On the hybrid composition and simulation of heterogeneous biochemical models

Models of biochemical systems presented as a set of formal reaction rules with kinetic expressions can be interpreted with different semantics: as either deterministic Ordinary Differential Equations, stochastic continuous-time Markov Chains, Petri nets or Boolean transition systems. While the formal composition of reaction models can be syntactically defined as the (multiset) union of the reactions, the hybrid composition of models in different formalisms is a largely open issue. In this paper, we show that the combination of reaction rules with conditional events, as the ones already present in SBML, does provide the expressive power of hybrid automata and can be used in a non standard way to give meaning to the hybrid composition of heterogeneous models of biochemical processes. In particular, we show how hybrid differentialstochastic and hybrid differential-Boolean models can be compiled and simulated in this framework, through the specification of a high-level interface for composing heterogeneous models. This is illustrated by a hybrid stochastic-differential model of bacteriophage T7 infection, and by a reconstruction of the hybrid model of the mammalian cell cycle regulation of Singhania et al. as the composition of a Boolean model of cell cycle phase transitions and a differential model of cyclin activation.