Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 768483, 11 pages

http://dx.doi.org/10.1155/2014/768483

Research Article

A Less Conservative Stability Criterion for Delayed Stochastic Genetic Regulatory Networks

Tingting Yu, Jing Wang, and Xian Zhang

School of Mathematical Science, Heilongjiang University, Harbin 150080, China

Received 18 January 2014; Accepted 3 March 2014; Published 8 April 2014

Academic Editor: Yuxin Zhao

Copyright © 2014 Tingting Yu 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.

This paper concerns the problem of stability analysis for delayed stochastic genetic regulatory networks. By introducing an appropriate Lyapunov-Krasovskii functional and employing delay-range partition approach, a new stability criterion is given to ensure the mean square stability of genetic regulatory networks with time-varying delays and stochastic disturbances. The stability criterion is given in the form of linear matrix inequalities, which can be easily tested by the LMI Toolbox of MATLAB. Moreover, it is theoretically shown that the obtained stability criterion is less conservative than the one in W. Zhang et al., 2012. Finally, a numerical example is presented to illustrate our theory.

1. Introduction

With the further progress of gene expression, researchers find that a gene expression is affected by other genes; conversely, it also influences others. Based on this reciprocal impact relation, gene expression forms a complex network—genetic regulatory network (GRN). GRNs are dynamical systems, which consist of an interaction of genes, proteins, and small molecules. In the past two decades, scholars have established mathematical models to represent GRNs. Basically, there are four types of GRN models, that is, Petri net model [1], Bayesian network model [2], Boolean model [3, 4], and (functional) differential equation model [5, 6]. The concentrations of mRNA and protein are described as the state variables in the functional differential equation model.

As dynamical systems, stability analysis is the first priority to explore GRNs. On the one hand, time delay inevitably occurs in GRNs due to the slow process of transcription, translation, and translocation [7]. On the other hand, internal noises of cells caused by random birth and death of the individual molecules and external noises from environmental fluctuations make the gene expression be best viewed as a stochastic process [8, 9]. So, it is very necessary to analyze the stability of GRNs with time-varying delays and stochastic disturbances [6, 10–18].

Recently, for a class of GRNs with interval time-varying delays and stochastic disturbances (see (17a), (17b) below), Wu et al. [18] established several delay-range-dependent and/or rate-dependent global stochastic asymptotical stability criteria in terms of linear matrix inequalities (LMIs) by using the stochastic analysis approach, employing some free-weighting matrices and introducing a type of Lyapunov-Krasovskii functional which includes the items like and , where and are real symmetric positive definite matrices. Furthermore, in [18], the item in the stochastic differential of was first estimated by employing Leibniz-Newton formula and the inequality

where is a free-weighting matrix, and then was enlarged to . Clearly, the conservatism would be produced because is enlarged to . In order to reduce the conservatism, Zhang et al. [17] first estimated the item by using Leibniz-Newton formula and the inequality

where and are free-weighting matrices and is an adjusting parameter with , and then a so-called convex combination technique was employed to obtain less conservative delay-range-dependent and/or rate-dependent global stochastic asymptotical stability criteria. It should be emphasized that the Lyapunov-Krasovskii functional used in [17] includes not only the items like and , but also the items like and involved in the adjustable parameter , which can be viewed as a generalized delay-partition approach due to the adjustable parameter . Generally, an appropriate delay-partition approach can bring less conservative stability criteria (see [19] and the references therein).

Note that free-weighting matrices (e.g., matrices , , and above) have been employed in both [17, 18] to obtain less conservative stability criteria. However, it is emphasized in [18, Remark  2] that free-weighting matrices may produce a super-high amount of computation for the feasible solutions of LMIs. In order to overcome the disadvantage, in this paper we will propose a delay-range partition (DRP) approach to estimate accurately the item (see (27) and (30)), where no free-weighting matrix is involved. By employing an appropriate Lyapunov-Krasovskii functional and introducing a DRP approach, a mean square stability criterion for GRNs with time-varying delays and stochastic disturbances is first established. Then it is theoretically shown that the proposed stability criterion is less conservative than [17, Theorem  1]. Finally, a numerical example is given to illustrate the theoretical results proposed here. The main contribution of this paper can be listed as follows: (i) the Lyapunov-Krasovskii functional employed in this paper does not include the items like which are required in [17, 18]; (ii) the items like in the stochastic differential of Lyapunov-Krasovskii functional are estimated accurately by proposing a DRP approach; (iii) theoretical comparison of the stability criterion [17, Theorem  1] and the one proposed in this paper is given; and (iv) there is no free-weighting matrix involved, which reduces the computational complexity.

The rest of the paper is organized as follows. In Section 2, the model of GRNs to be studied is described. A DRP-based mean square stability criterion (Theorem 3 below) for GRNs with time-varying delays and stochastic disturbances is established in Section 3. The theoretical comparison of Theorem 3 and [17, Theorem  1] is presented in Section 4. In Section 5, an example is given to show the validity of the obtained results. Finally, in Section 6, the conclusions are drawn.

Notation. For a positive integer , set . denotes the -dimensional Euclidean space. We denote by the set of matrices over . and represent the transpose and inverse of a matrix , respectively. For real symmetric matrices and , the notation means that the matrix is positive semidefinite (positive definite). is the identity matrix. In a symmetric matrix, denotes the entries implied by symmetry.

2. Model Description

The following differential equations have been used recently to describe GRNs [7]:

where and are the concentrations of the th mRNA and protein at time , respectively; , , and are constants, representing the degradation rate of the th mRNA, the degradation rate of the th protein, and the translation rate of the th mRNA to th protein, respectively; both and are transcriptional and translational delays, respectively; is the regulatory function of the th gene, which is generally a nonlinear function of the variables , but it is monotonic with each variable.

For convenience, we give the following assumptions throughout the paper.

Assumption 1. The delays and are differentiable functions satisfying

where , , , , , and are constants.

Assumption 2. The function is taken as

which is called SUM logic. Here, is a monotonic function of the Hill form; that is,

where is the Hill coefficient, is a scalar, and is a bounded constant, which denotes the dimensionless transcriptional rate of transcription factor to gene .

Clearly, GRN ((3a), (3b)) can be rewritten as

where

and is the set of all the transcription factors which is a repressor of gene .

Rewriting GRN ((7a), (7b)) into compact matrix form, we obtain

where

Let be an equilibrium point of ((9a), (9b)); that is, it is a solution of the following equation:

For convenience, we shift the equilibrium point to the origin by using the transformations and ; then we have

where .

From the relationship between and , one can easily find that satisfies the following sector condition:

where and are a pair of nonnegative scalars and is the th entry of . Since is a monotonically increasing and differentiable function with saturation, we have to choose as zero or a small positive number. Let and .

As shown in [15–17] the gene regulation is an intrinsically noisy process. For this reason, in this paper, we consider a class of GRNs with both time delays and noise disturbances by the following model:

where is an -dimensional Brown motion, , and is the noise intensity matrix at time such that

where are real symmetric positive semidefinite matrices.

For simplicity, set

Then, GRN ((14a), (14b)) can be represented as

3. Stability Criterion

In the following theorem, we will propose a DRP approach to present an asymptotical stability criterion in the mean square sense for GRNs with time-varying delays and stochastic disturbance.

Theorem 3. For given scalars , , , and , and positive integers and , under the conditions (15) and ((4a), (4b)), we can conclude that GRN ((14a), (14b)) is asymptotically stable in the sense of mean square, if there exist a scalar and matrices , , (), and such that the following LMIs hold:

where

Proof. Let and . Choose a Lyapunov-Krasovskii functional candidate as

where

and the matrices (), , and () are taken from a feasible solution to (18) and (19). By Itô’s formula, we can obtain the following stochastic differential:

where is the weak infinitesimal operator and

For any scalars , with , it follows from (16) that and , and hence

where represents the mathematical expectation operator. Next, from the sector condition (13), we can obtain that

When for some positive integer and for some positive integer , it is easy to see that

Then, the combination of (21)–(31) gives

where

Due to (19), we have , and hence GRN ((14a), (14b)) is asymptotically stable in the mean square sense.

Remark 4. In the above theorem a DRP approach has been proposed to establish an asymptotic mean square stability criterion for GRN ((14a), (14b)). Both DRP approach and the so-called piecewise analysis method (see, e.g., [20]) divide the delay-varying intervals into some parts with equal length. Then DRP approach enlarges the expectation of weak infinitesimal operator of the same Lyapunov-Krasovskii functional in every subinterval, while the piecewise analysis method constructs different Lyapunov-Krasovskii functional in every subinterval.

Remark 5. Comparing with the Lyapunov-Krasovskii functionals employed in Theorem 3 and [17, Theorem  1], we remove the items and , which is required in [17]. This will reduce the number of LMI variables to be solved, and hence Theorem 3 requires less computer time than [17, Theorem 1]. Furthermore, it will be shown in the next section that Theorem 3 is certainly less conservative than [17, Theorem  1].

4. Theoretical Comparisons

In this part we will offer a theoretical comparison on conservativeness of Theorem 3 and [17, Theorem  1]. For this reason, we introduce [17, Theorem  1] as follows.

Lemma 6 (see [17, Theorem  1]). When and , GRN ((14a), (14b)) subject to ((4a), (4b)) is asymptotically stable in the mean square sense, if there exist positive definite matrices , , , and , a diagonal positive matrix , matrices , , , , , , , and ) of appropriate sizes, and positive scalars , , and such that the following LMIs hold:

where

In order to show that Theorem 3 is less conservative than [17, Theorem 1], the following propositions are required.

Proposition 7. Let , , (), and be given real matrices of appropriate sizes. For given scalars and , set

If

then there exists a (sufficiently large) positive integer such that

Proof. It follows from (37) and the Schur complementary lemma [21] that

and hence there exists a (sufficiently large) positive integer such that

For an arbitrary but fixed , one can derive from (36) and (40) that

Since

we obtain from (41) that (38) holds. The proof is completed.

Proposition 8. Let , , ,   , (), , and be given real matrices of appropriate sizes and () and given scalars. Set

If

then there exists a pair of (sufficiently large) positive integers and such that

Proof. It follows from (44) that

and hence

that is,

where

Applying Proposition 7 to , , , , and , we obtain that

for some (sufficiently large) positive integer .

By the Schur complementary lemma, one can easily derive from (50) that

where

Again applying Proposition 7 to , , , , and , one can complete the proof.

Proposition 9. Let and satisfying . Then

Proof. One has

Now it is time to show that Theorem 3 is less conservative than [17, Theorem  1] in theory.

Theorem 10. Set and . If the LMIs in (34) are feasible, then the LMIs in (18) and (19) are feasible.

Proof. Set , , , , , , , and (). Then it follows from (34) and the Schur complementary lemma that (18) holds and

where

and , and are defined as noted previously.

By simple computation one can derive from (55) that

where

and are defined as in (19).

By Proposition 8, there exists a pair of (sufficiently large) positive integers and such that

where and are defined as in (19). This, together with Proposition 9, implies that

and hence the LMIs in (19) are feasible. The proof is completed.

Remark 11. In Theorem 10, it has been theoretically investigated that Theorem 3 is certainly less conservative than [17, Theorem  1]. On the other hand, the numbers of LMI variables to be solved in Theorem 3 and [17, Theorem  1] are and , respectively, which implies that Theorem 3 will require less computer time than [17, Theorem  1].

5. An Illustrative Example

In this section, a numerical example is given to illustrate the effectiveness and less conservativeness of our theoretical results.

We consider a delayed GRN with stochastic disturbances, with the parameters described as

Let and , which means that and . When , for , , , and , by using the MATLAB Toolbox, inequalities (18) and (19) have feasible solutions. By the theorem, the system is asymptotically stable in the mean square sense. Here, some of the solution matrices are given as follows:

and show the trajectories of the and in Figure 1.

Figure 1: State trajectories of the GRN in the considered example.

When and , the maximal allowable upper bounds of for different values of obtained by Theorem 3 and [17, Theorem  1] are shown in Table 1. It can be seen from Table 1 that Theorem 3 is the less conservative than [17, Theorem  1].

Table 1: The maximal allowable bounds of for different .

0.8
1
1.5

[17, Theorem 1]
1.5602
1.5500
1.5450

Theorem 3
2.095
2.053
2.047

6. Conclusions

In this paper, the stability problem for a class of GRNs with time-varying delays and stochastic disturbances has been investigated. By constructing an appropriate Lyapunov-Krasovskii functional and proposing a DRP approach, a mean square stability criterion is given in terms of LMIs, which can be easily tested by the LMI Toolbox of MATLAB. It is theoretically shown that the proposed result is less conservative than [17, Theorem  1]. Moreover, the number of LMI variables in this paper is more less than the one in [17, Theorem  1]. A numerical example has been provided to illustrate the theoretical results given in this paper.

Extending the idea of this paper to other system models, including singular delayed systems [19, 22–24], stochastic systems [25], Markovian jump systems [20, 26], and genetic regulatory networks [5, 10, 27], is under consideration.

Conflict of Interests

The authors declare that there is no commercial or associative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (11371006), the fund of Heilongjiang Province Innovation Team Support Plan (2012TD007), the fund of Heilongjiang University Innovation Team Support Plan (Hdtd2010-03), the fund of Key Laboratory of Electronics Engineering, College of Heilongjiang Province (Heilongjiang University), China, the Heilongjiang University Innovation Fund for Graduates, and the 2014 Scientific and Technological Research Programs of Education Commission of Heilongjiang Province. The authors thank the anonymous referees for their helpful comments and suggestions which improve greatly this paper.

References

L. J. Steggles, R. Banks, O. Shaw, and A. Wipat, “Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach,” Bioinformatics, vol. 23, no. 3, pp. 336–343, 2007.

N. Friedman, M. Linial, I. Nachman, and D. Pe'er, “Using Bayesian networks to analyze expression data,” Journal of Computational Biology, vol. 7, no. 3-4, pp. 601–620, 2000.

A. Graudenzi, R. Serra, M. Villani, C. Damiani, A. Colacci, and S. A. Kauffman, “Dynamical properties of a Boolean model of gene regulatory network with memory,” Journal of Computational Biology, vol. 18, no. 10, pp. 1291–1303, 2011.

S. Martin, Z. Zhang, A. Martino, and J.-L. Faulon, “Boolean dynamics of genetic regulatory networks inferred from microarray time series data,” Bioinformatics, vol. 23, no. 7, pp. 866–874, 2007.

X. Zhang, A. H. Yu, and G. D. Zhang, “M-matrix-based delay-range-dependent global asymptotical stability criterion for genetic regulatory networks with time-varying delays,” Neurocomputing, vol. 113, pp. 8–15, 2013.

F. Ren and J. Cao, “Asymptotic and robust stability of genetic regulatory networks with time-varying delays,” Neurocomputing, vol. 71, no. 4–6, pp. 834–842, 2008.

L. Chen and K. Aihara, “Stability of genetic regulatory networks with time delay,” IEEE Transactions on Circuits and Systems. I: Fundamental Theory and Applications, vol. 49, no. 5, pp. 602–608, 2002.

T. Tian, K. Burrage, P. M. Burrage, and M. Carletti, “Stochastic delay differential equations for genetic regulatory networks,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 696–707, 2007.

Z. Wang, X. Liu, Y. Liu, J. Liang, and V. Vinciotti, “An extended kalman filtering approach to modeling nonlinear dynamic gene regulatory networks via short gene expression time series,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 6, no. 3, pp. 410–419, 2009.

Y. T. Wang, A. H. Yu, and X. Zhang, “Robust stability of stochastic genetic regulatory networks with time-varying delays: a delay fractioning approach,” Neural Computing and Applications, vol. 23, no. 5, pp. 1217–1227, 2013.

C. Li, L. Chen, and K. Aihara, “Stability of genetic networks with SUM regulatory logic: Lur'e system and LMI approach,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 53, no. 11, pp. 2451–2458, 2006.

Q. Zhou, S. Xu, B. Chen, H. Li, and Y. Chu, “Stability analysis of delayed genetic regulatory networks with stochastic disturbances,” Physics Letters A: General, Atomic and Solid State Physics, vol. 373, no. 41, pp. 3715–3723, 2009.

Z. Wang, X. Liao, J. Mao, and G. Liu, “Robust stability of stochastic genetic regulatory networks with discrete and distributed delays,” Soft Computing, vol. 13, no. 12, pp. 1199–1208, 2009.

Z. Wang, G. Liu, Y. Sun, and H. Wu, “Robust stability of stochastic delayed genetic regulatory networks,” Cognitive Neurodynamics, vol. 3, no. 3, pp. 271–280, 2009.

H. Wu, X. Liao, S. Guo, W. Feng, and Z. Wang, “Stochastic stability for uncertain genetic regulatory networks with interval time-varying delays,” Neurocomputing, vol. 72, no. 13–15, pp. 3263–3276, 2009.

Y. He, L.-Y. Fu, J. Zeng, and M. Wu, “Stability of genetic regulatory networks with interval time-varying delays and stochastic perturbation,” Asian Journal of Control, vol. 13, no. 5, pp. 625–634, 2011.

W. Zhang, J.-A. Fang, and Y. Tang, “Robust stability for genetic regulatory networks with linear fractional uncertainties,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1753–1765, 2012.

H. Wu, X. Liao, S. Guo, W. Feng, and Z. Wang, “Stochastic stability for uncertain genetic regulatory networks with interval time-varying delays,” Neurocomputing, vol. 72, no. 13–15, pp. 3263–3276, 2009.

X. Zhang and H. Zhu, “Robust stability and stabilization criteria for discrete singular time-delay LPV systems,” Asian Journal of Control, vol. 14, no. 4, pp. 1084–1094, 2012.

F. B. Li, L. G. Wu, and P. Shi, “Stochastic stability of semi-Markovian jump systems with mode-dependent delays,” International Journal of Robust and Nonlinear Control, 2013.

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994.

H. Zhu, X. Zhang, and S. Cui, “Further results on control for discrete-time uncertain singular systems with interval time-varying delays in state and input,” Optimal Control Applications & Methods, vol. 34, no. 3, pp. 328–347, 2013.

F. Li and X. Zhang, “A delay-dependent bounded real lemma for singular LPV systems with time-variant delay,” International Journal of Robust and Nonlinear Control, vol. 22, no. 5, pp. 559–574, 2012.

F. B. Li, P. Shi, L. G. Wu, and X. Zhang, “Fuzzy-model-based D-stability and non-fragile control for discrete-time descriptor systems with multiple delays,” IEEE Transactions on Fuzzy Systems, 2013.

L. G. Wu, W. X. Zheng, and H. J. Gao, “Dissipativity-based sliding mode control of switched stochastic systems,” IEEE Transactions on Automatic Control, vol. 58, no. 3, pp. 785–791, 2013.

L. Wu, X. Su, and P. Shi, “Sliding mode control with bounded gain performance of Markovian jump singular time-delay systems,” Automatica, vol. 48, no. 8, pp. 1929–1933, 2012.

L. L. Chen, Y. Zhou, and X. Zhang, “Guaranteed cost control for uncertain genetic regulatory networks with interval time-varying delays,” Neurocomputing, vol. 131, pp. 105–112, 2014.