Potential landscape and flux of p53-Mdm2 oscillator mediated by Mdm2 degradation rate

The dynamics of the tumor suppressor p53 can play a crucial role in deciding cell fate after DNA damage. In this paper, we explore the dynamics and stability of p53 mediated by Mdm2 degradation rate in p53-Mdm2 oscillator through bifurcation, the potential landscape and flux. Based on the investigation of the bifurcation, we find that p53 can exhibit rich dynamics including monostability, bistability of two stable steady states and oscillation behaviors as well as bistability between a stable steady state and an oscillatory state. The stability of these states are further validated by the potential landscape. In addition, oscillatory behaviors of p53 are explored by means of the negative gradient of the potential landscape and the probability flux. It is shown that the negative gradient of the potential landscape can attract the system towards the oscillatory path and the flux can drive oscillation along the path. Moreover, the quicker the flux runs, the smaller the period is. Besides, stability and sensitivity of the system are explored by the barrier height and the entropy production rate in a single cell level, and we further compare the potential landscapes at single and population cell levels. Our results may be useful for understanding the regulation of p53 signaling pathways in response to DNA damage.