Bistability and chaos in the discrete two-gene Andrecut-Kauffman model

TL;DR

This paper analyzes the dynamics of a discrete two-gene model, revealing bifurcations, chaos, and bistability, with implications for understanding gene regulation unpredictability.

Contribution

It provides a detailed numerical analysis of the model with varying reaction rates, uncovering new bifurcation structures, chaos, and bistability not previously characterized.

Findings

Chaotic dynamics identified via Lyapunov exponents.

Bistability with two disjoint attractors found.

Parameter regions with mixed attractor types.

Abstract

We conduct numerical analysis of the 2-dimensional discrete-time gene expression model originally introduced by Andrecut and Kauffman (Phys. Lett. A 367: 281-287, 2007). In contrast to the previous studies, we analyze the dynamics with different reaction rates $\alpha_1$ and $\alpha_2$ for each of the two genes under consideration. We explore bifurcation diagrams for the model with $\alpha_1$ varying in a wide range and $\alpha_2$ fixed. We detect chaotic dynamics by means of the positive maximum Lyapunov exponent and we scan through selected parameters to detect those combinations for which chaotic dynamics can be found in the model. Moreover, we find bistability in the model, that is, the existence of two disjoint attractors. Both situations are interesting from the point of view of applications, as they imply unpredictability of the dynamics encountered. Finally, we show some specific values of parameters of the model in which the two attractors are of different kind (a periodic orbit and a chaotic attractor) or of the same kind (two periodic orbits or two chaotic attractors).