Abstract
In this paper, a p53-Mdm2 network model with Michaelis–Menten function involved in DNA damage repair is studied by using the dynamical system approach. Detailed bifurcations of the model including saddle-node bifurcation, Hopf bifurcation of codimension 3, and cusp-type Bogdanov–Takens bifurcation of codimension 3 are investigated. Meanwhile, the coexistence of three limit cycles and the coexistence of a homoclinic cycle and a limit cycle are also obtained. To our best knowledge, it is the first time that the coexistence of an isola and a cusp of limit cycles is observed for the p53 model, which indicates complex transitions of different oscillating dynamics. Furthermore, we find that oscillation could persist whether the damage is nonexistent, small or large, and multiple oscillations can be involved in the DNA repair process. In addition, the dynamics induced by the interaction between p53 and Mdm2 unveil the relationship between the positive feedback brought by p53 itself and the negative feedback brought by Mdm2, which triggers the digital oscillation modes for DNA damage. Of particular interest is that multiple oscillations and the threshold value of DNA damage are found to reveal the intrinsic mechanism of DNA damage repair.
