Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model

Biological cell networks exhibit complex combinations of both discrete and continuous behaviors: indeed, the dynamics that govern the spatial and temporal increase or decrease of protein concentration inside a single cell are continuous differential equations, while the activation or deactivation of these continuous dynamics are triggered by discrete switches which encode protein concentrations reaching given thresholds. In this paper, we model as a hybrid system a striking example of this behavior in a biological mechanism called Delta-Notch signaling, which is thought to be the primary mechanism of cell differentiation in a variety of cell networks. We present results in both simulation and reachability analysis of this hybrid system. We emphasize how the hybrid system model is computationally superior (for both simulation and analysis) to other nonlinear models in the literature, without compromising faithful modeling of the biological phenomena.