Ultrasensitive Dual Phosphorylation Dephosphorylation Cycle With Time Delayed State Feedback Control

Since the end of the 19th century, biochemistry as one of the important basic sub-ject of life science began to develop, and has become one of the most active and most rapid development disciplines. All life activities is maintained by the normal operation of metabolic, and metabolism in various reaction is conducted under the participation of enzyme. In this sense, studying the enzyme to clarify the mechanism of the phe-nomenon of life has important significance. At the same time, medicine, agriculture, some industry and defense departments have some problems in the production prac-tice, we can construct the model by ordinary differential equations and save it, it will have guiding significance to production practices.This paper firstly introduces a hypothesis in the process of enzyme catalysis in the biochemistry, intermediate complex theory. Then enumerates the most commonly used laws in the biochemical reaction process, the law of conservation of mass and Michael-Menten theory, and show the basic parameters, such as reaction flux, the equilibrium constant, etc. At the same time, the basic conception, feature, and stability theorem of delay equation is given. Huang and Qian's work introduced the ultrasensitive dual phosphorylation dephosphorylation cycle and the stability analysis of the corresponding singular equations.In this paper, the main work is add three kinds of reactant concentration related time delayed state feedback control in the ultrasensitive dual phosphorylation dephos-phorylation cycle, then give the stability switches of the system at the fixed points. After variable substitution, we can translate the system to (0,0), and use eigenvalue method to analyze the distribution of the characteristic equation roots, so that we can give the stability criterion at the fixed points. For the first control, it can only switch a single point from stable to unstable, or from unstable to stable,and we can't change the fixed points stability through one control at the same time. For the second control, we can switch the stable fixed points (0,1) and (1,0) to unstable state in the meantime. After analyzing the internal fixed point, we see it is possible to change a bistable system to monostable system. According to this two control, we consider the third control. When (0,1) and (1,0) are stable, there can be only k times switches between stability and instability. When (0,1) is unstable, we can choose the range of parameters so that one or more switches from instability to stability to instability occur. We can choose the range of parameters according to actual needs. According our research, we can give the appropriate control to change the stability.