Piecewise parametric chaotic model of p53 network based on the identified unifying framework of divergent p53 dynamics

Being the core of the large p53 protein (cancer) network, the ATM-p53-Wip1 sub-network is known to confer monostable oscillations and fixed levels of p53 to the p53 network. Using a 2-Dimensional parametric model of this sub-network proposed by [68], we seek to know what other dynamical patterns are possible. Not to miss out on any pattern, we analytically identify disjoint behavioral regions whose union covers the entire domain of equilibria in phase space. There exist five qualitatively different phase portraits leading to monostable dynamics, birhythmicity, single pulse dynamics and, typical and atypical bistability. Such patterns are known to exist in the p53 network in wet-lab settings, which evokes the idea that the underlying structure might be the ATM-p53-Wip1 sub-network. Additionally, the extension of the model by an accumulating variable, modeling the pro-apoptotic components of the p53 network, has the potential to construct a homoclinic orbit, which is shown in a piecewise fashion. Then, we propose a piecewise system by explicitly embedding the perceived HC orbit using a biologically meaningful sliding surface. Using the Shilnikov Theorem for piecewise smooth systems, we demonstrate mathematically the previously unobserved possibility of a chaotic mechanism in the p53 network, which awaits further biological confirmation. Chaos is observed under the stringent and pathological conditions of failed-apoptosis and unrepairable DNA damage. Thus, our results support the hypothesis that chaotic gene expressions accompany tumor formation.