Existence criteria of periodic oscillations in cyclic gene regulatory networks

In this paper, we derive graphical and analytic criteria for the existence of periodic oscillations in large-scale cyclic gene regulatory networks, and present quantitative biological insight based on the analytic result. Based on the Poincaré-Bendixson theorem for cyclic systems, it is first shown that local instability of an equilibrium point implies the existence of periodic oscillations. Then, we prove that local instability of the heterogeneous gene regulatory network system, where dynamics of gene expression is considerably different between genes, is satisfied, if the homogeneous system is locally unstable. From this observation, the simple graphical and its equivalent analytic criteria for the existence of periodic oscillations are derived by local instability analysis of homogeneous cyclic gene regulatory networks. These criteria have the remarkable feature that they are easily applied to a large-scale cyclic gene regulatory network systems consisting of arbitrary number of genes, and thus, dynamical properties of such large-scale systems can be systematically obtained. The latter part of this paper is devoted to the rigorous investigation of the nonlinear dynamical properties. Specifically, we examine the relation between an equilibrium state and biochemical parameters, and present the analytic criterion which takes the dependence of the equilibrium state on biochemical parameters into account. Since this rigorous analytic criterion depends only on the given biochemical parameters, one can easily obtain the relation between biochemical parameters and the existence of periodic oscillations. In particular, we propose the decisive physical quantities for the existence of periodic oscillations, and reveal the quantitative relation between such physical quantities and the dynamical behaviors. Finally, ∗Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 1138656, Japan. E-mail: Yutaka [email protected] †Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 1138656, Japan. E-mail: Shinji [email protected] ‡School of Mechanical Engineering, Chung-Ang University, 221 Heukseok-dong, Dongjak-gu, Seoul 156-756, Korea. E-mail: [email protected]