Stability of Uncertain Impulsive Stochastic Genetic Regulatory Networks with Time-Varying Delay in the Leakage Term
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.
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  firstly proposed a delay differential equation model for GRNs and studied its stability problem. In , Ren and Cao studied the asymptotic and robust stability of GRNs with time-varying delays. In , Zhang et al. investigated the stability analysis for GRNs with random discrete delays and distributed delays. Hu et al.  proposed a GRNs model with hybrid regulatory mechanism and studied its stability problem. Recently, Gopalsamy  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 . In , 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 , 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 . 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 ). 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 ). For any constant symmetric matrix , scalar γ > 0,
Lemma 5 (see ). 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):
Then, by Itô’s differential formula, taking the derivative of along the trajectories of the system (8), we can obtain the following stochastic differential :
where is the diffusion operator and
By Newton-Leibnitz formula, we have that where
By using Lemmas 4 and 5, we have