estimated param-

s was able to lead to a better t to the experimental

the NSRGA criteria to select the rst set of parame-r to avoid correlated parameters did not change therefore, the additional overhead to select the rst set of

seems to be unnecessary.

ion

ork, a numerical procedure for parameters selectionnary search was proposed in order to solve the problemrs identiability with minimum effort. Two examplesnted to illustrate the performance of the procedure.

example a simple case was presented showing thatcal procedure is able to select the parameters accord-r potential of estimation. The second example is arimental data scenario, using a complex mathemati-f E. coli metabolic network, containing 131 parametersnown initial conditions of metabolites concentra-h cases, the proposed procedure leads to signicantf the computational effort and similar results wheno the traditional procedures of one by one parameter

of NSRGA for previous evaluation of parameters corre-ot improve the results for the investigated examples.

results obtained with the NSRGA are overlapped byhe correlation parameter performed in each step of the

sion about the simultaneous parameters re-estimationth parameters identiability procedures was presented,shown that such simultaneous step is very importantg the inuence of uncertainties of initial parameters

provid

Appen

See

Table A1Nomenc

Metabglc g6p f6p fdp gap dhap pgp 3pg 2pg pep pyr 6pg ribu5pxyl5p sed7pxyl5p sed7prib5p e4p glp

Co-meatp adp nad nadh nadp nadphamp

EnzymALDO DAHPENO G1PATG3PDHG6PDHGAPDHMetSyMurSyPDH PEPCxPFK PGDHPGI PGK PGM PGluMPK PTS R5PI RPPK RU5P SerSynSynthSynthTA TIS TKa TKb TrpSynRibose phosphate pyrophosphokinaseRibulose phosphate epimeraseSerine synthesisSynthesis 1Synthesis 2TransaldolaseTriosephosphate isomeraseTransketolase aTransketolase bTryptophan synthesis

K.P.F.

Alberton

et al.

/ Com

puters and

Chemical

Engineering 55 (2013) 181 197

193

Table A2Forcing functions for co-metabolites concentrations (mM) (Chassagnole et al., 2002).

Catp = 4.27 4.163 t0.657 + 1.43t + 0.364t2 (A1)

Cadp = 0.582 + 1.73(2.7310.15t)(0.12t + 0.000214t3) (A2)Camp = 0.123 + 7.25 t

7.25 + 1.47t + 0.17t2 + 1.073t

1.29 + 8.05t (A3)Cnadph = 0.062 + 0.332(2.7180.46t)(0.0166t1.58 + 0.000166t4.73 + 1.16 1010t7.89 + 1.36 1013t11 + 1.23 1016t14.2) (A4)Cnadp = 0.159 + 0.00554

t

2.8 + 0.271t + 0.01t2 + 0.182t

4.81 + 0.526t (A5)Cnadh = 0.0934 + 0.0011(2.3710.123t)(0.844t + 0.104t3) (A6)Cnad = 1.314 + 1.314(2.73(0.0435t0.342))

(t + 7.871)(2.73(0.0218t0.171))8.481 + t (A7)

Table A3Kinetics rates for enzymes (Chassagnole et al., 2002).

rPTS =rmaxPTS C

extracelularglc

(Cpep/Cpyr)

(KPTS1 + KPTS2(Cpep/Cpyr) + KPTS,3Cextracelularglc + Cextracelularglc (Cpep/Cpyr))(1 + (CnPTS,g6pg6p /KPTS,g6p))

(A8)

rPGI =rmaxPGI (Cg6p (Cf6p/KPGI,eq))

KPGI,g6p(1 + (Cf6p/(KPGI,f6p(1 + (C6pg/KPGI,f6p,6pginh)))) + (C6pg/KPGI,g6p,6pginh)) + Cg6p(A9)

rPFK =rmaxPFK

[(eCf6p/KRf6p)(1 + (eCf6p/KRf6p))nPFK1(1 + (Cadp/KRadp))nPFK + L0[((1 + (Cpep/KTpep))(1 + (Cadp/KTadp))/((1 + (Cpep/KRpep))(1 + (Cadp/KRadp))))]nPFK ce(Cf6p/KRf6p)(1 + ce(Cf6p/KRf6p))nPFK1

](1 + (eCf6p/KRf6p))nPFK (1 + (Cadp/KRadp))nPFK + L0[((1 + (Cpep/KTpep))(1 + (Cadp/KTadp)))/((1 + (Cpep/KRpep))(1 + (Cadp/KRadp)))]nPFK (1 + ce(Cf6p/KRf6p))Cpep/KTpep

(A10)

rALDO =rmaxALDO(Cfdp (CgapCdhap/KALDO,eq))

KALDO,fdp + Cfdp + (KALDO,gapCdhap/KALDO,eqVALDO,blf) + (KALDO,dhapCgap/KALDO,eqVALDO,blf) + (CfdpCgap/KALDO,gap,inh) + (CdhapCgap/KALDO,eqVALDO,blf)(A11)

rTIS =rmaxTIS (Cdhap (Cgap/KTIS,eq))

KTIS,dhap(1 + (Cgap/KTIS,gap)) + Cdhap(A12)

rGAPDH =rmaxGAPDH(CgapCnad (CpgpCnadh/KGAPDH,eq))

(KGAPDH,gap(1 + (Cpgp/KGAPDH,pgp)) + Cgap)(KGAPDH,nad(1 + (Cnadh/KGAPDH,nadh)) + Cnad)(A13)

rPGK =rmaxPGK (CadpCpgp (CatpC3pg/KPGK,eq))

(KPGK,adp(1 + (Catp/KPGK,atp)) + Cadp)(KPGK,pgp(1 + (C3pg/KPGK,3pg)) + Cpgp)(A14)

rPGluMu =rmaxPGluMu

(C3pg (C2pg/KPGluMu,eq))KPGluMu,3pg(1 + (C2pg/KPGluMu,2pg) + C3pg

(A15)

rENO =rmaxENO (C2pg (Cpep/KENO,eq))

KENO,2pg(1 + (Cpep/KENO,pep)) + C2pg(A16)

rPK =rmaxPK CpepCadp((Cpep/KPK,pep) + 1)

(nPK1)

KPK,pep(LPK(((1 + (Catp/KPK,atp))/((Cfdp/KPK,fdp) + (Camp/KPK,amp) + 1)))nPK + ((Cpep/KPK,pep) + 1)nPK )(Cadp + KPK,adp)(A17)

194K.P.F.

Alberton

et al.

/ Com

puters and

Chemical

Engineering 55 (2013) 181 197

Table A3 (Continued)

rPDH =rmaxPDHC

nPDHpyr

KPDH,pyr + CnPDHpyr(A18)

rG6PDH =rmaxGPDHCg6p

(KG6PDH,g6p(1 + ktgn(Chtgnadp/(khtgtg + C

htgnadp

))) + Cg6p)(1 + (Cnadph/KG6PDH,nadphinh) + (Cnadh/KG6PDH,nadhinh))((Cnadp/KG6PDH,nadp)1+kdtn(Chdtnadh

/(khdtdt

+Chdtnadh

)))/

((1 + (Cnadp/KG6PDH,nadp)1+kdtn(Chdtnadh

/(khdtdt

+Chdtnadh

)) + (Cnadph/KG6PDH,nadphinh2)hnadph )(1 + (Cnadh/KG6PDH,nadh)hnadh ))

(A19)

rPGDH =rmaxPGDHC6pgCnadp

(C6pg + KPGDH,6pg)(Cnadp + KPGDH,nadp(1 + (Cnadph/KPGDH,nadph,inh))(1 + (Catp/KPGDH,atp,inh)))(A20)

rR5PI = rmaxR5PI(

Cribu5p Crib5pKR5PI,eq

)(A21)

rRU5P = rmaxRU5P(

Cribu5p Cxyl5p

KRU5P,eq

)(A22)

rTKa = rmaxTKa(

Crib5pCxyl5p Csed7pCgap

KTKa,eq

)(A23)

rTKb = rmaxTKb(

Cxyl5pCe4p Cf6pCgapKTKb,eq

)(A24)

rTA = rmaxTA(

CgapCsed7p Ce4pCf6pKTA,eq

)(A25)

rPEPCxylase =rmaxPEPCxylase

Cpep(1 + (Cfdp/KPEPCxylase,fdp)nPEPCxylase,fdp )KPEPCxylase,pep + Cpep

(A26)

rSynth1 =rmaxSynth1

Cpep

KSynth1,pep + Cpep(A27)

rSynth2 =rmaxSynth2

Cpyr

KSynth2,pyr + Cpyr(A28)

rSerSynth =rmaxSerSynth

C3pg

KSerSynth,3pg + C3pg(A29)

rRPPK =rmaxRPPKCrib5p

KRPPK,rib5p + Crib5p(A30)

rG3PDH =rmaxG3PDHCdhap

KG3PDH,dhap + Cdhap(A31)

rDAHPS =rmaxDAHPSC

nDAHPS,e4pe4p C

nDAHPS,peppep

(KDAHPS,e4p + CnDAHPS,e4pe4p )(KDAHPS,pep + C

nDAHPS,peppep )

(A32)

rPGM =rmaxPGM(Cg6p (Cg1p/KPGM,eq))

KPGM,g6p(1 + (Cg1p/KPGM,g1p)) + Cg6p(A33)

rG1PAT =rmaxG1PATCg1pCatp(1 + (Cfdp/KG1PAT,fdp)

nG1PAT,fdp )

(KG1PAT,g1p + Cg1p)(KG1PAT,atp + Catp)(A34)

rMurSynth = rmaxMurSynth (A35)rTrpSynth = rmaxTrpSynth (A36)rMetSynth = rmaxMetSynth (A37)


Table A4Kinetics parameters of the model.

Enzyme Parameters Description

Phosphotransferase system (PTS)

KPTS1 MM half-saturation constant (mM)KPTS2 Constant (mM)KPTS3 ConstantKPTS,g6p Inhibition constant (mM)nPTS,g6p ConstantrmaxPTS Maximum reaction rate (mM s

1)

Phosphoglucoisomerase (PGI)

KPGI,g6p MM half-saturation constant (mM)KPGI,f6p Inhibition constant (mM)KPGI,eq Equilibrium ConstantKPGI,g6p,6pg,inh Inhibition constant (mM)KPGI,f6p,6pg,inh Inhibition constant (mM)rmaxPGI Maximum reaction rate (mM s

1)

Phosphofructokinase (PFK)

KPFK,f6p,s MM half-saturation constant (mM)KPFK,atp,s MM half-saturation constant (mM)KPFK,adp,a Activation constant (mM)KPFK,adp,b Activation constant (mM)KPFK,adp,c Activation constant (mM)KPFK,amp,a Activation constant (mM)KPFK,amp,b Activation constant (mM)KPFK,pep Inhibition constant (mM)LPFK Allosteric constantnPFK Number of binding sitesrmaxPFK Maximum reaction rate (mM s

1)

Aldolase (ALDO)

KALDO,fdp MM half-saturation constant (mM)KALDO,dhap MM half-saturation constant (mM)KALDO,gap MM half-saturation constant (mM)KALDO,gap,inh Inhibition constant (mM)VALDO,blf Back-forward reaction rate relationKALDO,eq Equilibrium constant (mM)rmaxALDO Maximum reaction rate (mM s

1)

Triosephosphate isomerase (TIS)

KTIS,dhap MM half-saturation constant (mM)KTIS,gap MM half-saturation constant (mM)KTIS,eq Equilibrium constantrmaxTIS Maximum reaction rate (mM s

1)

Glyceraldehyde-3-phosphate dehydrogenase (GAPDH)

KGAPDH,gap MM half-saturation constant (mM)KGAPDH,pgp Inhibition constant (mM)KGAPDH,nad MM half-saturation constant (mM)KGAPDH,nadh Inhibition constant (mM)KGAPDH,eq Equilibrium constantrmaxGAPDH Maximum reaction rate (mM s

1)

Phosphoglycerate kinase (PGK)

KPGK,pgp MM half-saturation constant (mM)KPGK,3pg Inhibition constant (mM)KPGK,adp MM half-saturation constant (mM)KPGK,atp Inhibition constant (mM)KPGK,eq Equilibrium constantrmaxPGK Maximum reaction rate (mM s

1)

Phosphoglycerate mutase (PGluMu)

KPGluMu,3pg MM half-saturation constant (mM)KPGluMu,2pg Inhibition constant (mM)KPGluMu,eq Equilibrium constantrmaxPGluMu

Maximum reaction rate (mM s1)

Enolase (ENO)

KENO,2pg MM half-saturation constant (mM)KENO,pep Inhibition constant (mM)KENO,eq Equilibrium constantrmaxENO Maximum reaction rate (mM s

1)

Pyruvate kinase (PK)

KPK,pep MM half-saturation constant (mM)KPK,adp MM half-saturation constant (mM)KPK,atp Inhibition constant (mM)KPK,fdp Activation constant (mM)KPK,amp Activation constant (mM)LPK Allosteric constantnPK Number of binding sitesrmaxPK Maximum reaction rate (mM s

1)

Pyruvate dehydrogenase (PDH)KPDH,pyr MM half-saturation constant (mM)nPDH Number of binding sitesrmaxPDH Maximum reaction rate (mM s

1)

Phoenolpyruvate carboxylase (PEPCxylase)

KPEPCxylase,pep MM half-saturation constant (Mm)KPEPCxylase,fdp Activation constant (mM)nPEPCxylase,fdp Number of binding sitesrmaxPEPCxylase



Table A4 (Continued)

Enzyme Parameters Description

Phosphogluc

KPGM,g6p MM half-saturation constant (mM)KPGM,g1p Inhibition constant (mM)

Glucose-1-p

Ribose phos

Glycerol-3-p p

Serine synth pg

DAHP synth

Glucose-6-p

6-Phosphog

Ribulose pho

Ribose phos

Transketolas

Transketolas

Transaldolas

Synthesis 1

Synthesis 2

Mureine synTryptophan Methionine

References

Botelho, V. (20de parmeGrande do

Bristol, E. (196IEEE Trans

Brun, R., Reichenvironme

Cao, Y., & Rossselection.

Chassagnole, CDynamic mnology and

Chu, Y., & Hadynamic s

Chu, Y., Huanging for effeResearch, 5omutase (PGM)KPGM,eqrmaxPGM

hosphate adenyltransferase (G1PAT)

KG1PAT,g1pKG1PAT,atpKG1PAT,fdpnG1PAT,fdprmaxG1PAT

phate pyrophosphokinase (RPPK)KRPPK,rib5prmaxRPPK

hosphate dehydrogenase (G3PDH)KG3PDH,dharmaxG3PDH

esis (SerSynth)KSerSynth,3rmaxSerSynth

KDAHPS,e4pase (DAHPS)KDAHPS,pepnDAHPS,e4pnDAHPS,peprmaxDAHPS

hosphate dehydrogenase (G6PDH)

KG6PDH,g6pKG6PDH,nadpKG6PDH,nadph,nadpKG6PDH,nadph,g6phrmaxG6PDH

luconate dehydrogenase (PGDH)

KPGDH,6pgKPGDH,nadpKPGDH,nadph,inhKPGDH,atp,inhrmaxPGDH

sphate epimerase (RU5P)KRU5P,EQrmaxRU5PI

phate isomerase (R5PI)KR5PI,eqrmaxR5PI

e a (TKa)KTKa,eqrmaxTKa

e b (TKb)KTKb,eqrmaxTKb

e (TA)KTA,eqrmaxTA

KSynth1,peprmaxSynth1

KSynth2,pyrrmaxSynth2

thesis (MurSynth) rmaxMurSynth

synthesis (TrpSynth) rmaxTrpSynth

synthesis (MetSynth) rmaxMetSynth

12). Nova metodologia para anlise de identicabilidade e estimac otros de modelos fenomenolgicos. Brazil: Universidade Federal do Rio

Sul (Masters thesis, in Portuguese).6). On a new measure of interaction for multivariable process control.actions on Automatic Control, 11, 133134.ert, P., & Kunsch, H. R. (2001). Practical identiability analysis of largental simulation models. Water Resources Research, 37, 10151030.

iter, D. (1997). An input pre-screening technique for control structureComputers Chemical Engineering, 21, 563569.., Noisommit-Rizzi, N., Schimd, J. W., Mauch, K., & Reuss, M. (2002).odeling of the central carbon metabolism of Escherichia coli. Biotech-

Bioengineering, 79, 5372.hn, J. (2007). Parameter set selection for estimation of nonlinearystems. AIChE Journal, 53, 28582870., Z., & Hahn, J. (2011). Global sensitivity analysis procedure account-ct of available experimental data. Industrial & Engineering Chemistry0, 12941304.

Chu, Y., Huangset selectiocriterion. A

Diaz Ricci, J. Cior of phos145150.

Di Maggio, J.ysis in dy770781.

Di Maggio, J., models fogramming1570-7946

Goldberg, D. ENew York

Golub, G. H., &Hopkins U

Hiskens, I. A. (surementsEquilibrium constantMaximum reaction rate (mM s1)

MM half-saturation constant (mM)MM half-saturation constant (mM)Activation constant (mM)Number of binding sitesMaximum reaction rate (mM s1)

MM half-saturation constant (mM)Maximum reaction rate (mM s1)



MM half-saturation constant (mM)

MM half-saturation constant (mM)Number of binding sitesNumber of binding sitesMaximum reaction rate (mM s1)

MM half-saturation constant (mM)MM half-saturation constant (mM)

h,inh Inhibition constant (mM),inh Inhibition constant (mM)


MM half-saturation constant (mM)MM half-saturation constant (mM)Inhibition constant (mM)Inhibition constant (mM)Maximum reaction rate (mM s1)

Equilibrium constant (mM)Maximum reaction rate (mM s1)







Maximum reaction rate (mM s1)Maximum reaction rate (mM s1)Maximum reaction rate (mM s1)

, Z., Hahn, J., Chu, Y., & Hahn, J. (2012). Generalization of a parametern procedure based upon orthogonal projections and the D-optimalityIChE Journal, 58, 20852096.. (1996). Inuence of phosphoenolpyruvate on the dynamic behav-phofructokinase of Escherichia coli. Journal of Theoretical Biology, 178,

A., Ricci, J. D., & Daz, M. S. (2009). Global sensitivity anal-namic metabolic networks. Computers Chemical Engineering, 34,

Diaz Ricci, J. C., & Diaz, M. S. (2010). Parameter estimation in kineticr large scale metabolic networks with advanced mathematical pro-

techniques. Computer Aided Chemical Engineering, 28, 355360 (ISSN:).

. (1989). Genetic algorithms in search, optimization & machine learning.: Addison-Wesley.

van Loan, C. F. (1996). Matrix computations (3rd ed.). London: Johnsniversity Press.2001). Nonlinear dynamic model evaluation from disturbance mea-. IEEE Transactions on Power Systems, 16, 702710.


Hoque, M. A., Ushiyama, H., Tomita, M., & Shimizu, K. (2005). Dynamicresponses of the intracellular metabolite concentrations of the wild typeand pykA mutant Escherichia coli against pulse addition of glucose or NH3under those limiting continuous cultures. Biochemical Engineering Journal, 26,3849.

Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. In Proceedings of theIEEE international conference on neural networks Perth, Australia.

Kou, B., McAuley, K. B., Hsu, C. C., Bacon, D. W., & Yao, K. Z. (2005). Mathematicalmodel and parameter estimation for gas-phase ethylene/hexene copolymer-ization with supported metallocene catalyst. Macromolecular Materials andEngineering, 290, 537557.

Kravaris, C., Chu, Y., & Hahn, J. (2012). Advances and selected recent develop-ments in state and parameter estimation. Computers and Chemical Engineering,51, 111123.

Li, R. J., Henson, M. A., & Kurtz, M. J. (2004). Selection of model parameters for off-line parameter estimation. IEEE Transactions on Control Systems Technology, 12,402412.

Lund, B. F., & Foss, B. A. (2008). Parameter ranking by orthogonalizationAppliedto nonlinear mechanistic models. Automatica, 44, 278281.

Machado, V. C., Tapia, G., Gabriel, D., Lafuente, J., & Baeza, J. A. (2009). Systematicidentiability study based on the Fisher information matrix for reducing thenumber of parameters calibration of an activated sludge model. EnvironmentalModelling & Software, 24, 12741284.

Omlin, M., & Reichert, P. (1999). A comparison of techniques for the estimation ofmodel prediction uncertainty. Ecological Modelling, 115(1), 4559.

Petzold, L. R. (1989). DASSL: A differential/algebraic system solver (1st ed., pp. ).Computer and Mathematical Research Division, Lawrence Livermore NationalLaboratory.

Ratushny, A., Smirnova, O., Usuda, Y., & Matsui, K. (2006). Regulation of thepentose phosphate pathway in Escherichia coli: Gene network reconstructionand mathematical modeling of metabolic reactions. In Proceedings of the fth

international conference on bioinformatics of genome regulation and structure(BGRS) Novosibirsk, Russia.

Sandink, C. A., McAuley, K. B., & McLellan, P. J. (2001). Selection of parametersfor updating in on-line models. Industrial & Engineering Chemistry Research, 40,39363950.

Secchi, A. R., Cardozo, N. S. M., Almeida Neto, E., & Finkler, T. F. (2006). Analgorithm for automatic selection and estimation of model parameters. InProceedings of the international symposium on advanced control of chemical pro-cesses (ADCHEM2006) Gramado, Brazil.

Sun, C. L., & Hahn, J. (2006). Parameter reduction for stable dynamical systems basedon Hankel singular values and sensitivity analysis. Chemical Engineering Science,61, 53935403.

Schwaab, M., Alberton, A. L., & Pinto, J. C. (2011). ESTIMA&PLANEJA: Pacote com-putacional para estimaco de parmetros de planejamento de experimentos. InInternal Technical Report. Brazil: Universidade Federal do Rio de Janeiro.

Thompson, D. E., McAuley, K. B., & McLellan, P. J. (2009). Parameter estimationin a simplied MWD model for HDPE produced by a ZieglerNatta catalyst.Macromolecular Reaction Engineering, 3, 160177.

Velez-Reyes, M., & Verghese, G. C. (1995). Subset selection in identication, andapplication to speed and parameter estimation for induction machines. InProceedings of the 4th IEEE Conference on control applications New York, EUA.

Weijers, S. R., & Vanrolleghem, P. A. (1997). A procedure for selecting best identi-able parameters in calibrating activated sludge model no 1 to full-scale plantdata. Water Science and Technology, 36, 6979.

Yao, K. Z., Shaw, B. M., Kou, B., McAuley, K. B., & Bacon, D. W. (2003). Modeling ethy-lene/butene copolymerization with multi-site catalysts: Parameter estimabilityand experimental design. Polymer Reaction Engineering, 11, 563588.

Yue, H., Brown, M., Knowles, J., Wang, H., Broomhead, D. S., & Kellab, D. B. (2006).Insights into the behaviour of systems biology models from dynamic sensi-tivity and identiability analysis: A case study of an NF-kB signaling pathway.Molecular BioSystem, 2, 640649.

Accelerating the parameters identifiability procedure: Set by set selection1 Introduction2 Parameters identifiability3 Binary search4 Parameters identifiability based on the binary search5 Illustrative examples5.1 Example 1 a simple model5.1.1 Case a without the parameters re-estimation step5.1.2 Case b with the parameters re-estimation step

5.2 Example 2 a complex dynamic model5.2.1 Case a without the parameters re-estimation step5.2.2 Case b with the parameters re-estimation step

6 ConclusionAcknowledgmentsReferencesReferences

alberton etal 2013 parameters identifiability

Documents