Hybrid Simulations of Heterogeneous Biochemical Models in SBML

Models of biochemical systems presented as a set of formal reaction rules can be interpreted in different formalisms, most notably as either deterministic Ordinary Differential Equations, stochastic continuous-time Markov Chains, Petri nets, or Boolean transition systems. While the formal composition of reaction systems can be syntactically defined as the (multiset) union of the reactions, the composition and simulation of models in different formalisms remain a largely open issue. In this article, we show that the combination of reaction rules and events, as already present in SBML, can be used in a nonstandard way to define stochastic and Boolean simulators and give meaning to the hybrid composition and simulation of heterogeneous models of biochemical processes. In particular, we show how two SBML reaction models can be composed into one hybrid continuous--stochastic SBML model through a high-level interface for composing reaction models and specifying their interpretation. Furthermore, we describe dynamic strategies for automatically partitioning reactions with stochastic or continuous interpretations according to dynamic criteria. The performances are then compared to static partitioning. The proposed approach is illustrated and evaluated on several examples, including the reconstructions 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 with a continuous model of cyclin activation, the hybrid stochastic--continuous models of bacteriophage T7 infection of Alfonsi et al., and the bacteriophage λ model of Goutsias, showing the gain in both accuracy and simulation time of the dynamic partitioning strategy.