Qualitative Behavior of a Metabolic Pathway with Hybrid Feedback

TL;DR

This paper analyzes a metabolic pathway model with hybrid feedback, demonstrating its stability, oscillatory behavior, and the convergence of solutions from smooth approximations, providing insights into local regulation mechanisms.

Contribution

It introduces a differential inclusion model for metabolic regulation with hybrid feedback and proves stability, oscillation, and solution convergence properties.

Findings

The model converges to a globally stable equilibrium or oscillates depending on input.

Solutions of the switched system are limits of smooth or piecewise linear input solutions.

The system exhibits sliding mode behavior under certain conditions.

Abstract

We study the qualitative behavior of a model to represent local regulation in a metabolic network. The model is based on the end-product control structure introduced in [A. Goelzer, F. Bekkal Brikci, I. Martin-Verstraete et al., BMC Syst Biol 2 (2008), pp. 1--18]. In this class of regulation, the metabolite effector is the end-product of a metabolic pathway. We suppose the input to the pathway to switch between zero and a positive value according to the concentration of the metabolite effector. Considering the switched system as a differential inclusion, we prove that it converges to a globally uniformly asymptotically stable equilibrium point, reaches the sliding mode or oscillates around the sliding mode depending on the positive value of the input. Finally, we show that in any case the solution of the switched system is the limit of solutions of equation sequences with smooth or piecewise linear inputs.