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.
Highlights
In the past few decades, research on gene regulatory networks’ modeling has attracted many biologists and mathematicians. ere 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]
The study of stability requires the existence of a balance point
We consider the dissipative characteristics of the system
Summary
In the past few decades, research on gene regulatory networks’ modeling has attracted many biologists and mathematicians. ere 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]. Based on the enlightenment obtained from the abovementioned literature research, we discuss the characteristics of time-delay gene regulation networks with external inputs. Where mi(t), pi(t) ∈ R are the concentrations of the ith mRNA and protein, respectively, ai and ci are positive constants, representing the degradation rate of the ith mRNA and protein, respectively, di > 0 is the translation rate from the ith mRNA to the ith protein, hj(x) ((x/βj)Hj /1 + (x/βj)Hj ) is the regulatory functions of mRNA, where Hj is the Hill coefficient, βj is a scalar, σi(t) and τi(t) are transcriptional and translational delay with 0 < σi(t) ≤ σ, 0 < τi(t) ≤ τ, and qi j∈]iαij with ]i is the set of all the transcription factor j which is a repressor of gene i, and. For any A, B ∈ Rn×n and p 1, 2, ∞, the matrix measure μp(·) has the following properties: (1) − ‖A‖p ≤ μp(A) ≤ ‖A‖p; (2) μp(αA) αμp (A), ∀α > 0; (3) μp (A + B) ≤ μp(A) + μp(B)
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