A Calculus of Chemical Systems

We present the Calculus of Chemical Systems for the modular presentation of systems of chemical equations; it is intended to be a core calculus for rule-based modelling in systems biology. The calculus is loosely modelled after Milner’s Calculus of Communicating Systems, but with communication replaced by chemical reactions. We give a variety of compositional semantics for qualitative and quantitative versions of our calculus, employing a commutative monoid semantical framework. These semantics include (qualitative and quantitative) Petri nets, transition relations, ordinary differential equations (ODEs), and stochastic matrices. Standard semantics of Petri nets, whether of transition relations, ODEs, or stochastic matrices, fit within the framework as commutative monoid homomorphisms. We give complete equational axiomatisations and normal forms for all the semantics, and full abstraction results for the ODE and stochastic semantics. Definability can be characterised in some cases, as was already known for ODEs; other cases, including the stochastic one, remain open.