  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 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
universal 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 supercritical while others are subcritical, 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.
