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ABSTRACT: Autocatalytic pathways are frequently encountered in biological networks. One such pathway, the glycolytic pathway, is of special importance and has been studied extensively. Using tools from linear systems theory, our previous work on a simple two dimensional model of glycolysis demonstrated that autocatalysis can aggravate control performance and contribute to instability. Here, we expand this work and study properties of nonlinear autocatalytic pathway models (of which glycolysis is an example). Changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can perturb the system from the nominal operating point of the pathway. We investigate effects of such perturbations through the estimation of invariant subsets of the region of attraction around nominal operating conditions (i.e., a measure of the set of perturbations from which the cell recovers). Numerical experiments demonstrate that systems that are robust with respect to perturbations in parameter space have easily “verifiable” region of attraction properties in terms of proof complexity.
American Control Conference (ACC), 2010; 08/2010
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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 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.
American Control Conference (ACC), 2010; 08/2010
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ABSTRACT: Autocatalysis is necessary and ubiquitous in both engineered and biological systems but can aggravate control performance and cause instability. We analyze the properties of autocatalysis in the universal and well studied glycolytic pathway. A simple two-state model incorporating ATP autocatalysis and inhibitory feedback control captures the essential dynamics, including limit cycle oscillations, observed experimentally. System performance is limited by the inherent autocatalytic stoichiometry and higher levels of autocatalysis exacerbate stability and performance. We show that glycolytic oscillations are not merely a ldquofrozen accidentrdquo but a result of the intrinsic stability tradeoffs emerging from the autocatalytic mechanism. This model has pedagogical value as well as appearing to be the simplest and most complete illustration yet of Bode's integral formula.
American Control Conference, 2009. ACC '09.; 07/2009
