Bulletin of Mathematical Biology

, Volume 67, Issue 2, pp 339–367

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems


DOI: 10.1016/j.bulm.2004.07.005

Cite this article as:
Wang, R., Jing, Z. & Chen, L. Bull. Math. Biol. (2005) 67: 339. doi:10.1016/j.bulm.2004.07.005


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.

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