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.
Unable to display preview. Download preview PDF.
- Barbai, N., Leibler, S., 2000. Biological rhythms: circadian clocks limited by noise. Nature 403, 267.Google Scholar
- 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
- Goldbeter, A., 1995. A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B 261, 319–324.Google Scholar
- 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
- 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
- Kolmanovskii, V., Myshkis, A., 1999. Introduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers.Google Scholar
- 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
- Smith, H., 1995. Monotone Dynamical Systems, vol. 41. American Mathematical Society, Providence, RI.Google Scholar
- 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
- 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
- Wang, R., Jing, Z., Chen, L., 2004a. Periodic oscillators in genetic networks with negative feedback loops. WSEAS Trans. Math. 3, 150–156.Google Scholar
- Zhou, T., Chen, L., Aihara, K., 2004. Intercellular communications induced by random fluctuations (submitted for publication).Google Scholar