Domain-specific rule-based languages to represent the systems of reactions that occur inside cells, such as Kappa and BioNetGen, have attracted significant recent interest. For these models, powerful simulation and static analysis techniques have been developed to understand the behaviour of the systems that they represent, and these techniques can be transferred to other fields. The languages can be understood intuitively as transforming graph-like structures, but due to their expressivity these are difficult to model in 'traditional' graph rewriting frameworks. In this paper, we introduce pattern graphs and closed morphisms as a more abstract graph-like model and show how Kappa can be encoded in them by connecting its single-pushout semantics to that for Kappa. This level of abstraction elucidates the earlier single-pushout result for Kappa, teasing apart the proof and guiding the way to richer languages, for example the introduction of compartments within cells.