Hopf bifurcations in an atypical p53 protein oscillator model with discrete delay or distributed delay

Abstract

Human cells have a natural cancerous defense system, and p53 protein is at the core. It is well known that p53 relies on oscillatory dynamic behavior to perform tumor suppressor functions. However, the mechanism of its oscillation remains unclear. To reveal the effect of time lag on the onset of p53 oscillations, we analytically and numerically investigate a possible scenario in which p53 oscillations are driven by a time lag. The model has only three nodes, two negative loops, and one delay. We focus on two types of time delays: discrete and distributed, use Hopf bifurcation theory to derive conditions for the occurrence of limit cycle oscillations, and confirm the properties of Hopf bifurcations by normal form and central manifold approaches. The analytical results are supported by numerical results. We suggest that discrete time lag promote p53 oscillations, while distributed time lag have a dual effect on p53 oscillations. These results imply that controlling the time lag in an appropriate range favors the inhibition of p53 on cancer cells.