Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

TL;DR

This paper investigates minimally nonlinear models of genetic regulatory networks, revealing how positivity constraints induce complex dynamics like chaos and multistability, and analyzing their spectral and connectivity properties.

Contribution

It introduces a simple nonlinear modeling framework that captures key dynamical features of genetic regulatory networks, enhancing understanding of their complex behaviors.

Findings

Positivity constraints enable chaotic dynamics in minimal models

The Lyapunov spectrum characterizes stability and chaos

Connectivity influences oscillatory mode frequencies

Abstract

Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the average connectivity.