DELAY-INDEPENDENT STABILITY OF GENETIC REGULATORY NETWORKS WITH TIME DELAYS
In an organism, genes encode proteins, some of which in turn regulate other genes. Such interactions work in highly structured but incredibly complex ways, and make up a genetic regulatory network. Recently, nonlinear delay differential equations have been proposed for describing genetic regulatory networks in the state-space form. In this paper, we study stability properties of genetic regulatory networks with time delays, by the notion of delay-independent stability. We first present necessary and sufficient conditions for delay-independent local stability of genetic regulatory networks with a single time delay, and then extend the main result to genetic regulatory networks with multiple time delays. To illustrate the main theory, we analyze delay-independent stability of three genetic regulatory networks in E. coli or zebra fish. For E. coli, an autoregulatory network and a repressilatory network are analyzed. The results show that these two genetic regulatory networks with parameters in the physiological range are delay-independently robustly stable. For zebra fish, an autoregulatory network for the gene her1 is analyzed. The result shows that delay-independent stability of this network depends on the initial number of protein molecules, which is in agreement with the existing biological knowledge. The theories presented in this paper provide a very useful complement to the previous work and a framework for further studying the stability of more complex genetic regulatory networks.
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