On the existence of oscillatory behavior in gene regulatory networks with cyclic interconnections

In this paper, we derive an existence condition of periodic oscillations in cyclic gene regulatory networks, of which the dynamical behavior is described by nonlinear differential equations. For this purpose, we first point out that change of the equilibrium point with respect to biochemical parameters is important for comprehensive analysis, and show the properties which the equilibrium point should satisfy. Then, by employing these properties, we derive an analytic condition for the existence of periodic oscillations in cyclic gene regulatory networks where genes have homogeneous dynamics. Compared to preceding studies, our condition has the following distinctive features: (i)it is analytically written only in terms of the system parameters, (ii)it is applicable for cyclic gene regulatory networks consisting of any number of genes. Thus, biological interpretation of the result is easily obtained, and, in fact, it is concluded that cyclic gene regulatory networks with homogeneous gene dynamics tend to have periodic oscillations as the number of genes gets large. Finally, we discuss the case of cyclic gene regulatory networks with heterogeneous gene dynamics.