The p53 protein is involved in many biological processes including cell cycle regulation, cell differentiation and apoptosis. As a tumour suppressor it is one of the subjects of interests of scientists working on the designing and improvement of cancer therapies. It also plays an important role in the treatment of neurodegenerative diseases such as cerebral ischemia, traumatic brain injury, epilepsy and Alzheimer's disease. To understand the action of the p53 protein it is important to take into account the interaction between p53 and Mdm2 proteins. The p53 gene is crucial for cellular inhibition of the angiogenesis process, while Mdm2 is a negative regulator of the p53 tumour-suppressor.

In this talk, we present the generalisation of the p53-Mdm2 protein gene expression model introduced by Monk (2003), [2]. We study the asymptotic behaviour of the solutions of our model. We investigate the stability of the positive steady state and perform some numerical experiments. Moreover,

we formulate the conditions which guarantee the occurrence of the Hopf bifurcation and we prove its stability, [1].

This is joint work with Monika Piotrowska and Marek Bodnar.

References

[1] M.J. Piotrowska, A. Bart lomiejczyk, M. Bodnar, Mathematical analysis of a generalised p53-Mdm2 protein gene expression model, to appear in Appl. Math. Comput.

[2] N.A. Monk, Oscillatory expression of Hes1, p53, and NF-B driven by transcriptional time delays, Curr. Biol. 13(2003) 1409{1413.