Presentation Information
[C03-02]Harmonicity of circadian rhythm
*Yusuke Yamada1, Yutaro Kabata2, Ryoya Fukasaku1, Hiroshi Ito1 (1. Kyushu Univ. (Japan), 2. Nagasaki Univ. (Japan))
Keywords:
circadian rhythm,biological clock,curvature
Biological clocks exhibit self-sustained oscillations with waveform characteristics that vary across species and environmental conditions. In chronobiology, such waveforms are often described using physical terms such as “harmonic” and “relaxational” (Bünning, The Physiological Clock, 1964). However, these terms lack precise definitions in the context of biological oscillators.
In this study, we propose a quantitative definition of harmonicity based on the convexity of closed flows in phase plane. Specifically, an oscillator is harmonic if its phase plane flow is strictly convex—that is, it contains no points of zero curvature. For example, the phase flow of a simple harmonic oscillator is a perfect circle, maintaining convexity throughout. In contrast, relaxation oscillators such as the FitzHugh–Nagumo and Van der Pol models exhibit non-convex flows, with inflection points appearing along the flow.
This geometric criterion allows harmonicity to be directly evaluated from time-series data by calculating curvature. Using this framework, we analyzed the Goodwin model, a widely used circadian clock model. Numerical simulations demonstrated that the phase plane flows of this model do not exhibit convexity across a broad range of parameter settings. Additionally, we analytically derived the condition under which convex (harmonic) flows emerge in the limit of an infinite Hill coefficient and confirmed this behavior in numerically obtained limit cycles.
Furthermore, we observed that flows become more circular in variables positioned downstream from the nonlinear feedback. This tendency suggests that geometric features of phase plane flows may reflect the underlying hierarchical structure of regulatory networks in circadian clocks. In particular, properties such as the degree of roundness or the number of inflection points in each variable’s flow may serve as geometric markers for inferring the directionality and organization of control pathways. This study offers a novel perspective for understanding the internal structure of circadian clocks, aiming to bridge chronobiology, dynamical systems theory, and differential geometry.
In this study, we propose a quantitative definition of harmonicity based on the convexity of closed flows in phase plane. Specifically, an oscillator is harmonic if its phase plane flow is strictly convex—that is, it contains no points of zero curvature. For example, the phase flow of a simple harmonic oscillator is a perfect circle, maintaining convexity throughout. In contrast, relaxation oscillators such as the FitzHugh–Nagumo and Van der Pol models exhibit non-convex flows, with inflection points appearing along the flow.
This geometric criterion allows harmonicity to be directly evaluated from time-series data by calculating curvature. Using this framework, we analyzed the Goodwin model, a widely used circadian clock model. Numerical simulations demonstrated that the phase plane flows of this model do not exhibit convexity across a broad range of parameter settings. Additionally, we analytically derived the condition under which convex (harmonic) flows emerge in the limit of an infinite Hill coefficient and confirmed this behavior in numerically obtained limit cycles.
Furthermore, we observed that flows become more circular in variables positioned downstream from the nonlinear feedback. This tendency suggests that geometric features of phase plane flows may reflect the underlying hierarchical structure of regulatory networks in circadian clocks. In particular, properties such as the degree of roundness or the number of inflection points in each variable’s flow may serve as geometric markers for inferring the directionality and organization of control pathways. This study offers a novel perspective for understanding the internal structure of circadian clocks, aiming to bridge chronobiology, dynamical systems theory, and differential geometry.