Universality in Kinetic Models of Circadian Rhythms in Arabidopsis thaliana

Adapting to the 24-hour periodic environment on the Earth, plants have evolved sets of chemical reactions that regulate their circadian rhythms. Over the past fifteen years, researchers studying these circadian reactions in the common laboratory plant Arabidopsis thaliana have developed eleven, increasingly elaborate, chemical kinetic models based on genetic feedback loops. Each model consists of a system of coupled nonlinear ordinary differential equations. We find these models are all situated near a Hopf bifurcation in parameter space. This suggests that there may be some biological significance corresponding to this mathematical property.
To illustrate the special nature of these systems, we numerically compute the solutions to the kinetic models for Arabidopsis thaliana. Separately, we perform a weakly nonlinear analysis on each model near bifurcation to predict the amplitude and frequency of the oscillating concentration of chemical species from the Stuart-Landau amplitude equation. By scaling the numerical frequencies and amplitudes by our theoretical predictions, we show that the solutions to all these models collapse into a universal parameter-free form. We further comment on some implications of our results for improving future modeling efforts.