DELAY-INDEPENDENT STABILITY OF GENETIC REGULATORY NETWORKS WITH TIME DELAYS

Fang-Xiang Wu ()
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Fang-Xiang Wu: Division of Biomedical Engineering and Department of Mechanical Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, S7N 5A9, Canada

Advances in Complex Systems (ACS), 2009, vol. 12, issue 01, 3-19

Abstract: 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 inE. colior zebra fish. ForE. 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.

Keywords: Genetic regulatory networks; time delays; delay-independent stability; nonlinear differential equations; gene; protein; mRNA (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1142/S0219525909002040

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