Universality in Kinetic Models of Circadian Rhythms in Arabidopsis thaliana

TL;DR

This paper demonstrates that various kinetic models of Arabidopsis thaliana's circadian rhythms exhibit a supercritical Hopf bifurcation, revealing a universal dynamical structure near bifurcation points through nonlinear analysis.

Contribution

It shows that different models share a common bifurcation structure and introduces a scaling approach to unify their oscillatory behavior.

Findings

All models exhibit supercritical Hopf bifurcation.

Universal curves for amplitude and frequency were derived.

Models vary in proximity to bifurcation and parameter sensitivity.

Abstract

Biological evolution has endowed the plant Arabidopsis thaliana with genetically regulated circadian rhythms. A number of authors have published kinetic models for these oscillating chemical reactions based on a network of interacting genes. To investigate the hypothesis that the Arabidopsis circadian dynamical system is poised near a Hopf bifurcation like some other biological oscillators, we varied the kinetic parameters in the models and searched for bifurcations. Finding that each model does exhibit a supercritical Hopf bifurcation, we performed a weakly nonlinear analysis near the bifurcation points to derive the Stuart-Landau amplitude equation. To illustrate a common dynamical structure, we scaled the numerical solutions to the models with the asymptotic solutions to the Stuart-Landau equation to collapse the circadian oscillations onto two universal curves -- one for amplitude, and one for frequency. However, some models are close to bifurcation while others are far, some models are post-bifurcation while others are pre-bifurcation, and kinetic parameters that lead to a bifurcation in some models do not lead to a bifurcation in others. Future kinetic modeling can make use of our analysis to ensure models are consistent with each other and with the dynamics of the Arabidopsis circadian rhythm.