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Universality in kinetic models of circadian rhythms in Arabidopsis thaliana. (English) Zbl 1484.37115

The paper is concerned with the universality of the Arabidopsis thaliana circadian rhythms in kinetic models. Due to the rotation of the Earth, plants have evolved endogenous circadian rhythms to control many of their biological functions. Circadian rhythms are genetically regulated chemical reactions inside plants’ cells that cause chemical concentrations to rise and fall with daily periodicity. Thanks to the study of the Arabidopsis thaliana in laboratory, some dynamical models for the physiological rhythm and the involved oscillating chemical reactions (based on a network of interaction genes) have been proposed. With the increasing understanding of the circadian gene regulation, the dynamical models increased in their complexity. Recent efforts aim to reduce the mathematical complexity of these models while preserving their dynamical features. Other studies on circadian rhythms focused on the spatio-temporal patterns arising when the genetic expression of individual cells are coupled in the tissues of a live plant. Some studies used phenomenological models like the Stuart-Landau amplitude equation and the related Kuramoto coupled phase oscillator model, which reduce the complex dynamics of interacting gene networks with many rate constants to simpler dynamical models that contain only one or a few parameters.
In this paper the authors consider the Arabidopsis circadian rhythms models as sets of coupled nonlinear first-order ordinary differential equations. All of these models explicitly incorporate time dependence as a 24-h periodic function. The autonomous equations may be expressed in the general form, \[ \frac{dx}{dt}=f(x,\mu), \tag{1} \] where \(x\) is an \(n\)-dimensional vector of chemical concentrations associated with the circadian reactions, \(f\) is a nonlinear \(n\)-dimensional vector-valued function specifying the chemical reactions in a model, and \(\mu\) is a function of one of the rate constants in the reaction equations. By a weakly nonlinear analysis method – the reductive perturbation method – the authors obtain that each of the circadian rhythms models exhibit a supercritical Hopf bifurcation. At the bifurcation point \(\mu = 0\), the fixed point \(X_0\) of system (1) switches linear stability: in the pre-bifurcation region \(\mu < 0\), \(X_0\) is stable, and in the post-bifurcation region \(\mu > 0\), \(X_0\) is unstable. When the system (1) is in the post-bifurcation region, in addition to an unstable fixed point, it possesses a stable limit cycle, i.e., a linearly stable periodic solution \(X(t)\) that satisfies \[ \frac{dX}{dt}=f(X,\mu),\ \ \ X(t)=X(t+T) \tag{2} \] for some period \(T\). The limit cycle dynamics near a Hopf bifurcation belong to a dynamical universality class: for \(\mu \gtrapprox 0\), the amplitude of the limit cycle oscillations scales in proportion to \(\sqrt{\mu}\) and the frequency in proportion to \(\mu\). These properties enable an approximation to the amplitude and frequency of the limit cycle that is valid near the bifurcation point.
To illustrate a common dynamical structure, the authors scale the numerical solutions of the models with the asymptotic solutions of the Stuart-Landau equation to collapse the circadian oscillations onto two universal curves, one for the amplitude, and one for the 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 the authors’ analysis to ensure that models are consistent with each other and with the dynamics of the Arabidopsis circadian rhythm.

MSC:

37N25 Dynamical systems in biology
37G10 Bifurcations of singular points in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37M20 Computational methods for bifurcation problems in dynamical systems
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
92B25 Biological rhythms and synchronization
92D10 Genetics and epigenetics
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