MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 690587 10.1155/2012/690587 690587 Research Article Modeling and Parameter Identification Involving 3-Hydroxypropionaldehyde Inhibitory Effects in Glycerol Continuous Fermentation Gong Zhaohua Liu Chongyang Yu Yongsheng Weber Gerhard-Wilhelm 1 School of Mathematics and Information Science Shandong Institute of Business and Technology Yantai 264005 China sdibt.edu.cn 2012 14 11 2012 2012 09 07 2012 21 09 2012 2012 Copyright © 2012 Zhaohua Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Mathematical modeling and parameter estimation are critical steps in the optimization of biotechnological processes. In the 1,3-propanediol (1,3-PD) production by glycerol fermentation process under anaerobic conditions, 3-hydroxypropionaldehyde (3-HPA) accumulation would arouse an irreversible cessation of the fermentation process. Considering 3-HPA inhibitions to cells growth and to activities of enzymes, we propose a novel mathematical model to describe glycerol continuous cultures. Some properties of the above model are discussed. On the basis of the concentrations of extracellular substances, a parameter identification model is established to determine the kinetic parameters in the presented system. Through the penalty function technique combined with an extension of the state space method, an improved genetic algorithm is then constructed to solve the parameter identification model. An illustrative numerical example shows the appropriateness of the proposed model and the validity of optimization algorithm. Since it is difficult to measure the concentrations of intracellular substances, a quantitative robustness analysis method is given to infer whether the model is plausible for the intracellular substances. Numerical results show that the proposed model is of good robustness.

1. Introduction

Microbial conversion of glycerol to 1,3-propanediol (1,3-PD) is particularly attractive in that the process is relatively easy and does not generate toxic byproducts. 1,3-PD has numerous applications in polymers, cosmetics, foods, lubricants, and medicines. Industrial 1,3-PD production has attracted attention as an important monomer to synthesize a new type of polyester, polytrimethylene terephthalate (PTT) . However, compared with chemical routes, microbial production is difficult to obtain a high 1,3-PD concentration. The fermentation of glycerol by Klebsiella pneumoniae (K. pneumoniae) under anaerobic conditions is summarized in Figure 1. In the reductive pathway, 3-hydroxypropionaldehyde (3-HPA) is a toxic intermediary metabolite and its accumulation would arouse inhibitions to cells growth and to activities of the enzymes (such as glycerol dehydratase (GDHt) and 1,3-propanediol oxidoreductase (PDOR)) in glycerol metabolism .

Anaerobic metabolism pathways of glycerol by K. pneumoniae .

It is critical to formulate the fermentation process using a precise mathematical model in the optimization of biotechnological processes. An excess kinetic model for substrate consumption and product formation was established in previous studies . The models have been studied for parameter identification  and optimal control  in fed-batch fermentation process. However, the intermediate and intracellular substances or enzymes of glycerol metabolism are not taken into consideration in those models. In fact, some important intermediate substances (such as 3-HPA), intracellular substances (such as 1,3-PD), and enzymes GDHt and PDOR play significant roles in glycerol metabolism. A mathematical model of glycerol fermentation concerning enzyme-catalytic reductive pathway and transports of glycerol and 1,3-PD across cell membrane was established in . Although the achieved results are interesting, the effect of 3-HPA on cells growth is ignored. Moreover, that model is based on an assumption that 3-HPA inhibits the activities of the enzymes GDHt and PDOR all the time. In fact, there exists inhibitory effect of 3-HPA on cells growth owing to its toxicity. In addition, only when the accumulation of 3-HPA reaches some critical concentration, the inhibitions to enzymes can occur [3, 4].

Robustness is one of the fundamental characteristics of biological systems. By saying that a system is robust we imply that a particular function or characteristic of the system is preserved despite changes in the operating environment . For robust biological systems, we expect that mathematical models attempting to explain these systems should also be robust . In this paper, we are interested in the robustness to variations in kinetic parameters and use it to validate the plausibility of the mathematical model. This topic has been studied by the sensitivity analysis technique , that is, repeated simulations by varying one parameter while holding all others fixed. However, single parameter insensitivity may not be sufficient owing to interactions between several parameters. Therefore, new methods are needed for studying multiparameter robustness.

Considering 3-HPA inhibitions to cells growth and to activities of the enzymes GDHt and PDOR in glycerol metabolism, we propose a novel mathematical model to describe 1,3-PD production by K. pneumoniae in continuous cultures. Some properties of the model, such as existence and uniqueness of the solution, continuity of the solution in kinetic parameters, and compactness of the set of feasible parameters, are discussed. Furthermore, a parameter identification model is established to determine the kinetic parameters in the presented system. Basing on the penalty function technique and an extension of the state space method, an improved genetic algorithm (GA) is then constructed to solve the identification model. Numerical example shows the appropriateness of the proposed system and the validity of optimization algorithm. Finally, a quantitative robustness analysis method is given to infer whether the model is robust, and numerical result shows that the proposed system is of good robustness.

This paper is organized as follows. In Section 2, the kinetic model is formulated to describe continuous fermentation process, whose important properties are also discussed. In Section 3, a parameter identification model is presented and an optimization algorithm is developed. Section 4 explores the robustness analysis of the proposed dynamical system. Finally, conclusions are provided.

2. Mathematical Model in Continuous Culture and Its Properties 2.1. Mathematical Model

During continuous fermentation of glycerol metabolism by K. pneumoniae under anaerobic conditions, glycerol is fed to the reactor continuously. As stated in , we assume that

the transport of extracellular glycerol across cells membrane by passive diffusion and by glycerol transport facilitator;

intracellular 1,3-PD is expected to be diffused from the intracellular environment to the extracellular medium in the fermentative broth.

Let x(t):=(x1(t),x2(t),x3(t),x4(t),x5(t),x6(t),x7(t),x8(t))R8 be the state vector, and xi(t), i=1,2,8, respectively denote the concentrations of biomass, extracellular glycerol, extracellular 1,3-PD, extracellular acetic, extracellular ethanol, intracellular glycerol, intracellular 3-HPA, and intracellular 1,3-PD at time t in reactor. D is the dilution rate, and Cs0 is the initial glycerol concentration in feed.

Since 3-HPA is a toxic intermediary metabolite, its inhibition to the specific cellular growth rate is introduced besides substrate and products inhibitions. Therefore, the specific cellular growth rate μ  is modified as (2.1)  μ=μmx2x2+ks*(1-x2x2*)(1-x3x3*)(1-x4x4*)(1-x5x5*)(1-x7x7*). Under anaerobic conditions at 37°C and pH 7.0, the maximum specific growth rate, μm, and Monod constant, ks*, are 0.67h-1   and 0.28 mmol L-1, respectively. The critical concentrations xi*, i=2,3,4,5,7, are 2039, 1036, 1026, 360.9, and 300 mmolL-1 , respectively.

The specific consumption rate of substrate q2, and the specific formation rates of products q4 and q5 are expressed by the following equations based on previous works [6, 8, 10] (2.2)q2=k6+μk7+k8x2x2+k9,q4=-k10+μk11+k12x2x2+k13,q5=k14+μk15+k16x2x2+k17.

The governing equations based on mass balance [6, 10] can still be used to describe the concentrations of substrate and products for glycerol metabolism, for example, biomass, extracellular glycerol, extracellular 1,3-PD, acetate, ethanol, and intracellular glycerol which are described by (2.3)–(2.7), respectively (2.3)x˙1=(μ-D)x1,(2.4)x˙2=D(Cs0-x2)-q2x1,(2.5)x˙3=k1(x8-x3)x1-Dx3,(2.6)x˙i=qix1-Dxi,i=4,5,(2.7)x˙6=1k2(k3x2x2+k4+k5(x2-x6)-q2)-μx6.

The reductive pathway is emphasized because 3-HPA is the key intermediate for 1,3-PD production. 3-HPA accumulation during fermentation process can cause growth cessation and low product formation [2, 4]. Moreover, when the accumulation of 3-HPA reaches some critical concentration, the inhibitions to enzymes GDHt and PDOR can occur [3, 4]. So the intracellular concentration change of 3-HPA can be described by (2.8)  x˙7=k18u1x6km1*(1+IR+(x7-a*)((x7-a*)/k19))+x6-  k20u2x7km2*+x7(1+IR+(x7-a*)((x7-a*)/k21))-μx7, where km1*, km2* are 0.53 and 0.14 mmol L-1 [20, 21], respectively. a* is the critical concentration of 3-HPA beyond which the inhibitions to the activities of GDHt and PDOR occur. Moreover, in (2.8), (2.9)IR+(z)={1,z>0,0,z0.u1 and u2 are the specific activities of GDHt and PDOR in vitro, which can be described by the following equations: (2.10)u1=k22-k23μ-k24x7x7+k25,u2=k26-k27μ-k28x7x7+k29.

The intracellular 1,3-PD concentration depends on the conversion of 3-HPA catalyzed by PDOR whose activity is inhibited by the substrate, the diffusion from the intercellular to the extracellular and the dilution effect on cell growth, so whose variation can be formulated by (2.11)x˙8=k20u2x7km2*+x7(1+IR+(x7-a*)((x7-a*)/k21))-k30(x8-x3)-μx8.

Now, let k:=(k1,k2,,k30) be the kinetic parameter vector to be identified. Denote u:=(CS0,D) and let the right-hand sides of (2.3)–(2.7), (2.8), and (2.11) be f(x,u,k). Then, the proposed mathematical model can be rewritten as the following nonlinear dynamical system: (2.12)x˙=f(x,u,k)x(0)=x0,t[0,T], where T is the steady-state moment of the continuous fermentation process.

2.2. Properties of the Dynamical System

To begin with, we introduce some symbols which will be used below. Let KR30 be the admissible set of the kinetic parameter vector k. Let x*:=(0.001,100,0,0,0,0,0,0) and x*:=(5,2039,1036,1026,360.9,300,2039,1036) denote the lower and upper bounds of the state vector x, respectively. Let Wa:=i=18[xi*,xi*]R+8 be the admissible set of x, and let uU:=[110.96,1883]×[0.1,0.5] be the admissible set of initial glycerol concentration in feed medium Cs0 and dilution rate D.

For the system (2.12), we assume that

the set KR30 is a nonempty bounded closed set;

the absolute difference between extracellular and intracellular 1,3-PD and that of glycerol concentration is bounded, that is, M1>0 and M2>0 such that (2.13)|x8(t)-x3(t)|M1,t[0,T],|x2(t)-x6(t)|M2,t[0,T].

Under the assumptions (H3) and (H4), we can easily verify the following properties of the velocity vector field f(x,u,k).

Property 1.

For any kK and uU, the function f(x,u,k) is locally Lipschitz continuous in x on Wa.

Property 2.

For any kK and uU, the function f(x,u,k) satisfies linear growth condition, that is, there exist constants α,β>0 such that (2.14)f(x,u,k)α+βx,xWa, where · is Euclidean norm.

Proof.

For given kK and uU, let C1:=μm, C2:=|k6|+μm/|k7|+|k8|+D, C3:=|k1|M1, C4:=|k10|+μm|k11|+|k12|, C5:=|k14|+μm|k15|+|k16|, C6:=(1/|k2|)(|k3|+|k5|M2+C2-D), C7:=|k18|(|k22|+|k23|μm+|k24|)+|k20|(|k26|+|k27|μm+|k28|), C8:=|k20|(|k26|+|k27|μm+|k28|+|k30|M1. Let Li:=Ci+D, i=1,3,4,5, L2:=max{DCs0,C2}, Li:=max{Ci,μm}, i=6,7,8. Then, we can obtain (2.15)|f1(x,u,k)|(μ+D)|x1|L1(x+1),|f2(x,u,k)|DCs0+D|x2|+|q2||x1|L2(x+1),|f3(x,u,k)|M2|k1x1|+D|x3|L3(x+1),|fi(x,u,k)||qi||x1|+D|xi|Li(x+1),i=4,5,|fi(x,u,k)|Ci+μxiLi(x+1),i=6,7,8. Finally, set L:=max{Li,iI8:={1,2,,8}}, then we have (2.14) holds with α=β=22L. The proof is completed.

Then, the existence and uniqueness of the solution for the system (2.12) can be confirmed in the following theorem.

Theorem 2.1.

For any kK and uU, the system (2.12) with given initial state x0Wa has a unique solution denoted by x(·;u,k). Moreover, x(t;u,k) is continuous in k on K. (2.16)x(t;u,k)=x0+0tf(x(s),u,k)ds,t[0,T].

Proof.

The proof can be obtained from Properties 1, and 2 and the theory of ordinary differential equations .

Given x0Wa, we define the solution set 𝒮0 of the system (2.12) as follows: (2.17)𝒮0{x(·;u,k)x(t;u,k)isasolutionofthesystem(2.12)corresponding  tokK  and  uUforanyt[0,T]}. Since the concentrations of biomass, glycerol, 3-HPA, and products are restricted in Wa during the actual continuous cultures, the set of admissible solutions is (2.18)𝒮:={x(·;u,k)𝒮0x(t;u,k)Wat[0,T]}. Furthermore, let the set of feasible parameter vectors corresponding to 𝒮 be (2.19):={kKx(·;u,k)𝒮}.

From Theorem 2.1 and the above definitions, we have the following result.

Theorem 2.2.

The feasible parameter set defined by (2.19) is a compact set.

Proof.

In view of the compactness of K, we obtain that is a bounded set. Moreover, for any sequence {ki}i=1K, there exists at least a subsequence {k^i}{ki} such that k^ik^ as i. It follows from Theorem 2.1 that x(·;u,k^i)𝒮0 and x(·;u,k^)𝒮0. Since Wa is a compact set, we must conclude that x(·;u,k^)𝒮, which implies the closeness of set . The proof is completed.

3. Parameter Identification Model and Optimization Algorithm 3.1. Parameter Identification Model

Now, we determine the kinetic parameter k in K by constructing an identification problem as follows.

Let l be the total number of experiments carried out under different dilution rates and initial glycerol concentrations. For given uj=(Cs0j,Dj), jIl, we have the experimental steady-state data of extracellular substances in continuous cultures. Denote the steady-state concentrations of biomass, extracellular glycerol, extracellular 1,3-PD, acetate, and ethanol as y1j,y2j,y3j,y4j,y5j, correspondingly. Let yj:=(y1j,y2j,y3j,y4j,y5j)R+5,jIl.

In particular, the stable state of a dynamical system (2.12) in the following identification problem is actually referred to the approximate stability defined as follows.

Definition 3.1.

For given ujU, a state vector x(Tj;uj,k) is said to be an approximately stable solution within a precision ε>0 if there exists kK such that x(·;uj,k) is a solution of the system (2.12) satisfying (3.1)f(x(Tj),uj,k)<ε, where · is the Euclidean norm, and Tj=inf{tε:f(x(t),uj,k)<ε  for all t[tε,T]}.

Since the orders of magnitude for concentrations involved are different, we adopt the average relative error between the computational values x(Tj;uj,k) and the experimental data yj at steady-state moments Tj,jIl, as the criterion (3.2)J(k):=15m=15j=1l|xm(Tj;uj,k)-ymj|j=1lymj.

To determine the parameter values of the system (2.12), a parameter identification problem, in which J(k) is taken as the cost function, can be formulated as (PIP)minJ(k)s.t.f(x(Tj),uj,k)<ε,jIl,k.

Following the above properties, we can conclude the following theorem.

Theorem 3.2.

For ujU, jIl, there exists k* in (PIP) such that J(k*)J(k),for all k.

3.2. Optimization Algorithm

In (PIP), the constraint k actually involves the constraint of continuous state, that is, x*x(t;uj,k)x*,t[0,Tj],kK,jIl. In this section, we will develop a computational method for solving our proposed parameter identification problem (PIP). By means of the penalty function technique and an extension of state space method, we transcribe (PIP) into an optimization problem only with box constraint. First of all, we introduce a new state variable z satisfying (3.3)z˙=g(x(t)),t[0,Tj],jIlz(0)=0, where g(x(t)):=[x(t)-x*]++[x*-x(t)]+, and [x~]+:=i=18(max{x~i,0})2.

Obviously, the process x(t;uj,k) satisfies the constraint of continuous state if and only if z(Tj)=0,jIl. Furthermore, denote the cost function in (PIP) by (3.4)J~λ,μ(k):=J(k)+λj=1lf(x(Tj),uj,k)+μj=1lz(Tj), where λ and μ are penalty factors. Then (PIP) can be rewritten as (PIP(λ,μ))minJ~λ,μ(k)s.t.kK.

As a result, (PIP(λ,μ)) is an optimization problem only with the box constraint and equivalent to (PIP) as λ+ and μ+.

Since J~λ,μ(k) is nondifferentiable, we construct an improved genetic algorithm (GA) to solve (PIP(λ,μ)) taking advantage of the problem’s characteristic. Let k~:=(k1,k2,,k30)K. In the improved genetic algorithm, we take k~ as the individual, and (3.4) as the fitness function. Now, we describe the algorithm in detail as follows.

Algorithm 3.3.

We have the following steps:

Step 1. Initialize population size N, the maximal iterations H, penalty factors λ and μ, precision ε, parameters N1,α1, and α2. Set h=0 and randomly generate N individuals by uniform distribution, that is, k~i(h)K,i=1,2,N. Let P(h)={k~i(h)i=1,2,,N}.

Step 2. Compute the fitness value J~λ,μ(k~i(h)) by taking k~i(h) into the systems (2.12) and (3.3).

Step 3. Generate the crossover offspring dm(h)(m=1,2,,N1) by arithmetic crossover and evaluate their fitness values J~λ,μ(dm(h)).

Step 4. The individual produced by crossover is operated with normality variation until the produced one is in K. The mutation offspring is denoted by gm(h)(m=1,2,,N1). Compute their fitness values J~λ,μ(gm(h)).

Step 5. Form the next generation P(h+1) by selecting the best N individuals from the N+2N1 ones.

Step 6. Set h=h+1,λ=α1λ, and μ=α2μ, if either h>H or no progress is made in the last generations, then output the best individual and stop, otherwise go to Step 3.

3.3. Numerical Results

According to the actual continuous fermentation process, the initial state x0=(0.1115gL-1, 495mmolL-1,0,0,0,0,0,0) and 22 groups of the experimental steady-state data are used. Here, 10 groups of experimental steady-state data under substrate-limited conditions and 12 groups of experimental steady-state data under substrate-sufficient conditions. The critical value a* is taken as the value in . Moreover, the admissible set of parameter vectors K is taken as the decrements and increments of 0.9 times the kinetic parameter values in . By applying Algorithm 3.3, we obtained the optimal parameter vector k* of the system (2.12) under the substrate-limited and the substrate-sufficient conditions shown in Table 1. Accordingly, the cost function values, respectively, are 1.204  ×  10-2 and 1.585  ×  10-2 under the above two cases. Here, all the computations are performed in Visual C++ 6.0 and numerical results are plotted by MATLAB 7.10.0. In particular, the ODEs in the computation process are numerically calculated by improved Euler method  with the relative error tolerance 10-4. The parameters used in Algorithm 3.3 are H=1000, N=50, N1=30, λ=μ=1, ε=0.001, and α1=α2=1.1, respectively. It should be noted that these parameters are derived empirically after numerous experiments. The comparison of three extracellular substances concentrations, that is, biomass, extracellular glycerol, and extracellular 1,3-PD, between experimental steady-state data and computational results under substrate-limited conditions are shown in Figures 2, 3, and 4. Furthermore, The comparisons of three extracellular substance concentrations, that is, biomass, extracellular glycerol, and extracellular 1,3-PD, between experimental steady-state data and computational results under substrate-sufficient conditions are also shown in Figures 5, 6, and 7. From the above figures, we can see that the simulation results can approximate the experimental steady-state data well. Thus, the mathematical model considering 3-HPA inhibitions to cells growth and to activities of enzymes can well describe the continuous fermentation process.

The optimal parameter vector in the system (2.12).

Conditions Optimal parameter vector k
(16.93, 0.015, 375.9, 0.134, 9.5, 162.8,
0.013, 0.014, 0.518, 0.135, 6.239, 203.6,
Substrate-limited condition 37.5, 0.042, 18.43, 3.13, 0.006489,18.83,
12.82, 1.843, 52.03, 10.91, 5.045, 2.983,
0.567, 0.087, 1.852, 6.225, 0.346, 0.138)

(36.37, 0.287, 456, 0.878, 0.5, 162.8,
0.036, 0.087, 2.235, 2.216, 11.25, 282.1,
Substrate-sufficient condition  39.12, 0.629, 9.841, 3.854, 0.006678,
13.09, 10.77, 0.590, 62.83, 7.833, 17.39,
5.624, 0.353, 0.6145, 0.836, 8.094, 1.254, 0.138)

The concentrations of biomass change with respect to fermentation time under substrate-limited conditions for u=(295,0.25).

The concentrations of glycerol change with respect to fermentation time under substrate-limited conditions for u=(295,0.25).

The concentrations of 1,3-PD change with respect to fermentation time under substrate-limited conditions for u=(295,0.25).

The concentrations of biomass change with respect to fermentation time under substrate-sufficient conditions for u=(330,0.4).

The concentrations of glycerol change with respect to fermentation time under substrate-sufficient conditions for u=(330,0.4).

The concentrations of 1,3-PD change with respect to fermentation time under substrate-sufficient conditions for u=(330,0.4).

4. Robustness Analysis of the Model

For robust biological systems, we expect that mathematical models that attempt to explain these systems should also be robust. Robustness of the model is analyzed in this section. In view of the glycerol dissimilation mechanism, we assume that

for each x0Wa, k and given uU, there exists an approximate stable solution of the system (2.12).

4.1. Mathematical Measurement of Robustness

Robustness can be defined as a system’s characteristic that maintains one or more of its functions under external and internal perturbations . In this study, the substance concentrations at steady-state moments are viewed as the quantitative descriptors of the system and the perturbations are the parameters variations in .

Let U- be a finite set whose elements are drawn from U by random distribution. Since the state vector of the system (2.12) takes uU as the input parameter, we define the representative steady-state vector as (4.1)x-(tk;k):=1|U-|uU-x(tu,k;u,k), where (4.2)tu,k:=inf{tεf(x(t),u,k)<εfor  k,uU-,t[tε,T]},tk:=maxuU-{tu,kx(tu,k;u,k) isanapproximatelysteady-state vectorof(2.12)withk}, and |U-|  denotes the cardinal number of the set  U-.

Let PS(k,δ):={k~  |  k~-kδ,k} be the feasible space of the perturbation parameter vector, where δ>0 is sufficiently small positive number to measure the magnitude of the parameter disturbances. Furthermore, let 𝒫 be a perturbation set which is composed of elements randomly generated from PS(k,δ). Denote the set of the representative steady-state vector corresponding to the varied parameters by (4.3)𝒮-𝒫:={x-(tk;k)k𝒫}. For the representative steady-state vectors in 𝒮-𝒫, we are interested in their deviations from the representative steady-state vector corresponding to the optimal parameter vector k*. Define the expectation of the ith component of these deviations as follows: (4.4)U𝒫i:=1|𝒫|k𝒫(x-i(tk;k)x-i(tk*;k*)-1)2,      iI8.

Based on the above analyses, a mathematical definition of biological robustness can be stated as follows.

Definition 4.1.

The robustness measurement of a model with regard to steady states against a set of parameter perturbations 𝒫 is (4.5)𝒫:=1|Irob|iIrob1U𝒫i, where Irob is the state index set involved in robustness analysis.

4.2. Algorithm and Numerical Results

On the basis of the definition of robustness, we develop an Algorithm 4.2 to compute the robustness to variations of the optimal kinetic parameter vector k*.

Algorithm 4.2.

We have the following steps:

Step 1. Generate N2 perturbation parameter vectors ksPS(k*,δ),s=1,2,,N2 by random distributions domain.

Step 2. Compute the representative steady-state vectors corresponding to optimal parameters and to perturbation parameters x-(tk*;k*) and x-(tks;ks) according to (4.1).

Step 3. Evaluate the robustness by the mathematical measurement 𝒫 defined in (4.5).

Step 4. Repeat Step 1, Step 2, and Step 3 for N3 times and compute the expectation value of the generated sequence {𝒫}.

According to Algorithm 4.2, we have investigated the robustness of the system (2.12) for the intracellular substances. Here, the number of perturbation parameters N2, precision δ, the set Irob, and the repeating times N3 take values 200, 0.05, {6,7,8}, and 100, respectively. Computational results for the substrate-limited and substrate-sufficient cases are listed in Table 2. Furthermore, the simulation curves for intracellular glycerol, 3-HPA and intracellular 1,3-PD under the substrate-limited and the substrate-sufficient conditions are illustrated in Figures 8, 9, 10, 11, 12, and 13. From Table 2 and the numerical results, we can see that the proposed model is of good robustness.

The robustness of the system (2.12) under substrate-limited and substrate-sufficient conditions.

 Conditions Substrate-limited Substrate-sufficient Values 3.97268 2.70685

Robustness simulations of intracellular glycerol concentration under substrate-limited conditions for u=(295,0.25).

Robustness simulations of 3-HPA concentration under substrate-limited conditions for u=(295,0.25).

Robustness simulations of intracellular 1,3-PD concentration under substrate-limited conditions for u=(295,0.25).

Robustness simulations of intracellular glycerol concentration under substrate-sufficient conditions for u=(330,0.4).

Robustness simulations of 3-HPA concentration under substrate-sufficient conditions for u=(330,0.4).

Robustness simulations of intracellular 1,3-PD concentration under substrate-sufficient conditions for u=(330,0.4).

5. Conclusions

Glycerol bioconversion to 1,3-PD by K. pneumoniae in continuous cultures under anaerobic conditions was investigated. Contrasting with the existing models, the paper proposed a new mathematical model by considering 3-HPA inhibitory effects on cells growth and on the activities of the enzymes GDHt and PDOR. Then, we discussed some properties of the system. Furthermore, we presented a parameter identification model to determine the kinetic parameters in the presented system. Since the identification model is subject to constraint of continuous state, we developed a computational approach to solve the parameter identification model based on the penalty function technique and an extension of the state space method. More importantly, a quantitative robustness analysis method was given to infer whether the model is plausible. Numerical results showed that the proposed system is of good robustness.

Acknowledgments

The supports of the Natural Science Foundation for the Youth of China (nos. 11001153, 11201267), the Tianyuan Special Funds of the National Natural Science Foundation of China (no. 11126077), the Shandong Province Natural Science Foundation of China (nos. ZR2010AQ016, ZR2011AL003, and BS2012DX025) and the Fundamental Research Funds for the Central Universities (no. DUT12LK27) are gratefully acknowledged.

Biebl H. Menzel K. Zeng A.-P. Microbial production of 1,3-propanediol Applied Microbiology and Biotechnology 1999 52 3 289 297 Barbirato F. Grivet J.-P. Soucaille P. Bories A. 3-Hydroxypropionaldehyde, an inhibitory metabolite of glycerol fermentation to 1, 3-propanediol by enterobacterial species Applied and Environmental Microbiology 1996 62 4 1448 1451 Hao J. Lin R.-H. Zheng Z.-M. Sun Y. Liu D.-H. 3-Hydroxypropionaldehyde guided glycerol feeding strategy in aerobic 1,3-propanediol production by Klebsiella pneumoniae Journal of Industrial Microbiology and Biotechnology 2008 35 12 1615 1624 Barbirato F. Himmi E.-H. Conte T. Bories A. 1,3-propanediol production by fermentation: an interesting way to valorize glycerin from the ester and ethanol industries Industrial Crops and Products 1998 7 2-3 281 289 Wang W. Sun J. B. Hartlep M. Deckwer W.-D. Zeng A.-P. Combined use of proteomic analysis and enzyme activity assays for metabolic pathway analysis of glycerol fermentation by Klebsiella pneumoniae Biotechnology and Bioengineering 2003 83 5 525 536 Sun Y.-Q. Qi W.-T. Teng H. Xiu Z.-L. Zeng A.-P. Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane Biochemical Engineering Journal 2008 38 22 32 Zeng A.-P. Rose A. Biebl H. Tag C. Guenzel B. Deckwer W. D. Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation Biotechnology and Bioengineering 1994 44 8 902 911 Zeng A.-P. Deckwer W.-D. A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient conditions Biotechnology Progress 1995 11 1 71 79 Zeng A.-P. A kinetic model for product formation of microbial and mammalian cells Biotechnology and Bioengineering 1995 46 4 314 324 Xiu Z.-L. Zeng A.-P. An L.-J. Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol Journal of Dalian University and Technology 2000 40 428 433 Gong Z.-H. A multistage system of microbial fed-batch fermentation and its parameter identification Mathematics and Computers in Simulation 2010 80 9 1903 1910 10.1016/j.matcom.2009.12.011 2658522 Gong Z.-H. Liu C.-Y. Feng E.-M. Wang L. Yu Y.-S. Modelling and optimization for a switched system in microbial fed-batch culture Applied Mathematical Modelling 2011 35 7 3276 3284 10.1016/j.apm.2011.01.023 2785278 Liu C.-Y. Gong Z.-H. Feng E.-M. Modeling and optimal control of a nonlinear dynamical system in microbial fed-batch fermentation Mathematical and Computer Modelling 2011 53 1-2 168 178 10.1016/j.mcm.2010.08.001 2739254 Liu C.-Y. Gong Z.-H. Feng E.-M. Yin H.-C. Optimal switching control of a fed-batch fermentation process Journal of Global Optimization 2012 52 2 265 280 10.1007/s10898-011-9663-8 2886310 Kitano H. Towards a theory of biological robustness Molecular Systems Biology 2007 3 137 Tian T.-H. Robustness of mathematical models for biological systems The ANZIAM Journal 2004 45 C565 C577 2180323 Morohashi M. Winn A.-E. Borisuk M.-T. Bolouri H. Doyle J. Kitano H. Robustness as a measure of plausibility in models of biochemical networks Journal of Theoretical Biology 2002 216 1 19 30 10.1006/jtbi.2002.2537 1942207 Erb R.-S. Michaels G.-S. Sensitivity of biological models to errors in parameter estimates Pacific Symposium on Biocomputing 1999 4 53 64 Savageau M.-A. The behavior of intact biochemical control systems Current Topics in Cellular Regulation 1972 6 63 130 Honda S. Toraya T. Fukui S. In situ reactivation of glycerol inactivated coenzyme-B12-dependent glycerol dehydratase and dioldehydratase Journal of Bacteriology 1980 143 1458 1465 Daniel R. Boenigk R. Gottschalk G. Purification of 1,3-propanediol dehydrogenase from Citrobacter freundii and cloning, sequencing, and overexpression of the corresponding gene in Escherichia coli Journal of Bacteriology 1995 177 2151 2156 Arrowsmith D.-K. Place C.-M. Ordinary Differential Equations 1982 London, UK Chapman & Hall 684068 Shen B.-Y. Liu C.-Y. Ye J.-X. Feng E.-M. Xiu Z.-L. Parameter identification and optimization algorithm in microbial continuous culture Applied Mathematical Modelling 2012 36 2 585 595 10.1016/j.apm.2011.07.031 2845825 Hairer E. Nørsett S.-P. Wanner G. Solving Ordinary Differential Equations. I 1993 8 Berlin, Germany Springer 1227985