Existence conditions for oscillations in cyclic gene regulatory networks with time delay

In this paper, we present analytic criteria for the existence of oscillations in gene regulatory networks with cyclic structure. In particular, inherent time delays in transcription, translation and translocation process are explicitly treated, and the effects of such time delays are quantitatively revealed. We first show that local instability of the unique equilibrium state results in oscillations of protein concentrations based on the Poincaré-Bendixson theorem for cyclic time delay systems. Then, a scheme of local instability analysis for large-scale time delay systems is introduced, and the analytic criteria for the existence of oscillations are derived. The developed criteria can be applied to cyclic gene regulatory networks consisting of any number of genes, and they are expressed only in terms of biochemical parameters, which lead to fruitful biological insight.