Abstract
Genetic regulatory networks (GRNs) play an important role in the development and evolution of the biological system. With the rapid development of DNA technology, further research on GRNs becomes possible. In this paper, we discuss a class of time-delay genetic regulatory networks with external inputs. Firstly, under some reasonable assumptions, using matrix measures, matrix norm inequalities, and Halanay inequalities, we give the global dissipative properties of the solution of the time-delay genetic regulation networks and estimate the parameter-dependent global attraction set. Secondly, an error feedback control system is designed for the time-delay genetic control networks. Furthermore, we prove that the estimation error of the model is asymptotically stable. Finally, two examples are used to illustrate the validity of the theoretical results.
1. Introduction
In the past few decades, research on gene regulatory networks’ modeling has attracted many biologists and mathematicians. There have been many research studies in this area, such as models based on Boolean networks [1], models based on Bayesian methods [2], and models based on differential equations [3, 4]. In particular, the gene regulation networks’ model based on ordinary differential equation technology is welcomed by researchers because of its simple mathematical technology and clear biological meaning.
The concept of dissipative system originates from the research of control theory. In a sense, it is a generalization of Lyapunov’s stability theory. There are fruitful research results on stability issues in various research fields, such as neural network systems [5–10], biological systems [11, 12], and nonlinear systems [13, 14]. However, the study of stability requires the existence of a balance point. In some special cases, the balance point of the system may not exist. At this time, we consider the dissipative characteristics of the system. In the dissipative problem, we need to find a positive invariant set, and we only need to consider the dynamic characteristics of the system in the invariant set. So far, the dissipative characteristics of gene regulatory network models based on ordinary differential equation technology have not been seen in related publications.
In addition, the control system has been widely used in all aspects of human society. In the ecosystem, a large number of phenomena can be observed through the setting of the controller, and different control schemes will produce different kinds of effects. In [15], the author gives some characteristics of the linear time-varying system under the control of time-varying feedback conjugate. In [16], the authors establish the stability criterion and saddle node bifurcation of gene regulatory networks that have nothing to do with coupling delay. In [17], the authors discuss a delayed fractional-order two-gene regulatory network model that can more accurately reflect the memory and genetic characteristics of the gene network.
Based on the enlightenment obtained from the abovementioned literature research, we discuss the characteristics of time-delay gene regulation networks with external inputs. Compared with most previous related results, our novelty lies in the use of matrix measurement theory to carry out discussions. The matrix measurement strategy has the following two advantages. (1) Lyapunov function does not need to be constructed. (2) The result obtained by the matrix measurement is more accurate and less conservative than the result obtained by the matrix norm.
The rest of the paper is arranged as follows. In Section 2, we give the mathematical form of the gene regulatory network model and some basic assumptions. In Section 3, the global dissipativity of the model is discussed. In Section 4, we study the asymptotic stability of the error feedback control system. In Section 5, two examples are given to verify the validity of our conclusions.
Notations: in this paper, represents the -dimensional Euclidean space. represents the transpose of matrix . means but . For real symmetric matrices and , indicates that is positive semidefinite. represents . represents the maximum eigenvalue of matrix .
2. Preliminaries
The gene regulatory networks’ model has the following classic form:where are the concentrations of the th mRNA and protein, respectively, and are positive constants, representing the degradation rate of the th mRNA and protein, respectively, is the translation rate from the th mRNA to the th protein, is the regulatory functions of mRNA, where is the Hill coefficient, is a scalar, and are transcriptional and translational delay with , and with is the set of all the transcription factor which is a repressor of gene , and
We rewrite model (1) in the following matrix form:where , , , , , and .
Next, we give some basic assumptions and lemmas.
Assumption 1. The activation function meets the Lipschitz condition.
Remark 1. In most literature, satisfied , for all . The condition can be reduced to , where and are constants. Furthermore, the condition is reduced to Assumption 1, i.e., there exists a constant such that, for any , the following inequalities hold: . Clearly, Assumption 1 is more general and less conservative.
Let be the equilibrium point of (3), and it satisfiesLet , and we obtainwhere , and .
In the field of gene therapy, the effect of external forces or stimuli on the biological system is very important. For example, mRNA vaccine is to transfer RNA to human cells after relevant modification in vitro for expression and produce protein antigen, which can lead the body to produce immune response to antigens, and then expand the immune ability of the body. Therefore, we consider the following time-delay GRNs with external inputs described bywhere are control inputs for mRNA and protein, respectively.
The initial conditions of system (6) areBy the definition of and Assumption 1, it satisfies
Definition 1. If there exists a compact set , for any and any initial value , such that , when , where is the -neighborhood of ; then, the time-delay GRNs (6) are said to be a global dissipative system. In this case, is called a globally attractive set. If, for any initial value , there is , then the set is said to be positive invariant. Obviously, the globally attractive set is positive invariant.
Definition 2. If there exists a compact set , for any and any initial value , there exist and such thatThen, the time-delay GRNs (6) are said to be a globally exponentially dissipative system.
Definition 3. For constant matrix , the matrix norms are given asFor , the vector norms are defined as follows:
Definition 4. Suppose that is a real matrix; then, the matrix measure of is defined aswhere is a identity matrix, , and is the corresponding induced matrix norm.
It can be calculated directly by Definition 4:
Lemma 1 (see [18]). Let , if the continuous function , and
There exists a constant such that ; then,(1)When , the following formula is established:(2)When , the following formula is established:where , is the unique positive root of the transcendental equation , and is the upper-right Dini derivative.
Lemma 2 (see [19]). For any , the matrix measure has the following properties: .
3. Global Dissipativity
In this section, we give the global dissipative properties of the studied mathematical model.
Theorem 1. If satisfies (8), there exist positive constant , and the matrix measures , such that , where and . Then, the time-delay GRNs (6) are a globally dissipative system, and the set is a positive invariant set and a globally attractive set of (6).
Proof. LetWe calculate the upper-right hand Dini derivative of along the solution of (6):From (8), we can obtainBy (18) and (19), according to the definition of matrix measure, we deduce thatAccording to Lemma 1 and , we havewhere is the unique positive root of equation . Then, (6) is a globally dissipative system, and the globally attractive set isIf , by Lemma 2, we have , for all . Therefore, the set is also a positive invariant set of (6).
Remark 2. Obviously, from (22), we can easily get the following sets:
Corollary 1. Let , and other conditions are the same as Theorem 1. Then, the time-delay GRNs (6) are a globally exponentially dissipative system, and the globally attractive sets are .
Proof. By the proof of Theorem 1, if , we choose as the intersection point of the vector and the spherical surface of , where is the origin of coordinates. We haveAccordingly,Similarly,Thus, the time-delay GRNs (6) are a globally exponentially dissipative system and are globally exponentially attractive sets of (6).
In Theorem 1, parameter is considered only as the maximum value information. If the information of each component is utilized, the conservativeness of the result can be decreased.
DenoteThen, (6) can be written in the form of matrix blocks:where .
Theorem 2. If satisfies (8), there exist positive constant , and the matrix measures , such that . Then, the time-delay GRNs (6) are a globally dissipative system, and the sets,are the positive invariant sets and the globally attractive sets of (6).
Proof. Consider the upper-right hand Dini derivative of with respect to time along the trajectory of (6), and we obtainBy the definition of and (8), it is easy to acquire that holds for . Meanwhile, it also holds for . In fact, for , we have , i.e., . By Weyl’s monotonicity principle, we can obtainSince andWe can obtainand .
Accordingly, we haveCombining (30) and (34), we deriveLet , and the above inequality can reduce toBy inequality and Lemma 1, we obtainwhere is the unique positive root of equation . Similar to the proof of Theorem 1, we can obtain that the time-delay GRNs (6) are a globally dissipative system, and are positive invariant sets and globally attractive sets of (6).
4. State Estimation of Error Feedback Control System
In this section, we will discuss the global exponential stability of the following error dynamic system:and the considered state estimator is of the following form:where be an error feedback control term with the form , . and are feedback control gains, . Thus, the error dynamic system is given as
In order to facilitate the discussion later, we write (40) for the matrix block formwhere
Theorem 3. If meets (8), there exist matrix and matrix measure , such that . Then, the error dynamic system (40) is globally exponential stable under controller .
Proof. By inequality (30), we haveLet . If , that is, , we havewhere . Therefore, converges exponentially to zero with convergence rate of , i.e., the error system is globally exponential stable.
5. Numerical Simulations
In this part, we give a pair of examples to illustrate the validity of our theoretical results.
Example 1. Consider time-delay GRNs (6) withBy computation, we can find that , and hence,In order to make the calculation simple, we choose to be a diagonal matrix. At this time, takes any value in , and we have . Since is a nondiagonal matrix, it can be obtained through simple calculations: . Obviously, the parameters satisfy the conditions of Theorem 1, i.e., . This means that system (6) is globally dissipative. From the vector norm, we get the following results:When , . When , . When , . Figure 1 shows the time-dependent trajectory of mRNA and protein concentrations when the initial values are and . The verification of Theorem 2 conditions is similar to the verification in Example 2, so we will not state it here.
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Example 2. In this example, we consider the error dynamic system (40), in which the parameters are listed as follows:and is the same as Example 1. It is easy to obtain thatwhen takes any one of and there are . It can be directly obtained by simple matlab programming:The above calculation results are easy to see that the conditions in Theorem 3 are satisfied. That is, holds. This shows that system (40) is globally exponentially stable. Figure 2 shows the trajectory of error system (40) with time when the initial values are and .
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6. Conclusion
This paper proposes a sufficient condition for the global dissipative properties of GRN with external inputs. For the proof of dissipative properties, a large number of documents adopt the method of constructing Lyapunov function. However, this paper adopts the method of matrix measurement, which simplifies the calculation difficulty, and the result is satisfactory. At the same time, error feedback control is given to ensure the stability of the system. And, the limitation of this kind of control is very small, which can bring great convenience in practical application. In addition, the study of the influence of Brown process and Levy process on gene regulatory networks is a challenging problem, which requires more mathematical knowledge about stochastic processes, but it is a very meaningful topic.
Data Availability
All data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This study was supported by Yulin University Doctoral Research Foundation (17GK16), Yulin Science and Technology Bureau Project (2019-90-1), Shaanxi Provincial Department of Education Project (20JK1016), Shaanxi Educational Science Poverty Alleviation Project (JKYFP29), Yuyang District Basic Education Research Planning Project (YYGH19106), and Yulin City Educational Science Planning Project (YLKT19204).