8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 1/6

Computational Biology and Chemistry 33 (2009) 1–6

Contents lists available at ScienceDirect

Computational Biology and Chemistry

 j o u r n a l h o m e p a g e :   w w w . e l s e v i e r . c o m / l o c a t e / c o m p b i o l c h e m

Research Article

Computational method for inferring objective function of glycerol

metabolism in Klebsiella pneumoniae

Zhaohua Gong a,b,∗, Chongyang Liu a,b, Enmin Feng a, Qingrui Zhang c

a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, Liaoning, Chinab Mathematics and Information Science College, Shandong Institute of Business and Technology, Yantai 264005, Shandong, Chinac Department of Biotechnology, Dalian University of Technology, Dalian 116012, Liaoning, China

a r t i c l e i n f o

 Article history:

Received 18 June 2008

Accepted 22 June 2008

Keywords:

Optimization modelling

1,3-Propanediol

Metabolism

Klebsiella pneumoniae

Genetic algorithm

Robustness analysis

a b s t r a c t

Flux balance analysis (FBA) is an effective tool in the analysisof metabolic network. It canpredict the flux

distribution of engineered cells, whereas the accurate prediction depends on the reasonable objective

function. In this work, we propose two nonlinear bilevel programming models on anaerobic glycerol

metabolismin Klebsiellapneumoniae (K. pneumoniae)for 1,3-propanediol (1,3-PD) production. One intends

to infer the metabolic objective function, and the other is to analyze the robustness of the objective

function. In view of the models’ characteristic an improved genetic algorithm is constructed to solve

them, where some techniques are adopted to guarantee all chromosomes are feasible and move quickly

towards the global optimal solution. Numerical results reveal some interesting conclusions, e.g., biomass

production is the main force to drive K . pneumoniae metabolism, and the objective functions, which are

obtained in term of several different groups of flux distributions, are similar.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

1,3-Propanediol (1,3-PD) possesses potential applications on a

large commercial scale, especially as a monomer of polyesters or

polyurethanes, its biosynthesis has attracted worldwide interests

(Bibel et al., 1999; Nakamura and Whited, 2003),  etc. Among all

kinds of microbial production of 1,3-PD, dissimilation of glycerol

by Klebsiella pneumoniae (K. pneumoniae) has been widely investi-

gateddue to itshigh productivitysince 1980s(Zengand Biebl, 2002;

Menzel et al.,1997). However, comparedwith the competingchem-

ical process, the microbial production is difficult to obtain a high

1,3-PD concentration in the fermentation broth. Its an area of inter-

est to develop an improved technique to improve the productivity

of 1,3-PD.

The knowledge of cell physiology and metabolic regulation is

helpful to improve productivity of 1,3-PD by a metabolic engineer-ing approach on the strain (Stephanopoulos et al., 1998). Moreover,

computational methods for cellular metabolism play an impor-

tant role in understanding the complex biochemical interactions

within the cells. Among those methods, FBA has been proved to

be an effectively computational tool for understanding cell phys-

∗ Corresponding author at: Department of Applied Mathematics, Dalian Univer-

sity of Technology, Dalian 116024, Liaoning, China. Tel.: +86 41184708351 8025;

fax: +86 41184708116.

E-mail address: yt gzh@yahoo.com.cn (Z. Gong).

iology and regulation of metabolism, and has been successfully

applied to different microorganisms  (Maczek et al., 2006; Ozkan

et al., 2005; Sanchez et al., 2006; Shirai et al., 2005) . FBA can pre-

dict the metabolic fluxes by using a reduced set of measured fluxes

and mass balance equations around intracellular metabolites, and

thereby provide a better characterization of cellular phenotypes.

However, the accurate prediction depends on a reasonable objec-

tive function. Objective functions in practice can take on a linear

form, i.e.,   f   = c v , where   c   denotes the vector defining the coef-

ficients or weights for each flux in   v   (Beard et al., 2002).   The

elements of c enable the formulation of a number of diverse objec-

tives. Common objective functions include maximizing biomass or

cell growth, maximizing ATP production or maximizing the rate

of synthesis of a particular product  (Kauffman et al., 2003). Other

objective functions include minimizing ATP production in order

to determine conditions of optimal metabolic energy efficiency,and minimizing nutrient uptake in order to evaluate the condi-

tions under which a cell will perform its metabolic functions while

consuming the minimum amount of nutrients (Lee et al., 2006), etc.

Although many hypotheses have been put forward as surro-

gates for cellular objective functions, substantially less work has

been conducted toward systematically validating them with exper-

imentally derived flux distributions of metabolic networks. An

optimization-based framework is proved to be effective for infer-

ring the objective function of   E. coli  metabolism   (Burgard and

Maranas, 2003). For solving the bilevel problem, the authors trans-

form it into an equivalent nonlinear programming making use of 

1476-9271/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compbiolchem.2008.06.005

8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 2/6

2   Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1–6

the duality theory. However, up to date, no attempt has been made

to infer the objective function of anaerobic glycerol metabolism in

K. pneumoniae for 1,3-PD production.

In this work, we introduce a rigorous mathematical model

(NBP1) to infer whether a weighted sum of fluxes can explain

a set of experimental data observed from anaerobic glycerol

metabolism in   K. pneumoniae   for 1,3-PD production. The model

is a nonlinear bilevel programming, whose lower level is a con-

ventional FBA model with undetermined parameters and upper

level is a quadratic programming for evaluating the consistency

with observed fluxes.  Bard (1991)  proves that the bilevel linear

programming is NP-hard. So it is more difficult to solve NBP1.

To obtain the weight coefficients in the model, a fast convergent

genetic algorithm is constructed by adopting some techniques,

which greatly reduce the searching space and avoid the difficulty

to deal with the infeasible chromosomes. In this work, we also

propose a model (NBP2) to investigate the effect of deviations

between the flux distribution and the experimental ones on the

weight coefficients. Some meaningful conclusions are drawn by

several groups of data observed at different dilution rates and

initial glycerol concentrations.

The remainder of this paper is organized as follows. In Sec-

tion   2,   optimization models are formulated to determine the

objective function and analyze the robustness of the objective

function. In Section   3,   we construct an improved genetic algo-

rithm to solve NBP1(2). Section  4 gives the main results. Finally,

Section   5   concludes some remarks and future research direc-

tions.

2. Optimization Models

In this section, we propose a mathematically rigorous model to

determine whether the maximization of a weighted combination

of fluxes can explain the objective function of anaerobic glycerol

metabolism in K. pneumoniae for the production of 1,3-PD.

 2.1. Glycerol Metabolism in K. pneumoniae

K. pneumoniae   German Collection of Microorganisms (DSM)

2026 obtained from the DSM is used in this work. Culture medium

composition, culture conditions, and methods for the determina-

tion of fermentation products were reported previously (Ahrens et

al., 1998; Menzel et al., 1996).

It follows from Zeng et al. (1993) and Chen et al. (2003)  and

the related knowledge that the simplified metabolic network of 

bio-dissimilation of glycerol to 1,3-PD by   K. pneumoniae   under

anaerobic conditions is constructed by the author   Zhang et al.

(2006).   In the network, the intermediate metabolites without

branches are omitted. The network is composed of 22 reactions

and 11 intra-metabolites. In this work, we assume each reaction

flux associated with a metabolic drain or energy dissipation maybe related to the objective function driving cellular metabolism.

These fluxes are shown with bold arrows in   Fig. 1.   We also

list all mass balance equations of intracellular metabolites in

Table 1.

 2.2. Flux Balance Analysis (FBA)

In this paper, we make the following assumptions:

(H1). The metabolic system operates at a pseudo-steady state,

equivalently, considering the time-averaged behavior for sta-

tionary metabolic concentrations are zeros. In other words,

there is no intracellular metabolic accumulation in the net-

work;

(H2). Weight coefficients are assigned to each reaction flux associ-

ated with a metabolic drain or energy dissipation, the index

set of these reaction fluxes is denoted by P .

Basing on the above assumptions, flux balance analysis model

can be formulated as

max   c Tv 

s.t.   Sv = 00 ≤ v ≤ v max.

(1)

where cost vector c  = (c 1, . . . , c  n)T∈ C ⊂ Rn, and

C 

c ∈Rn|

n j=1

c  j  = 1 and

c  j  = 0 if  j /∈ P c  j ∈ (0, 1) if   j∈P 

.

In (1), S isan m × n matrix, Sij is the stoichiometric coefficient of 

metabolite  i in reaction j for all i = 1, . . . , m and j = 1, . . . , n.  v ∈Rn

denotes the vector of fluxes with elements corresponding to the

fluxes in given reactions (column) in S . c  j  is the weight associated

with reaction j. Furthermore, a high coefficient value implies that

the corresponding flux is maximized for the cellular metabolism,

whereas a low value implies the converse. 0 ≤ v ≤ v max   are boxconstraints, here   v max  is the maximal flux vector. Since the num-

ber of equations (m) is far less than that of unknown variables (n),

the equations Sv = 0 are under-determined and the solution is not

unique. Therefore, FBA optimizes the set of fluxes such that the

objective is maximized.

 2.3. Model for Inferring Objective Function

In this study, our purpose is to infer the objective function of 

the above FBA. That is, we need identify the weights  c  such that

the optimal solution of FBA (denoted by  v ∗) is consistent with the

experimental data. Let the experimental fluxes be  v ek

, k∈E , where

E  represents the index set of the fluxes with observed values. Here

we measure the consistence between  v ∗

and  v 

e

by the sum of thesquared flux deviations, i.e.,k∈E 

(v ∗

k− v 

ek

)2

(2)

Obviously, the sum is smaller,   v ∗ is more consistent with the

experimental fluxes. Hence the nonlinear bilevel programming

model to infer the objective of FBA can be formulated as

(NBP1) minc,v 

 J (c, v ) =k∈E 

(v k(c ) − v ek)

2

s.t.   c ∈ C 

where   v solves

maxv 

 f (c, v ) = c Tv 

s.t.   Sv = 0

0 ≤ v ≤ v max.

(3)

In above model, thelower level is a conventionalFBA modelwith

undetermined parameters and the upper level is a quadratic pro-

gramming for evaluating consistency with observed fluxes.  J (c, v )

and   f (c, v ) are the objective functions of upper level (termed as

leader) and lower level (termed as follower),respectively. c ∈Rn and

v ∈Rn are the decision variables of the upper level and lower level,

respectively.   v k(c ) represents the  k  th component of the optimal

solution of the lower level when c ∈ C  is given. In NBP1, the leader

makes its decision, taking account into the reaction of the follower.

In order to facilitate further discussion, some notations and

definitions are necessary to introduce following by  Bard (2006).

Let ˝ = {(c, v )∈Rn × Rn|c ∈ C , Sv = 0, 0 ≤ v ≤ v max} denote the con-

straint region of NBP. Let   ˝(c ) = {v ∈Rn

|Sv = 0, 0 ≤ v ≤ v max}   be

8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 3/6

 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1–6   3

Fig. 1.  Metabolic network of anaerobic glycerol metabolism in K. pneumoniae for 1,3-PD production. Reaction fluxes assigned with the weight coefficients are shown with

bold arrows.

the follower’s feasible region for fixed   c . In fact,   ˝(c ) is inde-

pendent of  c  in our NBP1 model. Let  ˝(C ) = {c ∈ C |∃v s.t.(c, v )∈˝}

be projection of  ˝   on the upper decision space, and let   M (c ) =

arg min{ f (c, v )|v ∈˝(c )}   be the follower’s rational reaction set for

a given   c . The union of all possible vectors   c   which the leader

may select and the corresponding rational reaction set   v ∈M (c )is called the inducible region. Let the inducible region denote by

IR = {(c, v )|c ∈˝(C ), v ∈M (c )}.

 Table 1

Mass balance equations of intracellular metabolites for glycerol bioconversion to

1,3-PD by K. pneumoniae in anaerobic continuous culture

No. I ntra-metabolite s Mas s balance equ ations un der

steady-state condition

1 Glycerol   v 1 − v 2 − v 3 − v 4  = 0

2 Phos.   v 4 − v 5 − v 6  = 0

3 Pyruvate   v 5 − v 8 − v 9 − v 10 − v 16 − v 19 = 0

4 Aceytl-CoA   v 9 + v 10 − v 12 − v 13 = 0

5 Acetoin 0.5v 16 − v 17 − v 7  = 0

6 Formate   v 9 − v 11 − v 18 = 0

7 CO2   v 10 + v 11 + v 16 − v 6 − v 20  = 0

8 H2   v 14 + v 11 − v 21 = 0

9 NADH2   v 3 − v 2 + 2v 4 − 2v 6 − v 8 − v 7 + v 15 −

2v 13 = 0

10 ATP   −7.5v 3 + v 5 + v 6 + v 12 − v 22  = 0

11 FADH2   v 10 − v 15 − v 14 = 0

Definition 1.   A point (c, v ) is feasible to NBP1 if (c, v )∈ IR.

(c ∗, v ∗)∈ IR   is called to be globally optimal to NBP1 if   J (c ∗, v 

∗) ≤

 J (c, v ) holds for any (c, v )∈ IR.

Definition 2.   A point-to-set mapping    : R p → 2Rqis called poly-

hedral if its graph  gr :={( x, y)∈R p × Rq| y∈˝( x)}   is equal to the

union of a finite number of polyhedral convex sets.

It can be shown that NBP1 has the following two properties.

Property 1.   The point-to-set mapping M (·) :  C → 2Rn

is a non-empty polyhedral.

Property 2.   NBP1 has a global optimal solution.

 2.4. Model for Robustness Analysis

Let the minimum value of the upper function of NBP1 be Min

and the optimal value of  c  be  c ∗. We assume the range of the sum

of the squared deviations between the identified and experimental

fluxes is from 0 to   r  Min due to experimental errors, where   r   is

a positive constant. Now we formulate a model to investigate the

effect of deviations on the robustness of the weight coefficient c  j in

(3), j ∈P , i.e., to determine the largest variation ranges of  c  j  due to

8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 4/6

4   Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1–6

experimental errors. The model can be written as

(NBP2) maxc,v 

 J (c, v ) = |c  j − c ∗ j|

s.t.   c ∈ C 

where   v solves

maxv 

 f (c, v ) = c Tv 

s.t.   Sv = 0k∈E 

(v k − v ek

)2 ≤   Min

0 ≤ v ≤ v max.

(4)

With respect to NBP2, the corresponding notations and defini-

tions can be introduced. It canalso be shown that NBP2 hasoptimal

solutions.

3. Algorithm for Solving NBP1(2)

Bilevel programming problemsare noteasy to solve.In fact they

have been proved to be NP hard. In recent years, many approaches

have been presented for solving linear bilevel programming, such

as branch and bound approach, penalty method, genetic algorithm

andmanyothers.Whilenonlinear bilevel programming is still in its

infancy with a hand full of algorithms such as branch and bound,

global optimization (Amouzegar, 1999; Fliege and Vicente, 2006;

Gümüs and Floudas, 2001; Lan et al., 2007; Wang et al., 2007, etc.).

Genetic algorithm (GA) is an iterative randomsearch algorithm.

GA maintains a population of candidate solutions where each

solution is called a chromosome. A set of chromosomes forms a

population which is ranked by a fitness function and evolves to the

next generation by randomly selecting “parents” from the popula-

tion and reproducing “children” using genetic operations. A set of 

selected individuals forms an improved population.

In this paper, we constructan improvedGA to solve theNBP1(2),

which takes advantage of the problem’s structural characteristic.

In the algorithm, the upper variable is coded as float, and the fit-

ness function is based on the objective function of the upper level.Designing the genetic operators, we adopt sometechniques to guar-

antee the offspring are still in the constraint region of NBP1(2) and

they move quickly towards the global optimal solution. Those tech-

niques reduce the searching space and avoid the difficulty to deal

with infeasible points. Now we describe the genetic operators in

detail.

Initial population: Randomly generate N  individuals by uniform

distribution from C , i.e., c i(0)∈ C , i = 1, 2, . . . , N  .

Fitness function: The fitness function is defined as

fit(c ) =  J (c, M (c )),   (5)

where J (·, ·) is the upper objective function of NBP1(2), and  M (c ) is

obtained by solving their corresponding lower level optimizationproblem.

Crossover operator : Let  pc  be the crossover probability. Repeat

the following operations from  i = 1 to  i = N : generate a random

number r i ∈ [0, 1], and if  r i < pc, then c i(k) is selected as a parent.

Furthermore, randomlymate all parents selected. Let c i(k) and c  j(k)

be two mated parents, whose offspring are denoted by  d i(k) and

d j(k). The formula of crossover is as follows:

di(k) = c i(k) + (1 − )c  j(k),   (6)

d j(k) = (1 − )c i(k) + c  j(k),   (7)

where ∈ [0, 1] is a random number. It is easy to find that the gen-

erated offspring are still in  C . Let the total number of all offspring

generated by crossover be N 1.

Mutation operator : Thebest individual of thepresent individuals

is denoted by d∗(k), so d∗(k) divide N 1 into two groups, i.e.,N 1+  = {l∈ {1, 2, . . . , N  1}|dl(k) − d∗(k) ≥ ε}

N 1−  = {l∈ {1, 2, . . . , N  1}|dl(k) − d∗(k) < ε}.   (8)

where ε isasmallpositivenumberand · represents the Euclidean

form. For each l∈N 1+, wemutate dl(k) bythe probability pm accord-

ing to the following formula

 g m(k) = ˛dl(k) + (1 − ˛)d∗(k),   (9)

where ˛ ∼ N (0, 1). For each l ∈N 1−, we mutate g l(k) by the proba-

bility pm according to the following formula:

 g m(k) = ˇdl(k) + (1 − ˇ)drand(k),   (10)

where ˇ ∼ N (0, 1) and  drand(k)∈ C  is generated randomly. Let the

total number of mutated offspring be N 2, then the present popula-

tion has N + N 1 + N 2 individuals, which are all in  C .

Selection operator : In order to guarantee the global convergence

of new GA, we select the best  N  individuals from the  N + N 1 + N 2ones to form the next generation.

So the steps of the improved GA for solving NBP1(2) can be

described as follows:

Step 1: Initialize crossover probability pc, mutation probability pm,

population size   N , the maximal iterations   K . Set   k = 0,

generate the initial population   p(k) = {c i(k)|i = 1, . . . , N  }.Compute each individual’s fitness value fit(c i(k)) by (5).

Step 2: Generate the crossover offspring dl(k)(l = 1, . . . , N  1)bythe

above crossover operator and evaluate their fitness value

fit(dl(k)).

Step 3: Generate the mutation offspring g m(k)(m = 1, . . . , N  2) by

the above mutation operator and evaluate their fitness

value fit( g m(k)).

Step 4: Form the next generation p(k + 1) by selection operator.

Step 5:  k = k + 1, if either k > K  or no progress is made in the last

generations, then output the best individual and stop, oth-erwise go to Step 2.

4. Main Results

Three groups of steady-state experimental data (Menzel et al.,

1996) at different dilution rates and initial glycerol concentrations

are used in this study. These data include the concentrations of 

glycerol, 1,3-PD, biomass, ethanol, lactate, etc. We may obtain 13

fluxes from each group of experimental data. The three groups of 

experimental fluxes (v ek

) and computed fluxes (v k) by NBP are all

listed in Table 2. In Table 2, both group 1 and group 2 are obtained

under substrate-limited conditions, whereas group 3 is got under

substrate-excess conditions. The coefficients identified are shown

in Fig. 2. The coefficients qualify the contribution of a given flux tothe objective function. That is, a high coefficient value implies that

the corresponding flux is maximized for the cellular metabolism,

whereas a low value implies the converse. The deviations of the

weight coefficients are shown in Fig. 3. The parameters chosen in

above algorithm is as follows: K  = 100, N  = 30, pc  = 0.8, pm  = 0.2.

In our study, we draw five conclusions. (i) The three groups

of weight coefficients are similar though the flux distributions

are quite different. The result is consistent with a conclusion

(Burgard and Maranas, 2003),   i.e., there exists a single cellular

driving force governs the distribution of metabolic fluxes under

different conditions. (ii) Biomass production is the main force to

drive K . pneumoniae metabolism because its weight coefficients is

the largest among all coefficients. (iii) The coefficient of biomass

obtained under substrate-excess conditions is a bit smaller than

8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 5/6

 Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1–6   5

 Table 2

Three sets of experimentally observed and calculated flux distributions under different D and C s0  conditions (anaerobic case)

Flux distributions

1 (0.15,809) a 2 (0.35,443) a 3 (0.35,870) a

v e1

(v 1) 25.28 (25.28) 58.69 (58.69) 112.50 (112.50)

v e2

(v 2) 10.72 (10.96) 28.77 (28.77) 65.21 (65.18)

v e3

(v 3) 1.485 (1.98) 3.465 (3.570) 3.465 (3.469)

v e

6(v 

6) 0.573 (0.891) 1.115 (1.836) 1.765 (2.590)

v e7

(v 7) 0.013 (0) 0 (0.0479) 11.73 (8.2)

v e8

(v 8) 0.464 (0.464) 0.532 (0.615) 3.935 (3.935)

v e12

(v 12) 3.768 (3.747) 10.032 (10.089) 10.543 (10.543)

v e13

(v 13) 6.992 (6.992) 12.459 (12.459) 10.130 (9.730)

v e17

(v 17) 0 (0) 0.047 (0) 0.498 (0)

v e18

(v 18) 0.541 (0.541) 2.226 (2.226) 6.017 (6.130)

v e19

(v 19) 0.004 (0.004) 0.021 (1.261) 0.096 (1.05)

v e20

(v 20) 9.452 (9.452) 18.71 (18.71) 22.90 (22.90)

v e21

(v 21) 9.306 (9.306) 18.58 (18.71) 11.55 (11.55)

Note: D, dilution rate (h−1); C s0, initial glycerol concentration (mmol L −1 h−1 );  v ek

, experimentally observed data;  v k, computated data according to NBP.a (D, C s0).

Fig. 2.  The values of weight coefficients obtained by three groups of flux distribu-

tions.

that obtained undersubstrate-limitedconditions. This is consistent

with the observed experimental fact that the growth of  K. pneu-

moniae is inhibited by the excess substrate. (iv) For a given flux

distribution, the range of allowable value c  j  is rather narrow, that

is, the weight coefficient  c  j  is of good robustness with respect to

experimental errors. (v) The errors between experimental fluxes

and computed ones are small, so FBA can predict the flux distribu-

tion moreaccurately making use of the obtained objective function.

These results can provide theoretical guidefor metabolicregulation

to improve the 1,3-PD production from glycerol by  K. pneumoniae

in anaerobic continuous culture.

5. Discussion and Further Studies

Undoubtedly, mathematical modelling of cellular metabolism

plays an important role to understand biological functions and to

provide identification of targets for biotechnological modification.

In this paper, we propose two NBP models on anaerobic glycerol

metabolism in K. pneumoniae for 1,3-propanediol production. One

intends to infer the metabolic objective function, and the other is

to analyze the robustness of the objective function. NBPis NP-hard,

andis noteasyto solve.We constructa fast convergent genetic algo-

rithm tosolvethe proposedmodels. Atlast, weget some interesting

results. The results are of significance. On the one hand, it can pro-

vide the theoretical guide for metabolic regulation to improve the

Fig.3.   Maximalabsolute variations of theweightcoefficients,i.e., max |c  j  − c ∗ j|, j∈P ,

forthe (a) group1,(b) group2 and(c) group3 experimental flux distributions when

solution constraint is relaxed. Where  F (c, v ) =

k∈E (v k  − v 

ek

)2

.

1,3-PD production. On the other hand,we can predict the intracellu-

lar fluxes more accurately. In addition, the optimization model can

also be used to determine the objective functions of other microor-

ganism’s metabolism.

8/12/2019 Gong Article

http://slidepdf.com/reader/full/gong-article 6/6

6   Z. Gong et al. / Computational Biology and Chemistry 33 (2009) 1–6

 Acknowledgements

This research is supported by National Natural Science Founda-

tion of China (No. 10671126) and National Basic Research Program

of China (No. 2007AA02Z208).

References

Ahrens, K., Menzel, K., et al., 1998. Kinetic, dynamic, and pathway studies of glyc-erol metabolism by  Klebsiella pneumoniae  in anaerobic continuous culture. III.Enzymes and fluxes of glycerol dissimilation and 1,3-propanediol formation.Biotechnol. Bioeng. 59, 544–552.

Amouzegar,M.A., 1999. A globaloptimization methodfor nonlinearbilevel program-ming problem. IEEE Trans. Syst. Man, Cybernet. B: Cybernet. 29 (6), 771–777.

Bard, J.F., 1991. Someproperties of the bilevel programming problem. J. Optim. The-ory Appl. 68, 371–378.

Bard, J.F., 2006. Practical Bilevel Optimization: Algorithm and Application (SeriesNonconvex Optimization and its Application). Springer Verlag.

Beard, D.A., Liang, S.D., Qian, H., 2002. Energy balance for analysis of complexmetabolic networks. Biophys. J. 8 (3), 79–86.

Bibel, H., Memzel,K., Zeng, Z.P., 1999. Microbial production of 1,3-propanediol. Appl.Microbiol. Biotechnol. 52, 289–297.

Burgard, A.P., Maranas, C.D., 2003. Optimization-based framework for inferring andtesting hypothesized metabolic objective functions. Biotechnol. Bioeng. 82 (6),670–677.

Chen, X., Xiu, Z.L., Wang, J.F., Zhang, D.J., Xu, P., 2003. Stoichiometric analysis and

experimental investigation of glycerolbioconversionto 1,3-propanediol by Kleb-siella pneumoniae under microaerobic conditions. Enzyme Microb. Technol. 33,386–394.

Fliege, J., Vicente,L.N.,2006. Multicriteriaapproachto bileveloptimization. J. Optim.Theory Appl. 131 (2), 209–225.

Gümüs, Z.H., Floudas, C.A., 2001. Global optimization of nonlinear bilevel program-ming. J. Global Optim. 20, 1–31.

Kauffman, K.J., Prakash, P., Edwards, J.S., 2003. Advances in flux balance analysis.Curr. Opin. Biotechnol. 14, 491–496.

Lan, K.W., Wen, U.P., Shin, H.S., Lee, E.S., 2007. A hybrid neural network approach tobilevel programming problems. Appl. Mater. Lett. 20 (8), 880–884.

Lee, J.M., Gianchandani, E.P., Papin, J.A., 2006. Flux balance analysis in the era of metabolomics. Briefings Bioinform. 7 (2), 140–150.

Maczek, J.,Junne, S.,et al., 2006. Metabolic flux analysisof thesterol pathwayin theyeast Saccharomyces cerevisiae. Bioprocess. Biosyst. Eng. 29, 241–252.

Menzel,K., Zeng, A.P., Deckwer, W.D.,1996. Kinetic,dynamic,and pathway studiesof glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous c ulture.I. The phenomena and characterizationof oscillation and hysteresis.Biotechnol.Bioeng. 52, 549–560.

Menzel, K., Zeng, A.P., Deckwer, W.D., 1997. High concentration and productivity of 1,3-propanediol fromcontinuousfermentationof glycerol by Klebsiella pneumo-niae. Enzyme Microb. Technol. 20, 82–86.

Nakamura, C.E., Whited, G.M., 2003. Metabolic engineering for the microbial pro-duction of 1,3-propanediol. Curr. Opin. Biotechnol. 14, 454–459.

Ozkan, P., Sariyar, B., et al., 2005. Metabolic flux analysis of recombinant proteinoverproduction in Escherichia coli. Biochem. Eng. J. 22, 167–195.

Sanchez, A.M., Bennett, G.N., et al., 2006. Batch culture characterization andmetabolic flux analysis of succinate-producing  Escherichia coli  strains. Metab.Eng. 8, 209–226.

Shirai, T., Nakato, A., et al., 2005. Comparative study of flux redistribution of metabolic pathway in glutamate production bytwo coryneform bacteria. Metab.Eng. 7, 59–69.

Stephanopoulos,G., Aristidou,A., Nielsen,J., 1998. MetabolicEngineering: Principlesand Methodologies. Academic Press, San Diego.

Wang, G.M., Wang, X.J., Wan, Z.P., Lv, Y.B., 2007. A globally convergent algorithm forsolving a class of bilevel nonlinearprogrammingproblem. Appl. Math. Comput.188 (1), 166–172.

Zeng, A.P., Biebl, H., Schlieker, H., Deckwer, W.D., 1993. Pathway analysis of glyc-erol fermentation by  Klebsiella pneumoniae: regulation of reducing equivalentbalance and product formation. Enzyme Microb. Technol. 15, 770–779.

Zeng, A.P., Biebl, H., 2002. Bulk-chemicals from biotechnology: the case of 1,3-propanediol production and the new trends. Adv. Biochem. Eng. Biotechnol. 74,239–259.

Zhang, Q.R., Xiu, Z.L., Zeng, A.P., 2006. Optimization of microbial production of 1,3-propanediol by   Klebsiella pneumoniae   under anaerobic and microaerobicconditions by metabolic flux analysis. J. Chem. Ind. Eng. 57, 1403–1409.