Mathematical and Computer Modelling 54 (2011) 618–624

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Mathematical and Computer Modelling

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Modeling and identification of a nonlinear hybrid dynamical system inbatch fermentation of glycerolJuan Wang a,b,∗, Jianxiong Ye a, Enmin Feng a, Hongchao Yin b, Zhilong Xiu c

a School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, PR Chinab School of Energy and Engineering, Dalian University of Technology, Dalian, Liaoning 116024, PR Chinac School of Environmental and Biological Science and Technology, Dalian University of Technology, Dalian, Liaoning 116012, PR China

a r t i c l e i n f o

Article history:Received 18 September 2010Received in revised form 3 March 2011Accepted 3 March 2011

Keywords:Batch cultureTransport mechanismsNonlinear hybrid dynamical systemSystem identificationOptimization algorithm

a b s t r a c t

A nonlinear hybrid dynamical system is established to describe the batch culture ofglycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae), in whichglycerol is assumed to pass the cell membrane by passive diffusion coupledwith facilitatedtransport and transport mechanism of 1,3-PD is assumed to be uncertain. The existence,uniqueness and continuity of solutions are also discussed. To infer the most reasonabletransport mechanism of 1,3-PD, a system identification model consisting of both discreteand continuous variables is proposed. Additionally, we construct a two-phase optimizationalgorithm on the basis of modified complex method together with very fast simulatedannealing technology. Numerical results show that it is most reasonable for 1,3-PD to passthe cell membrane by passive diffusion coupled with facilitated transport and that theoptimization algorithm is valid.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

As a by-product of the soap and detergent industry, glycerol can be converted to 1,3-PD by a number of bacteria suchas K. pneumoniae, Clostridium butyricum and Citrobacter freundii [1–3]. This bioconversion process is of technical interestsince the target product 1,3-PD has a wide range of potential uses, in particular as a monomer for polyesters, polyethers andpolyurethanes [4].

A great deal of research on this bioconversion process has been made since the 1980s [5], including experimentalinvestigation of the multiple inhibitions in the fermentation [6,7], modeling or parameter identification of this complexbio-process [8–11], and metabolic flux or metabolic pathway analysis [12,13].

Compared with continuous and fed-batch cultures, glycerol fermentation in batch culture can obtain the highestproduction concentration and molar yield 1,3-PD to glycerol [14]. So the nonlinear dynamical systems in this culture havebeen extensively studied in recent years [15–19].

Among all dynamical systems in batch culture, the latest one is proposed by Sun et al. in 2008 [15], in which intracellularsubstances (glycerol and 1,3-PD), intermediate substance (3-hydroxypropionaldehyde(3-HPA)), aswell as key enzymes (1,3-PDoxydoreductase (PDOR) and glycerol dehydratase (GDHt)) are all taken into consideration. Themodel is established underthe assumption that glycerol passes the cell membrane by passive diffusion coupled with facilitated transport and 1,3-PDby passive diffusion. However, since there is a scarce number of reports dealing with the transport mechanism of 1,3-PDand it is still not exactly known, we cannot guarantee the reliability of the model.

∗ Corresponding author at: School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, PR China. Tel.: +8641184708351x8025; fax: +86 41184708354.

E-mail addresses:whujuan@163.com (J. Wang), hcyin@dlut.edu.cn (H. Yin).

0895-7177/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2011.03.005

J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624 619

?

Intracellular environment

Extracellular environment (Glycerol) Glycerol 3-HPA 1,3-PD

Extracellular environment (1,3-PD)

Fig. 1. Reductive pathway of anaerobic glycerol metabolism in K. pneumoniae.

According to the literature [8], there are three possible transport mechanisms of substrate across cell membrane,i.e., passive diffusion, facilitated transport or passive diffusion coupled with facilitated transport. Among them, the passivediffusion of glycerol from the extracellular to intracellular environment, which complies with Fick’s law of diffusion [20],is active only when extracellular concentration is higher than its intracellular concentration. So the execution of passivediffusion is state dependent [21,22], and the metabolic process is actually a state-based switched process which can bedescribed by a hybrid dynamical system.

In the present work, aiming to determine the most possible transport mechanism of 1,3-PD, under the assumptionthat glycerol passes the cell membrane by passive diffusion coupled with facilitated transport, we established a nonlinearhybrid dynamical system to describe the batch fermentation of glycerol by K. pneumoniae. The existence, uniqueness andcontinuity of solutions are discussed. With the relative error of extracellular concentrations as performance index, a systemidentification model including discrete and continuous variables is established, which is subjected to the proposed hybriddynamic system. Finally, we develop an optimization algorithm based onmodified complex method together with very fastsimulated annealing algorithm [23,24] and apply it to the identification problem.

The rest of this paper is organized as follows. In Section 2, a nonlinear hybrid dynamical system of batch culture isestablished. In Section 3, some properties of solutions are discussed. A system identification model is proposed in Section 4.Section 5 is devoted to constructing a two-phase optimization algorithm. Numerical results are presented in Section 6.Discussions and conclusions are presented at the end of this paper.

2. Nonlinear hybrid dynamical system of batch culture

The fermentation of glycerol covers both extracellular and intracellular environments. The two environments are linkedby the transports of substrate and products across cell membranes. Since transport mechanism of 1,3-PD is still uncertain,we will take all three possible transport mechanisms, i.e., facilitated transport, passive diffusion or passive diffusioncoupled with facilitated transport, into consideration for mathematical modeling. Under anaerobic conditions, glycerol isdissimilated through coupled oxidative and reductive pathways and the latter generates the goal product 1,3-PD as shownin Fig. 1 [25]. In this paper, only the reductive pathway is emphasized because 3-HPA is the key intermediate for 1,3-PDproduction.

According to the actual experiment, we assume that(H1) No medium is pumped inside or outside the reactor in the process of batch fermentation.(H2) The concentration of reactants are uniform in the reactor, while time delay and nonuniform space distribution are

ignored.

Let x(t) := (x1(t), x2(t), . . . , x8(t))T ∈ R8 be the state vector, and x1(t), x2(t), . . . , x8(t) denote the concentrations ofbiomass, extracellular glycerol, extracellular 1,3-PD, acetate, ethanol, intracellular glycerol, 3-HPA and intracellular 1,3-PDat time t , respectively. To simplify notation, let In := {1, 2, . . . , n} and xi := xi(t), i ∈ I8.

Assumed that cell growth is always inhibited by the intermediate substance 3-HPA, the specific cell growth rate can beexpressed by Eq. (1) [3,15].

µ = µmx2

x2 + Ks

1 −

x7x∗

7

5∏i=2

1 −

xix∗

i

. (1)

While the uptake of extracellular glycerol is considered as a ‘‘black box’’ model, its specific consumption rate can beexpressed by Eq. (2) [15].

q20 = m2 +µ

Y2+1q2

x2x2 + K ∗

2. (2)

Similarly to the literature [15], extracellular glycerol is assumed to pass the cell membrane by passive diffusion coupledwith facilitated transport, so its specific consumption rate can be expressed by

q2 = k1x2

x2 + k2+ k3(x2 − x6)NR+

(x2 − x6) (3)

with kinetic parameters k1, k2, k3. The first term in the right-hand side of Eq. (3) represents facilitated transport and thesecond term represents passive diffusion, where

NR+(x2 − x6) =

1, x2 ≥ x6,0, x2 < x6.

620 J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624

Considering all three possible transport mechanisms of 1,3-PD, we can give the specific formation rate of extracellular1,3-PD by

q3 = l1k4x8

x8 + k5+ l2k6(x8 − x3)NR+

(x8 − x3) (4)

with l1, l2 ∈ {0, 1} and kinetic parameters k4, k5, k6. Here l1 = 0 (l2 = 0) indicates that passive diffusion (resp. facilitatedtransport) does not exist, whereas l1 = 1 (l2 = 1) represents the existence of passive diffusion (resp. facilitated transport).According to the factual fermentation, we have max{l1, l2} = 1. Let Dl := {(1, 0)T , (0, 1)T , (1, 1)T } and l := (l1, l2)T ∈ Dl.

The specific formation rates of acetate and ethanol can be expressed by Eqs. (5) and (6), respectively [16].q4 = m4 + µY4, (5)q5 = m5 + µY5. (6)

Then, under the above assumptions (H1) and (H2), the batch process can be formulated byx1(t) = µx1, (7)x2(t) = −q2x1, (8)x3(t) = q3x1, (9)x4(t) = q4x1, (10)x5(t) = q5x1, (11)

x6(t) =1k7

k8

x2x2 + k9

+ k10(x2 − x6)NR+(x2 − x6)− q20

− µx6, (12)

x7(t) = k11x6

KGm

1 +

x7k12

+ x6

− k13x7

K Pm + x7

1 +

x7k14

− µx7, (13)

x8(t) = k13x7

K Pm + x7

1 +

x7k14

− l1k15x8

x8 + k16− l2k17(x8 − x3)NR+

(x8 − x3)− µx8, (14)

x(0) = x0, t ∈ [0, T ], (15)where mi, Yi, i = 2, 4, 5,1q2 and K ∗

2 are kinetic parameters. µm is the maximum specific growth rate and Ks is a Monodsaturation constant. KG

m and K Pm are Michaelis–Menten constants. The critical concentrations of glycerol, 1,3-PD, acetate

and ethanol for cell growth are x∗

2 = 2039 mmol/L, x∗

3 = 939.5 mmol/L, x∗

4 = 1026 mmol/L and x∗

5 = 360.9 mmol/L,respectively. x0 is the initial state. [0, T ] is the total fermentation time.

Let u := (u1, . . . , u25)T

:= (k1, . . . , k17,m2,m4,m5, Y2, Y4, Y5,1q2, K ∗

2 )T

∈ R25 be continuous parameter vector andl ∈ Dl discrete parameter vector. Additionally, letDu ⊂ R25 be the admissible set of u ∈ R25, and denoteDu := [30.0, 70.0]×[1.0, 5.0] × [100.0, 5000.0] × [60.0, 150.0] × [1.0, 10.0] × [50.0, 100.0] × [5.0, 10.0] × [40.0, 70.0] × [0.5, 3.0] ×

[100.0, 5000.0]×[1.0, 50.0]×[100.0, 300.0]×[1.0, 50.0]×[0.01, 2.0]×[0.5, 20.0]×[1.0, 30.0]×[1.0, 100.0]×[1.1, 3.3]×[−1.455,−0.485]×[2.61, 7.83]×[0.0041, 0.0123]×[16.535, 49.605]×[5.83, 17.49]×[14.29, 42.87]×[5.715, 17.145].Then the nonlinear hybrid dynamical system of batch culture, denoted by NHDSBC, can be written as

x(t) = F(x, u, l), (u, l) ∈ Du × Dl, x(0) = x0, t ∈ [0, T ], (16)where F(x, u, l) := (f1(x, u, l), . . . , f8(x, u, l))T , and fi(x, u, l), i ∈ I8, denotes the right-hand side of the ith equation ofEqs. (7)–(14).

3. Properties of the hybrid dynamical system

Since biomass, glycerol and products cannot exceed their critical concentrations according to the factual fermentation,we consider properties of the system NHDSBC on subset of R8, Wa := [0.01, 15] × [15, 2039] × [0, 1036] × [0, 1026] ×

[0, 360.9] × [0, 2000] × [0, 80] × [0, 1000].Similarly to [18], it is easy to verify that the function F(x, u, l), (u, l) ∈ Du ×Dl, defined in (1)–(15) satisfies the following

property.

Property 1. For given l ∈ Dl, the function F(x, u, l) defined in (1)–(15) satisfies(1) F(x, u, l) is Lipschitzean in x on Wa and continuous in u on Du;(2) for given u ∈ Du, there exists positive constants a, b such that the linear growth condition holds, i.e.,

‖F(x, u, l)‖ ≤ a‖x‖ + b, ∀x ∈ Wa,

where ‖ · ‖ is the Euclidean norm;(3) for given u ∈ Du, there exists a unique solution to the system NHDSBC, denoted by x(·; u, l). Furthermore, x(·; u, l) is

continuous in u on Du.

J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624 621

Now, we define three sets as follows.

S(u, l) := {x(·; u, l) | x(·; u, l)is a solution to (16) with(u, l) ∈ Du × Dl},

Sw(u, l) := {x(·; u, l) | x(·; u, l) ∈ S(u, l) and x(t; u, l) ∈ Wa, ∀t ∈ [0, T ]},

Duw := {u ∈ Du | x(·; u, l) ∈ Sw(u, l)}.

Property 2. If S(u, l), Sw(u, l) and Duw are all nonempty sets, then S(u, l) and Sw(u, l) are compact in C([0, T ]; R8), and Duw iscompact in R25.

Proof. It follows from Property 1 and the compactness of Du that the mapping

ψ : u ∈ Du −→ x(·; u, l) ∈ C([0, T ]; R8)

is continuous. Therefore, the set S(u, l) is compact in C([0, T ]; R8). Similarly, we can prove that Sw(u, l) is compact inC([0, T ]; R8).

Let {ui} be any sequence in Duw . Since Duw ⊂ Du and Du is compact, there exists a subsequence of {ui}, labeled as {ui,r},and an element u∗

∈ Du such that ui,r −→ u∗ as r −→ ∞.Then, by the definition of Duw , we obtain

x(·; ui,r , l) ∈ S(u, l) and x(t; ui,r , l) ∈ Wa, ∀t ∈ [0, T ].

From Property 1, x(·; u, l) is continuous in u on Du, which implies that

x(·; u∗, l) ∈ S(u, l) and x(t; u∗, l) ∈ Wa, ∀t ∈ [0, T ].

Hence, it follows from the definition of Duw that u∗∈ Duw , which proves that Duw is compact in R25. �

4. System identification model

In this section, we will present a system identification model with (u, l) ∈ Duw × Dl as variables.In factual batch experimentwith given x0 ∈ Wa, wemeasuredQ groups of experimental data. Let yj := (yj1, yj2, yj3)T ∈ R3

be the extracellular experimental data, and yj1, yj2, yj3 denote the extracellular concentrations of biomass, glycerol and 1,3-PD at time tj, j ∈ IQ , respectively. Additionally, let x1(tj; u, l), x2(tj; u, l), x3(tj; u, l) be the corresponding computationalresults by the system NHDSBC. Then the system identification model, denoted by SIM, can be formulated by

SIM : min J(u, l) :=

Q−j=1

3−i=1

|xi(tj; u, l)− yji||yji|

s.t. x(·; u, l) ∈ Sw(u, l), (u, l) ∈ Duw × Dl.

(17)

According to the experiment process, we make the following assumptions.

(H3) For given x0 ∈ Wa, the system NHDSBC is controllable and observable.(H4) Duw is nonempty in R25.

Theorem 1. Under the above assumptions (H1)–(H4), for given l ∈ Dl, there exists a global optimal solution u∗ to SIM, that is,there exists u∗

∈ Duw such that

J(u∗, l) ≤ J(u, l), ∀u ∈ Duw. (18)

Proof. It follows from Property 1 that, for given l ∈ Dl, the mapping from u ∈ Duw to x(·; u, l) is continuous, so J(u, l) iscontinuous in u ∈ Duw . By the compactness of Duw ⊂ R25, we can conclude that SIM has an optimal solution, denoted by u∗,such that

J(u∗, l) ≤ J(u, l), ∀u ∈ Duw,

which completes our proof. �

Due to the coexistence of the continuous and discrete variables in the identification problem SIM, it is difficult to solveit directly. In the context that these two typical variables are independent, we can decompose the problem SIM into twosubproblems.

Given the discrete variable l ∈ Dl, denote u := u(l) and J(u) := J(u(l), l), then the subproblem with respect to u ∈ Duw ,denoted by SIM1(l), can be formulated as

SIM1(l) : min J(u) :=

Q−j=1

3−i=1

|xi(tj; u, l)− yji||yji|

s.t. x(·; u, l) ∈ Sw(u, l), u ∈ Duw ⊂ R25.

(19)

622 J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624

Table 1The optimal performance index of SIM1(l), l ∈ Dl .

l (1, 0)T (0, 1)T (1, 1)T

J(u∗(l), l) 7.23508 14.01192 5.0664

After that, let u∗= u∗(l) be the optimal solution to SIM1(l)with fixed l ∈ Dl. Then, the subproblemwith regard to l ∈ Dl,

denoted by SIM2, can be formulated as

SIM2 : min{J(u∗(l), l) | l ∈ Dl}. (20)Let l∗ := argmin{J(u∗(l), l) | l ∈ Dl}. Then the transport mechanism corresponding to l∗ is the most reasonable.

Meanwhile, u∗(l∗) and J(u∗(l∗), l∗) are the corresponding optimal parameter vector and performance index, respectively.

5. Optimization algorithm

In this section, we construct a two-phase optimization algorithm based on modified complex method and very fastsimulated annealing algorithm [23,24] to solve the subproblem SIM1(l), l ∈ Dl.Phase 1. Generate an initial point u0 by modified complex method.

Step 1.1. Randomly generate Q sample points u(1), . . . , u(Q ) from Duw . Set parametersM1, α,△α, σ , ε.Step 1.2. Compute the point ubad

:= u(b) := argmax1≤i≤Q J(u(i)). And compute the weighted centroid of theremaining points ucen

:=1

Q−1

∑Qi=1,i=b λ

iu(i), where λi := J(u(i))/∑Q

j=1,j=b J(u(j)). If Q > M1, goto

Step 1.3; else let u0= u(Q ), stop.

Step 1.3. If α‖ucen− ubad

‖ < ε, let u0:= ubad, stop; else goto Step 1.4.

Step 1.4. If α > σ , compute unew:= ucen

+ α(ucen− ubad), goto Step 1.5; else let u(i) = u(i+1), b ≤ i ≤ Q − 1, and

set Q := Q − 1, goto Step 1.2.Step 1.5. If unew

∈ Duw , compute J(unew), goto Step 1.5; else set α := α − △α, goto Step 1.3.Step 1.6. If J(unew) < J(ubad), let ubad

:= unew , goto Step 1.2; else set α := α − △α, goto Step 1.3.Phase 2. Simulated annealing with the initial point u0 obtained in Phase 1.

Step 2.1. Compute J(u0) and given initial temperature T0. Set parameters M2,M3,M4, β , γ ,△γ , γ0, η, and setu∗

= unext= ucurrent

:= u0, J(u∗) = J(unext) = J(ucurrent) := J(u0), k := 0,m := 0.Step 2.2. If k < M2, stop, output u∗ and J(u∗); else goto Step 2.3.Step 2.3. Ifm < M3, goto Step 2.4; else set k := k + 1,m := 0, and reduce temperature by setting Tk := T0βk, goto

Step 2.2.Step 2.4. Generate a random variable ξk from the interval (0, 1], and set γ := γ0.Step 2.5. If γ < M4, compute y := Tksgn(ξk−0.5)[(1+

1Tk)|2ξk−1|

−1], and the point unext:= ucurrent

+γ y(uup−ulow),

where uup and ulow are the upper and lower bounds of the parameter sample u, respectively, goto Step 2.6.Else goto Step 2.4.

Step 2.6. If unext∈ Duw , compute J(unext) and △J := J(unext)− J(ucurrent), goto Step 2.7; else set γ := γ + △γ , goto

Step 2.5.Step 2.7. If △J < 0, compute u∗

:= argmin{J(u∗), J(unext)}, and let ucurrent:= unext . Else compute the acceptance

probability P := [1 − (1 − η)△JTk

]1/(1−η), and let ucurrent

:= Punext+ (1 − P)ucurrent .

Step 2.8. Set m := m + 1, goto Step 2.3.

Then, u∗ is the optimal parameter vector of the subproblem SIM1(l), l ∈ Dl, and J(u∗) is the corresponding optimalperformance index, i.e., J(u∗(l), l) defined in the formula (18).

6. Numerical example

In this example, we consider a factual batch experiment with the initial state x0 = (0.102 g/L, 418.2609 mmol/L, 0,0, 0, 0, 0, 0)T . The parameters chosen in the above algorithm are as follows: Q = 1000, α = 1.3,△α = 0.01, σ =

−0.5, T0 = 1000,M1 = 5,M2 = 1000,M3 = 500,M4 = 1.0, β = 0.98, γ = 0.01, γ0 = 0.01,△γ = 0.01, η = −5.In Table 1, J(u∗(l), l) denotes the optimal performance index of the subproblem SIM1(l)with given l ∈ Dl.As is shown in Table 1, the index of the subproblem SIM1(l) with l = (1, 1)T is the smallest, which indicates that the

transport mechanism corresponding to l = (1, 1)T of 1,3-PD is the most reasonable, that is, it is most possible that 1,3-PDpasses the cell membrane by passive diffusion coupled with facilitated transport. The corresponding optimal kinetic param-eter vector is u∗(l = (1, 1)T ) = (56.1633, 4.6239, 203.853, 128.468, 1.4704, 50.5692, 6.5778, 56.9537, 1.0529, 2100.57,16.8822, 249.53, 32.9252, 0.7654, 13.1999, 29.8004, 85.789, 2.9292,−0.8212, 5.2296, 0.0103, 22.391, 12.715, 40.183,9.0077)T . Fig. 2 shows the comparison between experimental and computational results of extracellular concentrationsof biomass, glycerol and 1,3-PD in batch culture, where real lines denote computational curves and the scattergrams denoteexperimental results, which indicates that the proposed hybrid dynamical system can describe the factual batch experimentproperly.

J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624 623

Fig. 2. Comparison of simulated curves and experimental results of extracellular concentrations of biomass, glycerol and 1,3-PD in batch culture.

7. Discussions and conclusions

In this paper, we establish a nonlinear hybrid dynamical system to describe the batch fermentation of glycerol by K.pneumoniae, inwhich three possible transportmechanisms of 1,3-PD are all taken into consideration. A system identificationmodel is proposed and solved by using a two-phase optimization algorithm. Through numerical calculation, we concludethat it is most possible for 1,3-PD to pass the cell membrane by passive diffusion coupled with facilitated transport. Besides,numerical simulation with the identified parameters is carried out, which shows that our algorithm is efficient and feasible.

The current work deals with the problem of modeling and system identification of the hybrid dynamical system in batchculture. In a futurework, wewill consider sensitivity analysis of kinetic parameters in ourmodel, so as to reduce the numberof thesemodel parameters to be identified and concentrate on identifying these relatively sensitive parameters. Additionally,our effort will focus on multistage analysis and optimal control of the hybrid dynamical system.

Acknowledgements

This work was supported by 863 Program (Grant No. 2007AA02Z208), 973 Program (Grant No. 2007CB714304), theNational Natural Science Foundation of China (Grant Nos. 10471014, 10671126 and 10871033) and Natural ScienceFoundation of Department of Education, Henan (Grant No. 2008B110010).

References

[1] H. Biebl, S.Marten, H. Hippe,W.D. Deckwer, Glycerol conversion to 1,3-propanediol by newly isolated clostridia, Appl.Microbiol. Biotechnol. 36 (1992)592–597.

[2] G. Gottschalk, B. Averhoff, Process for the microbial preparation of 1,3-propanediol from glycerol, European patent NO. EP 0373230 A1, 1990.[3] T. Homann, C. Tag, H. Biebl,W.D. Deckwer, B. Schink, Fermentation of glycerol to 1,3-propanediol by Klebsiella and Citrobacter strains, Appl. Microbiol.

Biotechnol. 33 (1990) 121–126.[4] U.Witt, R.J. Miiller, J. Augusta, H.Widdecke,W.D. Deckwer, Synthesis, properties and biodegradability of polyesters based on 1,3-propanediol, Makro-

Mol. Chem. Phys. 195 (1994) 793–802.[5] Z. Xiu, Research progress on the production of 1,3-propanediol by fermentation, Microbiology 27 (2000) 300–302.[6] A.-P. Zeng, A. Ross, H. Biebl, C. Tag, B. Guenzel,W.-D. Deckwer,Multiple probuct inhibition and growthmodeling of Clostridium butyricum and Klebsiella

pneumoniae in glycerol fermentation, Biotechnol. Bioeng. 44 (1994) 902–911.[7] A.-P. Zeng, H. Biebl, Bulk-chemicals from biotechnology: the case of microbial production of 1,3-propanediol and the new trends, Adv. Biochem. Eng.

Biotechnol. 74 (2002) 237–433.[8] A.-P. Zeng,W.D. Deckwer, A kineticmodel for substrate and energy consumption ofmicrobial growth under substrate-sufficient condition, Biotechnol.

Prog. 11 (1995) 71–79.[9] Z. Xiu, A. Zeng, L. An, Mathematical modelling of kinetics and research on multiplicity of glycerol bioconversion to 1,3-propanediol, J. Dalian Univ.

Technol. 40 (2000) 428–433.[10] C. Gao, E. Feng, Z. Wang, Z. Xiu, Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures, Appl. Math.

Comput. 169 (2005) 476–484.[11] G. Wang, E. Feng, Z. Xiu, Vector measure for explicit nonlinear impulsive system of glycerol bioconversion in fed-batch cultures and its parameter

identification, Appl. Math. Comput. 188 (2007) 1151–1160.

624 J. Wang et al. / Mathematical and Computer Modelling 54 (2011) 618–624

[12] Z. Gong, C. Liu, E. Feng, Q. Zhang, Computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae, Comput.Biol. Chem. 33 (1) (2008) 1–6.

[13] Q. Zhang, Z. Xiu, Metabolic pathway analysis of glycerol metabolism in Klebsiella pneumoniae incorporating oxygen regulatory system, Biotechnol.Prog. 25 (1) (2009) 103–115.

[14] B. Gtinzel, Mikrobielle Herstellung von 1,3-Propandiol durch Clostridium butyricum und adsorptive Abtremutng von Diolen, Ph.D. Dissertation, TUBraunschweig, Germany, 1991.

[15] Y. Sun, W. Qi, H. Teng, Z. Xiu, A. Zeng, Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme-catalyticreductive pathway and transport of glycerol and 1,3-propanediol across cell membrane, Biochem. Eng. J. 38 (2008) 22–32. B.

[16] C. Gao, Z. Wang, E. Feng, Z. Xiu, Parameter indetification and optimization of process for bio-dissimilation of glycerol to 1,3-propanediol in batchculture, J. Dalian Univ. Technol. 46 (5) (2006) 771–774.

[17] X.H. Li, E.M. Feng, Z. Xiu, Optimal control and property of nonlinear dynamic system for microorganism in batch culture, OR Trans. 9 (4) (2005) 89–96.[18] L. Wang, J. Ye, E. Feng, Z. Xiu, An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification,

Nonlinear Anal. Hybrid Syst. 3 (2009) 455–462.[19] F. Shan, E. Feng, Y. Li, Z. Xiu, Multin stage identification model, algorithm and appication of nonlinear dynaic system, Operations Research and

Management Science 15 (3) (2006) 6–12.[20] A.-P. Zeng, Quantitative Zellphysiologie, Metabolic Engineering und Modellierung der Glycerinfermentation zu 1,3-Propandiol, Habilitationschrift,

Technical University of Braunschweig, Germany, 2000.[21] Y. Gao, John Lygeros, Marc Quincampoix, Nicolas Seube, Approximate stabilisation of uncertain hybrid systems, Hybrid Syst.: Comput. Control 2623

(2003) 203–215.[22] S. Engell, G. Frehse, E. Schnieder, Nonlinear hybrid dynamical systems: modelling, optimal control, and applications, Modelling, Analysis ans Design

of Hybrid Systems 279 (2002) 311–335.[23] L. Zhang, et al., Fast simulated annealing algorithm and its application, OGP 32 (5) (1997) 654–660.[24] Mohammad-Taghi Vakil-Baghmisheh, Alireza Navarbaf, A modified very fast simulated annealing algorithm, in: International Symposium on

Telecommunications, 2008, pp. 61–66.[25] E.C.C. Lin, Glycerol dissimilation and its regulation in bacteria, Annu. Rev. Microbiol. 30 (1976) 535–578.