Oscillations in a cAMP signalling model for cell aggregation -- a geometric analysis
type: Item
instance of: scholarly article
title: Oscillations in a cAMP signalling model for cell aggregation -- a geometric analysis
zbMATH Open document ID: 1431.34067
author: Zhouqian Miao
author: Peter Szmolyan
author: Nikola Popović
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: In this article, the authors perform a geometric analysis of a singularly perturbed model for a cyclic adenosine monophosphate (cAMP) signalling system that controls aggregation of the amoeboid microorganism \textit{Dictyostelium discoideum} and which can be rewritten into the form of a planar system of ordinary differential equations with two small parameters \(\kappa\) and \(\varepsilon,\) \[ R'=\kappa(U+\mathcal{P}\varepsilon)\frac{(U+\varepsilon)^2+\Lambda R^2 U^2}{(U+\varepsilon)(U+\frac{\varepsilon}{c})}\left[\tilde\mu\varepsilon^{\frac12}(U+\varepsilon)-(U+\tilde d\varepsilon^{\frac32})R \right], \] \[ U'=\tilde b\varepsilon^2(U+\varepsilon)^2+\Theta R^2 U^2 -U\left[(U+\varepsilon)^2 +\Lambda R^2 U^2\right], \] where the state variables \(R\) and \(U\) correspond to the total proportion of receptors in the active state and the concentration of extracellular cAMP, respectively. Using the geometric singular perturbation theory and blow-up techniques, the authors prove that for \(\varepsilon\in(0,\varepsilon_0]\) there exists \(\kappa_0=\kappa_0(\varepsilon_0)\) such that for \(\kappa\in(0,\kappa_0]\) the above system admits a unique family of attracting periodic orbits \(\Gamma_{\kappa\varepsilon}\) which tends to a singular cycle \(\Gamma_{0\varepsilon}\) as \(\kappa\to0\) uniformly for \(\varepsilon\in(0,\varepsilon_0],\) and to a singular cycle \(\Gamma_{00}\) as \((\kappa,\varepsilon)\to(0,0).\)
reviewed by: Robert Vrabel
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C26
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 92C37
zbMATH DE Number: 7138437
zbMATH Keywords: cAMP signalling
zbMATH Keywords: singular perturbation
zbMATH Keywords: geometric singular perturbation theory
zbMATH Keywords: blow-up technique
zbMATH Keywords: relaxation oscillation
zbMATH Keywords: periodic orbit
MaRDI profile type: Publication
OpenAlex ID: W2980071864
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Wikidata QID: Q127084318
DOI: 10.1016/J.JMAA.2019.123577
description: scientific article; zbMATH DE number 7138437