Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

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Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

Complex metabolic network of glycerol fermentation by Klebsiellapneumoniae and its system identification via biological robustnessJuan Wang a,b,∗, Jianxiong Ye a, Enmin Feng a, Hongchao Yin b, Bing Tan a

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 China

a r t i c l e i n f o

Article history:Received 19 February 2010Accepted 5 October 2010

Keywords:Complex metabolic networkNonlinear hybrid dynamical systemBiological robustnessSystem identification

a b s t r a c t

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K.pneumoniae) can be characterized by an intricatemetabolic network of interactions amongbiochemical fluxes, metabolic compounds, key enzymes and genetic regulation. Sincethere are some uncertain factors in the fermentation, especially the transport mechanismsof substances across cell membrane, the metabolic network contains multiple possiblemetabolic systems. In this paper, we establish a complex metabolic network and thecorresponding nonlinear hybrid dynamical system aiming to determine the most possiblemetabolic system. The existence, uniqueness and continuity of solutions are discussed. Wequantitatively describe biological robustness and present a system identification modelon the basis of robustness performance. The identification problem is decomposed intotwo subproblems and a procedure is constructed to solve them. Numerical results showthat it is most possible that both glycerol and 1,3-PD pass the cell membrane by activetransport coupling with passive diffusion under substrate-sufficient conditions, whereas,under substrate-limited conditions, glycerol passes cell membrane by active transportcoupling with passive diffusion and 1,3-PD by active transport.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

1, 3-Propanediol (1, 3-PD) is a monomer widely used in the modern plastic industry [1]. Its microbial production isrecently paid attention to throughout the world because of its low cost, high production and no pollution [2]. Among allkinds of microbial productions of 1, 3-PD, dissimilation of glycerol by K. pneumoniae has been widely investigated since the1980s due to its high productivity [3].

In 1995, Zeng and Deckwer [4] proposed a substrate-sufficient kinetic model to describe the substrate consumption andextracellular products formation, inwhich the concentrations of extracellular substances (biomass, glycerol, 1, 3-PD, acetateacid and ethanol) were considered. Thereafter, a large number of further investigations have been carried out based on thismodel [5–8]. Nevertheless, the concentrations of intracellular substances were not considered in the above works.

In 2008, Sun et al. [9] proposed a novel mathematical model, in which the concentration changes of both extracellularsubstances (biomass, glycerol, 1, 3-PD, acetic acid and ethnol) and intracellular substances (glycerol, 1, 3-PD and 3-hydroxypropionaldehyde (3-HPA)) were all taken into consideration. In the model, it was assumed that glycerol passesthe cell membrane by both active transport and passive diffusion and 1, 3-PD by passive diffusion. However, the transportmechanisms of glycerol and 1, 3-PD in K. pneumoniae are still not exactly known. So the reliability of the model cannot beguaranteed. Although there are many additional works dealing with the metabolic process of glycerol by K. pneumoniae by

∗ 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), yejianxiong128@yahoo.cn (J. Ye), hcyin@dlut.edu.cn (H. Yin).

1751-570X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2010.10.002

J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112 103

the metabolic flux analysis [10,11] or metabolic pathway analysis [12,13], no attempt has been made to determine thetransport mechanisms.

Since the transport mechanisms of substances across cell membranes are uncertain, the system of glycerol metabolismcannot be determined. The problem is to find the true metabolic system from all possible ones. In the context that theconcentrations of intracellular metabolites cannot be measured, we shall take the basic features of biological systems intoconsideration. In other words, systems that poorly own the common features of biological systems should be excluded.

One of the fundamental characteristics of biological systems is robustness [14–17]. It is a property that allows a systemto maintain its functions despite external and internal perturbations. A biological function produced by a biochemicalreaction network can be defined to correspond to a particular stationary behavior of a dynamical systemwhich can faithfullyrepresent the biochemical reactions [14]. It is alsomentioned that stable steady-states of the dynamical system is one of localstationary behaviors and that the traditional and current approach to robustness analysis of biochemical network models isbased on perturbing the model parameter, i.e., parametric robustness. Additionally, Barkai and Leibler [15] argued that thekey properties of biochemical networks are robust, that is, they are relatively insensitive to the precise values of biochemicalparameters. This point of view, which has been observed for a wide variety of experiments [17–19], is being graduallyaccepted by experts in the field of systemsbiology. Stelling et al. [19] and Steuer [20] qualitatively described the robustness ofbiological systems and the topological structure ofmetabolic networks, respectively. However, the quantitative descriptionsof biological robustness are still seldom found.

There are, respectively, three possible transport mechanisms for glycerol and 1, 3-PD, i.e., active transport, passivediffusion or passive diffusion coupled with active transport [4]. Among them, the passive diffusion of glycerol from anextracellular environment to an intracellular environment, which complies with the Fick diffusion law [21], is active onlywhen the extracellular glycerol concentration is higher than its intracellular concentration. Similarly, the passive diffusion of1, 3-PD from an intracellular environment to an extracellular environment is active onlywhen its intracellular concentrationis higher than its extracellular concentration. In aword, whether the passive diffusion is active or not is state dependent [22].So themetabolic process consisting of the passive diffusion of substances is actually a hybrid process, which can be describedby a state-based hybrid dynamical system.

In this paper, considering all possible transport mechanisms of glycerol and 1, 3-PD, we propose a complex metabolicnetwork and the corresponding nonlinear hybrid dynamical system to describe the continuous fermentation of glycerol byK. pneumoniae. The Lipschitz continuity and linear growth condition of the velocity vector field are proved, as well as theexistence, uniqueness and continuous dependence of solutions with respect to parameters. We then present a quantitativedescription of biological robustness, which is taken as the performance index to establish an identification model withpossible metabolic systems as discrete optimized variables, kinetic parameters as continuous optimized variables, therelative error of extracellular substances and the approximate stability of dynamical systems as constraint conditions.Through numerical computation, we identify the discrete variables to infer the most possible metabolic system of thecomplex metabolic network and estimate the kinetic parameters of the corresponding dynamical system under substrate-sufficient and substrate-limited cultures, respectively.

This paper is organized as follows. In Section 2, the complex metabolic network and the nonlinear hybrid dynamicalsystem of glycerol metabolism are proposed. Section 3 is devoted to exploring some properties of the system. A quantitativedescription of biological robustness is presented in Section 4. In Section 5, we establish a system identification model andconstruct a numericalmethod to solve it. In Section 6, we infer themost possiblemetabolic system of the complexmetabolicnetwork and estimate the unknown kinetic parameters under substrate-sufficient and substrate-limited conditions throughnumerical calculation. Discussions and conclusions are presented at the end of this paper.

2. Complex metabolic network and nonlinear hybrid dynamical system

It is well known that metabolic pathways in biology can be represented by networks, called metabolic networks, whichare composed of metabolic substrates and products with directed edges joining them if a known metabolic reaction existsthat acts on a given substrate and produces a given product [23].

The fermentations of glycerol cover both extracellular and intracellular environments. The two environments are linkedby the transports of substrate and products across cell membranes. Under anaerobic conditions, glycerol is dissimilatedthrough coupled oxidative and reductive pathways and the latter generates the goal product 1, 3-PD as shown in Fig. 1 [24].Due to the complexity of the oxidative pathway and lack of experimental data, we considered the oxidative pathway as a‘black box’ model, i.e., only the input and output in this pathway are considered but the details in the intermediate processare ignored. The reductive pathway will be emphasized in this paper because 3-HPA is the key intermediate for 1, 3-PDproduction.

Since the transport mechanisms of glycerol and 1, 3-PD across cell membrane in the reductive pathway are stillunclear, we shall regard the metabolic pathway with a combination of possible transport mechanisms of glycerol and 1,3-PD as a possible metabolic system, which is convenient to be represented by a simple metabolic network. Taking allpossible transport mechanisms of glycerol and 1, 3-PD into consideration, we can obtain nine possible combinations oftransportsmechanisms, which correspond to nine possible simplemetabolic networks (or nine possiblemetabolic systems),denoted by S1, S2, . . . , S9. All nine possible simple metabolic networks form a complex metabolic network, denoted byS , {S1, S2, . . . , S9}.

104 J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

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.

In view of the property that the execution of passive diffusion is state dependent, the temporal behavior of eachsimple possible metabolic network can be described by a nonlinear hybrid dynamical system. We call the nonlinear hybriddynamical system,which describes thewhole complexmetabolic network S of glycerolmetabolism, a generalized nonlinearhybrid dynamical system, denoted by NHDS.

In the following, we shall build the hybrid dynamical system NHDS to describe the complex metabolic network S incontinuous fermentation of glycerol by K. pneumoniae.

In continuous culture, the composition of culturemedium, cultivation conditions and analytical methods of fermentativeproducts were similar to those reported by Zeng [25]. Glycerol is added to the reactor continuously, the broth in reactorpours out at the same rate and the volume of the fermentation broth keeps constant. According to the experiment process,we assume that

(H1) The concentrations of reactants are uniform in the reactor, time delay and nonuniform space distribution are ignored.(H2) During the process of continuous culture, the substrate added to the reactor only includes glycerol.

Let x(t) = (x1(t), x2(t), . . . , x8(t))T be the state vector, where x1(t), x2(t), . . . , x8(t) are the concentrations of biomass,extracellular glycerol, extracellular 1, 3-PD, acetate, ethanol, intracellular glycerol, intracellular 3-HPA and intracellular 1,3-PD at time t , respectively. To simplify notation, let In , {1, 2, . . . , n} and xi = xi(t), i ∈ I8. Let Q be the total number ofexperiments carried out under different dilution rates and initial glycerol concentrations (Dj, Cs0j), j = 1, 2, . . . ,Q .

According to the previous work [9], the specific cell growth rate can be expressed by

µ = µmx2

x2 + Ks

1 −

x2x∗

2

1 −

x3x∗

3

1 −

x4x∗

4

1 −

x5x∗

5

. (1)

While the uptake of extracellular glycerol is considered as a ‘‘black box’’ model, its specific consumption rate can beexpressed by Eq. (2) [9].

q20 = m2 +µ

Y2+ 1q2

x2x2 + K ∗

2. (2)

The consumption rate of glycerol is determined by the rate at which extracellular substrate enters the cell and how fastthe substrate ismetabolized intracellularly [4]. So the specific consumption rate of extracellular glycerol of the kthmetabolicsystem in the jth experiment can be expressed by

q2(k) = l1,ku1j,k

x2x2 + u2

j,k+ l2,ku3

j,k(x2 − x6)NR+(x2 − x6) (3)

with l1,k, l2,k ∈ {0, 1} and kinetic parameters u1j,k, u

2j,k, u

3j,k. The first term of Eq. (3) represents active transport and the second

term represents passive diffusion. So l1,k = 0 (l2,k = 0) indicates that the passive diffusion (resp. active transport) doesn’texist, whereas l1,k = 1 (l2,k = 1) represents the existence of passive diffusion (resp. active transport). In addition, theindicator function

NR+(x2 − x6) =

1, x2 > x6,0, x2 ≤ x6,

means that the structures of equations which contain this term are variant.Similarly, the specific formation rate of extracellular 1, 3-PD of the kth metabolic system in the jth experiment can be

given by

q3(k) = l3,ku4j,k

x8x8 + u5

j,k+ l4,ku6

j,k(x8 − x3)NR+(x8 − x3) (4)

with li,k ∈ {0, 1}, i = 3, 4, k ∈ I9.According to the actual fermentation process, we have max{l1,k, l2,k} = 1 and max{l3,k, l4,k} = 1. Let lk , (l1,k, l2,k,

l3,k, l4,k)T , k ∈ I9, then lk corresponds to the kth metabolic system Sk in the complex metabolic network S. For details, seeTable 1.

J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112 105

Table 1Transport mechanisms of glycerol and 1, 3-PD of metabolic system Sk , k ∈ I9 , corresponding to parameter vector lk , k ∈ I9 .Abbreviations: A, active transport; P, passive diffusion; AP, passive diffusion coupled with active transport.

Sk −→ lk Glycerol 1, 3-PD

S1 −→ l1 : (0, 1, 0, 1)T P PS2 −→ l2 : (0, 1, 1, 0)T P AS3 −→ l3 : (0, 1, 1, 1)T P APS4 −→ l4 : (1, 0, 0, 1)T A PS5 −→ l5 : (1, 0, 1, 0)T A AS6 −→ l6 : (1, 0, 1, 1)T A APS7 −→ l7 : (1, 1, 0, 1)T AP PS8 −→ l8 : (1, 1, 1, 0)T AP AS9 −→ l9 : (1, 1, 1, 1)T AP AP

The specific formation rate of extracellular acetate and ethanol can be expressed by Eqs. (5) and (6) [9], respectively.

q4 = m4 + µY4 + 1q4x2

x2 + K ∗

4, (5)

q5 = m5 + µY5 + 1q5x2

x2 + K ∗

5. (6)

Under the assumptions (H1) and (H2), the continuous fermentation process of glycerol by K. pneumoniae of the kthmetabolic system in the jth experiment can be formulated as

x1(t) = (µ − Dj)x1, (7)

x2(t) = Dj(Cs0j − x2) − q2(k)x1, (8)

x3(t) = q3(k)x1 − Djx3, (9)

x4(t) = q4x1 − Djx4, (10)

x5(t) = q5x1 − Djx5, (11)

x6(t) =1u7j,k

l1,ku8

j,kx2

x2 + u9j,k

+ l2,ku10j,k(x2 − x6)NR+

(x2 − x6) − q20

− µx6, (12)

x7(t) = u11j,k

x6

KGm

1 +

x7u12j,k

+ x6

− u13j,k

x7

K Pm + x7

1 +

x7u14j,k

− µx7, (13)

x8(t) = u13j,k

x7

K Pm + x7

1 +

x7u14j,k

− l3,ku15j,k

x8x8 + u16

j,k− l4,ku17

j,k(x8 − x3)NR+(x8 − x3) − µx8. (14)

Under anaerobic conditions at 37 °C and PH 7.0, the maximum specific growth rate µm and Monod saturation constantKs are 0.67 h−1 and 0.28 mmol/L, respectively. KG

m and K Pm are Michaelis–Menten constants, which are of 0.53 mmol/L and

0.14 mmol/L, respectively. 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. Yi, 1qi, K ∗

i , i = 2, 3, 4, 5, areall given constants whose concrete biological meanings and values can be referred to in previous literature [5].

For a given (j, k) ∈ IQ × I9, let f (x, u, j, k) = (f1(x, u, j, k), . . . , f8(x, u, j, k))T , where fi(x, u, j, k), i ∈ I8, denotes theright-hand side of the ith equation of Eqs. (7)–(14), and u = u(j, k) = (u1

j,k, . . . , u17j,k)

T∈ R17

+is the kinetic parameter vector

to be determined.The continuous fermentation of glycerol begins with batch fermentation. In the batch phase, no medium is fed into

and poured out from the bio-reactor. Denote the time intervals of batch and continuous periods by [t0, tb] and [tb, tf ],respectively, where t0 < tb < tf < ∞. Then, the hybrid dynamical systemof the kthmetabolic system in the jth experiment,denoted by NHDS(j, k), can be described by the combination of the following two systems.

x(t) = f (x, u, j, k), t ∈ [t0, tb]x(t0) = x0, (15)x(t) = f (x, u, j, k), t ∈ [tb, tf ]x(tb) = xb. (16)

Here, the system (15) describes the batch fermentation process, in which x0 is the initial state, xb is the terminal state, andthe dilution rates are set to be zero for all j ∈ IQ . The system (16) describes the continuous fermentation process startingfrom the terminal state of the system (15) with prescribed dilution rate Dj = 0.

106 J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

The generalized nonlinear hybrid dynamical system NHDS of the whole complex metabolic system S can be denoted by

NHDS , {NHDS(j, k) : (j, k) ∈ IQ × I9},

which is composed of |IQ | × |I9| = 9 × Q hybrid dynamical systems.In this work, the stable state of a dynamical system is actually referred to the approximate stability defined as follows.

Definition 1. Given (j, k) ∈ IQ × I9, let σ > 0 be sufficiently small. If there exists tσ ∈ [tb, tf ] such that

tσ = inf{ts : ‖f (x, u, j, k)‖ < σ, ∀t ∈ [ts, tf ]},

where ‖ · ‖ is the Euclidean norm, we say that the system NHDS(j, k) reaches approximately stable state at time tσ , and thatx(·; u, j, k) is an approximately stable solution to NHDS(j, k) under the accuracy σ .

3. Existence, uniqueness and continuous dependence of solutions

To begin with, we introduce some symbols which will be used below.Let Du(j, k) ⊂ R17

+be the allowable set of the kinetic parameter vector u. Let x∗ , (0.01, 0, 0, 0, 0, 0, 0, 0)T and

x∗ , (15, 2039, 1036, 1026, 360.9, 2000, 80, 1000)T be the lower and upper bounds of the state vector x. Let Wa ,Π8

i=1[xi∗, x∗

i ] ⊂ R8+, Dd , [0.1, 0.5] ⊂ R+ and Dc , [500, 2000] ⊂ R+ respectively be the admissible sets of state vector x,

dilution rate Dj and initial glycerol concentration in feed medium Cs0j, j ∈ IQ .According to factual experiments, we assume that

(H3) For given (j, k) ∈ IQ × I9, the set Du(j, k) ⊂ R17+

is a nonempty bounded closed set.(H4) The absolute difference between extracellular and intracellular glycerol concentration and that of 1, 3-PD concentra-

tion are bounded, i.e., ∃M1 > 0,M2 > 0 such that

|x2(t) − x6(t)| ≤ M1, ∀t ∈ [t0, tf ], (17)

|x8(t) − x3(t)| ≤ M2, ∀t ∈ [t0, tf ]. (18)

Under the assumptions (H3) and (H4), for given (j, k) ∈ IQ × I9, we can easily verify the following properties of thevelocity vector field f (x, u, j, k).

Property 1. The velocity vector field f (x, u, j, k) defined in (1)–(14) is continuous in (x, u) on Wa × Du(j, k). Particularly,f (x, u, j, k) is Lipschitz continuous in x on Wa.

Property 2. For given u ∈ Du(j, k), the velocity vector field f (x, u, j, k) satisfies the linear growth condition, i.e., ∃a, b > 0 suchthat

‖f (x, u, j, k)‖ ≤ a‖x‖ + b. (19)

Proof. For given (j, k) ∈ IQ × I9 and u ∈ Du(j, k), let C1 , |m2| + µm|1/Y2| + |1q2|, C2 , l1,ku1j,k + l2,ku3

j,kM1,C3 , l3,ku4

j,k + l4,ku6j,kM2, Ci , |mi| + µm|Yi| + |1qi|, i = 4, 5, then it follows from |x2/(x2 + K ∗

i )| < 1, i = 2, 4, 5 that|q20| ≤ C1, |q2(k)| ≤ C2, |q3(k)| ≤ C3, |q4| ≤ C4, |q5| ≤ C5.

Letting L1 , µm + Dj, L2 , max{DjCs0j, C2 + Dj} and Li , Ci + Dj, i = 3, 4, 5, we can obtain

|f1(x, u, j, k)| ≤ (µ + Dj)|x1| ≤ L1‖x‖ ≤ L1(‖x‖ + 1),|f2(x, u, j, k)| ≤ |DjCs0j| + Dj|x2| + |q2(k)||x1| ≤ L2(‖x‖ + 1),|fi(x, u, j, k)| ≤ Ci|x1| + |Dj||xi| ≤ Li(‖x‖ + 1).

Similarly, let C6 , (l1,ku8j,k + l2,ku10

j,kM1 + C2)/u7j,k, C7 , u11

j,k + u13j,k, C8 , u13

j,k + l3,ku15j,k + l4,ku17

j,kM2 and Li , max{Ci, µm},i = 6, 7, 8, and we have

|f6(x, u, j, k)| ≤ (l1,ku8j,k + l2,ku10

j,k|x2 − x6| + |q20|)/u7j,k + µ|x6| ≤ L6(‖x‖ + 1),

|f7(x, u, j, k)| ≤ C7 + µ|x7| ≤ L7(‖x‖ + 1),|f8(x, u, j, k)| ≤ u13

j,k + l3,ku15j,k + l4,ku17

j,k|x8 − x3| + µ|x8| ≤ L8(‖x‖ + 1).

Finally, set L′ , max{Li, i ∈ I8}, then we have that (19) holds with a = b = 2√2L′, which completes our proof. �

Under the above assumptions, applying the classical theory of differential equations and the above properties, we canobtain the following property of the hybrid system NHDS.

Property 3. For given (j, k) ∈ IQ × I9 and u ∈ Du(j, k), there exists a unique solution to the system NHDS(j, k), denoted byx(·; u, j, k). Furthermore, x(·; u, j, k) is continuous in u on Du(j, k).

J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112 107

Now, we define the following four sets:

S(j, k) , {x(·; u, j, k) | x(·; u, j, k) is a solution to NHDS(j, k) with u ∈ Du(j, k)},Duw(j, k) , {u ∈ Du(j, k) | x(·; u, j, k) ∈ S(j, k) and x(t; u, j, k) ∈ Wa, ∀t ∈ [t0, tf ]},Sw(j, k) , {x(·; u, j, k) | x(·; u, j, k) is a solution to NHDS(j, k) with u ∈ Duw(j, k)},Dus(j, k) , {u ∈ Duw(j, k) | x(·; u, j, k) is an approximately stable solution to NHDS(j, k)}.

Theorem 1. Under the assumptions (H3) and (H4), given (j, k) ∈ IQ × I9, if the sets S(j, k), Sw(j, k), Duw(j, k) and Dus(j, k) areall nonempty, then S(j, k) and Sw(j, k) are compact in C([t0, tf ], R8) and Duw(j, k) together with Dus(j, k) are compact in R17

+.

Proof. Using the assumption (H3) and (H4), by Property 3 one can easily verify that the mapping T j,k:

(Dj, Cs0j, u(j, k)) ∈ Dd × Dc × Du(j, k) −→ x(·; u, j, k) ∈ C([t0, tf ], R8)

is continuous. Therefore, the set S(j, k) is compact in C([t0, tf ], R8).Similarly, we can prove that Sw(j, k) is compact in C([t0, tf ], R8).Let {ui} be any sequence in Duw(j, k). Since Duw(j, k) ⊂ Du(j, k) and Du(j, k) is compact, there exists a subsequence of

{ui}, labelled as {ui,r}, and an element u∗∈ Du(j, k) such that ui,r −→ u∗ as r −→ ∞.

Then, by the definition of Duw(j, k), we obtain

x(·; ui,r , j, k) ∈ S(j, k) and x(t; ui,r , j, k) ∈ Wa, ∀t ∈ [t0, tf ].

From Property 3, x(·; u, j, k) is continuous in u on Du(j, k), which implies that

x(·; u∗, j, k) ∈ S(j, k) and x(t; u∗, j, k) ∈ Wa, ∀t ∈ [t0, tf ].

Hence, it follows from the definition of Duw(j, k) that u∗∈ Duw(j, k), which proves that Duw(j, k) is compact in R17

+.

Similarly, we can obtain that Dus(j, k) is compact in R17+. �

4. Biological robustness index

Given (j, k) ∈ IQ × I9, to determine the reliability of the dynamical system NHDS(j, k) or to evaluate the reliability ofcomputational values by NHDS(j, k), an evaluation criterion for this system should be given. Generally, the computationalresults are respected to be consistent with experimental data. However, our experimental data only include concentrationsof extracellular substances. So, for the extracellular concentrations, we define the relative error between computationalconcentrations and experimental data.

For given (j, k) ∈ IQ × I9 and u ∈ Dus(j, k), assume that the system NHDS(j, k) reaches an approximately stable stateat time tσ . Let xi(·; u, j, k), i ∈ I5, be the approximately stable solution and yi(j), i ∈ I5, the corresponding steady-stateextracellular concentration measured in the jth experiment.

Definition 2. The relative error of extracellular substances is defined as

SSD(u, j, k) ,15

5−i=1

|xi(tσ ; u, j, k) − yi(j)||yi(j)|

. (20)

On the other hand, for the intracellular concentrations, due to the difficulties in measurement and the inaccuracy of theexperimental data, we present the biological robustness on the basis of local stationary behavior and parametric robustnessaccording to the previous literature [14]. Taking account of intracellular variations over the disturbances of parametervectors, we define the relative deviation of intracellular substances with regard to parameter vector u ∈ Dus(j, k) andquantitatively describe the biological robustness of the dynamical system NHDS(j, k) and that of the metabolic systemSk, k ∈ I9.

For a given (j, k) ∈ IQ × I9 and u ∈ Dus(j, k), let B(u; δ) , {v|v ∈ Dus(j, k), ‖v −u‖ ≤ δ}, where δ > 0 is sufficiently smallto measure themagnitude of the parameter disturbances. Obviously, B(u; δ) is compact in R17

+. Randomly generate n sample

points from B(u; δ) by uniform distribution, denoted by vi, i ∈ In, where n is the size of the Monte Carlo experiments.Assume that the system NHDS(j, k) reaches an approximately stable state at time t1σ and t2σ with the kinetic parameter

vector taking values v′, v′′∈ B(u; δ), respectively. Denote the corresponding approximately stable solutions by xi(·; v′, j, k)

and xi(·; v′′, j, k), i = 6, 7, 8, respectively.

Definition 3. The relative deviation of intracellular substances with regard to parameter vector v′ against the disturbancev′′ is defined as

MSD(v′, v′′; j, k) ,

−i=6,7,8

|xi(t1σ ; v′, j, k) − xi(t2σ ; v′′, j, k)||xi(t2σ ; v′, j, k)|

. (21)

108 J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

The biological robustness of the system NHDS(j, k) at parameter vector vi, i ∈ In is defined as

MSDav(vi; j, k) ,

1n − 1

n−l=1

MSD(vl, vi; j, k). (22)

Definition 4. Given u ∈ Dus(j, k), the biological robustness of the system NHDS(j, k) over B(u; δ) is defined as

MSDmin(u; j, k) , min{MSDav(vi; j, k) : i ∈ In}. (23)

The optimal biological robustness of the system NHDS(j, k) to parameter vector u over Dus(j, k) is defined as

J(u∗; j, k) , min{MSDmin(u; j, k) | u ∈ Dus(j, k)} (24)

where

u∗= u∗(j, k) , arg min{MSDmin(u; j, k) | u ∈ Dus(j, k)}. (25)

By Property 3 and the above definitions, we can easily get the following property.

Property 4. For given (j, k) ∈ IQ × I9 and u ∈ Dus(j, k), the mapping MSD(·, ·; j, k) : B(u; δ) × B(u; δ) −→ R+,MSDav(·; j, k) : B(u; δ) −→ R+ and MSDmin(·; j, k) : Dus(j, k) −→ R+ are continuous in their domains, respectively.

It follows from Property 4 that biological robustness defined by the expression (24) is meaningful.Let

U∗(k) , (u∗(1, k), u∗(2, k), . . . , u∗(Q , k))T ∈

Q∏j=1

Dus(j, k), k ∈ I9. (26)

Definition 5. The biological robustness of the metabolic system Sk to U∗(k) is defined as

J(U∗(k), k) ,1Q

Q−j=1

MSDmin(u; j, k), k ∈ I9. (27)

We can conclude that the smaller the value of J(U∗(k), k) is, the more robust the metabolic system Sk, k ∈ I9, is.

5. System identification model and algorithm

In the above hybrid system NHDS, there are two types of parameters to be determined, i.e., kinetic parameter vectoru ∈ Dus(j, k) ⊂ R17

+and parameter k ∈ I9. Taking u as continuous parameter vector and k as discrete parameter, we establish

an identification problem corresponding to the generalized hybrid system NHDS, which is subjected to some conditionsincluding the hybrid dynamical system NHDS(j, k), (j, k) ∈ IQ × I9, the approximate stability of the system NHDS(j, k), therestriction of relative error of extracellular concentrations and the bounded constraints of states and optimized variables.

Given (j, k) ∈ IQ × I9, denote

Dssd(j, k) , {u | u ∈ Dus(j, k) and SSD(u, j, k) ≤ d},

where d is a given positive real number. For a given metabolic system Sk, k ∈ I9, if the setQ

j=1 Dssd(j, k) is empty, weargue that the metabolic system Sk, k ∈ I9, is very poor in characterizing the actual fermentation process and set its optimalrobustness index to be +∞. So, to ensure the optimal robustness index of Sk, k ∈ I9, finite-valued, we assume that the setDssd(j, k), (j, k) ∈ IQ × I9 is nonempty by setting d large enough.

The identification problem corresponding to the hybrid system NHDS, denoted by IP, can be formulated as

IP : min

J(U(k), k) | U(k) ∈

Q∏j=1

Dssd(j, k), k ∈ I9

. (28)

Because of the coexistence of the continuous and discrete optimized variables in the identification problem IP, it isdifficult to solve it directly. However, due to the independence of these two typical variables, we can decompose the problemIP into two subproblems, i.e., an identification subproblem on the continuous parameter vector u ∈ Dssd(j, k) and anothersubproblem with the discrete parameter k ∈ I9 as design variable.

Firstly, given (j, k) ∈ IQ × I9, let J1(u, j, k) , MSDmin(u; j, k). The identification subproblem corresponding to NHDS(j, k),denoted by IP1(j, k), can be formulated as

IP1(j, k) : min{J1(u; j, k) | u ∈ Dssd(j, k)}. (29)

Since (j, k) ∈ IQ × I9 is given, there is only continuous parameter vector u ∈ Dssd(j, k) in IP1(j, k).

J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112 109

Table 2Biological robustness index of the kth metabolic system under substrate-sufficient and substrate-limited conditions, respectively.

k 1 2 3 4 5 6 7 8 9

Jks +∞ 0.1764 0.1757 +∞ 0.1813 0.2040 +∞ 0.1238 0.1027Jkl +∞ 0.1554 0.3201 +∞ 0.1280 0.1213 +∞ 0.09383 0.1274

Table 3Experimental data and the corresponding numerical results under substrate-sufficient conditions.

D = 0.25 h−1 , Cs0 = 675 mmol/L, y = (3.21, 5.85, 343.86, 104.1, 114.47)T

x∗= (3.7069, 5.81339, 317.095, 113.611, 120.62, 5.80025, 143.58, 314.822)T

u∗= (45.218, 1.71436, 776.695, 68.98, 1.25001, 20.817, 9.35261, 55.0715, 1.88857,

1089.06, 43.767, 107.232, 27.3344, 0.869602, 13.8719, 4.68963, 40.5647)T

Secondly, given k ∈ I9, let

J2(U∗(k), k) ,1Q

Q−j=1

MSDmin(u∗; j, k).

where u∗ andU∗(k) are defined in (25) and (26), respectively. Then, the identification subproblemconcerning k ∈ I9, denotedby IP2(k), can be formulated as

IP2(k) : min{J2(U∗(k), k) | k ∈ I9}. (30)Let k∗ , arg min{J2(U∗(k), k)|k ∈ I9}. Then the k∗th metabolic system is the most robust one and J2(U∗(k∗), k∗) is the

optimal value of IP. The optimal continuous parameter vectors of IP are u∗(1, k∗), u∗(2, k∗), . . . , u∗(Q , k∗).When we intend to solve the problem IP, our main difficulty lies in the solution of the subproblem IP1(j, k). The method

proposed in the previous work [26] can be applied to solving our problem based on Properties 3 and 4. Let M and N be thesizes of the Monte Carlo experiments. Then, a procedure to solve IP1(j, k) is constructed as follows.Step 1. Given d > 0, δ > 0 and σ > 0. Randomly generate M sample points from Du(j, k) and select m sample points ur ,

r ∈ Im, such that ur∈ Dssd(j, k). Set r := 1.

Step 2. Randomly generateN sample points from B(ur; δ) and select n−1 sample points vri, i ∈ In−1, such that vri

∈ Dssd(j, k).Let vrn

:= ur .Step 3. Compute MSDav(v

ri; j, k), i = 1, . . . , n, from (22) and (23). Let

J1(ur; j, k) := MSDmin(ur

; j, k) := min{MSDav(vri; j, k) : i ∈ In},

ur:= arg min{MSDav(v

ri; j, k) : i ∈ In}.

Step 4. If r < m, let r := r + 1, go to step 2; else compute

J1(u∗; j, k) = min{J1(ur

; j, k) : r ∈ Im},

u∗= u∗(j, k) = arg min{J1(ur

; j, k) : r ∈ Im}.

6. Numerical example and results

In this example, the above method is carried out 100 times based on 9 groups of experiments under substrate-sufficientconditions and 7 groups under substrate-limited conditions, respectively. The parameters chosen in the above method areas follows: x0 = (0.1, 400, 0, 0, 0, 0, 0, 0)T , tb = 5, tf = 100,M = 10,000, N = 1000, d = 0.3, σ = 0.01, δ = 0.05,m = 10and n = 20. Every robustness index in Table 2 is calculated as the average of the 100 groups of computational results ofthe corresponding robustness index. Let Jks and Jkl , k ∈ I9 be the averaged robust performance of the metabolic systemSk under substrate-sufficient and substrate-limited conditions, respectively. For the metabolic systems Sk, k = 1, 4, 7,their robustness index are set to be +∞ because the sets Dssd(j, k), k = 1, 4, 7, are empty for all j ∈ IQ even if thenumber of random samples is large enough, which indicates that these metabolic systems (S1, S4 and S7) are all very poor inrepresenting the actual fermentation process. Table 2 shows that S9 and S8 are the most robust under substrate-sufficientand substrate-limited conditions, respectively. So we conclude that the most possible metabolic system, under substrate-sufficient and substrate-limited conditions, are S9 and S8, respectively.

Tables 3 and 4 give the numerical results corresponding to one group of experimental data under substrate-sufficientand substrate-limited conditions from S9 and S8, respectively, including the computational stable state vector and theidentified continuous parameters. Using the identified parameters in Tables 3 and 4, we carry out numerical simulation forthe metabolic system S9 under substrate-sufficient conditions and for S8 under substrate-limited conditions, respectively.Figs. 2 and 3 show the comparison of the first three extracellular concentrations between experimental data andcomputational results under substrate-sufficient and substrate-limited conditions, respectively, where the dashed linesdenote the computational curves and the real lines denote the experimental steady values. Figs. 4 and 5 show the simulatedresults of three intracellular concentrations under two conditions, respectively.

110 J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

Table 4Experimental data and the corresponding numerical results under substrate-limited conditions.

D = 0.1 h−1, Cs0 = 616 mmol/L, y = (3.73, 0.43, 227.96, 86.86, 209.32)T

x∗= (3.75326, 0.482661, 217.264, 88.9257, 239.655, 0.481121, 132.778, 0.777525)T

u∗= (67.9523, 1.60993, 471.737, 16.0973, 1.38442, 23.7976, 9.20826, 62.7407, 2.30795,

3341.33, 35.8174, 268.708, 24.1471, 1.4359, 7.21178, 30.2732, 80.1388)T

Fig. 2. Comparison of experimental data and simulation results of biomass concentration, extracellular glycerol and 1, 3-PD concentration in continuousculture under substrate-sufficient conditions at initial glycerol concentration of 675 mmol/L and dilution rate of 0.25 h−1 .

Fig. 3. Comparison of experimental data and simulation results of biomass concentration, extracellular glycerol and 1, 3-PD concentration in continuousculture under substrate-limited conditions at initial glycerol concentration of 616 mmol/L and dilution rate of 0.1 h−1 .

J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112 111

Fig. 4. Model simulations of main intracellular substance concentration in continuous culture under substrate-sufficient conditions at initial glycerolconcentration of 675 mmol/L and dilution rate of 0.25 h−1 .

Fig. 5. Model simulations of main intracellular substance concentration in continuous culture under substrate-limited conditions at initial glycerolconcentration of 616 mmol/L and dilution rate of 0.1 h−1 .

7. Conclusion

In this paper, we established a complex metabolic network composed of nine metabolic systems and formulated thecorresponding nonlinear hybrid system to describe the continuous fermentation of glycerol by K. pneumoniae. We thendiscussed the existence, uniqueness and continuous dependence of solutions with respect to parameters. Moreover, wequantitatively described the biological robustness of the metabolic system and presented a system identification problemon the basis of robustness performance. Numerical results show that it is most possible that both glycerol and 1, 3-PD passthe cell membrane by active transport coupled with passive diffusion under substrate-sufficient conditions, whereas, under

112 J. Wang et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 102–112

substrate-limited conditions, glycerol passes the cell membrane by active transport coupled with passive diffusion and 1,3-PD by active transport.

Next, it is intended to search for a grounded theoretical basis for the biological robustness definition proposed in thispaper and construct some optimization algorithms for the identification problem.

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).

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