Modeling the cell cycle: from deterministic models to hybrid systems.

The cell cycle is a complex biological system frequently investigated from a mathematical perspective. In fact, over the past years a huge number of deterministic mathematical models describing the dynamics and the regulation of this process have been proposed. A crucial point concerning the cell cycle modeling is the combination of continuous and discrete dynamics in order to obtain results which are coherent with the biological context. To face with this problem we propose a novel approach to the mathematical modeling of biological processes based on the use of hybrid systems. This new methodology essentially consists in a model reduction (using the modified Prony's method) which allows to define the crucial features of the dynamical system. The final aim is to implement a corresponding hybrid system which preserves the properties of the starting deterministic model. Thus, we implemented a methodology which allows to describe the cellular system by combining continuous behavior with discrete events by using the hybrid automata technology. In this way we try to overcome some drawbacks of the deterministic approach, especially regarding the possibility to introduce new variables during simulation and the associated variation of parameters in a more efficient way than the continuous method can do. We applied this innovative methodology to the reconstruction of a simplified hybrid model concerning one of the crucial mammalian cell cycle control point. In particular, we investigated the role of the transcription factors E2F in the R-point transition. The resulting hybrid model preserve the properties of the deterministic one and it allows the identification of the parameter which controls the transition from the inactive (quiescent) to the active state (R-point transition) after the mitogenic stimulation. At the best of our knowledge no hybrid model for the R-point transition are available in literature.