Abstract
This paper is concerned with the stability problem for a class of uncertain impulsive stochastic genetic regulatory networks (UISGRNs) with time-varying delays both in the leakage term and in the regulator function. By constructing a suitable Lyapunov-Krasovskii functional which uses the information on the lower bound of the delay sufficiently, a delay-dependent stability criterion is derived for the proposed UISGRNs model by using the free-weighting matrices method and convex combination technique. The conditions obtained here are expressed in terms of LMIs whose feasibility can be checked easily by MATLAB LMI control toolbox. In addition, three numerical examples are given to justify the obtained stability results.
1. Introduction
Genetic regulatory networks (GRNs) which govern many essential functions of living cells have received much attention due to their extensive applications in many practical systems, especially in the biology, engineering, and other research fields [1–6]. That is why GRNs have become a hot topic of research recently. Several computational models have been applied to investigate the behaviours of GRNs: Petri net models [7–9], Bayesian network models [10–12], the Boolean models [13–15], the differential equation models [16–18], and so forth. In this paper, we will use differential equation models to encode genetic regulatory networks. The rate of change in concentration of a particular transcript is given by an influence function of other RNA concentrations.
Time delay is an interesting feature of signal transmission and becomes one of the main sources for causing divergence, instability, and poor performances for networks stability. So, it is important to consider the delay effects on the dynamical behavior of GRNs. Up to now, in almost all existing works on modeling GRNs [5, 19–21], time delay is included in the regulator function to describe the existing time delays peculiar to transcription, translation, and translocation processes in genetic networks. Chen and Aihara [5] firstly proposed a delay differential equation model for GRNs and studied its stability problem. In [19], Ren and Cao studied the asymptotic and robust stability of GRNs with time-varying delays. In [20], Zhang et al. investigated the stability analysis for GRNs with random discrete delays and distributed delays. Hu et al. [21] proposed a GRNs model with hybrid regulatory mechanism and studied its stability problem. Recently, Gopalsamy [22] put forward a neural network model with the incorporation of time delays in the leakage terms (i.e., negative feedback or decay terms which widely appeared in the models of neural networks, population dynamics, and GRNs). Along this line, a time delay will be taken into consideration in the decay terms of our GRNs model and we also call it “leakage delay.”
When modeling the GRNs, stochastic disturbance should be taken into consideration since molecular noise plays important roles in biological functions of GRNs in practice. In [23, 24], the authors studied the model of GRNs with stochastic disturbances. Moreover, impulsive effects are also likely to exist in the genetic networks systems [25]. In [26], Li and Sun researched the stability of GRNs under impulsive control. On the other hand, it is well known that the stability of well-designed GRNs may often be destroyed by its unavoidable uncertainty in practice. In [27, 28], the authors investigated the stability for uncertain GRNs with interval time-varying delays. In [29, 30], the authors researched the stability problem of GRNs with stochastic disturbance and parameter uncertainties, simultaneously. In [31], Sakthivel et al. dealt with the asymptotic stability of delayed GRNs with both stochastic disturbance and impulsive effects. However, so far there has been very little published concerning the stability problem for GRNs with leakage delay, impulsive effects, stochastic disturbances, and parameter uncertainties, simultaneously.
Motivated by the above discussion, the stability analysis for UISGRNs with time-varying delays in the leakage term requires further consideration. By constructing a suitable Lyapunov-Krasovskii functional which uses the information on the lower bound of all the delays, the derived conditions are expressed in terms of LMIs whose feasibility can be easily checked by using numerically efficient MATLAB LMI control toolbox. It is believed that the result is meaningful and useful for the design and applications of UISGRNs. Finally, numerical examples are provided to show the usefulness of the derived LMI-based stability conditions.
Notations. Throughout this paper, and denote, respectively, the -dimensional Euclidean space and the set of all real matrices. The superscript denotes the transposition and the notation (resp., ), where and are symmetric matrices, and it means that is positive semidefinite (resp., positive definite). denotes the diagonal matrix, and means a column vector. In symmetric block matrices, we use an asterisk (*) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
2. Problem Formulation and Preliminaries
In this paper, we consider the following model: where , are concentrations of mRNA and protein of the th node at time , respectively, and are positive real numbers that are the degradation rates of the mRNA and protein, is a positive constant that represents the translation rate, and is the regulatory function of the th gene. The first term in the first and third equations of the right side of (1) is called decay term and , , is called “leakage delay” as discussed in the Introduction. The regulatory function is of the form , which is called SUM logic [32]. The stochastic disturbance is one-dimensional Brownian motion defined on a complete probability space with a natural filtration and is the noise intensity.
The function is a monotonic function of the Hill form as follows: where is the Hill coefficient, is a positive constant, and is the dimensionless transcriptional rate of transcription factor to gene , which is a bounded constant. Therefore, (1) can be rewritten into the following form: where , is defined as a basal rate, and is the set of all the which is a repressor of gene . The matrix of the genetic network is defined as follows:
Rewriting system (3) into compact matrix form, we obtain where , , , , , , , , and .
Let be a nonnegative equilibrium point of the system (5). In the following, we will always shift the equilibrium point to the origin by letting , . Hence, system (5) can be transformed into the following form: where , , , , , the function , and obviously .
Due to the fact that is a monotonically increasing function with saturation, from the relationship of and , we know that, for any , where and are known constant scalars.
Taking parameter uncertainties into the GRNs model (6), we consider the following UISGRNs model: where and are the initial function which are continuously differentiable on with . We extend on to satisfy with , , where .
Moreover, the noise intensity satisfies where and are constant matrices with appropriate dimensions.
In order to obtain our main theorem, the following assumptions and lemmas for the system (8) are always made throughout this paper.
Assumption 1. The parametric uncertainties , , , and satisfy where , , , , , and are some given constant matrices with appropriate dimensions and satisfies , , for any .
Assumption 2. , , , and are the time-varying delays satisfying
Lemma 3 (Schur complement, see [30]). For a given matrix where and a vector function such that the intergrals concerned as well defined, then the following holds:(i), ,(ii), .
Lemma 4 (see [33]). For any constant symmetric matrix , scalar γ > 0,
Lemma 5 (see [29]). For any vectors and any positive matrix satisfying:
3. Main Result
In this section, mean square stability result for model (8) is summarized in the following theorem.
Theorem 6. If (7), (9), and Assumptions 1 and 2 hold, there exist , , , , , , and , such that the impulsive operator satisfies . The system (8) is stable in the mean square if there exist real matrices , , (), (), , and , diagonal matrices (), and any matrices , , , , , , , , , , , , , , , , , , , , , and to satisfy the following ten linear matrix inequalities: where
Proof. We consider the following Lyapunov functional candidate for system (8):
where
Then, by Itô’s differential formula, taking the derivative of along the trajectories of the system (8), we can obtain the following stochastic differential [29]:
where is the diffusion operator and
with
By Newton-Leibnitz formula, we have that
where
By using Lemmas 4 and 5, we have
It follows from (7) that
Next, it follows from (9) and (27) that
Add both sides of (34) and (38) to both sides of (32) and apply (36)–(37); one can obtain that
where , with .
Noting Assumption 2, can be seen as the convex combination of and on , and on , and on , and and on . Therefore, holds if
where
By Schur complements, (40) is equivalent to , , respectively. Then
On the other hand, from (27) and Theorem 6 conditions, we note that
Moreover, it is obvious that , . Hence, we get . By Lyapunov-Krasovskii stability theorem, the equilibrium point of (8) is stable in the mean square. The proof is completed.
Remark 7. For the UISGRNs (8) without stochastic disturbances, leakage delay, and impulsive effects, it reduces to the model of [27]. And when the model of (8) without impulsive, it reduces to the model of [30]. In addition, it is easy to see that the main theorem obtained above covers the sparse results available in the literature in the concern of only one or two of the complex dynamics generally being involved with GRNs, leakage delays, parameter uncertainties, impulsive effects, and stochastic disturbances.
We give a couple of corollaries below in order to show further that our main result is general enough to cover two cases that have not been investigated in the literature. Hence, they are new and significant. Firstly, for model (6) or the UISGRNs (8) without parameter uncertainties (i.e., ), we have the following corollary.
Corollary 8. If (7), (9), and Assumption 1 hold, there exist , , , , , , , , such that the impulsive operator satisfies . The system (6) is stable in the mean square if there exist real matrices , , (), (), , and , diagonal matrices (), and any matrices , , , , , , , , , , , , , , , , , , , , , and , to satisfy conditions (16)–(25) replaced accordingly by the following:
In the second case, we suppose that there are no stochastic disturbances in the UISGRNs model (8). Hence, model (8) can be reduced to
Then we have the following new result.
Corollary 9. If (7) and Assumptions 1 and 2 hold, there exist , , , , , such that the impulsive operator satisfies . The system (45) is stable if there exist real matrices , , (), (), , and , diagonal matrices (), and any matrices , , , , , , , , , , , , , , , , , , , , , and , to satisfy conditions (18)–(25) replaced accordingly by the following:
4. Numerical Examples
In this section, we present three numerical examples so as to illustrate the usefulness of our results derived in this paper.
Example 1. Let us firstly consider the system (8) with parameters as follows:
For convenience, we assume that , , , , , and ; then we can obtain
Using MATLAB LMI control toolbox and by solving the LMIs (16) in Theorem 6, we can obtain the feasible solutions. Due to space limitations, we do not list them here. We find that the delayed UISGRNs (8) are stable in the mean square which is shown in Figure 1.
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Example 2. Consider the system (6) with the parameters that are the same as in Example 1 other than
We can find feasible solutions for the LMIs in Corollary 8, so the impulsive stochastic GRNs (6) are stable in the mean square which can be shown in Figure 2.
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Example 3. When the parameters in (45) are the same as in Example 1 in addition to
It is easy to see that the uncertain impulsive GRNs (45) are stable in the mean square by checking Corollary 9 conditions. Figure 3 shows that the result is valid.
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5. Conclusions
In this paper, we have investigated the stability problem for a new UISGRNs model with the introduction of leakage delay. By employing the Lyapunov stability theory, free-weighting matrix, and convex combination technique combining with stochastic stability approach and the LMI framework, we have obtained a sufficient condition to justify the stability of the proposed UISGRNs model. The obtained stability condition is expressed in terms of LMIs which can be easily solved by the efficient MATALB LMI toolbox. Finally, numerical examples have been provided to illustrate the usefulness of the derived stability results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was jointly supported by the Fundamental Research Funds for the Central Universities (JUSRP51317B, JUSRP211A21), the National Natural Science Foundation of China under Grants 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012741, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grants 20110092110017 and 20130092110017.