{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T15:36:03Z","timestamp":1776267363579,"version":"3.50.1"},"reference-count":21,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advs. Complex Syst."],"published-print":{"date-parts":[[2009,2]]},"abstract":"<jats:p> 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. <\/jats:p>","DOI":"10.1142\/s0219525909002040","type":"journal-article","created":{"date-parts":[[2009,3,17]],"date-time":"2009-03-17T07:24:00Z","timestamp":1237274640000},"page":"3-19","source":"Crossref","is-referenced-by-count":27,"title":["DELAY-INDEPENDENT STABILITY OF GENETIC REGULATORY NETWORKS WITH TIME DELAYS"],"prefix":"10.1142","volume":"12","author":[{"given":"FANG-XIANG","family":"WU","sequence":"first","affiliation":[{"name":"Division of Biomedical Engineering and Department of Mechanical Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK, S7N 5A9, Canada"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1332628100"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1038\/35002131"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1038\/35002125"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1002\/cplx.6130010612"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/bth093"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0767-3"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1006\/jtbi.1998.0790"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1006\/jtbi.1993.1179"},{"key":"rf13","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1038\/77589","volume":"18","author":"Fussenegger M.","journal-title":"Nature Biotechnol."},{"key":"rf14","first-page":"2369","volume":"13","author":"Zak D. 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Biol."},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1109\/9.412637"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1093\/imamci\/2.2.153"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1016\/0005-1098(93)90175-S"}],"container-title":["Advances in Complex Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219525909002040","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T16:26:40Z","timestamp":1565108800000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0219525909002040"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,2]]},"references-count":21,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2009,2]]}},"alternative-id":["10.1142\/S0219525909002040"],"URL":"https:\/\/doi.org\/10.1142\/s0219525909002040","relation":{},"ISSN":["0219-5259","1793-6802"],"issn-type":[{"value":"0219-5259","type":"print"},{"value":"1793-6802","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,2]]}}}