Pushouts in Computational Systems Biology

Rule-based formalisms for modelling biochemical pathways such as Kappa, BioNetGen and PySB have emerged as powerful tools for the analysis of biochemical pathways. They allow concise, intuitive models to be developed, and by avoiding through the use of rules the combinatorial explosion in the number of biochemical species that befalls traditional techniques, they make in-silico experimentation and simulation more feasible. However, the growing number of rule-based modelling languages calls for a general, language-independent theory. The present paper describes an abstract categorical approach based on singlepushout rewriting, in which mixtures of biochemical species are modelled as objects in a category of graph-like objects and system evolution corresponds to taking a pushout over a rule and one of its “redexes” or “matchings”, both of which are formalized as special forms of partial map. The main theorem establishes a sufficient condition for the existence of pushouts of partial maps in arbitrary categories; the condition is necessary locally, i.e. it is necessary in all slice categories. The relevance of this theoretical result is illustrated by showing that the condition can be lifted to arbitrary full subcategories, yielding a general method to develop categorical accounts of models in a way that has already been applied successfully to Kappa.