Dynamic Analysis of the Time-Delayed Genetic Regulatory Network Between Two Auto-Regulated and Mutually Inhibitory Genes
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Dynamic Analysis of the Time-Delayed Genetic Regulatory Network Between Two Auto-Regulated and Mutually Inhibitory Genes Guiyuan Wang1,2 · Zhuoqin Yang1 · Marc Turcotte3 Received: 18 August 2019 / Accepted: 16 March 2020 © Society for Mathematical Biology 2020
Abstract Time delays play important roles in genetic regulatory networks. In this paper, a gene regulatory network model with time delays and mutual inhibition is considered, where time delays are regarded as bifurcation parameters. In the first part of this paper, we analyze the associated characteristic equations and obtain the conditions for the stability of the system and the existence of Hopf bifurcations in five special cases. Explicit formulas are given to determine the direction and stability of the Hopf bifurcation by using the normal form method and the center manifold theorem. Numerical simulations are then performed to illustrate the results we obtained. In the second part of the paper, using time-delayed stochastic numerical simulations, we study the impact of biological fluctuations on the system and observe that, in modest noise regimes, unexpectedly, noise acts to stabilize the otherwise destabilized oscillatory system. Keywords Time delays · Hopf bifurcation · Oscillation · Stability · Genetic regulatory network model · Noise
1 Introduction Genetic regulatory networks (GRNs) describe interactions between DNA, RNA, proteins and small molecules in living cells and play fundamental roles in many life
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Marc Turcotte [email protected] Zhuoqin Yang [email protected]
1
School of Mathematics and Systems Science and LMIB, Beihang University, Beijing 100191, China
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College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, Hebei, P.R. China
3
Biological Sciences Department, The University of Texas at Dallas, Richardson, TX 75080, USA 0123456789().: V,-vol
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processes (Parmar et al. 2015; Ling et al. 2017). Because the processes of gene transcription and messenger RNA translation are not co-located in space and consequently not completed instantaneously, time delays are inevitable in GRNs. The effect of time delay on the dynamic behavior of a GRN model has attracted extensive attention of many scholars, and some research results have been obtained (Parmar et al. 2015; Ling et al. 2017; Wu and Eshete 2011; Bodnar and Bartłomiejczyk 2012; Lewis 2003; Wu 2011; Wang et al. 2010; Wu 2011; Zhang et al. 2017; Lai 2018; Monk 2003; Verdugo and Rand 2008; Sun et al. 2018; Yue et al. 2017; Huang et al. 2016). In Wu and Eshete (2011), the authors considered a model of gene express with two delays and showed effect of time delays on the model. In Bodnar and Bartłomiejczyk (2012), the Hes 1 genetic oscillator system was presented, and the authors showed that the stability of the steady state depends on the sum of time delays. The conditions for the occurrence of Hopf bifurcation and the stability of equilibrium point were proved. Zhang et al. (2017) investigated oscillatory e
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