Compositional semantics and behavioural equivalences for reaction systems with restriction

Reaction systems are an abstract model of interactions among biochemical reactions, developed around two opposite mechanisms: facilitation and inhibition. The evolution of a reaction system is driven by the external objects which are sent into the system by the environment at each step. In order to increase the modelling expressiveness of the calculus, we consider an extension of reaction systems with restriction, which allows the hiding of entities, such as those occurring inside membranes. To this purpose, we recently developed the Reaction Algebra, a calculus resembling reaction systems extended with a restriction operator. In the present paper, three equivalent semantics for the Reaction Algebra are presented: a reduction semantics, and two state-abstract compositional semantics. The reduction semantics is meant to capture the behaviour of Reaction Algebra models at a high-level, while the two compositional semantics make the interactive nature of reaction systems explicit. The difference between the two compositional semantics lies in how the behaviour with respect to the contextual entities is described: one uses an extensional description, while the other uses an intensional one. We also define, in the settings of both compositional semantics, a behavioural equivalence subsuming the functional equivalence of reaction systems, which is also shown to be congruence, thus providing a formal ground to the modular description of models. Finally, as an example of application of the techniques developed in the paper, we compare the semantics of two different Reaction Algebra models of the functioning of the lac operon in the E. coli bacterium.