Theoretical analysis of the delay on the p53 micronetwork

Abstract Time delay plays a crucial role in p53 dynamic. However, the theoretical understanding is still lower. Thus we construct a micro-differential equation model and introduce the time delay τ based on the regulation process. Firstly, we linearize the system and analyze the associated characteristic equation. We can conclude that there exists the delay threshold τ 0 $\tau _{0}$ such that when the delay τ is less than τ 0 $\tau _{0}$ , the system is asymptotically stable and otherwise stable oscillations occur. Secondly, we use the normal form method and the center manifold theorem to derive the direction and stability of the Hopf bifurcation. Finally, by numerical simulations we verify our theoretical results. We also find that the effect of noise on the amplitude is more severe than that of the period, which well agrees with the experimental results.