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Journal of Chemistry
Volume 2013 (2013), Article ID 105765, 13 pages
http://dx.doi.org/10.1155/2013/105765
Research Article

Theoretical Study and Optimization of the Biochemical Reaction Process by Means of Feedback Control Strategy

1School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China
2Bioinformatics Factory, Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora, Szafrana 4a, 65-516 Zielona Gora, Poland
3School of Information Engineering, Dalian University, Dalian 116622, China

Received 23 June 2012; Accepted 2 September 2012

Academic Editor: Jorge Barros-Velazquez

Copyright © 2013 Kaibiao Sun 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.

Abstract

The aim of this work is to present a theoretical analysis and optimization of a biochemical reaction process by means of feedback control strategy. To begin with, a mathematical model of the biochemical reaction process with feedback control is formulated. Then, based on the formulated model, the analysis of system's dynamics is presented. The optimization of the bioprocess is carried out, in order to achieve maximal biomass productivity. It is shown that during the optimization, the bioprocess with impulse effects loses the possibility of synchronization and strives for a simple continuous bioprocess. The analytical results presented in the work are validated by numerical simulations for the Tessier kinetics model.

1. Introduction

Chemical and bioprocess engineering play an important role in the production of many chemical products. In particular, bioreactor engineering as a branch of chemical engineering and biotechnology is an active area of research on bioprocesses, including among others development, control, and commercialization of new technology [1]. The reaching of optimal results and the obtainment of maximal profits require modern control strategies based on mathematical models or artificial intelligence methods [2]. There are many advantages with the use of mathematical models, and one of which is the possibility of testing process stability [3]. According to different reactions and differential control technologies, many dynamic biochemical models in a chemostat have been established [412]. However, there are many factors which affect the growth and reproduction of microorganisms in the bioprocess. The key variable is the biomass concentration, because, among other things, it provides information concerning biomass productivity. For that reason, almost all mathematical growth models contain the biomass as an important variable [13]. Moreover, for aerobic microbes, the dissolved oxygen concentration (DOC) is an important growth factor. During the growth of microorganisms, the dissolved oxygen concentration depends on several factors, among them the biomass concentration, the concentration and type of substrate used, and the bioprocess conditions. Because a low level of dissolved oxygen concentration decreases the biomass yield and the specific growth rate, it is necessary to monitor and control the DOC level so that it stays within the appropriate range [14].

Since many biological phenomena such as bursting rhythm models in, for example, medicine, biology, pharmacokinetics, and frequency modulated systems exhibit impulsive effects [15], impulsive differential equations, which appear as a natural description of observed evolution phenomena of several real world problems, have been introduced in different kind of biological systems, for example, in population dynamics [1619] and in biochemical process [20, 21]. It should be pointed out that in these works, the authors were concerned about the fixed time impulse effects, which has the rationality in describing the biological phenomena. While in some cases, using the state-dependent impulse effects to describe the biological phenomena is more appropriate. As far as the state-dependent impulse effect is concerned, Tang and Chen [22] introduced a Lotka-Voterra model, which is constructed according to the practices of IPM. By using analytical method, it is shown that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given economic threshold. Further, the complete expression of period of the periodic solution is given. More researches on the applications of the state-dependent impulse effects can be found in [2336]. In these researches, the authors analyzed the proposed system's dynamic behavior (e.g., the existence and stability of period-1 solution and the existence of period-2 solution) by applying the Poincaré principle and Poincaré-Bendixson theorem of the impulsive differential equation. Recently, in microbioprocess engineering, the dynamic properties of the kinetic models with impulse effects characterized by the universal microorganism growth rate and two different kinds of biomass yield are also analyzed theoretically by Sun et al. [3739].

The objective of this work is to illuminate theoretical and practical aspects of the nonlinear analysis of a universal mathematical model of the biochemical reaction process. The paper is organized as follows. In Section 2, a universal mathematical model of the biochemical reaction process with any characteristics of growth kinetics is formulated under impulse effects. In Section 3, a qualitative analysis of the proposed model is presented. In Section 4, simulations with Tessier's kinetics are presented in order to verify the theoretical results and discuss the biological essence. Moreover, in Section 4, in order to optimize the biochemical reaction process, the conception of objective function is presented, and next the bioprocess optimization is put forward. Finally, in Section 5, we offer our conclusions.

2. Mathematical Model and Preliminaries

The general model of continuously culturing microorganism in a chemostat is given by the following form of differential equations [40]: where denotes the biomass concentration , and the substrate concentration in the bioreactor medium at time , , and denotes the initial biomass concentration and substrate concentration, respectively; is the dilution rate (); is the influent substrate concentration ; is the biomass yield defined as the ratio of the biomass produced to the amount of substrate consumed; for biological constraints; the function describes the biochemical kinetics, which are characterized by the cell concentration , depending on one limiting substrate with concentration .

Crooke et al. [6] pointed out that the biochemical kinetics expression plays an important role in the intrinsic oscillation mechanisms. So many different assumptions for the kinetics models are given in the literature, for example, Monod-type kinetics [3], that is, , where is the maximum specific growth rate and is the substrate saturation constant; Tessier-type kinetics [41], that is, ; Moser-type kinetics [41], that is, , where is a positive constant. It can be easily seen that these kinetic models satisfy the following properties: (i)(regularity) : is continuously differential and , (ii)(monotonicity) is monotonically increasing, that is, for all ; for some means that for , (iii)(convexity) for all and .

Thus, in the study that the kinetic models only satisfy the regularity, monotonicity, and convexity are considered.

According to the Herbert's and Pirt's models [41], if the substrate concentration is big enough (i.e., ), the biomass yield is constant. Because in continuous bioprocesses the optimal biomass productivity is obtained for bigger substrate concentration, in this work the constant biomass yield is assumed, that is, . In addition, according to the design ideas of the bioreactor, the biomass concentration in the chemostat should be controlled to a set level , where , and is the critical level of biomass concentration in the bioreactor medium. When the biomass concentration reaches the set level, part of the medium containing biomass and substrate is discharged from the bioreactor, followed by the input of the “clear” medium (i.e., the medium without substrate). Therefore, system (1) can be modified by introducing the impulsive state feedback control as follows: where is the part of “clear” medium inputted into the bioreactor in each biomass oscillation cycle, which satisfies , where is the maximum part of “clear” medium inputted into the bioreactor in each biomass oscillation cycle.

Before presenting the main results, we recall the following definitions and lemmas first [31, 42, 43].

Definition 1. is said to be period-1 solution if in a minimum cycle time there is one impulse effect. Similarly, is said to be period-2 solution if in a minimum cycle time there are two impulse effects.

Definition 2. is said to be orbitally stable, if for any , there exists , with the proviso that every solution of system (2) whose distance from is less than at , will remain within a distance less than from for all . Such a is said to be orbitally asymptotically stable if, in addition, the distance of from tends to zero as . Moreover, if there exist positive constants , and a real constant such that for , then is said to be orbitally asymptotically stable and enjoys the property of asymptotic phase.

Definition 3. Lambert function is defined to be a multivalued inverse of the function satisfying

In the followings, we simply denote . Then, it follows from (3) that when and . First of all, the function has the positive derivative if . Define the inverse function of restricted on the interval to be . Similarly, we define the inverse function of restricted on the interval to be . For this study, both and will be employed only for due to our practical problem (see Figure 1). For more details of the concepts and properties of the Lambert function, see Corless et al. [43].

105765.fig.001
Figure 1: The property of Lambert function.

Lemma 4. Assume that there exists a bounded closed region , as shown in Figure 2. The boundaries and are the no cut-arcs of model (5), and on and , the direction of the orientation field determined by model (5) is pointing to the interior of . In addition, there is no singularity in the interior of , and the boundary. One of the boundary is the impulse set of the model (5); the corresponding phase set satisfies ; is also the no cut-arcs of model (5), and on , the direction of the orientation field determined by model (5) is pointing to the interior of . Then, there must exist at least one period-1 solution of model (5) in region .

105765.fig.002
Figure 2: Illustration of Bendixson field.

Lemma 5 (analogue of Poincaré’ criterion). The -periodic solution , of system is orbitally asymptotically stable and enjoying the property of asymptotic phase if the multiplier satisfies the condition and unstable if , where , , and are calculated at the point .

3. Properties Analysis of the Bioprocess

We focus our discussion on the case , , and .

3.1. Existence of Period-1 Solution

The line intersects the isoclinal line at the point , intersects the curve at the point , and intersects the line at the point , where satisfies that The line intersects the line at the point and intersects the line at the point . Denote and , where and . The impulsive set , and the phase set . Let the point be the primary image of , and denote the trajectories of the system starting from the points and that intersect the segment at and respectively. Denote the trajectory of the system between and by , the trajectory of the system between and by .

Theorem 6. If one of the conditions holds (i)   and ; (ii) If , and ; (iii) If , and , then system (2) has a period-1 solution with .

Proof. Firstly, it is obvious that all trajectories of system (2) starting from the region must interest with the segment and then jump to the segment if and . Next, we construct the closed region such that all solutions of system (2) starting from enter into and retain there.
Case (i) (). It is obvious that the straight line is described by , and is described by , and both lines pass through the point . The derivative of the straight line passing through the points and along the trajectories of system (2) is in terms of , for the segment . From the qualitative characteristic of system (2), we know that for and for . Besides, for . On the other hand, it follows from system (2) that and for . Especially, when , is the semitrivial solution of system (2). Therefore, we have found a closed region , the boundary of which consists of , , , , , and ; see Figure 3(a).
Case (ii) ( and ). Similar to the discussion of Case (i), the boundary of can be constructed by , , , and ; see Figure 3(b).
Case (iii) ( and ). In this case, the boundary of can be constructed by , , , , and ; see Figure 3(c).
In addition, there is no singularity in the region . Therefore, it follows from the qualitative characteristics of the system that all the trajectories satisfying the conditions of theorem enter the closed region and retain there. Then it follows from Lemma 4 that system (2) has a period-1 solution.

fig3
Figure 3: Illustration of system (2) when and : (a) ; (b) and ; (c) and .
3.2. Position of Period-1 Solution

Suppose the period of the period-1 solution is . Let ,   be such a -periodic solution. Then is also -periodic. Denote ,   ,  , and . Then from the -periodicity of the periodic solution and the third and fourth equations of system (2), we have ,  , and . Without loss of generality, for , satisfies the relation Then, we have In particular, for , we have .

By (10), if , then we have for all , else we have for all .

Let be the root of the following equation: Then, we have the following results.

Theorem 7. Suppose that and . Then the initial values of period-1 solution of system (2) satisfies and , where and are determined by (7) and (11), respectively.

Proof. Since . Then, , which implies that that is, .
Next, we consider the following comparison system of system (2) without impulsive feedback control: It is easy to verify that system (13) has one positive equilibrium , which is a center point. Then all of its solutions are closed trajectories which satisfy where is an arbitrary constant. The derivative of along the trajectories of system (2) is for . This implies that the trajectories of system (2) intersecting with the trajectories of system (13) pass through the trajectories of system (13) from the left to the right. In addition, the trajectory of system (13) passing through (i.e., in (14)) intersects the line at two points and , where satisfy that that is, Since , then we have . Thus, we have and , where denotes Lambert function defined in Definition 3.
If , then the closed region constructed in Theorem 6 can be reduced to , the boundary of which consists of , , , , , , and . Thus, the period-1 solution satisfies that . Else, we have . Then, the trajectory of system (13) passing through intersects the segment at the point . In this case, the closed region constructed in Theorem 6 can be reduced to , the boundary of which consists of , , , , , , and . Thus, the period-1 solution satisfies that .

Corollary 8. Suppose that and . Then, the initial values of the period-1 solution of system (2) satisfies that and .

Corollary 9. Suppose that and . Then the initial values of the period-1 solution of system (2) satisfies that and .

Corollary 10. Suppose that , and . Then, the initial values of the period-1 solution of system (2) satisfies that and .

3.3. Stability of Period-1 Solution

Let . Denote where , and denotes Lambert function defined in Definition 3. Then, we have the following result.

Theorem 11. Suppose that , . Then, the period-1 solution of system (2) is orbitally asymptotically stable and enjoys the property of asymptotic phase.

Proof. Let . Denote where , and denotes Lambert function defined in Definition 3. According to Lemma 5, we calculate the multiplier of system (2) in variations corresponding to the -periodic solution . Denote , , where , , , and . In system (2), since , , , , and , then Therefore, Note that , that is, . If , that is, the periodic solution lies in the region, where , then we have , and the periodic solution () is orbitally asymptotically stable and enjoying the property of asymptotic phase. But for , that is, , we have . Since the expression of is hard to be given, we cannot determine whether the periodic solution is stable or not. However, we can give some sufficient conditions to guarantee when , that is, the periodic solution is orbitally asymptotically stable and enjoying the property of asymptotic phase.
Since , then we have , that is, , which implies that . If , then we have , which is equivalent to or . Thus, we get , that is, the periodic solution is orbitally asymptotically stable and enjoying the property of asymptotic phase. In the following, we will give the conditions under which .
From the expression of the isoclinal line , it follows that for , we have .
Consider the following comparison system of system (2) without impulsive feedback control It is easy to verify that system (23) has one positive equilibrium , which is a center point. Then, all of its solutions are closed trajectories which satisfy where is an arbitrary constant. The derivative of along the trajectories of system (2) is for . This implies that the trajectories of system (2) intersecting with the trajectories of system (23) pass through the trajectories of system (23) from the left to the right. In addition, the trajectory of system (23) passing through (i.e., in (24)) intersects the line at two points and , where satisfy thatthat is, Since , then we have . Thus, by Definition 3, we have and .
Next, we will prove that when the conditions of theorem are satisfied. Let be the intersection point of the line and , , and let be the intersection points of the line and the trajectory of system (23) passing through the point . If , then we have . Since , , , , , and is semi-trivial solution of system (2). Therefore, we get a closed region , the boundary of which consists of , , , , , , and . In addition, there is no singularity in the region . Therefore, it follows from the qualitative characteristics of the system that all the trajectories satisfying the conditions of theorem enter the closed region and retain there. Then, it follows from Theorem 6 that system (2) has a period-1 solution in . Furthermore, we have . If , then , in this case, we have . Else , in this case, we have . Therefore, if the conditions of the theorem are satisfied, then the period-1 solution of system (2) is orbitally asymptotically stable and enjoying the property of asymptotic phase.

3.4. Discussion on the Period- () Solution

For the predator-prey model concerning IPM strategies, Tang and Cheke [23] proved that there is no periodic solution with order larger than or equal to three, except for one special case, by using the properties of the Lambert function and Poincaré map. Moreover, They showed that the existence of an order two periodic solution implies the existence of an order one periodic solution. Next, the existence of the period- () solution is discussed.

Theorem 12. Suppose that and . Then, system (2) does not have period- () solution.

Proof. Let be an arbitrary solution of system (2). Denote the first intersection point of the trajectory to the impulsive set by (here without loss of generality, we assume that ), and the corresponding consecutive points are , , , respectively. Consequently, under the effect of impulsive function , the corresponding points after pulse are . By the qualitative analysis of system (2), we know that for and for . If there exists the th impulse effect such that for , then the sequence will be monotone, that is, or . In this case, system (2) has no period- solution. Let be the line passing the point with the slope : Since , then the trajectory starting from the point on the segment either does not intersect or retain on before reaching the segment . Therefore, when , the period-1 solution determined in Theorem 6 satisfies that , which also implies that all .
The tr
ajectory of the solution jumps to the point from the point . By the dynamics of system (2), we have the following two sequences according to lies on the right or left of : (a)  ; (b)  . Since the subsequences and    are monotone and bounded, then there exists a limit for them, respectively. Let and . When sequence (a) occurs, it can be shown that . Therefore, system (2) has a period-2 solution. When sequence (b) occurs, we have . If , then system (2) has a period-2 solution. Else we have , in this case, the trajectory tends to the period-1 solution . But it can be obtained that there is no order solution in system (2), thus the system is not chaos by Li-Yorke chaos Theorem [44].

Remark 13. From the proof of Theorem 12, it can be seen that there is a possibility to exist a period-2 solution. Since it is difficult to obtain the exact expression of the solution to (2), then it will be challenging to give the sufficient conditions under which system (2) has a period-2 solution or not.

4. Applied Instance

Here, we will take the Tessier kinetics model [41], that is, , as the example of specific growth rate to verify the theoretical results. In addition, , , and are used for the demonstration of system behavior.

4.1. Numerical Simulations

The simulations are carried out by changing one main parameter and fixing all other parameters. The following numerical simulations are given by the programs of Maple and Matlab softwares. The time interval is set two days, that is, .

We assume in the following that and . The vector graph of system (1) for and are shown in Figure 4, from which it can be seen that the equilibrium is a stable node.

105765.fig.004
Figure 4: Illustration of vector graph of system (1) when , , , , and .

The time series and phase portrait for is shown in Figure 5, it can be seen that no impulse occurs when .

fig5
Figure 5: The time series and phase portrait when , and .

Figure 6 gives the time series and phase portrait when and . It can be seen that the trajectory tends to be periodic.

fig6
Figure 6: The time series and phase portrait when , , , and .

Figure 7 shows the different positions of the period-1 solution for and .

fig7
Figure 7: Dependence of the biomass and substrate concentration on . The phase portraits when , , , and .

Figure 8 shows the different positions of the period-1 solution for , , and .

fig8
Figure 8: Dependence of the biomass and substrate concentration on . The phase portraits when , , , and .

The numerical simulations are consistent with the theoretical results obtained and presented in Section 3.

4.2. Optimization of the Bioprocess

Next, we discuss aspects of the bioprocess optimization.

4.2.1. The Proposed Objective Function and the Constraints

The Objective Function where is the biomass productivity in the steady state.

The Constraints (a)—the part of “clear” medium inputted into the bioreactor in each biomass oscillation cycle, —the maximum part of “clear” medium inputted into the bioreactor in each biomass oscillation cycle. (b): —the concentration of the inputted substrate, —the minimum concentration of the inputted substrate, —the critical level of the dosaged substrate concentration. (c): —the dilution rate of the chemostat, —the minimum dilution rate. If the used microorganisms growth kinetics model satisfies regularity and monotonicity, then for there is where , —the maximum specific growth rate, —the substrate saturation constant.(d): —the set level of the biomass concentration in the bioreactor medium, —the critical level of biomass concentration in the bioreactor medium.

The aim of the optimization is to find the maximum of the objective function (30) under constraints (a)–(d), that is,

4.2.2. The Analysis of the Optimization

(A) Case of a Chemostat with Impulse Effect. For and , a continuous bioprocess (a chemostat with impulse effect) is obtained, that is, In the steady state, that is, for the period-1 solution , by the second equation of system (33), we have Then, travelling along from the point , with to the point , with in the counterclockwise direction yields the period , that is, It is obviouse that when . In addition, by (34), for , we have , thus, Therefore, the objective function can be formulated as For given and , we have where and .

(B) Case of Synchronization Loss. In this case the optimization of the function (37) for any value of is presented. We should find the maximum of the objective function (37) under the constraints , , , and .

Let . Take the first derivative of with respect to , we have Let . Then, we have . Take the first derivative of with respect to , we have then , which implies that , that is, is decreasing on .

Therefore, for given , we have where .

5. Conclusions

In the paper the dynamic behavior of a universal biochemical reaction process with feedback control was analyzed, and it was shown that the stability of the bioprocess (i.e., the existence of the positive period-1 solution), depended on both the biomass yield and the microorganism growth rate. Furthermore, the conditions for the existence and stability of the system's period-1 solution were obtained (i.e., Theorems 6 and 11). It also pointed out that the system (2) was not chaotic according to an analysis of period-2 solution existence. According to the theoretical results, the production of the microorganisms tended to be periodic. The microorganisms in the considered chemostat always kept the suitable growth rate. The biomass concentration could have been controlled to a given level between the critical and the optimal dissolved oxygen concentration. Moderation of the main bioprocess parameters in the selected biomass oscillation cycles for the Tessier kinetics model was presented.

After the analysis of the system stability, bioprocess optimization was covered in the work. In the first case, the optimization of a simple continuous bioprocess (i.e., bioprocess in which and ) was presented. In the second case of optimization, it was demonstrated that theoretically it was possible to obtain almost optimal biomass productivity without lost of synchronization of simultaneously performed bioprocesses (see (38) for small values of ). This was possible, when the substrate was dosed by both continuous (i.e., ) and pulse (i.e., ) method. In this case, the impulsive chemostat was obtained, where the biomass productivity mainly depended on , and the synchronization of simultaneously performed bioprocesses depended on . The third case of optimization showed that during the optimization, the impulsive chemostat lost possibility of synchronization and strived for a simple continuous bioprocess (see (41)). Moreover, it was shown that improving the biomass productivity and synchronization of simultaneously performed bioprocesses were conflict criteria.

Acknowledgment

This research is supported in part by National Natural Science Foundation of China (Grant no. 11101066) and the Fundamental Research Funds for the Central Universities.

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