Abstract
In this chapter, we consider the simplified GRN model, with the assumption that it is under delayed negative feedback. By analyzing the fixed points of a single function determined from the nonlinear connections, we show that in this case the system has a unique equilibrium point in the positive cone. Then, delay independent global stability, and instability, conditions are derived. For a delay dependent stability condition the secant condition is extended to cover systems with time delays. Special stability conditions are derived for homogenous GRNs where nonlinearities are Hill functions.
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References
H. Özbay, Introduction to Feedback Control Theory. CRC Press, 1999.
G. A. Enciso, “On the asymptotic behavior of a cyclic biochemical system with delay,” in 45th IEEE Conference on Decision and Control, pp. 2388–2393, 2006.
J. Mallet-Paret and G. R. Sell, “The Poincaré–Bendixson theorem for monotone cyclic feedback systems with delay,” Journal of Differential Equations, vol. 125, no. 2, pp. 441–489, 1996.
Y. Hori, M. Takada, and S. Hara, “Biochemical oscillations in delayed negative cyclic feedback: Existence and profiles,” Automatica, vol. 49, no. 9, pp. 2581–2590, 2013.
D. Allwright, “A global stability criterion for simple control loops,” Journal of Mathematical Biology, vol. 4, no. 4, pp. 363–373, 1977.
M. Mackey, “Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis,” Blood, vol. 51, no. 5, p. 941, 1978.
E. D. Sontag, “Passivity gains and the secant condition for stability,” Systems & Control Letters, vol. 55, no. 3, pp. 177–183, 2006.
M. Arcak and E. D. Sontag, “Diagonal stability of a class of cyclic systems and its connection with the secant criterion,” Automatica, vol. 42, no. 9, pp. 1531–1537, 2006.
T.-H. Kim, Y. Hori, and S. Hara, “Robust stability analysis of gene–protein regulatory networks with cyclic activation–repression interconnections,” Systems & Control Letters, vol. 60, no. 6, pp. 373–382, 2011.
R. Wang, Z. Jing, and L. Chen, “Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems,” Bulletin of Mathematical Biology, vol. 67, no. 2, pp. 339–367, 2005.
J. Wagner and G. Stolovitzky, “Stability and time-delay modeling of negative feedback loops,” Proceedings of the IEEE, vol. 96, no. 8, pp. 1398–1410, 2008.
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Ahsen, M.E., Özbay, H., Niculescu, SI. (2015). Gene Regulatory Networks Under Negative Feedback. In: Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays. SpringerBriefs in Electrical and Computer Engineering(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15606-4_5
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DOI: https://doi.org/10.1007/978-3-319-15606-4_5
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