Efficient reduction of Kappa models by static inspection of the rule-set

When designing genetic circuits, the typical primitives used in major existing modelling formalisms are gene interaction graphs, where edges between genes denote either an activation or inhibition relation. However, when designing experiments, it is important to be precise about the low-level mechanistic details as to how each such relation is implemented. The rule-based modelling language Kappa allows to unambiguously specify mechanistic details such as DNA binding sites, dimerisation of transcription factors, or co-operative interactions. However, such a detailed description comes with complexity and computationally costly execution. We propose a general method for automatically transforming a rule-based program, by eliminating intermediate species and adjusting the rate constants accordingly. Our method consists of searching for those interaction patterns known to be amenable to equilibrium approximations (e.g. Michaelis-Menten scheme). The reduced model is efficiently obtained by static inspection over the rule-set, and it represents a particular theoretical limit of the original model. The Bhattacharyya distance is proposed as a metric to estimate the reduction error for a given observable. The tool is tested on a detailed rule-based model of a $\lambda$-phage switch, which lists $96$ rules and $16$ agents. The reduced model has $11$ rules and $5$ agents, and provides a dramatic reduction in simulation time of several orders of magnitude.