Topological-numerical analysis of global dynamics in the discrete-time two-gene Andrecut-Kauffman model

TL;DR

This paper applies topological and numerical methods to analyze the complex global dynamics of a discrete-time two-gene gene regulation model, revealing multistability and chaos across parameter ranges.

Contribution

It introduces a topological-numerical framework using Morse decomposition and Conley indices to study gene regulatory dynamics, providing new schematic symbols for interpretation.

Findings

Identification of multistability in the model

Detection of chaotic attractors under certain parameters

Demonstration of topological methods' effectiveness in biological systems

Abstract

We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use rigorous numerical methods to construct Morse decomposition of the system across a wide range of parameters. We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We introduce new symbols to convey the information provided by the Conley index in an easy to understand schematic way. We additionally conduct numerical simulations aimed at confirming the presence of complex dynamical phenomena, including multistability and the existence of chaotic attractors. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics in a gene regulatory model and highlight the richness of dynamics that can be observed in such a system when parameter values change.