Bulletin of Mathematical Biology

, Volume 67, Issue 2, pp 339–367 | Cite as

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems



In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.


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  1. Barbai, N., Leibler, S., 2000. Biological rhythms: circadian clocks limited by noise. Nature 403, 267.Google Scholar
  2. Becskei, A., Serrano, L., 2000. Engineering stability in gene networks by autoregulation. Nature 405, 590–593.CrossRefGoogle Scholar
  3. Belair, J., Campbell, S.A., Driessche, P., 1996. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56, 245–255.MathSciNetCrossRefGoogle Scholar
  4. Campbell, S.A., Ruan, S., Wei, J., 1999. Qualitative analysis of a neural network model with multiple time delays. Int. J. Bifurcation Chaos 9, 1585–1595.MathSciNetCrossRefGoogle Scholar
  5. Chen, L., Aihara, K., 2002a. Stability of genetic regulatory networks with time delay. IEEE Trans. Circuits Syst. I. 49, 602–608.MathSciNetCrossRefGoogle Scholar
  6. Chen, L., Aihara, K., 2002b. A model of periodic oscillation for genetic regulatory systems. IEEE Trans. Circuits Syst. I. 49, 1429–1436.MathSciNetCrossRefGoogle Scholar
  7. Chen, L., Wang, R., Kobayashi, T., Aihara, K., 2004. Dynamics of gene regulatory networks with cell division cycles. Phys. Rev. E 70, 011909, 1–13.MathSciNetGoogle Scholar
  8. Crosthwaite, S.K., Dunlap, J.C., Loros, J.J., 1997. Neurospora wc-1 and wc-2: transcription, photoresponses, and the origin of circadian rhythmicity. Science 276, 763–769.CrossRefGoogle Scholar
  9. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O., 1995. Delay Equations Functional-, Complex-, and Nonlinear Analysis. Springer-Verlag.Google Scholar
  10. Dunlap, J.C., 1999. Molecular bases for circadian clocks. Cell 96, 271–290.CrossRefGoogle Scholar
  11. Elowitz, M.B., Leibler, S., 2000. A synthetic oscillatory network of transcriptional regulators. Nature 403, 335–338.CrossRefGoogle Scholar
  12. Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S., 2002. Stochastic gene expression in a single cell. Science 297, 1183–1186.CrossRefGoogle Scholar
  13. Gardner, T.S., Cantor, C.R., Collins, J.J., 2000. Construction of a genetic toggle switch in Escherichia Coli. Nature 403, 339–342.CrossRefGoogle Scholar
  14. Glass, L., Kauffman, S.A., 1973. The logical analysis of continuous, non-linear biochemical control networks. J. Theor. Biol. 39, 103–129.CrossRefGoogle Scholar
  15. Goldbeter, A., 1995. A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B 261, 319–324.Google Scholar
  16. Golden, S.S., Ishiura, M., Johnson, C.H., Kondo, T., 1997. Cyanobacterial circadian rhythms. Ann. Rev. Plant Physiol. Plant Mol. Biol. 48, 327–354.CrossRefGoogle Scholar
  17. Gonze, D., Leloup, J.-C., Goldbeter, A., 2000. Theoretical models for circadian rhythms in Neurospora and Drosophila. C. R. Acad. Sci. Paris, III 323, 57–67.Google Scholar
  18. Goodwin, B.C., 1965. Oscillatory behavior in enzymatic control process. Adv. Enzyme Regua. 3, 425–438.CrossRefGoogle Scholar
  19. Gouze, J.-L., 1998. Positive and negative circuits in dynamical systems. J. Biol. Syst. 6, 11–15.MATHCrossRefGoogle Scholar
  20. Hall, J.C., 1998. Genetics of biological rhythms in Drosophila. Adv. Genet. 38, 135–184.CrossRefGoogle Scholar
  21. Hasty, J., Isaacs, F., Dolnik, M., McMillen, D., Colins, J.J., 2001. Designer gene networks: towards fundamental cellular control. Chaos 11, 207–220.CrossRefGoogle Scholar
  22. Hunding, A., 1974. Limit cycles in enzyme systems with nonlinear negative feedback. Biophys. Struct. Mech. 1, 47–54.CrossRefGoogle Scholar
  23. Kobayashi, T., Chen, L., Aihara, K., 2002. Design of genetic switches with only positive feedback loops. In: Proceedings of 2002 IEEE Computer Society Bioinformatics Conference. pp. 151–162.Google Scholar
  24. Kobayashi, T., Chen, L., Aihara, K., 2003. Modelling genetic switches with positive feedback loops. J. Theor. Biol. 221, 379–399.MathSciNetCrossRefGoogle Scholar
  25. Kolmanovskii, V., Myshkis, A., 1999. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers.Google Scholar
  26. Leloup, J.-C., Goldbeter, A., 1998. A model for circadian rhythms in Drosophila incorporating the formation of a complex between the PER and TIM proteins. J. Biol. Rhythms 13, 70–87.CrossRefGoogle Scholar
  27. Leloup, J.-C., Goldbeter, A., 2000. Modelling the molecular regulatory mechanism of circadian rhythms in Drosophila. BioEssays 22, 84–93.CrossRefGoogle Scholar
  28. Mallet-Paret, J., Sell, G.R., 1996a. Systems of differential delay equations: floquet multipliers and discrete Lyapunov Functions. J. Differ. Eqns. 125, 385–440.MathSciNetCrossRefGoogle Scholar
  29. Mallet-Paret, J., Sell, G.R., 1996b. The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay. J. Differ. Eqns. 125, 441–489.MathSciNetCrossRefGoogle Scholar
  30. Mikkar, A.J., Carre, I.A., Strayer, C.A., Chua, N.H., Kay, S.A., 1995. Circadian clock mutants in Arabidopsis identified by iuciferase imaging. Science 267, 1161–1163.Google Scholar
  31. Smith, H., 1991. Periodic tridiagonal competitive and cooperative systems of differential equations. SIAM J. Math. Anal. 22, 1102–1109.MATHMathSciNetCrossRefGoogle Scholar
  32. Smith, H., 1995. Monotone Dynamical Systems, vol. 41. American Mathematical Society, Providence, RI.Google Scholar
  33. Swain, P.S., Elowitz, M.B., Siggia, E.D., 2002. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc. Natl. Acad. Sci. 99, 12795-12800.Google Scholar
  34. Tei, H., Okamura, H., Shigeyoshi, Y., Fukuhara, C., Ozawa, R., Hirose, M., Sakaki, Y., 1997. Circadian oscillation of a mammalian homologue of the Drosophila period gene. Nature 389, 512–516.CrossRefGoogle Scholar
  35. Walther, H.O., 1995. The 2-dimensional attractor of x(t) = −μx(t) + ƒ (x(t–1)). Memoirs of the Amer. Math. Soc., 544. Amer. Math. Soc, Providence, RI.Google Scholar
  36. Wang, R., Jing, Z., Chen, L., 2004a. Periodic oscillators in genetic networks with negative feedback loops. WSEAS Trans. Math. 3, 150–156.Google Scholar
  37. Wang, R., Zhou, T., Jing, Z., Chen, L., 2004b. Modelling periodic oscillation of biological systems with multiple time scale networks. Syst. Biol. 1, 71–84.CrossRefGoogle Scholar
  38. Wu, J., Zou, X., 1995. Patterns of sustained oscillations in neural networks with delayed interactions. Appl. Math. Comput. 73, 55–75.MathSciNetCrossRefGoogle Scholar
  39. Zhou, T., Chen, L., Aihara, K., 2004. Intercellular communications induced by random fluctuations (submitted for publication).Google Scholar

Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  1. 1.Osaka Sangyo UniversityOsakaJapan
  2. 2.Hunan Normal UniversityChangsha, HunanChina
  3. 3.Department of Electrical Engineering and ElectronicsOsaka Sangyo UniversityOsakaJapan
  4. 4.Academy of Mathematics and System SciencesCASBeijingChina

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