Research Article  Open Access
Zhaohua Gong, Chongyang Liu, Yongsheng Yu, "Modeling and Parameter Identification Involving 3Hydroxypropionaldehyde Inhibitory Effects in Glycerol Continuous Fermentation", Mathematical Problems in Engineering, vol. 2012, Article ID 690587, 18 pages, 2012. https://doi.org/10.1155/2012/690587
Modeling and Parameter Identification Involving 3Hydroxypropionaldehyde Inhibitory Effects in Glycerol Continuous Fermentation
Abstract
Mathematical modeling and parameter estimation are critical steps in the optimization of biotechnological processes. In the 1,3propanediol (1,3PD) production by glycerol fermentation process under anaerobic conditions, 3hydroxypropionaldehyde (3HPA) accumulation would arouse an irreversible cessation of the fermentation process. Considering 3HPA 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 1,3propanediol (1,3PD) is particularly attractive in that the process is relatively easy and does not generate toxic byproducts. 1,3PD has numerous applications in polymers, cosmetics, foods, lubricants, and medicines. Industrial 1,3PD production has attracted attention as an important monomer to synthesize a new type of polyester, polytrimethylene terephthalate (PTT) [1]. However, compared with chemical routes, microbial production is difficult to obtain a high 1,3PD concentration. The fermentation of glycerol by Klebsiella pneumoniae (K. pneumoniae) under anaerobic conditions is summarized in Figure 1. In the reductive pathway, 3hydroxypropionaldehyde (3HPA) 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 1,3propanediol oxidoreductase (PDOR)) in glycerol metabolism [2–5].
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 [7–10]. The models have been studied for parameter identification [11] and optimal control [12–14] in fedbatch 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 3HPA), intracellular substances (such as 1,3PD), and enzymes GDHt and PDOR play significant roles in glycerol metabolism. A mathematical model of glycerol fermentation concerning enzymecatalytic reductive pathway and transports of glycerol and 1,3PD across cell membrane was established in [6]. Although the achieved results are interesting, the effect of 3HPA on cells growth is ignored. Moreover, that model is based on an assumption that 3HPA inhibits the activities of the enzymes GDHt and PDOR all the time. In fact, there exists inhibitory effect of 3HPA on cells growth owing to its toxicity. In addition, only when the accumulation of 3HPA 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 [15]. For robust biological systems, we expect that mathematical models attempting to explain these systems should also be robust [16]. 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 [17–19], 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 3HPA inhibitions to cells growth and to activities of the enzymes GDHt and PDOR in glycerol metabolism, we propose a novel mathematical model to describe 1,3PD 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 [6], we assume that(H1) the transport of extracellular glycerol across cells membrane by passive diffusion and by glycerol transport facilitator;(H2) intracellular 1,3PD is expected to be diffused from the intracellular environment to the extracellular medium in the fermentative broth.
Let be the state vector, and , , respectively denote the concentrations of biomass, extracellular glycerol, extracellular 1,3PD, extracellular acetic, extracellular ethanol, intracellular glycerol, intracellular 3HPA, and intracellular 1,3PD at time in reactor. is the dilution rate, and is the initial glycerol concentration in feed.
Since 3HPA 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 [6] is modified as Under anaerobic conditions at 37°C and pH 7.0, the maximum specific growth rate, , and Monod constant, , are 0.67 and 0.28 mmol , respectively. The critical concentrations , , are 2039, 1036, 1026, 360.9, and 300 mmol [6], respectively.
The specific consumption rate of substrate , and the specific formation rates of products and are expressed by the following equations based on previous works [6, 8, 10]
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 1,3PD, acetate, ethanol, and intracellular glycerol which are described by (2.3)–(2.7), respectively
The reductive pathway is emphasized because 3HPA is the key intermediate for 1,3PD production. 3HPA accumulation during fermentation process can cause growth cessation and low product formation [2, 4]. Moreover, when the accumulation of 3HPA reaches some critical concentration, the inhibitions to enzymes GDHt and PDOR can occur [3, 4]. So the intracellular concentration change of 3HPA can be described by where , are 0.53 and 0.14 mmol [20, 21], respectively. is the critical concentration of 3HPA beyond which the inhibitions to the activities of GDHt and PDOR occur. Moreover, in (2.8), and are the specific activities of GDHt and PDOR in vitro, which can be described by the following equations:
The intracellular 1,3PD concentration depends on the conversion of 3HPA 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
Now, let be the kinetic parameter vector to be identified. Denote and let the righthand sides of (2.3)–(2.7), (2.8), and (2.11) be . Then, the proposed mathematical model can be rewritten as the following nonlinear dynamical system: where is the steadystate 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 be the admissible set of the kinetic parameter vector . Let and , denote the lower and upper bounds of the state vector , respectively. Let be the admissible set of , and let be the admissible set of initial glycerol concentration in feed medium and dilution rate .
For the system (2.12), we assume that(H3) the set is a nonempty bounded closed set;(H4) the absolute difference between extracellular and intracellular 1,3PD and that of glycerol concentration is bounded, that is, and such that
Under the assumptions (H3) and (H4), we can easily verify the following properties of the velocity vector field .
Property 1. For any and , the function is locally Lipschitz continuous in on .
Property 2. For any and , the function satisfies linear growth condition, that is, there exist constants such that where is Euclidean norm.
Proof. For given and , let , , , , , , , . Let , , , , . Then, we can obtain Finally, set , then we have (2.14) holds with . 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 and , the system (2.12) with given initial state has a unique solution denoted by . Moreover, is continuous in on .
Proof. The proof can be obtained from Properties 1, and 2 and the theory of ordinary differential equations [22].
Given , we define the solution set of the system (2.12) as follows: Since the concentrations of biomass, glycerol, 3HPA, and products are restricted in during the actual continuous cultures, the set of admissible solutions is Furthermore, let the set of feasible parameter vectors corresponding to be
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 , we obtain that is a bounded set. Moreover, for any sequence , there exists at least a subsequence such that as . It follows from Theorem 2.1 that and . Since is a compact set, we must conclude that , 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 in by constructing an identification problem as follows.
Let be the total number of experiments carried out under different dilution rates and initial glycerol concentrations. For given , , we have the experimental steadystate data of extracellular substances in continuous cultures. Denote the steadystate concentrations of biomass, extracellular glycerol, extracellular 1,3PD, acetate, and ethanol as , correspondingly. Let .
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 , a state vector is said to be an approximately stable solution within a precision if there exists such that is a solution of the system (2.12) satisfying where is the Euclidean norm, and .
Since the orders of magnitude for concentrations involved are different, we adopt the average relative error between the computational values and the experimental data at steadystate moments , as the criterion
To determine the parameter values of the system (2.12), a parameter identification problem, in which is taken as the cost function, can be formulated as
Following the above properties, we can conclude the following theorem.
Theorem 3.2. For , , there exists in such that .
3.2. Optimization Algorithm
In , the constraint actually involves the constraint of continuous state, that is, . In this section, we will develop a computational method for solving our proposed parameter identification problem . By means of the penalty function technique and an extension of state space method, we transcribe into an optimization problem only with box constraint. First of all, we introduce a new state variable satisfying where , and .
Obviously, the process satisfies the constraint of continuous state if and only if . Furthermore, denote the cost function in by where and are penalty factors. Then can be rewritten as
As a result, is an optimization problem only with the box constraint and equivalent to as and .
Since is nondifferentiable, we construct an improved genetic algorithm (GA) to solve taking advantage of the problem’s characteristic. Let . In the improved genetic algorithm, we take 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 , the maximal iterations , penalty factors and , precision , parameters , and . Set and randomly generate individuals by uniform distribution, that is, . Let . Step 2. Compute the fitness value by taking into the systems (2.12) and (3.3). Step 3. Generate the crossover offspring by arithmetic crossover and evaluate their fitness values . Step 4. The individual produced by crossover is operated with normality variation until the produced one is in . The mutation offspring is denoted by . Compute their fitness values . Step 5. Form the next generation by selecting the best individuals from the ones. Step 6. Set , and , if either 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 g, mmol and 22 groups of the experimental steadystate data are used. Here, 10 groups of experimental steadystate data under substratelimited conditions and 12 groups of experimental steadystate data under substratesufficient conditions. The critical value is taken as the value in [23]. Moreover, the admissible set of parameter vectors is taken as the decrements and increments of 0.9 times the kinetic parameter values in [6]. By applying Algorithm 3.3, we obtained the optimal parameter vector of the system (2.12) under the substratelimited and the substratesufficient conditions shown in Table 1. Accordingly, the cost function values, respectively, are and 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 [24] with the relative error tolerance . The parameters used in Algorithm 3.3 are , , , , , and , 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 1,3PD, between experimental steadystate data and computational results under substratelimited conditions are shown in Figures 2, 3, and 4. Furthermore, The comparisons of three extracellular substance concentrations, that is, biomass, extracellular glycerol, and extracellular 1,3PD, between experimental steadystate data and computational results under substratesufficient 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 steadystate data well. Thus, the mathematical model considering 3HPA inhibitions to cells growth and to activities of enzymes can well describe the continuous fermentation process.

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(H5) for each , and given , 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 [15]. In this study, the substance concentrations at steadystate moments are viewed as the quantitative descriptors of the system and the perturbations are the parameters variations in .
Let be a finite set whose elements are drawn from by random distribution. Since the state vector of the system (2.12) takes as the input parameter, we define the representative steadystate vector as where and .
Let be the feasible space of the perturbation parameter vector, where is sufficiently small positive number to measure the magnitude of the parameter disturbances. Furthermore, let be a perturbation set which is composed of elements randomly generated from . Denote the set of the representative steadystate vector corresponding to the varied parameters by For the representative steadystate vectors in , we are interested in their deviations from the representative steadystate vector corresponding to the optimal parameter vector . Define the expectation of the th component of these deviations as follows:
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 is where 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 .
Algorithm 4.2. We have the following steps: Step 1. Generate perturbation parameter vectors by random distributions domain. Step 2. Compute the representative steadystate vectors corresponding to optimal parameters and to perturbation parameters and according to (4.1). Step 3. Evaluate the robustness by the mathematical measurement defined in (4.5). Step 4. Repeat Step 1, Step 2, and Step 3 for times and compute the expectation value of the generated sequence .
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 , precision , the set , and the repeating times take values 200, 0.05, , and 100, respectively. Computational results for the substratelimited and substratesufficient cases are listed in Table 2. Furthermore, the simulation curves for intracellular glycerol, 3HPA and intracellular 1,3PD under the substratelimited and the substratesufficient 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.

5. Conclusions
Glycerol bioconversion to 1,3PD 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 3HPA 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.
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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.