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Computers and Chemical Engineering 33 (2009) 1851–1860

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

ybrid differential evolution with geometric mean mutation in parameterstimation of bioreaction systems with large parameter search space

ang-Kai Liu, Feng-Sheng Wang ∗

epartment of Chemical Engineering, National Chung Cheng University, 168 University Rd., Min-Hsiung, Chia-yi 621-02, Taiwan

r t i c l e i n f o

rticle history:eceived 24 August 2008eceived in revised form 8 May 2009ccepted 11 May 2009

a b s t r a c t

Problems of parameter estimation of nonlinear bioreaction systems are in general formulated as func-tion optimization problems and are known to be frequently ill-conditioned and multimodal. While theoptimization problems are defined on a large parameter search space, only few evolutionary algorithms

vailable online 21 May 2009

eywords:nverse problemlobal optimizationystems biology

are able to find a global solution to the large parameter search problem. In this study, a geometric meanmutation was embedded to hybrid differential evolution to replace a gene of the selected individual out-side the assigned region. The replaced individuals were then applied to a differential mutation strategyto yield a perturbed individual. The benefit of using a large parameter search space to an inverse problemis to reduce the kinetic model complexity to yield a more compact formulation. Two inverse problemsand twelve static benchmark problems with the large parameter search space are employed to illustrate

ropos

volutionary algorithm the effectiveness of the p

. Introduction

Mathematical models describing behaviors of biological systemsrom time-course data are one of the most issues in systems biology.he ultimate goal of mathematical modeling is to obtain an expres-ion that quantitatively describes the dynamic behaviors of theystem under consideration. Such mathematical models can be pro-ided for analysis, design, optimization and control to the biologicalystems. In biological reaction systems, various kinds of kineticodels, such as Michaelis–Menten model or power law model, have

een proposed to quantitatively describe the dynamic behaviors ofiological reactions. The kinetic parameters are, in general, unableo directly measure so they are estimated from experimental datan vitro or in vivo (Gonzalez, Asenjo, & Andrews, 2001; Heijnen

Vissera, 2003; Schwacke & Voit, 2005; Vissera, Schmid, Mauch,euss, & Heijnen, 2004; Wang, Su, & Jang, 2001). Evaluation of theinetic parameters is a central point in biochemical modeling, buthe graphical methods, commonly suggested to determine thesearameters, have some limitations.

Various optimization algorithms such as gradient-based meth-ds (Mendes & Kell, 1998), branched and bound method (Esposito &

loudas, 2000; Polisetty, Voit, & Gatzke, 2006), alternating regres-ion (Chou & Voit, 2006) and stochastic optimization (Edwards,dgar, & Maousiouthakis, 1998; Gonzalez, Kuper, Jung, Naval, &endoza, 2007; McKay, Willis, & Barton, 1997; Moles, Mendes, &

∗ Corresponding author. Tel.: +886 5 2720411x33404; fax: +886 5 2721206.E-mail address: chmfsw@ccu.edu.tw (F.-S. Wang).

098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2009.05.008

ed method.© 2009 Elsevier Ltd. All rights reserved.

Banga, 2003) have been applied to determine parameters in bio-chemical models. Gradient methods have the possibility of gettingtrapped at local optima, depending upon the degree of system non-linearity and the initial starting point (Mendes & Kell, 1998). Thebranch and bound algorithm (Esposito & Floudas, 2000; Polisettyet al., 2006) is applied to convert the inverse problem into a con-vex optimization problem to generate a lower and upper bound onthe problem. The global solution is obtained while lower and upperbound converge as the algorithm iterates. Several state-of-the-artstochastic optimization methods are applied to solve parameterestimation problems. Such stochastic methods have high probabil-ity converging to global estimates as time goes to infinity. Moleset al. (2003) have reviewed and compared several evolutionaryalgorithms to the problem of parameter estimation of a three-steppathway. The parameter estimation problem consisted of the 36parameters, which were divided into two different classes: Hillcoefficients, allowed to vary within the range [0.1, 10] and theothers, allowed varying within the range [10−12, 1012]. Indeed, dif-ferential evolution (DE) was unable to solve such large search spaceproblems as discussed in Moles et al. (2003).

The mutation of DE as well as its variant methods (Angira &Santosh, 2007; Babu & Angira, 2006; Chiou, 2007; Chiou, Chang, &Su, 2004; Chiou & Wang, 1999; Srinivas & Rangaiah, 2007) used thedifference between two or four randomly selected individuals as

a search direction. Such a difference perturbed from large param-eter values is unable to generate more diversified individuals sothe solution results in yielding a premature solution. In this study,a geometric mean mutation is introduced as a novel strategy intoDE and hybrid differential evolution (HDE; Chiou & Wang, 1999) to

1 hemical Engineering 33 (2009) 1851–1860

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Table 1Basic operations for evolutionary algorithms and hybrid differential evolution.

Differential evolution Hybrid differential evolution

1. Representation and initialization 1. Representation and initialization2. Mutation operation 2. Mutation operation3. Crossover operation 3. Crossover operation4. Selection and evaluation 4. Restriction operation5. Repeat steps 2–4 5. Selection and evaluation

852 P.-K. Liu, F.-S. Wang / Computers and C

olve optimization problems with a large parameter search space.or evaluating the performance of the improved strategy, HDE withhe geometric mean mutation (refer to HDE-GM) is applied to solve

dry-lab benchmark discussed by Moles et al. (2003), a wet-labystem discussed by Wang et al. (2001) and benchmark static opti-ization problems.

. Method

.1. Parameter estimation

The problem of parameter estimation (or inverse problem) isormulated as a function optimization, i.e. to find the model param-ters � to minimize the error criterion

k = 1nNs

n∑i=1

Ns∑j=1

(Xei(tj) − Xi(tj))

2

X2eimax

, k = 1, . . . , Nexp (1)

ubject to the model constraints

dXi

dt= fi(X, �)

=r∑

j=1

Sijvj(X, �), i = 1, . . . , n(2)

LB ≤ � ≤ �UB (3)

here X represents n-dimensional components, � denotes n�-imensional parameters, Xei

(tj) is the measured data for the ithomponent at t = tj, Xi(tj) is the computed concentration for the ithomponent at t = tj, and Xeimax is the maximum measured concen-ration of the ith component. Here, Ns is the number of sampledata points and Nexp is the number of experiments. Sij is the stoi-hiometric coefficient and vj is the reaction rate for the jth pathway.ichelis–Menten like models are commonly applied to express

ach rate equation so the model parameters, Hill coefficients andate constants, should be determined within the parameter searchpace [�LB, �UB]. HDE is able to solve most nonlinear optimizationith a moderate parameter search space. On the other hand, a large

arameter search space can be employed to the inverse problem tonvestigate whether a substrate inhibits on a rate equation of inter-st. Under such a situation, a novel mutation strategy for HDE will bentroduced to surmount the problem with large parameter searchpace.

The solution quality of an inverse problem depends on efficiencyor an optimization method and a differential equation solver. Mostonlinear regressions are gradient-based optimization methods sohat the solution quality strongly depends on the provided initialtarting point. Moreover, gradient-based methods may yield a localinimum, not a global solution. Evolutionary algorithms can be

pplied to overcome such drawbacks. Numerical integration fail-re is the major problem during the evolutionary search progress.

n addition, numerical integration is time consuming. Many tech-iques have been employed to alleviate numerical integrationurden. Voit and Almeida (2004) utilized a decoupling schemeo estimate the slopes of the dynamic processes. The decompos-ng method (Kimura et al., 2005; Maki et al., 2002) is employedo convert the large ordinary differential equations into decoupledlgebraic equations. Tsai and Wang (2005) used the modified col-ocation method to convert differential equations into algebraicquations to approximate dynamic profiles at sampling points. Such

n approximation not only reduces computation time, but also con-erts the coupled algebraic equations into a set of uncoupled oneso that parallel computation can be straightforwardly applied inarameter estimation. In the modified collocation approach, the

ormulation of the converted algebraic equations depends on the

6. Acceleration if necessary7. Migration naturally or enforced ifnecessary8. Repeat steps 2–6

employed shape polynomials (Wang, 2000). Here, we consider thepiecewise linear Lagrange polynomial as the shape polynomials toobtain the approximate equations:

X(tl) = X(tl−1) + 0.5�l{f(X(tl), �) + f(X(tl−1), �)}, l = 1, . . . , Ns (4)

where X(tl) is the expansion coefficient vector at the lth collocationpoint and is equal to the solution X(t) at time t = tl, f(X(tl), �) theexpansion coefficient vector for the rate functions at the lth col-location point, �l is the time interval between the lth collocationpoint and the (l − 1)th point.

2.2. Geometric mean mutation

Hybrid differential evolution (HDE) is a quite simple populationbased stochastic function method and has extended from the origi-nal algorithm of DE introduced by Storn and Price (1996, 1997). Theoriginal algorithm of DE is used to solve the unconstrained nonlin-ear programming problems. Wang and Chiou (1997) have extendedthe original DE to solve optimal control problems. The basic opera-tions of DE are similar to the conventional EAs as listed in Table 1.HDE includes three additional operations, restriction, accelerationand migration, as shown in Table 1. The DE and HDE structureis a parallel direct search algorithm which utilizes Np vectors ofthe decision parameters � in optimization problems as a popula-tion in the generation G. The initialization randomly generates Np

individuals and tries to cover the entire search space uniformly.The mutation operation of DE and HDE is the essential com-

ponent, compared with the other evolutionary algorithms. Themutation process in DE and HDE uses the difference between twoor four randomly selected individuals to create a search direc-tion. The mutation process at the (G − 1)th generation begins byrandomly selecting either two or four mutually independent indi-viduals, �j, �k, �l, and �m. Each selected individual pair, �j and �k, atthe (G − 1)th generation is employed to define a difference vectorDjk as

Djk = �G−1j − �G−1

k

=

⎡⎣ �1

...�n�

⎤⎦

G−1

j

−

⎡⎣ �1

...�n�

⎤⎦

G−1

k

(5)

Four mutually independent individuals are then combined to forma difference vector Djklm as

Djklm = Djk + Dlm = (�G−1j − �G−1

k ) + (�G−1l − �G−1

m ) (6)

A perturbed or mutant individual �̂G−1i is therefore generated on

the basis of the parent individual �G−1p in the mutation process by

�̂G−1i = �G−1

p + FDjklm, i = 1, . . . , Np (7)

Here, we refer to the mutation process (7) by the difference of theselected individuals as the differential mutation strategy.

P.-K. Liu, F.-S. Wang / Computers and Chemic

Table 2The differential mutation strategies in differential evolution and hybrid differ-ential evolution. The linear combination factor � in the sixth strategy is a randomnumber between [0, 1].

Item Mutation strategy

1 �̂G−1

i = �G−1best + F(�G−1

j − �G−1k )

2 �̂G−1

i = �G−1i + F(�G−1

j − �G−1k )

3 �̂G−1

i = �G−1i + F[(�G−1

best − �G−1i ) + (�G−1

j − �G−1k )]

4 �̂G−1

i = �G−1best + F[(�G−1

j − �G−1k ) + (�G−1

l − �G−1m )]

tmodsmiitcp2i

mfTotipsibsti

[bms�m

�

TmbbtciTt

5 �̂G−1

i = �G−1i + F[(�G−1

j − �G−1k ) + (�G−1

l − �G−1m )]

6 �̂G−1

i = [��G−1best + (1 − �)�G−1

i ] + F(�G−1j − �

G−1k )

In DE, a differential mutation factor F ∈ (0, 1, 2] is fixed and set byhe user to ensure the fastest possible convergence. However, the

utation factor in HDE is randomly selected at every generation tobtain a more perturbed individual. Table 2 lists five common usedifferential mutation strategies in DE and HDE. The sixth mutationtrategy was introduced by Liao, Tzeng, and Wang (2001) to solveixed-integer optimization problems. A linear crossover for the ith

ndividual and the best individual is first applied to yield the parentndividual, and then the parent individual is employed to generatehe next mutant individual. This mutation strategy has been suc-eeded to solve several mixed-integer and real-valued optimizationroblems (Cheng & Wang, 2008; Chiou & Wang, 2001; Liao et al.,001; Lin & Wang, 2007). In this study, this strategy is also included

n DE and HDE to solve real-value optimization problems.As observed from Eqs. (5) or (6), the difference of two or four

utually independent individuals is served as a search directionor each mutation strategy in the solving parameter search space.he mutant individual in (7) is essentially a perturbed replicaf the parent individual. While the population diversity is small,he candidate individuals will rapidly gather together so that thendividuals cannot be further improved. This fact may result in aremature convergence. On the other hand, if the parameter searchpace is very large, some genes (or components) for each mutuallyndependent individual may be located nearby the lower or upperoundary. This situation is very difficult to generate a realizableearch direction from the difference vector in Eqs. (5) and (6) dueo a numerical deviation. In this study, a geometric mean mutations introduced to overcome such a drawback.

Suppose that the domain of the large parameter search space�LB, �UB] is positive, i.e. both small lower bound and large upperound are positive. We definite a region for the geometric meanutation operation as [�LB

GM, �UBGM]. This region must be a complete

ubspace of the parameter search space, i.e. �LB < �LBGM and �UB >

UBGM . For a gene beyond the region [�LB

GM, �UBGM], the geometric mean

utation is therefore carried out to replace this gene by:

G−1i

=

⎧⎪⎪⎨⎪⎪⎩

�(�G−1i

�G−1i,best

�UBi

)1/3

, if �G−1i

< �LBi,GM

�(�G−1i

�G−1i,best

�LBi

)1/3

, if �G−1i

> �UBi,GM

�G−1i

, if �LBi,GM

≤ �G−1i

≤ �UBi,GM

(8)

his equation indicates that the replaced gene �G−1i

is the geometricean of the current gene, the corresponding best gene and its upper

ound if the current gene is smaller than �LBi,GM

. Here, a random num-er � ∈ [0, 1] is applied to yield a more diversified gene. In contrast,he replaced gene is the geometric mean of the current gene, the

orresponding best gene and its lower bound if the current genes greater than �UB

i,GM; otherwise, the gene is kept the original value.

he two or four replaced individuals are then employed to computehe difference vector in order to generate a perturbed individual.

al Engineering 33 (2009) 1851–1860 1853

The parameter search space for some optimization problemsis not positive, but the lower bound is very negative large, i.e.[−�LB, �UB]. Under this situation, the machine precision ε, e.g.2.22045E−16 on a personal computer, is applied to substitute thelower and upper bound to yield the replaced gene as follows:

�G−1i

=

⎧⎪⎪⎨⎪⎪⎩

−�(∣∣�G−1

i

∣∣ �G−1i,best

ε)1/3

, if �G−1i

< −�LBi,GM

�(�G−1i

�G−1i,best

ε)1/3

, if �G−1i

> �UBi,GM

�G−1i

, if − �LBi,GM

≤ �G−1i

≤ �UBi,GM

(9)

The region for the geometric mean mutation operation is definedas [−�LB

GM, �UBGM], where �LB

GM and �UBGM are large positive vectors.

The crossover operation in HDE is employed to increase the localpopulation diversity, which is similar to the conventional evolu-tionary algorithms. The original code of the DE algorithm (Storn &Price, 1997) did not check if the new generated individuals werewithin their bound regions, therefore the restriction for each deci-sion parameter has added into the HDE algorithm. In DE and HDE,the fitness of an offspring competes one to one with that of itsparent. This competition, which is also different from the conven-tional evolutionary algorithms, gives rise to a faster convergencerate. However, this faster convergence also leads to a higher proba-bility of obtaining a local optimum because the population diversitydescends faster during the solution progress. This drawback can beovercome by using a larger population size, although much compu-tation time is expended to evaluate the fitness function. This fact isparticularly serious when using DE to solve optimal control prob-lems (Wang & Chiou, 1997). The HDE algorithm can use a smallerpopulation size and has been applied to solve biochemical processsystems engineering problems (Chen & Wang, 2003; Cheng & Wang,2008; Chiou & Wang, 1999; Ko, Wang, Chao, and Chen, 2006; Lin &Wang, 2007; Liu & Wang, 2008a, 2008b; Tsai & Wang, 2005; Wang,2000; Wang, Jing, & Tsao, 1998; Wang et al., 2001; Wang & Lin,2003).

When using an evolutionary algorithm to optimize a function, anacceptable trade-off between convergence and population diversitymust generally be determined. Convergence implies a fast conver-gence even though it may be to a local optimum. In contrast, thepopulation diversity guarantees a high probability of obtaining theglobal optimum. When the population diversity is small, the can-didate individuals are closely clustered. Therefore, the mutationand crossover operations can no longer generate the next betterindividual because a premature solution is obtained. The HDE algo-rithm is embedded acceleration and migration into the originalDE. These two operations serve as a trade-off operator for balanc-ing the convergent rate and population diversity. The accelerationis used to speed up convergence. This operation is similar to asimulated annealing method to avoid in yielding a premature solu-tion. According to our experience, using DE to solve optimizationproblems, the best fitness does not descend continuously from gen-eration to generation. It usually descends toward a better fitnessafter several generations. Under this situation, the acceleration canbe used to speed up convergence. When the best fitness in thepresent generation is not improved any longer by the mutation andcrossover operations, a descent method is then applied to push thebest individual toward obtaining a better solution.

The rate of convergence can be improved by the acceleration.However, faster descent usually results in yielding a prematuresolution. In addition, performing this operation frequently canmake the candidate individuals gradually cluster around the best

individual so that the population diversity decreases. Furthermore,the closely clustered individuals cannot reproduce better individ-uals by mutation and crossover operations. Under tins situation, amigration operation is turned on to escape this local cluster. Thenew candidate individuals are regenerated on the basis of the best

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3

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Fpls

of 6,603,890, equivalent to the CPU time of about 17.44 min. Theoptimal estimates shown in Table 3 are quite close to the true val-ues. The greatest error for the estimated parameters is Km6 that has

854 P.-K. Liu, F.-S. Wang / Computers and C

ndividual at the current generation. Correspondingly, the diversityf the candidates can be retained by using such a regeneration pro-edure. The migration in HDE is performed only if a measure of theopulation diversity fails to satisfy the desired tolerance. Chiou andang (1999) proposed two measures called the degree of popula-

ion diversity and gene diversity to check whether the migrationhould be turned on. The degree of gene diversity is accounted howany genes of the candidate individuals close to that of the best

ndividual. The total figures of the diversified genes are then appliedo evaluate the degree of population diversity so its value is betweenero and one. A zero value implies that all of the genes are clusteredround the best individual. In contrast, the value of one indicateshat the current candidate individuals are a completely diversifiedopulation. The desired tolerance for population diversity is accord-

ngly assigned within this region. A zero tolerance implies that theigration in HDE is turned off. A tolerance of one implies that theigration is performed at every generation.

. Applications

To show the effectiveness of the proposed method, we appliedhe method to two problems of parameter estimation of a three-tep pathway (Moles et al., 2003) and ethanol fermentation (Wangt al., 2001), and static benchmark optimization problems. All theomputations were performed on a Pentium IV 3.0 GHz computersing Microsoft Windows XP. The HDE algorithm is implemented

n Compaq Visual Fortran, and has to provide four required settingactors by the user. The four setting factors used for all runs in thease studies are listed as follows: The crossover factor is set to be 0.5.wo tolerances used in the migration are set to be 0.05. The popu-

ation size of 5 is used in the computation. The maximum iterationf 200,000 is served as the termination criterion for the computa-ion in the case studies 1 and 2. In static benchmark problems, thealue-to-reach is used as the termination criterion.

.1. Three-step biochemical pathway

The system of the three-enzyme pathway with gene expressionhown in Fig. 1 is the challenging benchmark problem recentlyresented by Mendes (2001), Moles et al. (2003) and Rodriguez-ernandez, Mendes, and Banga (2006). The nonlinear mathematicalormulation of the system consists of 8 ordinary differential equa-

ions with 36 parameters. We employ the same initial conditionsiscussed by Moles et al. (2003) to generate the 16 different exper-

mental data. For the sake of brevity, the dynamic equations shownn Appendix A and all the additional needed data (including thexperimental data set) are not given here.

ig. 1. Three-step biochemical pathway scheme. The pathway substrate (S) and theroduct (P) are held at constant concentrations; Ml and M2 are intermediate metabo-

ites of the pathway; El , E2, and E3 are the enzymes and G1, G2, and G3 are the mRNApecies for the enzymes.

cal Engineering 33 (2009) 1851–1860

We first consider the same parameter search space discussedby Moles et al. (2003), that is divided into two different classes:Hill coefficients, allowed varying within the range [0.1, 10] and theothers, allowed varying within the range [10−12, 1012]. The origi-nal HDE using various mutation strategies listed in Table 2 is firstapplied to solve the estimation problem with this large parametersearch space. However, HDE is incapable of finding a convergentsolution, but the almost identical estimates can be obtained if theparameter search space is changed to [0.01, 100].

The HDE-GM algorithm is then applied to solve this large param-eter search problem. In this work, we introduce a global/localsearch to determine optimal estimates. In the global search phase,the HDE-GM algorithm is first applied to minimize the error cri-terion (1) with the large parameter search space. In addition,the modified collocation equations (4) are employed to approx-imate dynamic profiles for the model. The best error criterionvalue of 2.3655E−5 is obtained using the total function call of2,756,001 which is equivalent to the CPU time of about 8.19 minon a Pentium 3.0 GHz computer. This computation uses the migra-tion operations of 1006 and acceleration operations of 9318. Botherror criterion value and objective function evaluations obtainedby HDE-GM are smaller than those of Moles et al. (2003). To yieldmore accurate results, the optimal estimates obtained from theglobal search are then served as the initial starting point for agradient-based optimization method, a subroutine BCONF in IMSLMath/Library, to obtain the refined solution. The local optimizationphase employs Runge–Kutta pairs of various orders, a subroutineIVMRK in IMSL Math/Library, to solve the full differential equationstowards obtaining time-course profiles of the model. The refinederror criterion value of 2.7179E−6 is obtained. The computationaltime required for the local search about 3.74 min on a Pentium IV3.0 GHz. The local search is able to obtain more accurate solution.

The power orders for the above large parameter search spaceare the symmetric values. We also consider the parameter searchspace with asymmetric power orders for Hill coefficients as [10−2,104] and for the others as [10−12, 1015] to investigate whether thegeometric mean mutation strategy can handle this asymmetricregion. Following similar procedures, the best error criterion valueof 2.8312E−5 is obtained by HDE-GM using the total function call

7% different to the “true” value, as observed from Table 3. We have

Table 3The true parameter values and the estimated values obtained by HDE-GM in the casestudy system of the three-enzyme pathway with gene expression. HDE-GM uses theparameter search space with asymmetric power orders for Hill coefficients as [10−2,104] and for the others as [10−12, 1015] in parameter estimation.

Parameter True Estimated Parameter True Estimated

V1 1 0.9930 V4 0.1 0.1003Ki1 1 0.9988 K4 1 0.9999ni1 2 1.9978 k4 0.1 0.1003Ka1 1 1.0013 V5 0.1 0.1000na1 2 2.0067 K5 1 1.0128k1 1 0.9927 k5 0.1 0.0993V2 1 0.9950 V6 0.1 0.1003Ki2 1 1.0009 K6 1 1.0068ni2 2 1.9987 k6 0.1 0.0998Ka2 1 0.9984 kcat1 1 0.9980na2 2 2.0054 Km1 1 1.0135k2 1 0.9963 Km2 1 1.0556V3 1 1.0026 kcat2 1 1.0181Ki3 1 0.9988 Km3 1 1.0389ni3 2 1.9925 Km4 1 1.0374Ka3 1 1.0024 kcat3 1 1.0306na3 2 1.9949 Km5 1 1.0591k3 1 0.9989 Km6 1 1.0665

hemical Engineering 33 (2009) 1851–1860 1855

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3

dhtbr(ma

wotycf

�

q

q

Nm1tTwt

ucfbfaooiopei

Fig. 2. The experimental data and computational profiles for ethanol fermentationsusing the various initial glucose concentrations. The experimental data for the initialglucose concentrations of 100 and 150 g/L are served as the training data. The initial

P.-K. Liu, F.-S. Wang / Computers and C

pplied the parametric sensitivity analysis to evaluate the problem.e found that the relative sensitivity of the enzyme E1 with respect

o changes in the parameter of Km6 is the most relative sensitivealue of 0.0586 through sensitivity analysis. This situation indi-ates that the computed profiles are insensitive to Km6. For the localptimization phase, the refined error criterion value of 7.6767E−6

s obtained. The computational time required for the local searchbout 3.15 min.

.2. Kinetics model of ethanol fermentation

In this wet-lab study, we reanalyze an experimental systemescribed by Wang et al. (2001). The fermentation process usesigh ethanol tolerance yeast, Saccharomyces diastaticus LORRE 316,o produce ethanol. The batch experiments were carried out on a 5 Lioreactor to collect the time-course data. The experimental mate-ials, methods and kinetic models were illustrated in Wang et al.2001). The material balance equations that describe the growth of

icroorganisms, consumption of glucose and formation of productsre given as follows:

dx

dt= �x (10)

ds

dt= − qp1

Yp1/sx − qp2

Yp2/sx (11)

dp1

dt= qp1 x (12)

dp2

dt= qp2 x (13)

here x is the concentration of cell mass, s is the concentrationf glucose, p1 is the concentration of ethanol, p2 is the concentra-ion of glycerol, Yp1/s is the ethanol yield factor, Yp2/s is the glycerolield factor, and the unstructured kinetic model for the specificell growth and product formation are, respectively expressed asollows:

= �ms

Ks + s + s2/KSI

Kp1

Kp1 + p1 + p21/Kp1I

Kp2

Kp2 + p2 + p22/Kp2I

(14)

p1 = �p1 s

K ′s + s + s2/K ′

sI

K ′p1

K ′p1

+ p1 + p21/K ′

p1I

(15)

p2 = �p2 s

K ′′s + s + s2/K ′′

sI

K ′p2

K ′p2

+ p2 + p22/K ′

p2I

(16)

ineteen parameters including yield coefficients have to be esti-ated from experimental observations. The time-course data with

-h sampling were obtained from two repeated batch fermenta-ions using the initial glucose concentrations of 100 and 150 g/L.hese repeated time-course data (error bars) shown in Fig. 2a and bere noisy so a curve-fitting method was first employed to smooth

he observed data for evaluating the error criterion.The global/local search approach is applied to solve the problem

sing the parameter search space of [10−12, 1015], except the yieldoefficient for ethanol, Yp1/s ∈ [0.1, 0.51], and the yield coefficientor glycerol, Yp2/s ∈ [0.1, 0.2]. For the global optimization phase, theest error criterion value of 3.5687E−1 is obtained using the total

unction call of 1,694,102 which is equivalent to the CPU time ofbout 1.69 min. HDE-GM in this computation uses the migrationperations of 315 and acceleration operations of 6734. For the localptimization phase, the refined error criterion value of 8.9587E−3

s obtained by using the computational time about 0.51 min. Theptimal estimates are listed in Table 4, and the optimal com-uted profiles shown in the dashed curves of Fig. 2a and b fit thexperimental data satisfactorily. One saturation coefficient and fivenhibition coefficients are very large observed from this table. These

glucose of 200 g/L is used for model validation. Data points indicate the repeatedexperimental data, which is adopted from Wang et al. (2001). The solid curves arethe computed profiles using the optimal estimates from the pruned kinetic model.The dashed curves are the computed profiles using the original kinetic model.

large parameters should contribute insignificant effects on theircorresponding specific rates so these coefficients can be deletedfrom the rate equations. The pruned kinetic models are thereforeexpressed as follows:

� = �ms

Ks + s + s2/KSI

Kp1

Kp1 + p1 + p21/Kp1I

(17)

qp1 = �p1 s

K ′s + s

K ′p1

K ′p1

+ p1(18)

qp2 = �p2 s

K ′′s + s

K ′p2

K ′p2

+ p2(19)

Following the similar procedures, the pruned models are appliedto refit the training data again. For the global optimization phase,the best error criterion value of 3.3832E−1 is obtained using thetotal function call of 530,277 which is equivalent to the CPU timeof about 0.88 min. HDE-GM in this computation uses the migration

1856 P.-K. Liu, F.-S. Wang / Computers and Chemical Engineering 33 (2009) 1851–1860

Table 4The optimal estimates for the original model and pruned model, and the statistic results using different random seeds for HDE-GM to find optimal estimates of the prunedmodel.

Parameter Optimal estimates Statistic results

Original model Pruned model Mean Stardard deviation Variance

�m 6.2840E−1 6.7925E−1 6.6521E−1 3.0142E−2 9.0856E−4�p1 1.1967 1.1773 1.2123 6.7634E−2 4.5743E−3�p2 1.1228E−1 1.4122E−1 1.4402E−1 3.0857E−2 9.5217E−4Yp1/s 5.0177E−1 4.9231E−1 4.9826E−1 4.5316E−3 2.0535E−5Yp2/s 1.9938E−1 2.1350E−1 2.1815E−1 1.4934E−2 2.2302E−4Ks 2.0688 7.2408 5.4135 2.3206 5.3851Ks1 4.7392E2 3.5111E2 3.8352E2 6.1296E1 3.7572E3Kp1 3.8528E1 4.7655E1 4.5970E1 4.7691 2.2744E1Kp1I

1.3419E1 1.0973E1 1.0934E1 1.5040 2.2621Kp2 1.3112E10 – – – –Kp2I

1.3611E9 – – – –K ′

s 7.4002 9.2993 1.1110E1 4.0952 1.6771E1K ′

sI3.8445E9 – – – –

K ′p1

1.3327E2 1.7095E2 1.9928E2 8.1454E1 6.6348E3K ′

p1I1.1058E4 – – –

K ′′ 3.5187E1 6.2061E1 6.4008E1 2.5875E1 6.6950E2K

K

K

ooietpfitreseTaa52tveslt

tdtpcoiT8tr

3

rfpA

s′′sI

6.8223E9 –′p2

7.4745E1 2.3003E2′p2I

5.9065E12 –

perations of 477 and acceleration operations of 1698. For the localptimization phase, the refined error criterion value of 8.1527E−3

s obtained with using the computational time about 0.65 min. Therror criteria for both global and local searches are nearly equalo those from the original model. The optimal estimates from theruned model are also listed in Table 4. The optimal computed pro-les shown in the solid curves of Fig. 2a and b are nearly identical to

hose obtained from the original model. We also apply 10 differentandom seeds for the HDE-GM algorithm in order to, respectivelystimate the parameter values for the pruned model. The mean,tandard deviation and variation are computed from the 10 optimalstimates, and are shown in the fourth, fifth and sixth columns ofable 4. The three statistical measures for the error criteria, whichre, respectively obtained from HDE-GM and the refined search,re also computed. The mean, standard deviation and variation are.8963E−1, 3.0820E−2 and 9.4989E−4 for HDE-GM and 8.3862E−3,.0257E−3 and 4.1034E−6 for the refined search. Both variances forhe error criteria obtained from HDE-GM and the refined search areery small even though some parameter values are very different,.g. standard deviations for Ks1 , K ′

p1and K ′

p2are rather large. This

ituation indicates that the inverse problem is a multi-model prob-em. As a result, we are incapable of finding a unique solution forhe web-lab experiment due to the measured noise.

To validate the inferred model, an additional experiment withhe initial glucose of 200 g/L, which is 33% greater than the trainingata, are employed to predict the dynamic behavior. Fig. 2c showshe computed results from both models and experimental data. Theruned model is to neglect the inhibition of glycerol on the specificell growth and on the specific ethanol formation, and the inhibitionf ethanol on the specific glycerol formation. The prediction power

s still quite well as same as that obtained from the original model.he error criterion values for the original and pruned models are.0138E−2 and 8.2035E−2, respectively. HDE-GM can apply to solvehe inverse problem with the large parameter search space and alsoeduces the model complexity to yield a more compact formulation.

.3. Static benchmark problems

To further illustrate the effectiveness of the HDE-GM algo-ithm, twelve static benchmark optimization problems selectedrom Storn and Price (1997), and Schwefel (1994) are solved by theroposed algorithm. These original benchmark problems shown inppendix B were defined on the moderate parameter space. In this

– – –2.2522E2 2.4355E2 5.9318E4– – –

work, a value-to-reach, VTR, is assigned for each run to serve asa stopping criterion to inspect whether the optimization searchshould be terminated. The setting factors used for all runs are assame as the mentioned-above case studies.

The original DE and HDE algorithms are able to find a global solu-tion for each original 10-dimensional benchmark problem. Notethat the original Corana’s parabola function is a four-dimensionalproblem. We modify the dj values those are those are randomlyselected from the set {1, 1000, 10, 100} so the problem becomes aD-dimensional problem. While we consider each benchmark prob-lem with the large parameter search space as [−1010, 1010], somefunctions cannot be found a convergent solution by DE and HDE.Table 5 shows the best fitness value, CPU time in seconds and num-ber of function evaluation (nfe) obtained by DE and HDE. In thiswork, DE uses the population size of 50 to solve the large searchspace benchmark problems. Eight functions cannot be achievedto VTR by DE. However, HDE uses the population size of 5, canfind a global solution, except Ackley, Griewangk, Rosenbrock andSchwefel functions.

HDE-GM is then applied to solve these 10-dimensional bench-mark problems with the large parameter search space [−1010, 1010].Six mutation strategies shown in Table 2 are, respectively appliedto solve the problems. Notice that the geometric mean mutation in(9) should be performed in advance if a gene of the selected individ-ual is outside the assigned region [�LB

GM, �UBGM]. The assigned region

is [−103, 103] for the Schwefel function, and [−108, 108] for theothers. The computational results for various mutation strategiesare also shown in Table 5. The best fitness values for all problemscan be achieved to its VTR, except the first and fifth strategies forsolving the Schwefel function. While we turn off the accelerationand migration operations in HDE-GM, the algorithm becomes DE-GM, which is also applied to solve these problems for comparison.Table 5 shows the computational results for DE-GM using the sixthmutation strategy. DE-GM has to use more function calls to reacha global solution. Moreover, three problems, F23, Rosenbrock andSchwefel functions, cannot be achieved to their VTR.

We also apply HDE-GM using the sixth strategy to solve theselarge search space problems with the 100-dimensional parameter

vector. HDE-GM can still solve these 100-dimensional problems,except the Schwefel function. The Ackley function solved byHDE-GM spends the shortest CPU time of 0.32 s to achieve theassigned VTR. The Rosenbrock function using 63.5 min is the longestone.

P.-K.Liu,F.-S.W

ang/Com

putersand

ChemicalEngineering

33 (2009) 1851–18601857

Table 5The computational results by various strategies for 12 benchmark static optimization problems. The large parameter search space is [−1010, 1010] for all examples. VTR denotes the value-to-reach. CPU time denotes the computationtime in seconds. nfe denotes number of function evaluation.

Function name Ackley Hyper-Ellipsoid Corana’s parabola F21 F22 F23Dimension 10 10 10 10 10 10VTR 1.0E−7 1.0E−15 1.0E−7 1.0E−10 1.0E−10 1.0E−10

HDE-GM strategy Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe

1 9.95E−8 3.13E−2 17,861 8.22E−16 1.56E−2 7,970 9.49E−8 3.13E−2 21,058 7.46E−11 3.13E−2 17,736 9.60E−11 3.89 2,029,471 7.85E−11 3.13E−2 24,4222 9.58E−8 1.56E−2 3,747 8.11E−16 3.13E−2 36,557 9.79E−8 1.56E−2 18,419 6.70E−11 2.81E−1 144,013 6.40E−11 1.95 1,064,180 7.54E−11 1.88E−1 105,2613 9.99E−8 1.56E−2 7,861 9.92E−16 1.56E−2 7,754 9.89E−8 4.69E−2 52,582 7.19E−11 1.09E−1 66,467 8.99E−11 3.44E−1 1,538,200 8.22E−11 2.03E−1 110,1794 7.83E−8 1.56E−2 6,142 7.45E−16 1.56E−2 7,002 9.38E−8 1.56E−2 17,146 3.33E−11 2.03E−1 107,242 8.79E−11 4.53E−1 1,526,375 9.05E−11 4.53E−1 241,8095 8.77E−8 1.56E−2 4,412 9.21E−16 1.56E−2 7,269 7.64E−8 3.13E−2 25,629 9.63E−11 1.0 505,515 7.21E−11 2.13 1,109,049 4.90E−11 1.78 868,4456 8.78E−8 1.56E−2 7,611 9.88E−16 1.56E−2 11,479 9.77E−8 1.56E−2 18,313 3.62E−11 1.09E−1 56,916 5.97E−11 2.03E−1 110,272 8.89E−11 1.41E−1 78,649

DE−GM 9.29E−8 4.69E−2 28,550 9.21E−16 3.13E−2 21,150 8.41E−8 3.13E−2 20,750 9.12E−11 1.25E−1 64,550 9.21E−11 2.58 1,350,679 6.59E−5 1.76E1 1.00E7

HDE 2.00E1 1.40E1 5,010,807 9.49E−16 6.25E−2 34,963 9.09E−8 1.41E−1 111,843 6.45E−10 1.31 524,206 2.63E−9 3.69 1,530,016 5.04E−10 7.34 1,389,151

DE 2.00E1 1.31E2 50,000,050 4.97E−16 3.13E−2 34,850 8.11E−8 4.69E−2 35,250 9.52E37 1.15E1 5,000,050 1.84E−10 9.27E1 50,136,374 9.52E37 6.97E1 13,500,050

Function name F24 Griewangk Rastrigin Rosenbrock Katsuura SchwefelDimension 10 10 10 10 10 10VTR 1.0E−10 1.0E−15 1.0E−10 1.0E−4 1.01 1.0E−4

HDE-GM strategy Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe Fitness cpu time nfe

1 8.85E−11 6.25E−2 31,407 7.77E−16 1.56E−2 2,943 6.86E−11 1.56E−2 13,656 9.98E−5 1.88 2,229,698 1.01 4.69E−2 3,110 4.12E1 1.38E2 50,820,4992 3.12E−11 2.19E−1 116,736 9.99E−16 1.56E−2 1,054 9.18E−11 6.25E−2 45,244 9.31E−5 1.49E1 14,176,379 1.01 2.66E−1 18,080 9.21E−5 3.05E 1,128,8513 6.22E−11 1.41E−1 68,672 6.66E−16 1.56E−2 4,064 8.72E−11 4.69E−2 35,671 9.93E−5 2.02 2,433,766 1.01 1.33 74,438 9.90E−5 3.36E 1,300,8974 6.70E−11 3.13E−1 167,660 7.77E−16 1.56E−2 9,377 9.13E−11 1.56E−2 13,392 6.26E−5 2.59 3,002,178 1.01 1.11 66,418 9.90E−5 1.23E2 41,559,9355 8.43E−11 3.81E 1,931,722 6.66E−16 1.56E−2 1,336 8.89E−11 1.56E−2 15,096 7.78E−5 3.13E1 24,773,005 1.01 3.75E−1 18,877 2.55E−1 1.43E2 50,108,4796 7.01E−11 1.25E−1 66,024 9.99E−16 1.56E−2 3,553 9.16E−11 3.13E−2 24,465 1.0E−5 1.45 1,371,498 1.01 3.13E−1 22,825 9.98E−5 7.50E−1 292,411

DE-GM 8.18E−11 9.38E−1 469,031 6.66E−16 1.25E−1 59,200 6.84E−11 3.13E−2 25,300 9.49E1 5.62E2 500,000,050 1.01 3.88 187,750 1.94E3 1.34E3 500,000,050

HDE 6.18E−10 2.00 820,666 6.64E−2 9.53E−1 432,264 0.00E0 2.81E−1 136,094 9.92E9 3.14E1 22,154,885 1.05 2.31 46,197 2.11E2 1.56E2 45,077,625

DE 9.52E37 1.86E1 8,000,050 1.23E−2 4.77E1 25,000,050 9.72E−11 7.81E−2 48,950 1.47E38 2.76E2 200,000,050 1.01 1.10E1 265,900 1.2971 1.45E3 450,000,050

1 hemi

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858 P.-K. Liu, F.-S. Wang / Computers and C

. Conclusions

The simulation of bioreaction systems is a powerful approachhat can be used for: (i) maximizing the flux of interesting productsnd minimizing the undesired by-products, and (ii) calibrating theodel to reproduce the experimental results in the best possibleay. Establishing a validated model, its appropriate model param-

ters need to be determined by experiments in order to achieveroper simulations. Optimization methods are routinely used inarameter estimation to find suitable model parameters. However,here are still many challenges to determine the global estimatesor solving an inverse problem of nonlinear dynamic biochemi-al systems from time-course data. A large parameter search spaceould be employed to the inverse problem to investigate whethersubstrate/product inhibits on its corresponding rate equation of

nterest. Many evolutionary algorithms could not yield a conver-ent result to the inverse problem with the large parameter searchpace. In this study, a geometric mean mutation was embedded intoDE to surmount such a problem. The geometric mean mutation

hould be performed in advance if a gene of the selected indi-idual was outside the assigned region. The replaced individualsere then applied to a differential mutation strategy to yield a per-

urbed individual. From the first case study, we observed that theroposed HDE-GM algorithm was more efficient than the reportedethods. HDE-GM could apply to solve an inverse problem withlarge parameter search space and also reduce the kinetic model

omplexity to yield a more compact formulation as discussed inhe second case study. In addition, twelve static benchmark opti-

ization problems with the large parameter search space weremployed to illustrate the effectiveness of the proposed approach.

cknowledgement

The financial support from the National Science Council, Taiwan,OC (Grant NSC97-2627-B-194-001 and NSC97-2221-E-194-010-Y3), is highly appreciated.

ppendix A.

The mathematical formulation of the system of the three-nzyme pathway is listed as follows:

dG1

dt= V1

1 + (P/Ki1 )ni1 + (Ka1 /S)na1− k1G1

dG2

dt= V2

1 + (P/Ki2 )ni2 + (Ka2 /M1)na2− k2G2

dG3

dt= V3

1 + (P/Ki3 )ni3 + (Ka3 /M2)na3− k3G3

dE1

dt= V4G1

K4 + G1− k4E1

dE2

dt= V5G2

K4 + G2− k5E2

dE3

dt= V6G3

K6 + G3− k6E3

dM1

dt= kcat1 E1(S − M1)/Km1

1 + S/Km1 + M1/Km2

− kcat2 E2(M1 − M2)/Km3

1 + M1/Km3 + M2/Km4

dM2

dt= kcat2 E2(M1 − M2)/Km3

1 + M1/Km3 + M2/Km4

− kcat3 E3(M2 − P )/Km5

1 + M2/Km5 + P/Km6

cal Engineering 33 (2009) 1851–1860

where M1, M2, E1, E2, E3, G1, G2, and G3 represent the concentrationsof the species involved in the different biochemical reactions andS and P keep fixed initial values for each experiment. The “true”values of Hill coefficients and the others coefficients are shown inTable 3.

Appendix B.

Twelve benchmark test functions are listed as follows. Thedimension D for each problem is, respectively considered as 10 and100 in the computations.

(1) Ackley function

f1(x) = − 20 exp

⎛⎝−0.02

√√√√D−1

D∑j=1

x2j

⎞⎠

− exp

⎛⎝D−1

D∑j=1

cos(2�xj)

⎞⎠+ 20 + exp(1)

The initial parameter range, IPR, is assigned by xj ∈ [−30, 30].The global minimum is f1(0) = 0.

(2) Hyper-Ellipsoid function f2(x) =D∑

j=1

j2x2j

; IPR : xj ∈ [−1, 1].

The global minimum is f2(0) = 0.(3) Corana’s parabola function

f3(x) =D∑

j=1

{0.15(zj − 0.05 sgn(zj))

2dj, if∣∣xj − zj

∣∣< 0.05djx

2j, otherwise

with zj = 0.2⌊∣∣∣ xj

0.2

∣∣∣+ 0.49999⌋

sgn(xj); IPR : xj ∈ [−1000,

1000]. The original function is a four-dimensional optimiza-tion problem with dj = {1, 1000, 10, 100}. The global minimumis f3(x) = 0, with |xj| < 0.05. In this study, we modify the djvalues randomly selected from the set {1, 1000, 10, 100} asfollows:

dj =

⎧⎪⎨⎪⎩

1, for rand < 0.251000, for 0.25 ≤ rand < 0.510, for 0.5 ≤ rand < 0.75100, for 0.75 ≤ rand; j = 1, . . . , D

where rand is a random number between 0 and 1. As a result,the problem becomes a D-dimensional problem.

(4) F21 function f4(x) = g1(x) +D∑

i=1

u(xi, 10, 100, 4) with

g1(x) = (�/D){

10 sin2((� + (�/4)(x1 − 1))

+D−1∑i=1

0.125(xi−1)2[1+10 · sin2((�+(�/4)(xi+1 − 1))]

+ 0.125(xD − 1)2}

where the penalization function u(z, a, k, m) is defined by

u(z, a, k, m) ={

k(z − a)m, z > a0, −a ≤ z ≤ ak(−z − a)m, z < −a

hemic

(

(

P.-K. Liu, F.-S. Wang / Computers and C

The initial parameter range, IPR, is assigned by xj ∈ [−10, 10].The global minimum is f4(1) = 0.

(5) F22 function f5(x) = g2(x) +D∑

i=1

u(xi, 10, 100, 4) with

g2(x) = (�/D){

10 sin2(�x1)

+D−1∑i=1

(xi − 1)2[1 + 10 sin2(�xi+1 − 1)] + (xD − 1)2}

The initial parameter range, IPR, is assigned by xj ∈ [−10, 10].The global minimum is f5(1) = 0.

(6) F23 function f6(x) = g3(x) +D∑

i=1

u(xi, 10, 100, 4) with

g3(x) = 0.1{

sin2(3�x1) +D−1∑i=1

(xi − 1)2[1 + sin2(3�xi+1)

+ (xD − 1)2[1 + sin2(2�xD)]}

The initial parameter range, IPR, is assigned by xj ∈ [−10, 10].The global minimum is f6(1) = 0.

(7) F24 function f7(x) = g3(x) +D∑

i=1

u(xi, 5, 100, 4)

The initial parameter range, IPR, is assigned by xj ∈ [−10, 10].The global minimum is f7(1) = 0.

(8) Griewangk’s function f8(x) =D∑

j=1

x2j

4000 −D∏

j=1

cos(

xj√j

)+ 1

IPR: xj ∈ [−600, 600]. The global minimum is f8(0) = 0.(9) Rastrigin’s function

f9(x) = 10D +D∑

i=1

(x2j − 10 cos(2�xj))

IPR: xj ∈ [−600, 600]. The global minimum is f9(0) = 0.10) Rosenbrock’s saddle function

f10(x) =D−1∑j=1

[100(x2j − xj+1)

2 + (1 − xj)2]

IPR: xj ∈ [−2.048, 2.048]. The global minimum is f10(0) = 0.(11) Katsuura’s function

f11(x)=D∏

j=1

(1 + j

ˇ∑k=0

|2kxj − n int(2kxj)|2−k

), ˇ is set to 32;

IPR: xj ∈ [−2.048, 2.048]. The global minimum is f11(0) = 0.12) Schwefel function

f12(x) =D∑

i=1

(D∑

j=1

(aij sin ˛j + bij cos ˛j) −n∑

j=1

(aij sin xj + bij cos xj)

)2

;

IPR: xj ∈ [−65.536, 65.536]. The coefficients aij and bij areinteger random numbers between −100 and 100, and ˛j arereal random numbers between −� and �. The global minimumis f12(0) = 0.

al Engineering 33 (2009) 1851–1860 1859

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