Existence of Oscillations in Cyclic Gene Regulatory Networks with Time Delay

TL;DR

This paper develops systematic analysis tools to determine when oscillations occur in cyclic gene regulatory networks with time delays, providing biological insights and conditions based on biochemical parameters.

Contribution

It introduces a graphical and analytical framework for predicting oscillations in gene networks with delays, applicable to various biological systems.

Findings

Oscillations linked to local instability of equilibrium.

Graphical condition for oscillation existence.

Application to Repressilator and Hes7 networks.

Abstract

This paper is concerned with conditions for the existence of oscillations in gene regulatory networks with negative cyclic feedback, where time delays in transcription, translation and translocation process are explicitly considered. The primary goal of this paper is to propose systematic analysis tools that are useful for a broad class of cyclic gene regulatory networks, and to provide novel biological insights. To this end, we adopt a simplified model that is suitable for capturing the essence of a large class of gene regulatory networks. It is first shown that local instability of the unique equilibrium state results in oscillations based on a Poincare-Bendixson type theorem. Then, a graphical existence condition, which is equivalent to the local instability of a unique equilibrium, is derived. Based on the graphical condition, the existence condition is analytically presented in terms of biochemical parameters. This allows us to find the dimensionless parameters that primarily affect the existence of oscillations, and to provide biological insights. The analytic conditions and biological insights are illustrated with two existing biochemical networks, Repressilator and the Hes7 gene regulatory networks.