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Dynamic Analysis of the Time-Delayed Genetic Regulatory Network Between Two Auto-Regulated and Mutually Inhibitory Genes

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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.

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Acknowledgements

The authors wish to express sincere thanks to anonymous referee number 2 for very valuable suggestions. ZY and WG acknowledge support from NSFC 11872084.

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Authors and Affiliations

  1. School of Mathematics and Systems Science and LMIB, Beihang University, Beijing, 100191, China

    Guiyuan Wang & Zhuoqin Yang

  2. College of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, Hebei, P.R. China

    Guiyuan Wang

  3. Biological Sciences Department, The University of Texas at Dallas, Richardson, TX, 75080, USA

    Marc Turcotte

Authors
  1. Guiyuan Wang
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  2. Zhuoqin Yang
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  3. Marc Turcotte
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Corresponding author

Correspondence to Marc Turcotte.

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Methods

In numerical simulations, ODEs and DDEs were separately integrated by using ODE45 and DDE23 in MATLAB R2014a.

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Appendix

Appendix

$$\begin{aligned} {m_0}= & {} 4B_0^2{A_0}{C_0},\\ {m_1}= & {} 2{C_0}\left( {2{B_0}{A_0}{B_1} - {A_1}B_0^2}\right) ,\\ {m_2}= & {} 4{C_0}\left( {2{A_1}{B_1}{B_0} - 2{A_0}{B_2}{B_0} - {A_0}B_1^2 - {A_1}B_0^2} \right) ,\\ {m_3}= & {} 2{C_0}\left( {{A_3}B_0^2 - 2{B_2}{B_1}{A_0} - 2{B_0}{B_1}{A_2} + {A_1}B_1^2 + 2{B_0}{B_2}{A_1}} \right) ,\\ {m_4}= & {} 4{C_0}\left( { - 2{A_3}{B_1}{B_0} - 2{A_1}{B_1}{B_2} + 2{A_2}{B_2}{B_0} + 4B_2^2{A_0} + {A_2}B_1^2 + B_0^2} \right) ,\\ {m_5}= & {} 2{C_0}\left( {2{A_2}{B_1}{B_2} - {A_3}B_1^2 - 2{A_3}{B_0}{B_2} + 2{B_1}{B_0} - {A_1}B_2^2} \right) ,\\ {m_6}= & {} 4{C_0}\left( {2{A_3}{B_1}{B_2} + B_2^2{A_2} - B_1^2 - 2{B_2}{B_0}} \right) ,\\ {m_7}= & {} 2{C_0}\left( {B_2^2{A_3} - 2{B_1}{B_2}} \right) ,\\ {m_8}= & {} 4{C_0}B_2^2,\\ {n_1}= & {} 2A_3^2 - 4{A_2},\\ {n_2}= & {} A_3^4 + 6A_2^2 - B_2^2 - 4{A_1}{A_3} - 4{A_2}A_3^2 + 4{A_0},\\ {n_3}= & {} 2A_1^2 - 12{A_0}{A_2} - 4A_2^3 + 2B_2^2{A_2} - A_3^2B_2^2 + 2{B_2}{B_0} + 4{A_0}A_3^2 + 8{A_1}{A_2}{A_3} \\&+2A_2^2A_3^2 - B_1^2 - 4A_3^3{A_1},\\ {n_4}= & {} 2B_1^2{A_2} - 8{A_0}{A_1}{A_3} - B_0^2 - A_3^2B_1^2 + 12{A_0}A_2^2 - A_2^2B_2^2 - 2C_0^2 + 6A_1^2A_3^2 \\&+ A_2^4 +6A_0^2 - 2B_2^2\left( {{A_0} + {C_0} - A_3^2{B_0} - {A_1}{A_3}} \right) \\&- 4{A_2}\left( {{A_1}{A_2}{A_3} + {B_0}{B_2} + {A_0}A_3^2 + A_1^2} \right) ,\\ {n_5}= & {} 4{A_2}C_0^2 - 12{A_2}A_0^2 + 2A_1^2\left( {2{A_0} + A_2^2} \right) + 2A_3^2\left( {A_0^2 - C_0^2} \right) - 4A_2^3{A_0} \\&- A_1^2\left( {4{A_1}{A_3} + B_2^2} \right) \\&+ B_0^2\left( {2{A_2} - A_3^2} \right) + B_1^2\left( {2{C_0} - A_2^2 - 2{A_0} + 2{A_1}{A_3}} \right) \\&- 4{A_3}\left( {{B_1}{B_2}{C_0} + {B_0}{B_2}{A_1}} \right) \\&+8{A_0}{A_1}{A_2}{A_3} + 4{B_0}{B_2}\left( {{A_0} + {C_0}} \right) + 2{A_2}{B_2}\left( {{B_2}{C_0} + {A_2}{B_0}} \right) ,\\ {n_6}= & {} 6A_2^2A_0^2 - 2C_0^2\left( {2{A_0} - A_2^2} \right) - A_1^2B_1^2 - B_2^2\left( {A_0^2 + C_0^2} \right) \\&- B_0^2\left( {2{A_0} + 2{C_0} + A_2^2} \right) \\&+ 4{B_1}{C_0}\left( {{B_2}{A_1} + {B_0}{A_3}} \right) - 4{B_2}\left( {{A_2}{B_0}{A_0} + {A_2}{B_0}{C_0}} \right) \\&+ 2{B_0}{B_2}A_1^2 + 2{A_1}{A_3}B_0^2 \\&+2B_1^2{A_2}\left( {{A_0} - {C_0}} \right) - 4{A_0}{A_2}A_1^2 + 4{A_1}{A_3}\left( {C_0^2 - A_0^2} \right) - 2{A_0}{C_0}B_2^2 \\&+ 4A_0^3 + A_1^4,\\ {n_7}= & {} 2B_0^2{A_2}{C_0} + 2A_1^2A_0^2 + 2C_0^2{B_0}{B_2} - 4{A_1}{B_1}{B_0}{C_0} - A_1^2\left( {B_0^2 + 2C_0^2} \right) \\&+ 2B_1^2{A_0}{C_0} \\&+4{B_2}{A_0}{B_0}{C_0} + A_0^2\left( {2{B_0}{B_2} - B_1^2} \right) - C_0^2\left( {B_1^2 - 4{A_0}{A_2}} \right) - 4{A_2}A_0^3 \\&+ 2B_0^2{A_0}{A_2},\\ {n_8}= & {} - 2B_0^2{A_0}{C_0} - A_0^2\left( {2C_0^2 + B_0^2} \right) - B_0^2C_0^2 + A_0^4 + C_0^4,\\ M_1^ *= & {} {{{c_1}P_1^ * } /{{d_1}}},\\ P_2^ *= & {} \left( - b{b_{21}}{d_1}\left( {{s^{2n}} + {s^n}{{\left( {p_1^ * } \right) }^n}} \right) \right. \Big /\\&\left. {\left( { - {r_1}{c_1}P_1^ * {s^n} - {r_1}{c_1}{{\left( {P_1^ * } \right) }^{n + 1}} + a{a_1}{{\left( {P_1^ * } \right) }^n}{d_1}} \right) - {s^n}} \right) ^{\left( {1/n} \right) },\\ M_2^ *= & {} {{{c_2}P_2^ * } / {{d_2}}}. \end{aligned}$$

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Wang, G., Yang, Z. & Turcotte, M. Dynamic Analysis of the Time-Delayed Genetic Regulatory Network Between Two Auto-Regulated and Mutually Inhibitory Genes. Bull Math Biol 82, 46 (2020). https://doi.org/10.1007/s11538-020-00722-1

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