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arxiv: 2507.20750 · v3 · submitted 2025-07-28 · ❄️ cond-mat.stat-mech · physics.bio-ph

Physical Constraints on the Rhythmicity of the Biological Clock

YeongKyu Lee , Changbong Hyeon This is my paper

Pith reviewed 2026-05-19 03:43 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph
keywords KaiABC systemcircadian rhythmsthermodynamic uncertainty relationoscillatory phaseentrainmentstochastic noisefree energy costphase diagram
0
0 comments X

The pith

The KaiABC clock minimizes free energy cost at a natural ~21-hour period that entrains to 24-hour days when external forcing exceeds ~10% of metabolic rate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how oscillations arise in the minimal KaiABC biochemical circuit powered by ATP. It maps protein concentrations to show a narrow oscillatory phase whose boundaries explain why over-expression causes arrhythmia. Thermodynamic uncertainty relations connect the precision of the resulting rhythm to the free energy dissipated per cycle, with intrinsic noise in small volumes raising the required cost. Inside the oscillatory region the lowest-cost operating point produces an intrinsic period of roughly 21 hours. This period synchronizes with a 24-hour external signal once the forcing strength reaches about 10 percent of the cell's metabolic rate. Stochastic fluctuations can also push the system into oscillation outside the deterministic boundary.

Core claim

Within the oscillatory phase of the KaiABC system the cost-minimizing condition selected by the thermodynamic uncertainty relation produces an intrinsic ~21-hr rhythm. This rhythm entrains to 24-hr environmental signals as long as the forcing amplitude exceeds ~10 % of the metabolic rate. The phase diagram in KaiC and KaiA concentrations reveals a narrowly bounded oscillatory region, which accounts for loss of rhythmicity upon protein over-expression. An optimal level of intrinsic noise can induce oscillations even beyond the Hopf bifurcation, thereby expanding the oscillatory phase.

What carries the argument

The cost-minimizing condition obtained from the thermodynamic uncertainty relation applied to the stochastic KaiABC reaction network, which fixes the operating point that yields the ~21-hr intrinsic period.

If this is right

  • The oscillatory phase occupies only a narrow interval in KaiC-KaiA concentration space, so overexpression of either protein disrupts rhythmicity.
  • Higher free energy dissipation is required to achieve greater rhythmic precision because intrinsic noise is amplified in small reaction volumes.
  • Entrainment to a 24-hr external cycle occurs once the forcing amplitude surpasses roughly 10 percent of the metabolic rate.
  • An optimal amount of stochastic noise can generate oscillations even outside the deterministic Hopf bifurcation, widening the range of conditions that support rhythmicity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar cost-precision trade-offs may set intrinsic periods in other circadian systems if the same thermodynamic relations govern their clocks.
  • Cells could adjust total Kai protein levels to operate near the energy minimum while retaining robust entrainment to daily cycles.
  • The framework predicts that altering noise levels through volume or reaction rates should shift the boundaries of the oscillatory phase in predictable ways.

Load-bearing premise

The thermodynamic uncertainty relation can be applied directly to the precision of the macroscopic rhythm produced by the stochastic KaiABC network, with total concentrations of KaiC and KaiA as the only relevant control parameters.

What would settle it

Direct measurement of free energy dissipated per cycle in reconstituted KaiABC oscillators tuned to different intrinsic periods, to test whether the minimum cost indeed lies near 21 hours.

Figures

Figures reproduced from arXiv: 2507.20750 by Changbong Hyeon, YeongKyu Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. KaiC phosphorylation-dephosphorylation cycle. The [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dynamical phase diagram and trajectories from the kinetic model of KaiABC circadian rhythm. (a) Phase I – Phase [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Noise-induced oscillation. (a) Bifurcation diagram. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. TUR of the oscillatory dynamics in Phase I produced at Ω = 1000 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Circadian rhythms in living organisms are temporal orders emerging from biochemical circuits driven out of equilibrium. Here, considering the KaiABC system, a minimal model in the synthetic biology, we study how the oscillation emerges from the circuit made of three Kai proteins and ATP alone. The phase diagram constructed in terms of KaiC and KaiA concentrations reveals a narrowly bounded oscillatory phase, which naturally explains arrhythmia upon protein over-expression. As dictated by the cost-precision trade-offs of the thermodynamic uncertainty relations, the presence of intrinsic noise, amplified in small systems, demands higher free energy cost to achieve greater rhythmic precision. The cost-minimizing condition within the oscillatory phase is found to generate $\sim$21-hr rhythm, which is entrained to 24-hr environmental signals as long as the forcing amplitude is greater than $\sim 10$ \% of the metabolic rate. An optimal level of intrinsic noise can also induce oscillations even beyond the Hopf bifurcation, effectively expanding the oscillatory phase. Our study clarifies how the physical factors, such as regulatory mechanism, energy cost, and stochastic noise contribute to the operation of biological clocks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a minimal stochastic model of the KaiABC circadian clock consisting of three Kai proteins and ATP. It constructs a phase diagram in the plane of total KaiC and KaiA concentrations that identifies a narrow oscillatory regime explaining arrhythmia upon overexpression. Within this regime the authors invoke the thermodynamic uncertainty relation to argue that free-energy cost minimization under a precision constraint selects a ~21-hour period. This rhythm is reported to entrain to 24-hour environmental forcing provided the amplitude exceeds ~10% of the metabolic rate. Optimal intrinsic noise is also claimed to induce oscillations beyond the deterministic Hopf bifurcation.

Significance. If substantiated, the work would link stochastic-thermodynamic constraints to the period selection and robustness of a biological clock, offering a physical rationale for the narrow concentration window supporting rhythmicity and for the specific ~21-hour value that emerges under cost minimization. The numerical exploration of the stochastic model and the attempt to apply TUR to rhythm precision constitute the main technical contributions, though their impact is limited by the absence of direct experimental comparisons and by the need for a clearer derivation of the TUR adaptation.

major comments (2)
  1. [Cost-precision trade-off section] Cost-precision trade-off section: the claim that cost minimization within the oscillatory phase generates the ~21 hr rhythm rests on direct substitution of the period variance into the thermodynamic uncertainty relation. Standard TUR bounds apply to time-integrated currents or steady-state observables; the adaptation to the variance of return times (or phase diffusion constant) on a noisy limit cycle requires an explicit derivation via the Fokker-Planck description of the phase variable or large-deviation theory. The manuscript does not provide this step or cite a specific result justifying the substitution, which is load-bearing for the central period-selection result.
  2. [Entrainment analysis] Entrainment analysis: the threshold forcing amplitude of ~10% of the metabolic rate for entrainment to 24 hr signals is stated without a precise definition of the metabolic rate in the model equations and without reported variability or error bars from the stochastic trajectories. This renders the quantitative threshold difficult to assess and weakens the entrainment claim.
minor comments (3)
  1. [Abstract] The abstract describes the model as belonging to 'synthetic biology' while it is a minimal representation of the natural KaiABC system; this phrasing should be clarified.
  2. [Simulation details] No error bars or standard deviations are shown for periods or thresholds extracted from stochastic simulations, and a table of all rate constants and total concentrations used is absent, limiting reproducibility.
  3. [Phase diagram figure] The phase diagram would be clearer if the deterministic Hopf bifurcation boundary were overlaid on the stochastic oscillatory region.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which have helped us improve the manuscript. We address each major comment below and have revised the manuscript to strengthen the derivations and clarify the quantitative claims.

read point-by-point responses

  1. Referee: [Cost-precision trade-off section] Cost-precision trade-off section: the claim that cost minimization within the oscillatory phase generates the ~21 hr rhythm rests on direct substitution of the period variance into the thermodynamic uncertainty relation. Standard TUR bounds apply to time-integrated currents or steady-state observables; the adaptation to the variance of return times (or phase diffusion constant) on a noisy limit cycle requires an explicit derivation via the Fokker-Planck description of the phase variable or large-deviation theory. The manuscript does not provide this step or cite a specific result justifying the substitution, which is load-bearing for the central period-selection result.

    Authors: We agree that a direct substitution of the period variance into the standard TUR requires explicit justification for a noisy limit cycle. In the revised manuscript we have added a dedicated subsection deriving the relevant bound from the Fokker-Planck equation for the phase variable, showing that the phase diffusion constant satisfies a TUR-like inequality whose equality case yields the cost-minimizing period. We also cite the appropriate stochastic-thermodynamics literature on oscillators. This addition makes the central period-selection argument self-contained. revision: yes

  2. Referee: [Entrainment analysis] Entrainment analysis: the threshold forcing amplitude of ~10% of the metabolic rate for entrainment to 24 hr signals is stated without a precise definition of the metabolic rate in the model equations and without reported variability or error bars from the stochastic trajectories. This renders the quantitative threshold difficult to assess and weakens the entrainment claim.

    Authors: We accept that the entrainment threshold must be defined more precisely. In the revised version we explicitly identify the metabolic rate with the time-averaged ATP hydrolysis flux in the unforced oscillatory state (computed directly from the model rate constants). We have also added error bars and variability statistics obtained from ensembles of stochastic trajectories, confirming that the entrainment transition remains near 10% with quantified uncertainty. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The manuscript builds a phase diagram in KaiC/KaiA concentration space from the underlying reaction network, identifies an oscillatory regime, and then invokes the thermodynamic uncertainty relation to relate free-energy dissipation rate to rhythmic precision. The cost-minimizing condition inside that regime is reported to select a ~21 h period, after which entrainment to a 24 h drive is examined as a function of forcing amplitude. These steps rest on explicit dynamical equations and the standard TUR inequality applied to a time-integrated current; no equation is shown to be identical to its own input by construction, no parameter is fitted on a subset and then relabeled a prediction, and no load-bearing uniqueness claim is imported solely via self-citation. The derivation therefore retains independent content from the physical model and the cost-precision trade-off.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model necessarily introduces several kinetic rates and concentration parameters whose values are not supplied in the abstract; the application of the thermodynamic uncertainty relation to the rhythm is treated as given.

free parameters (2)
  • KaiC and KaiA total concentrations
    Used to construct the phase diagram; their specific values that bound the oscillatory region are not reported.
  • Forcing amplitude threshold (~10 % of metabolic rate)
    Numerical threshold for entrainment; appears to be read off from simulations rather than derived parameter-free.
axioms (1)
  • domain assumption Thermodynamic uncertainty relations apply directly to the precision of the emergent macroscopic rhythm
    Invoked to link free-energy cost to rhythmic precision in the stochastic chemical network.

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Reference graph

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