Abstract

This study seeks to address the finite-time robust stability of delayed genetic regulatory networks (GRNs) with uncertain parameters and reaction-diffusion terms. We employ an appropriate Lyapunov-Krasovskii functional to derive some less conservative stability criteria for GRNs under Dirichlet boundary conditions, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. The time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. In addition, we define some new variables to deal with uncertain parameters. Moreover, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed to handle integral terms. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the obtained stability criteria.

1. Introduction

Genetic regulatory networks (GRNs) are collections of DNA segments, which interact with each other and with other substances in the cell to control the gene expression levels of mRNA and protein. To better understand the complex structure of biological mechanism is very important in biological and biomedical sciences [13]. To date, it has attracted considerable attention for its extensive applications.

In the practical GRNs, due to the slow speed of transcription and translation, time delay is inevitable in modeling gene regulation process. The existence of time delay is a primary factor impacting the dynamic behavior of whole GRNs, which may result in poor performance or even system instability, bifurcation, and oscillation. To describe the mathematic model more accurately, time delay has been considered in the genetic regulatory processes in the past few years [410]. For example, [4] presents the asymptotic stability analysis of delayed GRNs with SUM regulatory logic. The sampled-data stabilization problem for Takagi-Sugeno fuzzy GRNs with leakage delays is concerned in [5]. Reference [8] reports the stability analysis of Markovian switching GRNs with leakage and mode-dependent time-varying delays along Brownian motions.

In addition to the time delay, it should be noticed that parameter uncertainties are also unavoidable in the modelling process. This is owing to the intrinsic fluctuations, extrinsic disturbances, data errors, using an approximate system model for simplicity, and so on. Hence, an increasing number of studies have investigated the robust problem of uncertain GRNs with time delays [1124]. For instance, the robust stability analysis for various kinds of delayed GRNs with parametric uncertainties are studied in [1120], including T-S fuzzy GRNs, Markovian jumping GRNs, stochastic GRNs, etc. Besides, by using control and sample-data method, [2124] investigate the robust control and state estimation problem for uncertain delayed GRNs.

Moreover, it is necessary to consider the diffusion of regulatory proteins or metabolites from one compartment to another [25, 26] since the concentrations of mRNAs and proteins are not homogenous in space at all times. However, the reaction-diffusion term has not been taken into consideration in the above-mentioned works [424]. As far as we know, the stability problems for GRNs with reaction-diffusion terms are first proposed in [27, 28]. Due to the reaction-diffusion terms considered in GRNs, it is much more difficult to obtain stability criteria in terms of linear matrix inequalities (LMIs) than it would be without them. A new inequality introduced in [29] solves this problem and makes the diffusion-dependent stability criteria in terms of LMIs. However, there are still a few papers [3034] that have considered the dynamic properties of GRNs with reaction-diffusion terms. References [30, 31] investigate the asymptotic stability problem for delayed GRNs with reaction-diffusion terms under Dirichlet boundary conditions, Neumann boundary conditions, and Robin boundary conditions. From [28, 30, 31], we can find that the reaction-diffusion terms have no effect on the stability of GRNs under Neumann and Robin boundary conditions. However, it can be still retained in the stability criterion under Dirichlet boundary conditions, which reduces the conservatism. The finite-time stability problem of the delayed GRNs with reaction-diffusion terms under Dirichlet boundary conditions is concerned in [32] by employing the Wirtinger-type integral inequality, Gronwall inequality, convex technique, and reciprocally convex technique to obtain less conservative results. In [27, 28, 3034], the lower bound of time-varying delay is all regarded as zero, which should be improved. Meanwhile, the reaction-diffusion GRNs with parameters uncertainty are only considered in [27, 28, 31].

Furthermore, the stability analysis of GRNs is a significant field for understanding the living organisms at both molecular and cellular levels. In real biological system, the GRNs regulate the concentrations of mRNA and protein in a steady state, which is fatal for normal life activities. Otherwise, the instability of those molecular concentrations can lead to very serious consequences. Take cancer, for example; an important factor that causes cancer is the high amount of antiapoptosis proteins in the cancer patient’s body, which exceeds the ability of regulation by GRNs; see [35]. However, in the healthy human body, there is a fine-tuned balance in the process of apoptosis, which can force the cells containing genetic mutations to die and protect the whole living organisms. Therefore, the stability analysis of GRNs is an important area of research in the biological and biomedical sciences.

Motivated by the work discussed above, the main purpose of this study is to investigate the finite-time robust stability of delayed GRNs with uncertain parameters and reaction-diffusion terms. In order to obtain less conservative stability criteria, the time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. Besides, the approach for conducting uncertain parameters is to define parts of them as some new variables. Moreover, the Lyapunov stability theory, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed in the proof of Theorem 11. Finally, sufficient stability conditions are formed for GRNs in terms of LMIs, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. The numerical example is provided to illustrate the effectiveness of the proposed theoretical results.

The rest of the paper is organized as follows. The problem formulation and some preliminaries are given in Section 2 firstly. Second, Section 3 presents the finite-time robust stability analysis for GRNs with reaction-diffusion terms under Dirichlet boundary conditions. Then, a numerical example is given to illustrate the effectiveness of the main stability criteria in Section 4. Finally, we conclude this paper in Section 5.

Notations. The notations used throughout the paper are fairly standard. The superscript “” stands for matrix transposition; denotes the -dimensional Euclidean space; is the set of all real matrices; the notation means that is a positive definite matrix; and represent identity matrix and zero matrix with dimension , respectively; denotes the diagonal matrix. and denote the minimum and maximum eigenvalue of matrix , respectively. In symmetric block matrices, we use an asterisk to represent a term which is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. is a compact set in space with smooth boundary and denotes the measure of ; dS is the element of . represents the space of derivable -order real functions with compact support on . Let denotes the space of real Lebesgue measurable functions on . The norm is defined for any , and is defined on ,

2. Problem Formulation and Preliminaries

We consider the following delayed GRNs presented in [3]: are concentrations of mRNA and protein in the th node at time t, respectively. and are the degradation rates of mRNA and protein, and is the translation rate. and are the time-varying delays. is the regulatory function of the th gene with the form , which is called SUM logic [36].The function is a monotonic function of the Hill form. is the Hill coefficient, is a positive constant, and the bounded constant is the dimensionless transcriptional rate of transcription factor to gene . Therefore, (1) can be rewritten into the following form:, is defined as a basal rate, and is the set of all the which are repressors of gene .

Rewriting system (3) into compact matrix form, we can obtainwhere

Let be an equilibrium point of system (4). In the following, we will always shift the equilibrium point to the origin by letting . Hence, system (4) can be transformed into the following form:where

In this paper, we take parameter uncertainties and reaction-diffusion terms into account, and then the GRNs model (6) can be expressed as follows:where , is the diffusion range of mRNAs and proteins, and is a constant. and denote the transmission diffusion rate matrices of mRNA and protein, respectively. . The initial functions .

and are the time-varying delays satisfying , ; , , where , and are constants.

The parametric uncertainties , and satisfy , , , , where , and are some given constant matrices with appropriate dimensions, and satisfies for any

In order to conduct the stability analysis for the aforementioned systems, it is necessary to make the following definition, assumptions, and lemmas.

Definition 1 (see [27]). The uncertain GRNs (8) are said to be finite-time robustly stable with respect to positive real numbers , , and , if

Assumption 2. The GRNs (8) satisfy the Dirichlet boundary conditions; that is,

Assumption 3 (see [6]). For some constants and , we assume that satisfies the sector condition: , which implies Let , , , .

Remark 4. In [27, 28, 3034], is a monotonically increasing function satisfying . The constants and in Assumption 3 are allowed to be positive, negative, or zero. Obviously, it relaxes the restriction on this condition.

Lemma 5 (see [29]). For the compact set , if satisfies , that is, it vanishes on the boundary of , then

Lemma 6 (Jensen’s integral inequality [37]). For any positive definite matrix , scalars , vector function such that the integrations concerned are well defined, the following inequality holds:

Lemma 7 (Wirtinger-type integral inequality [38]). For a given matrix , the following inequality holds for all continuously differentiable function in : where .

Lemma 8 (reciprocally convex combination inequality [39]). For all vector , the function is given by where , , and are matrices with appropriate dimensions. Then, the following inequality holds,if there exists a matrix such that .

Lemma 9 (see [30]). For real diagonal matrices and , system (8) satisfies

Lemma 10 (Gronwall inequality [40]). If the functions and are nonnegative and integrable over , and exists, then implies that

3. Finite-Time Robust Stability Analysis for Genetic Regulatory Networks

In this section, we firstly provide a finite-time robust stability criterion for GRNs (8) under Dirichlet boundary conditions and then derive three corollaries based on Theorem 11.

Theorem 11. For given constants , , , , , , , and positive constant , the GRNs (8) are finite-time robust stable if there exist positive definite matrices , , diagonal matrices , and any matrices , with appropriate dimension, any constants , such that , , and the following LMIs (21) hold: , , , and are the particular matrices of matrix when , ; , ; , and , , respectively. are elementary matrices; for example, .

Proof. Define vectors:Then, the uncertain delayed GRNs (8) can be transformed into the following form:For systems (24), we construct the following Lyapunov-Krasovskii functional:whereAs a convenience, we defineThen, taking derivative of along the trajectories of systems (24), we can obtain the following differential:withNote thatAccording to Lemma 6, we haveThus,Similarly,Combining (33), (39), and (40), we can obtainNext, we deal with the integral terms in (35) by applying Lemmas 7 and 8,Similarly,The combination of (35), (42), and (43) givesBesides, for diagonal matrices and , the following equations are true from the model (24):In accordance with Green formula, Lemma 9, and Dirichlet boundary conditions, we havewhere is an outward pointing normal vector, is the element of , Analogously,Furthermore, we can obtain the following inequalities from Assumption 3 for any diag:Moreover, the following inequalities are true for any constants according to (23) and :Combining (29), (30), (31), (32), (33), (34), (35), (36), (37), (38), (39), (40), (41), (42), (43), (44), (45), (46), (47), (49), (50), (51), (52), (53), (54), and (55), we can obtainAccording to the inequalities in (21) and the convex combination method, we have , which implies Integrating the two sides of inequality (57) from 0 to , we haveThen, we can obtain the following inequality by the well-known Gronwall inequality (Lemma 10),whereNote that ; that is, there exist nonnegative real numbers , , , and such that Sequentially, the following inequalities hold:Thus, there exist nonnegative real numbers and satisfying the following inequalities:Consequently, we havewhereOn the other hand,Together with (65) and (67), we haveAccording to Definition 1, it is easy to see that the trivial solution of GRNs (8) under Dirichlet boundary conditions is finite-time robust stable. This completes the proof.

Remark 12. The time-varying delays and in [27, 28, 3034] are supposed to meet the conditions and , respectively. However, the lower bound of time-varying delay is not always zero. Therefore, we set and in this paper, where , and are constants.

Remark 13. In this study, we consider the delayed GRNs with reaction-diffusion and uncertain parameters. To handle with the uncertain parameters, , , , and are defined as new variables, which can be seen as another way to deal with the uncertainties. In this way, the model (8) is rewritten as (24). And then, we present the finite-time robust stability criteria for this system in Theorem 11.

Remark 14. In order to get less conservative stability conditions, Jensen’s inequality and Wirtinger-type integral inequality together with convex combination theory are employed to handle integral terms. For example, the processing method of integral term is in . Firstly, the integral interval is divided into two parts: and . Then, we use the Wirtinger-type integral inequality to estimate these two parts separately. Finally, they have been managed by convex combination method.

When the time-varying delays and are satisfying ,, . , and are constants. The following corollary can be used to check the stability of GRNs (8).

Corollary 15. For given constants , , , and positive constant , the GRNs (8) are finite-time asymptotic stable if there exist positive definite matrices , , diagonal matrices , and any matrices , with appropriate dimension, any constants , such that ,, and the following LMIs (69) hold: , , , and are the particular matrices of matrix when , ; , ; , and , , respectively. are elementary matrices; for example,

Furthermore, we consider the GRNs (8) without uncertain parameters; that is,The following corollary can be used to check the stability of GRNs (71).

Corollary 16. For given constants , , , and positive constant , systems (71) are finite-time asymptotic stable if there exist positive definite matrices , , diagonal matrices , and any matrices , with appropriate dimension, such that , , and the following LMIs (72) hold: , , , and are the particular matrices of matrix when , ; , ; , and , , respectively. are elementary matrices with appropriate dimension, and the expressions of in are equal to the corresponding terms in Corollary 15.

Remark 17. Theorem 11 and Corollary 15 are obtained based on the Dirichlet boundary conditions. When the boundary conditions are Neumann boundary conditions or Robin boundary conditions, we can also give the similar finite-time stability criteria by employing the method proposed in [28, 30, 31].

When there are no reaction-diffusion terms, the GRNs (8) can be simplified as follows:The following corollary can be used to check the stability of GRNs (74).

Corollary 18. For given constants , , , , , , , and positive constant , systems (74) are finite-time robust stable if there exist positive definite matrices , diagonal matrices , and any matrices with appropriate dimension, any constants , such that , and the following LMIs (75) hold: , , , and are the particular matrices of matrix when , and , respectively.The other elements in are equal to the corresponding terms in Theorem 11.

4. Numerical Simulations

In this section, the following numerical simulation example is given to demonstrate the effectiveness and correctness of the stability criteria obtained above.

Example 1. Consider the uncertain reaction-diffusion GRNs (8) with Dirichlet boundary conditions and the following parameters [28]:

In [28], the derivatives of time delays must be less than one. However, such strict constraint is removed in this paper. For example, when , , we can obtain the following feasible solutions for LMIs (69) in Corollary 15 (to save space, we only list parts of our feasible solutions):

Figures 1 and 2 show the trajectories of variables (mRNA concentrations) and (protein concentrations) for GRNs (8) with the above parameters and Dirichlet boundary conditions.

5. Conclusion

This study has investigated the finite-time robust stability problem of uncertain GRNs with interval time-varying delays and reaction-diffusion terms. The time-varying delays and their derivatives are both bounded with lower and upper bounds, where the lower bound of them can be zero or non-zero. Besides, the approach for conducting uncertain parameters is to define parts of them as some new variables. In addition, we employ an appropriate Lyapunov-Krasovskii functional to derive some less conservative stability criteria for GRNs under Dirichlet boundary conditions, which are delay-dependent, delay-derivative-dependent, and reaction-diffusion-dependent. Moreover, Jensen’s integral inequality, Wirtinger-type integral inequality, reciprocally convex combination inequality, Gronwall inequality, and Green formula are employed to handle with the integral terms in the derivative of Lyapunov-Krasovskii functional. Finally, a numerical example is presented to illustrate the feasibility and effectiveness of the obtained stability criteria. The foregoing results obtained in this study hold the potential to be extended to the other problems of delayed GRNs, including the decentralized event-triggered exponential stability, state estimation, sampled-data control, and so on. Meanwhile, the approaches can be used for the other time-delay systems, such as complex networks, neural networks, and multiagent systems.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the Science & Technology Development Fund of Tianjin Education Commission for Higher Education (Grant No. 2017KJ096), the National Natural Science Foundation of China (Grant Nos. 61603272, 11526149, 81501451, and 11672207), and the Youth Fund Project of Tianjin Natural Science Foundation (Grant Nos. 16JCQNJC03900, 18JCQNJC10900).