Hybrid systems and biology. continuous and discrete modeling for systems biology

Hybrid Systems are dynamical systems presenting both discrete and continuous evolution. Hybrid Automata are a formal model for hybrid systems, originally proposed to study embedded systems, where a discrete control acts on a continuously changing environment. The presence of both discrete and continuous dynamics makes this formalism appealing also for modeling biological systems. However, the situation in this case is subtler, as there is no natural separation into discrete and continuous components. No surprise, then, that hybrid automata have been used in systems biology in rather different ways. Some approaches, like the description of biological switches, concentrate on the use of model-checking routines. Other applications, like the switching between continuous and discrete/stochastic simulation, focus on the exploitation of the interplay between discreteness and continuity in order to reduce the computational burden of numerical simulation, yet maintaining an acceptable precision. We will survey the use of hybrid automata in systems biology, through a series of cases studies that we deem interesting and paradigmatic