Physical Constraints on the Rhythmicity of the Biological Clock

TL;DR

This work analyzes how the KaiABC biochemical clock generates circadian rhythmicity under nonequilibrium driving, revealing how energy dissipation and intrinsic noise constrain timing via the thermodynamic uncertainty relation. Using a minimal three-variable KaiC phosphorylation model, the authors map a deterministic phase diagram with a narrowly bounded oscillatory phase and then incorporate finite-size noise to uncover noise-assisted phenomena, including coherence resonances near a Hopf boundary. A Stuart-Landau reduction near the Hopf bifurcation captures the essential limit-cycle dynamics and reproduces the observed scaling $[\text{KaiA}]\propto[\text{KaiC}]^{2/3}$ for ~24-hr rhythms, linking network structure to timing. The results establish physical design principles for biochemical clocks, highlighting how energy cost, stochasticity, and regulatory interactions shape robust entrainment and offering guidance for synthetic oscillator design.

Abstract

Circadian rhythms in living organisms are temporal orders emerging from biochemical circuits driven out of equilibrium. Here, we study how the rhythmicity of KaiABC clock is generated from the underlying circuit. The phase diagram in terms of KaiC and KaiA concentrations reveals a narrowly bounded oscillatory phase. As dictated by the cost-precision trade-offs of the thermodynamic uncertainty relations, the presence of intrinsic noise, amplified in small systems, demands higher energy cost to achieve greater rhythmic precision. The cost-minimizing condition giving rise to $\sim$21-hr rhythm is identified close enough to entrain the system to 24-hr environmental signals. An optimal level of intrinsic noise can induce oscillations beyond the Hopf bifurcation, effectively expanding the oscillatory phase. Our study clarifies how the physical factors, such as energy cost, stochastic noise, and regulatory mechanism, contribute to the operation of biological clocks.

Figures (12)

• Figure 1: A kinetic scheme of the KaiC phosphorylation-dephosphorylation cycle for the ODE model studied here (see the main text for the details).
• Figure 2: Dynamical phase diagram and trajectories from the model of KaiABC clock. (a) Phase I -- III classified based on the eigenvalue characteristics of the fixed point. More complicated structure of Phase IV -- VII is explained in the SM. (b) Trajectories of $T$, $D$, and $S$ states of KaiC hexamer generated at $([{\rm KaiC}] (\mu{\rm M}),[{\rm KaiA}] (\mu{\rm M}))=(3.4,1.4)$ (green star) in Phase I, at $(10,10)$ (magenta star) in Phase II, and at $(20,3)$ (blue star) in Phase III.
• Figure 3: Dynamics in the oscillatory phase (Phase I) produced at $\Omega = 1000~\mu\mathrm{m}^3$. (a) Period of oscillation and (b) its variance, (c) entropy production, and (d) the uncertainty product. The yellow stars in Fig. \ref{['fig:TUR']}a and \ref{['fig:TUR']}d mark the $\mathcal{Q}$-minimizing condition [KaiC]=5.71 $\mu$M and [KaiA]=1.87 $\mu$M, which leads to $\langle T_{\rm os}\rangle\simeq 21$ hr in (a).
• Figure 4: Noise-induced oscillation. (a) Bifurcation diagram at $[{\rm KaiA}]=1.3$$\mu$M. Solid black line, blue, green, and red dots depict the minimum and the maximum values of $(T+D+S)$ for $\Omega\rightarrow\infty$, 1000, 100, and $10 \, \mu m^{3}$, respectively. (b) Time evolutions of $(T+D+S)$ at $[{\rm KaiC}]=6.07$$\mu$M. (c) Noise-induced oscillations generated at [KaiC]$=6.07$$\mu$M for varying $\Omega$s: $(i)$$\Omega=0.10$, $(ii)$$0.43$, $(iii)$$3.79$, and $(iv)$$69.5$$\mu{\rm m}^3$ (left) and the corresponding power spectra $P(\nu)$s (right). (d) Height ($H$) and width ($\Delta\nu/\nu_0$) of the resonant peak versus $\Omega$. (e) SNR ($\beta[=H/(\Delta \nu/\nu_0)]$) versus $\Omega$ for varying [KaiC]. The values of $\beta$ calculated at different $\Omega$s in (c) are marked with $i$ to $iv$.
• Figure S1: Restoration of the normal oscillatory dynamics after a transient time of adjustment upon mixing two out-of-phase KaiABC oscillators. The dotted line in grey depicts the concentration of free KaiA in the solution, [KaiA]$_{\rm free}=A(S)=\max{(0,[{\rm KaiA}]-S)}$.