A Less Conservative Stability Criterion for Delayed Stochastic Genetic Regulatory Networks

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