Compositional analysis of autocatalytic networks in biology

Autocatalytic pathways are a necessary part of core metabolism. Every cell consumes external food/resources to create components and energy, but does so using processes that also require those same components and energy. Here, we study effects of parameter variations on the stability properties of autocatalytic pathway models and the extent of the regions of attraction around nominal operating conditions. Motivated by the computational complexity of optimization-based methods for estimating regions of attraction for large pathways, we take a compositional approach and exploit a natural decomposition of the system, induced by the underlying biological structure, into a feedback interconnection of two input-output subsystems: a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics. This decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input-output properties of the subsystems. It enables us to numerically construct block-diagonal Lyapunov functions for families of pathways that are not amenable to direct analysis. Furthermore, it leads to analytical construction of Lyapunov functions for a large family of autocatalytic pathways.