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Oscillatory dynamics in a gene regulatory network mediated by small RNA with time delay

Abstract

A genetic regulatory network mediated by small RNA with two time delays is investigated. We show by mathematical analysis and simulation that time delays can provide a mechanism for the intracellular oscillator. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

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References

  1. Bartel, D.P.: MicroRNAs: genomics, biogenesis, mechanism, and function. Cell 116, 281–297 (2004)

    Article  Google Scholar 

  2. Gottesman, S.: Micros for microes: non-coding regulatory RNA in bacteria. Trends Genet. 21, 399–404 (2005)

    Article  Google Scholar 

  3. Storz, G., Altuvia, S., Wassarman, K.M.: An abundance of RNA regulators. Annu. Rev. Biochem. 74, 199–217 (2005)

    Article  Google Scholar 

  4. Zamore, P.D., Haley Ribo-gnome, B.: The big world of small RNAs. Science 309, 1519–1524 (2005)

    Article  Google Scholar 

  5. Mitarai, N., Benjamin, J.M., Krishna, S., Semsey, S., Csiszovszki, S., et al.: Dynamic features of gene expression control by small regulatory RNAs. Proc. Natl. Acad. Sci. USA 106, 10655–10659 (2009)

    Article  Google Scholar 

  6. Mehta, P., Goyal, S., Wingreen, N.: A quantitative comparison of sRNAbased and protein-based gene regulation. Mol. Syst. Biol. 221, 1–10 (2008)

    Google Scholar 

  7. Shimoni, Y., Friedlander, G., Hetzroni, G., Niv, G., Altuvia, S., Biham, O, Margalit, H: Regulation of gene expression by small non-coding RNAs: a quantitative view. Mol. Syst. Biol. 3, 1789–3699 (2007)

    Article  Google Scholar 

  8. Tsang, J., Zhu, J., van Oudenaarden, A.: MicroRNA-mediated feedback and feedforward loops are recurrent network motifs in mammals. Mol. Cell 26, 753–767 (2007)

    Article  Google Scholar 

  9. Ambros, V.: The functions of animal microRNAs. Nature 431, 350–355 (2004)

    Article  Google Scholar 

  10. Chen, L., Wang, R., Zhang, X.: Biomolecular Networks: Methods and Applications in Systems Biology. Wiley, New York (2009)

    Book  Google Scholar 

  11. Shimoni, Y., Friedlander, G., Hetzroni, G., Niv, G., Altuvia, S., Biham, O, Margalit, H: Regulation of gene expression by small non-coding RNAs: a quantitative view. Mol. Syst. Biol. 3(138), 1–9 (2007)

    Google Scholar 

  12. Levine, E., Zhang, Z., Kuhlman, T., Hwa, T.: Quantitative characteristics of gene regulation by small RNA. PLoS Biol. 5(9), e229 (2007)

    Article  Google Scholar 

  13. Liu, D., Chang, X., Liu, Z., Chen, L., Wang, R.: Bistability and oscillations in gene regulation mediated by small noncoding RNAs. PLoS ONE 6, e17029 (2011)

    Article  Google Scholar 

  14. Xiao, M., Cao, J.: Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays. Math. Biosci. 215, 55–63 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Monk, N.: Oscillatory expression of Hes1, p53, and NF-kappaB driven by transcriptional time delays. Curr. Biol. 13, 1409–1413 (2003)

    Article  Google Scholar 

  16. Zhang, D., Song, H., Yu, L., Wang, Q., Ong, C.: Set-values filtering for discrete time-delay genetic regulatory networks with time-varying parameters. Nonlinear Dyn. 69, 693–703 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang, D., Yu, L.: Passivity analysis for stochastic Markovian switching genetic regulatory networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simul. 16(8), 2985–2992 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Vembarasan, V., Nagamani, G., Balasubramaniam, P., Park, J.H.: State estimation for delayed genetic regulatory networks based on passivity theory. Math. Biosci. 244(2), 165–175 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhang, D., Yu, L.: H output tracking control for neutral systems with time-varying delay and nonlinear perturbations. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3284–3292 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhang, D., Yu, L.: H filtering for linear neutral systems with mixed time-varying delays and nonlinear perturbations. J. Franklin Inst. 347(7), 1374–1390 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ren, F., Cao, J.: Asymptotic and robust stability of genetic regulatory networks with time-varying delays. Neurocomputing 71(4), 834–842 (2008)

    Article  Google Scholar 

  22. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  23. Song, Y., Han, M., Wei, J.: Stability and Hopf bifurcation on a simplified BAM neural network with delays. Physica D 200, 185–204 (2005)

    MATH  MathSciNet  Google Scholar 

  24. Meng, X., Huo, H., Zhang, X., Xiang, H.: Stability and Hopf bifurcation in a three-species system with feedback delays. Nonlinear Dyn. 64, 349–364 (2011)

    Article  MathSciNet  Google Scholar 

  25. Ruan, S., Wei, J.: On the zeros of a third degree exponential polynomial with application to a delayed model for the control of testosterone secretion. IMA J. Math. Appl. Med. Biol. 18, 41–52 (2001)

    Article  MATH  Google Scholar 

  26. Zhang, C., Yan, X., Cui, G.: Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay. Nonlinear Anal., Real World Appl. 11, 4141–4153 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hale, J.K.: Theory of Functional Differential Equations. Spring, New York (1977)

    Book  MATH  Google Scholar 

Download references

Acknowledgement

The authors of this paper express their grateful gratitude for any helpful suggestions from reviewers and the partial support of the National Natural Science Foundation of China (11172158) and Yunnan Natural Science Foundation (2011FZ086) and the First-class Discipline of Universities in Shanghai

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Correspondence to Zengrong Liu.

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Liu, H., Yan, F. & Liu, Z. Oscillatory dynamics in a gene regulatory network mediated by small RNA with time delay. Nonlinear Dyn 76, 147–159 (2014). https://doi.org/10.1007/s11071-013-1117-z

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  • DOI: https://doi.org/10.1007/s11071-013-1117-z

Keywords

  • Small RNA
  • Time delay
  • Hopf bifurcation
  • Periodic solution
  • Oscillator