Compositional analysis of autocatalytic networks in biology

Abstract

Abstract-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 blockdiagonal 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.

Quantitative nonlinear analysis of autocatalytic pathways with applications to glycolysis

“…Let U 0 and η 0 denote the optimal values of η and U in (12) and α 0 be the largest of α such that Ω U0,α0 ⊆ Ω ξ T ξ,η0 . Then, U 0 and α are feasible for (11) ( [10]).…”

“…We therefore limit our analysis to small dimensional state space models. For larger dimensional systems we emphasize that exploiting the underlying structure of the pathway (i.e., a chain of intermediate reaction coupled with a autocatalytic component) leads to more efficient compositional analysis tools (e.g., see [12]).…”

L<inf>2</inf> gain verification for interconnections of locally stable systems using integral quadratic constraints

“…Individual analysis on the isolated subsystems reveals coarse properties of the subsystems (eg., passive, small-gain or generalizations, such as dissipativeness with respect to various supply rates, etc.). In some cases, the coarse properties, coupled with the interconnection topology is enough to verify the overall behavior, [15], [7], [4]. In special cases, optimization can select which coarse properties are most important [13].…”