Insights from a qualitative analysis of a gene expression model with delays

Hakki Ulas Unal (1) , Marc Roussel (2) , Islam Boussaada (3, 4) , Silviu-Iulian Niculescu (3, 4)

1 Eskişehir Technical University
2 University of Lethbridge
3 L2S - Laboratoire des signaux et systèmes
4 DISCO - Dynamical Interconnected Systems in COmplex Environments

Delays appear in the dynamics of many systems due to non-vanishing reaction times of control systems. In biochemical systems, long sequences of repeated steps, especially in biopolymerization processes, can be modeled by delays. However, modelling systems with delays is often complicated by physical constraints, such as the requirement that solutions representing concentrations of chemical species remain positive. In this work, we consider a model for a detoxifying enzyme whose synthesis is controlled by its substrate. The model includes bindingsite clearance delays, caused by the time required for an RNA polymerase or ribosome to clear its binding site before another such machine can bind. The existence of a positive equilibrium and the positivity and boundedness of solutions of the corresponding delay-differential equations are proven. In addition, the stability of the model is studied using the "small-gain" theorem.

Mots clés: Delay systems; biochemical systems; gene-regulatory systems

Domaines: Automatique / Robotique

Hakki Ulas Unal, Marc Roussel, Islam Boussaada, Silviu-Iulian Niculescu. Insights from a qualitative analysis of a gene expression model with delays. MTNS 2021 -24th International Symposium on Mathematical Theory of Networks and Systems, organisée par I. BOUSSAADA, G. MAZANTI, S.-I. NICULESCU, J. CHEN, Aug 2021, Cambridge, United Kingdom. ⟨10.1016/j.ifacol.2021.06.174⟩. ⟨hal-03108576⟩