Analysis Of Stability For A Class Of Genetic Regulatory Network Based On Differential Equation Model

With the rapid development of bio-informatics and the appearance of the DNA chips, it has become possible to measure gene expression levels on a genomic scale and furthermore the analyze the genetic regulatory networks. Genetic regulatory networks is a new research field in bio-informatics and biochemical, considerable attention has been contributed to theoretical analysis and experimental investigation on genetic regulatory networks. And genetic regulatory network is the key to understand how genes and proteins to form a network that performs complicated biological functions. As the genetic regulatory network is a dynamical system to describe the main gene productions such as mRNA and proteins, it is natural to model the genetic regulatory network with dynamical systems. Several computational models have been applied to investigate the dynamical behavior of genetic regulatory networks:Boolean network, Bayesian network, Differential equation networks and so on. In the differential equation network models, the variables describe the concentrations on gene productions, such as mRNA and proteins, as continuous value of the genetic regulatory systems. Meanwhile the differential equation model has been more widely used to investigate the genetic regulatory networks, and certain quantitative results have been presented. As a dynamical systems, it is of significant to investigate the stability of the genetic regulatory networks, and the researches on stability of genetic regulatory networks is also an important research field in bio-informatics nowadays.In this thesis, we mainly focus on one class of genetic regulatory network with SUM regulatory logic, which can be described by differential equation model. The existence and stability of equilibrium point of the system is discussed. Taking the delays and uncertainties of the system into account, several stability criterias have been obtained.The main research work of this paper includes two aspects:1. Firstly, we discuss the existence of the equilibrium points of the system and study the stability of genetic regulatory systems with continuous varying delays based on the Lyapunov stability theory. The derivative of the delays has been expanded to greater than 1, which is more flexible than existing results. Meanwhile, in the application and design of genetic networks, there are often some unavoidable uncertainties, such as modeling error, external perturbation and parameters fluctuations, the robust asymptotical stability of the system is also studied. And the results obtained above are given in the form of LMIs.2. From a dynamical system point of view, globally stable networks in Lyapunov sense are monostable systems, which have a unique equilibrium point attracting all trajectories asymptotically. But in many real applications, biological systems are no longer globally stable, thus more appropriate notions of stability are needed to deal with multistable networks. The main difference between Lagrange stability and Lyapunov stability is that Lagrange stability refers to the stability of total systems, not the stability of equilibrium points. To the best of our knowledge, the current results of stability analysis for genetic regulatory networks are all based on the Lyapunov global stability. The Lagrange stability for genetic regulatory networks has not been touched. Hence, from the perspective of theory and application, it is of significance to study the Lagrange stability for genetic regulatory networks In this paper, for the first time, we study the globally exponentially stability in Lagrange sense for genetic regulatory networks with time delay and presents some criteria for global exponential attractive set of genetic regulatory networks. It is believed that our results are useful for comprehensive analysis of genetic regulatory networks and have certain applicable value.At last, a compact summary of this paper is given by combining the advances of the previous researches in this fields and our work. The prospect for future study is also given.