Equilibrium, stability and dynamical response in a model of the extrinsic apoptosis pathway

In the past few years several mathematical models have been proposed to formally represent the biochemical processes that lead to the programmed death of the cell (apoptosis) starting from an intrinsic or extrinsic stimulus. In this paper we consider the model proposed by Eissing and colleagues in 2004 and, compared to the previously published results, provide several original contributions. We prove formally that the model can have one, two or three equilibrium states; one of these (the life equilibrium) represents the normal state of the cell: we state a stability criterion for this equilibrium and prove that its stability/instability is related to the number of equilibrium states. A large sample of models with randomly generated parameter vectors (representative of a population of cells) is numerically analyzed as regards both the equilibria and respective stability properties, and the dynamical behavior. Many patterns of stable/unstable equilibrium states, and different types of bifurcations are discovered. Correlations between model parameters, equilibrium patterns and equilibrium concentration of a critical protein are also caxried out. The analysis of the dynamical time responses shows that the richness of behaviors accounted by the model is much larger than that implied by the classification into life-monostable, bistable, death- monostable models.