MATHEMATICAL ENGINEERING TECHNICAL REPORTS Oscillations in Cyclic Gene Regulatory Networks with Time Delay: Analytic Existence Conditions with Biological Insight

In this paper, we present analytic conditions 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 revealed. We first show that local instability of a unique equilibrium state results in oscillations of protein concentrations based on the PoincaréBendixson theorem for cyclic time delay systems. Then, an analysis scheme of local instability for large-scale time delay systems is introduced. Using this scheme, we derive graphical and analytic conditions for the existence of oscillations. The developed analytic conditions are represented only in terms of biochemical parameters, thus they are suitable for obtaining biological insights. The results are applied to existing genetic networks, and biological insights are demonstrated with illustrative numerical simulations.