| Summary: | The p53 oscillator is an essential cell fate controller after DNA damage. In general, the delayed p53-Mdm2 negative feedback loop is thought to be the foundational structure responsible for p53 oscillations. However, it is experimentally found that Mdm2 also has a positive effect on p53. Therefore, we redevelop a p53 model to investigate the effect of the p53-Mdm2 positive feedback loop and time delay on p53 oscillations. To study the effect of delay on the Hopf bifurcation, we analyze the characteristic equation of the linearized system and apply the central manifold method to determine the properties of the Hopf bifurcation. The theoretical results are proved numerically. Furthermore, we reveal that the delay induces not only oscillatory birth but also oscillatory death, depending on the parameters in the positive feedback. Finally, considering the biological background, we discuss the effects of the p53-Mdm2 positive feedback parameters on the amplitude of p53 oscillations. This work may be helpful for artificially altering p53 dynamics and even cell fate.
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