A model of periodic oscillation for genetic regulatory systems

In this paper, we focus on modeling and explaining periodic oscillations in gene-protein systems with a simple nonlinear model and on analyzing effects of time delay on the stability of oscillations. Our main model of genetic regulation comprises of a two-gene system with an autoregulatory feedback loop. We exploit multiple time scales and hysteretic properties of the model to construct periodic oscillations with jumping dynamics and analyze the possible mechanism according to the singular perturbation theory. As shown in this paper, periodic oscillations are mainly generated by nonlinearly negative and positive feedback loops in gene regulatory systems, whereas the jumping dynamics is generally caused by time scale differences among biochemical reactions. This simple model may actually act as a genetic oscillator or switch in gene-protein networks because the dynamics are robust for parameter perturbations or environment variations. We also explore effects of time delay on the stability of the dynamics, showing that the time delay generally increases the stability region of the oscillations, thereby making the oscillations robust to parameter changes. Two examples are also provided to numerically demonstrate our theoretical results.