Multi-level modelling via stochastic multi-level multiset rewriting

We present a simple stochastic rule-based approach to multilevel modelling for computational systems biology. Populations are modelled using multilevel multisets; these contain both species and agents, with the latter possibly containing further such multisets. Rules are pairs of such multisets, but now allowing variables to occur (as well as species and agents), together with an associated stochastic rate. We give two illustrative examples. The first is an extracellular model of virus infection, coupled with an intracellular model of viral reproduction; this model can demonstrate successive waves of infection. The second is a model of cell division in which a repressor protein is diluted in successive generations, when repression no longer occurs. The multilevel multiset approach can also be seen in terms of stochastic term rewriting for the theory of a commutative monoid, equipped with extra constants (for the species) and unary operations (for the agents). We further discuss the relationship of this approach with two others: Krivine et al.’s stochastic bigraphs, restricted to Milner’s place graphs, and Coppo et al.’s Stochastic Calculus of Wrapped Compartments. These various relationships provide evidence for the fundamental nature of the approach.