Multiset reaction systems with saturation and permanence

We propose an extension to multisets of the model of reaction systems, classically based on sets. In the proposed model, four main variations are present with respect to the classical model: (1) reactions (here called m-reactions) have multisets rather than sets in their reactant, inhibitor, and product components; (2) states are multisets of entities, rather than sets; (3) based on the numerosity of the entities in the state, several instances of non-conflicting m-reactions can be simultaneously applied, forming a block; and (4) entities not consumed in the application of a block remain in the state, to which the cumulatively produced multiset of entities is added. Threshold phenomena (reaching a critical mass enabling new m-reactions or saturating the state with entities inhibiting m-reactions to be enabled) can therefore be observed, so that some context-independent behaviours can be observed, which are not possible in the classical case. The paper explores some formal consequences of the proposed model and illustrates its relevance by presenting a model of the dynamics of the heat shock response cellular regulatory mechanism.