Transient dynamics around unstable periodic orbits in the generalized repressilator model

TL;DR

This paper analyzes the dynamics of the generalized repressilator, revealing conditions for unstable periodic orbits that cause long-lived oscillations relevant in biological and synthetic systems.

Contribution

It provides analytical conditions for unstable periodic orbits and links their properties to traveling wave solutions and symmetries in the model.

Findings

Unstable periodic orbits lead to long-lived oscillatory transients.

Spatial symmetries characterize the family of unstable orbits.

Analogies with electronic systems like magnetic flux gates are established.

Abstract

We study the spatio-temporal dynamics of the generalized repressilator, a system of coupled repressing genes arranged in a directed ring topology, and give analytical conditions for the emergence of a cascade of unstable periodic orbits (UPOs) that lead to reachable long-lived oscillating transients. Such transients dominate the finite time horizon dynamics that is relevant in confined, noisy environments such as bacterial cells (see our previous work [Strelkowa and Barahona, 2010]) and are therefore of interest for bioengineering and synthetic biology. We show that the family of unstable orbits possesses spatial symmetries and can also be understood in terms of traveling wave solutions of kink-like topological defects. The long-lived oscillatory transients correspond to the propagation of quasistable two-kink configurations that unravel over a long time. We also assess the similarities between the generalized repressilator model and other unidirectionally coupled electronic systems, such as magnetic flux gates, which have been implemented experimentally.