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Bifurcation analysis of a mathematical model for genetic regulatory network with time delays

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  • Zang, Hong
  • Zhang, Tonghua
  • Zhang, Yanduo

Abstract

In this paper, we aim to investigate the dynamics of a gene regulatory network which is a time-delayed version of the model proposed by Elowitz and Leibler [Nature 403 (2000) 335–338]. Based on the normal form theory and center-manifold reduction, Hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated. We also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network. It shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation. More precisely, (i) the amplitude increases with small time delay, while the change of amplitude is not sensitive to relatively large time delay. However, the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter, where amplitude always increases quickly and linearly with the transcriptional rate; (ii) the period of oscillation increases as the time delay increases, but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value, which implies that the robustness of period to the transcriptional rate variations occurs. Our numerical simulations also support the theoretical conclusions, namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks.

Suggested Citation

  • Zang, Hong & Zhang, Tonghua & Zhang, Yanduo, 2015. "Bifurcation analysis of a mathematical model for genetic regulatory network with time delays," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 204-226.
    • Handle: RePEc:eee:apmaco:v:260:y:2015:i:c:p:204-226
    DOI: 10.1016/j.amc.2015.03.041
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    References listed on IDEAS

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    1. Shen, Jianwei & Liu, Zengrong & Zheng, Weixing & Xu, Fengdan & Chen, Luonan, 2009. "Oscillatory dynamics in a simple gene regulatory network mediated by small RNAs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2995-3000.
    2. Attila Becskei & Luis Serrano, 2000. "Engineering stability in gene networks by autoregulation," Nature, Nature, vol. 405(6786), pages 590-593, June.
    3. Stefano Ciliberti & Olivier C Martin & Andreas Wagner, 2007. "Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology," PLOS Computational Biology, Public Library of Science, vol. 3(2), pages 1-10, February.
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    Cited by:

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    2. Yuan, Jun & Zhao, Lingzhi & Huang, Chengdai & Xiao, Min, 2019. "Novel results on bifurcation for a fractional-order complex-valued neural network with leakage delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 868-883.
    3. Zarei, Amin & Tavakoli, Saeed, 2016. "Hopf bifurcation analysis and ultimate bound estimation of a new 4-D quadratic autonomous hyper-chaotic system," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 323-339.
    4. Huang, Chengdai & Cao, Jinde & Xiao, Min & Alsaedi, Ahmed & Hayat, Tasawar, 2017. "Bifurcations in a delayed fractional complex-valued neural network," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 210-227.
    5. Li, Peiluan & Gao, Rong & Xu, Changjin & Ahmad, Shabir & Li, Ying & Akgül, Ali, 2023. "Bifurcation behavior and PDγ control mechanism of a fractional delayed genetic regulatory model," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    6. Kaslik, Eva & Rădulescu, Ileana Rodica, 2022. "Stability and bifurcations in fractional-order gene regulatory networks," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    7. Gao, Bo & Deng, Zheng-hong & Zhao, Da-wei & Song, Qun, 2017. "State analysis of Boolean control networks with impulsive and uncertain disturbances," Applied Mathematics and Computation, Elsevier, vol. 301(C), pages 187-192.
    8. Ayachi, Moez, 2022. "Dynamics of fuzzy genetic regulatory networks with leakage and mixed delays in doubly-measure pseudo-almost periodic environment," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    9. Wang, Xinhe & Lu, Junwei & Wang, Zhen & Li, Yuxia, 2020. "Dynamics of discrete epidemic models on heterogeneous networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    10. Yang, Juanping & Sheng, Yuhong & Li, Hong-Li & Hu, Cheng, 2023. "Stability and adaptive control-based synchronization of delayed uncertain fractional-order gene regulatory networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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