CDS Thesis Seminar: Control Theoretic Analysis of Autocatalytic Networks in Biology With Applications to Glycolysis

Gentian Buzi

ABSTRACT: Autocatalytic networks (pathways) are a necessary part of core metabolism. In every cell, food and resources are broken down to create energy and components via processes that also require the use of those same components and energy. Indeed at a certain level all biological networks are massively autocatalytic. The simplest and most widely studied autocatalytic network is the glycolytic pathway. This metabolic pathway is used by the cell to produce energy anaerobically (without oxygen) by breaking down glucose. It is probably the most common autocatalytic pathway on the planet, found in every cell of living organisms, from bacteria to humans. Its special autocatalytic structure, like the structure of many similar autocatalytic networks, can lead to instabilities and makes it particularly hard to control.

In this thesis, we study autocatalytic metabolic networks, and specifically glycolysis, to investigate fundamental performance tradeoffs in these network topologies. Using classical linear control theory, we hypothesize that the instabilities in glycolysis, which are exhibited in the form of oscillation are a result of performance tradeoffs that stem from the structure of the pathways and a conservation law mathematically described by a special form of the Bode Sensitivity Integral. We show that the size of the pathway and the consumption of the intermediate metabolites by other processes in the cell adversely affect the performance of the pathway, while reversibility of chemical reactions improves performance. We also establish tight stability bounds for the feedback control gains, which guarantee stability of pathways of arbitrary size and arbitrary parameter values of the intermediate reactions.

In addition 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 address the question of whether the controller can restore the system to normal operating conditions from these perturbations, and the maximum perturbations a cell can sustain before it dies. We investigate effects of such perturbations through the estimation of invariant subsets of the region of attraction (RoA) around nominal operating conditions (i.e., a measure of the set of perturbations from which the cell recovers). The numerical procedure for estimating the RoA is composed of system theoretic characterizations and optimization-based formulations. For large, computationally intractable systems we employ a different technique based on the underlying biological structure, which offers a natural decomposition of the system 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 and leads to analytical construction of Lyapunov functions for a large family of autocatalytic pathways.

The results of our analysis of these autocatalytic networks reveal fundamental tradeoffs between performance and robustness, energy efficiency, evolvability of the pathway, and computational complexity.

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