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Physical constraints on the rhythmicity of the biological clock

YeongKyu Lee and Changbong Hyeon*

  • *Contact author: hyeoncb@kias.re.kr
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Phys. Rev. Research 8, 023033 – Published 9 April, 2026

DOI: https://doi.org/10.1103/9byx-f411

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 overexpression. 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 ∼21-h rhythm, which is entrained to 24-h environmental signals as long as the forcing amplitude is greater than ∼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.

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References (78)

  1. M. Nakajima, K. Imai, H. Ito, T. Nishiwaki, Y. Murayama, H. Iwasaki, T. Oyama, and T. Kondo, Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro, Science 308, 414 (2005).
  2. H. Kageyama, T. Nishiwaki, M. Nakajima, H. Iwasaki, T. Oyama, and T. Kondo, Cyanobacterial circadian pacemaker: Kai protein complex dynamics in the KaiC phosphorylation cycle in vitro, Mol. Cell 23, 161 (2006).
  3. K. Terauchi, Y. Kitayama, T. Nishiwaki, K. Miwa, Y. Murayama, T. Oyama, and T. Kondo, ATPase activity of KaiC determines the basic timing for circadian clock of cyanobacteria, Proc. Natl. Acad. Sci. USA 104, 16377 (2007).
  4. M. Ishiura, S. Kutsuna, S. Aoki, H. Iwasaki, C. R. Andersson, A. Tanabe, S. S. Golden, C. H. Johnson, and T. Kondo, Expression of a gene cluster kaiABC as a circadian feedback process in cyanobacteria, Science 281, 1519 (1998).
  5. J. A. Swan, C. R. Sandate, A. G. Chavan, A. M. Freeberg, D. Etwaru, D. C. Ernst, J. G. Palacios, S. S. Golden, A. LiWang, G. C. Lander, and C. L. Partch, Coupling of distant ATPase domains in the circadian clock protein KaiC, Nat. Struct. Mol. Biol. 29, 759 (2022).
  6. E. Emberly and N. S. Wingreen, Hourglass model for a protein-based circadian oscillator, Phys. Rev. Lett. 96, 038303 (2006).
  7. J. S. van Zon, D. K. Lubensky, P. R. Altena, and P. R. ten Wolde, An allosteric model of circadian KaiC phosphorylation, Proc. Natl. Acad. Sci. USA 104, 7420 (2007).
  8. T. S. Hatakeyama and K. Kaneko, Generic temperature compensation of biological clocks by autonomous regulation of catalyst concentration, Proc. Natl. Acad. Sci. USA 109, 8109 (2012).
  9. M. Sasai, Role of the reaction-structure coupling in temperature compensation of the KaiABC circadian rhythm, PLoS Comput. Biol. 18, e1010494 (2022).
  10. D. Zhang, Y. Cao, Q. Ouyang, and Y. Tu, The energy cost and optimal design for synchronization of coupled molecular oscillators, Nat. Phys. 16, 95 (2020).
  11. H. Fu, C. Fei, Q. Ouyang, and Y. Tu, Temperature compensation through kinetic regulation in biochemical oscillators, Proc. Natl. Acad. Sci. USA 121, e2401567121 (2024).
  12. M. J. Rust, J. S. Markson, W. S. Lane, D. S. Fisher, and E. K. O’Shea, Ordered phosphorylation governs oscillation of a three-protein circadian clock, Science 318, 809 (2007).
  13. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  14. R. Marsland III, W. Cui, and J. M. Horowitz, The thermodynamic uncertainty relation in biochemical oscillations, J. R. Soc. Interface 16, 20190098 (2019).
  15. Y. Song and C. Hyeon, Thermodynamic uncertainty relation to assess biological processes, J. Chem. Phys. 154, 130901 (2021).
  16. P. Mehta and D. J. Schwab, Energetic costs of cellular computation, Proc. Natl. Acad. Sci. USA 109, 17978 (2012).
  17. Y. Cao, H. Wang, Q. Ouyang, and Y. Tu, The free-energy cost of accurate biochemical oscillations, Nat. Phys. 11, 772 (2015).
  18. P. Sartori and Y. Tu, Free energy cost of reducing noise while maintaining a high sensitivity, Phys. Rev. Lett. 115, 118102 (2015).
  19. G. Lan, P. Sartori, S. Neumann, V. Sourjik, and Y. Tu, The energy-speed-accuracy trade-off in sensory adaptation, Nat. Phys. 8, 422 (2012).
  20. E. Marañón, C. Fernández-González, and G. A. Tarran, Effect of temperature, nutrients and growth rate on picophytoplankton cell size across the atlantic ocean, Sci. Rep. 14, 28034 (2024).
  21. A. Hahn, M. Stevanovic, O. Mirus, and E. Schleiff, The TolC-like protein HgdD of the cyanobacterium Anabaena sp. PCC7120 is involved in secondary metabolite export and antibiotic resistance, J. Biol. Chem. 287, 41126 (2012).
  22. H. Gang, T. Ditzinger, C.-Z. Ning, and H. Haken, Stochastic resonance without external periodic force, Phys. Rev. Lett. 71, 807 (1993).
  23. A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system, Phys. Rev. Lett. 78, 775 (1997).
  24. O. Ushakov, H.-J. Wünsche, F. Henneberger, I. Khovanov, L. Schimansky-Geier, and M. Zaks, Coherence resonance near a Hopf bifurcation, Phys. Rev. Lett. 95, 123903 (2005).
  25. H. Iwasaki, T. Nishiwaki, Y. Kitayama, M. Nakajima, and T. Kondo, KaiA-stimulated KaiC phosphorylation in circadian timing loops in cyanobacteria, Proc. Natl. Acad. Sci. USA 99, 15788 (2002).
  26. Y. Kitayama, H. Iwasaki, T. Nishiwaki, and T. Kondo, KaiB functions as an attenuator of KaiC phosphorylation in the cyanobacterial circadian clock system, EMBO J. 22, 2127 (2003).
  27. K. Eguchi, M. Yoda, T. P. Terada, and M. Sasai, Mechanism of robust circadian oscillation of KaiC phosphorylation in vitro, Biophys. J. 95, 1773 (2008).
  28. Y. Xu, T. Mori, R. Pattanayek, S. Pattanayek, M. Egli, and C. H. Johnson, Identification of key phosphorylation sites in the circadian clock protein KaiC by crystallographic and mutagenetic analyses, Proc. Natl. Acad. Sci. USA 101, 13933 (2004).
  29. M. Egli, R. Pattanayek, J. H. Sheehan, Y. Xu, T. Mori, J. A. Smith, and C. H. Johnson, Loop–loop interactions regulate KaiA-stimulated KaiC phosphorylation in the cyanobacterial KaiABC circadian clock, Biochemistry 52, 1208 (2013).
  30. F. Hayashi, H. Ito, M. Fujita, R. Iwase, T. Uzumaki, and M. Ishiura, Stoichiometric interactions between cyanobacterial clock proteins KaiA and KaiC, Biochem. Biophys. Res. Commun. 316, 195 (2004).
  31. C. Brettschneider, R. J. Rose, S. Hertel, I. M. Axmann, A. J. Heck, and M. Kollmann, A sequestration feedback determines dynamics and temperature entrainment of the KaiABC circadian clock, Mol. Syst. Biol. 6, 389 (2010).
  32. See Supplemental Material at http://link.aps.org/supplemental/10.1103/9byx-f411 for the linear stability analysis and Jacobian matrix for the model, stochastic simulation using Gillespie algorithm. synchronization of two KaiABC clocks upon mixing, dynamics in the phases [IV–VII in Fig. 2] with three fixed points, effects of mutation on the KaiABC dynamics, approximation of limit cycle dynamics in phase I to the Stuart-Landau equation, Figs. S1–S8, and Table S1.
  33. N. Kawamoto, H. Ito, I. T. Tokuda, and H. Iwasaki, Damped circadian oscillation in the absence of KaiA in Synechococcus, Nat. Commun. 11, 2242 (2020).
  34. Y. Xu, T. Mori, and C. H. Johnson, Circadian clock-protein expression in cyanobacteria: Rhythms and phase setting, EMBO J. 19, 3349 (2000).
  35. X. Qin, M. Byrne, T. Mori, P. Zou, D. R. Williams, H. Mchaourab, and C. H. Johnson, Intermolecular associations determine the dynamics of the circadian KaiABC oscillator, Proc. Natl. Acad. Sci. USA 107, 14805 (2010).
  36. M. Nakajima, H. Ito, and T. Kondo, In vitro regulation of circadian phosphorylation rhythm of cyanobacterial clock protein KaiC by KaiA and KaiB, FEBS Lett. 584, 898 (2010).
  37. Y.-G. Chang, N.-W. Kuo, R. Tseng, and A. LiWang, Flexibility of the C-terminal, or CII, ring of KaiC governs the rhythm of the circadian clock of cyanobacteria, Proc. Natl. Acad. Sci. USA 108, 14431 (2011).
  38. K. Ito-Miwa, Y. Furuike, S. Akiyama, and T. Kondo, Tuning the circadian period of cyanobacteria up to 6.6 days by the single amino acid substitutions in KaiC, Proc. Natl. Acad. Sci. USA 117, 20926 (2020).
  39. Y. Ouyang, C. R. Andersson, T. Kondo, S. S. Golden, and C. H. Johnson, Resonating circadian clocks enhance fitness in cyanobacteria, Proc. Natl. Acad. Sci. USA 95, 8660 (1998).
  40. R. Pattanayek, D. R. Williams, S. Pattanayek, Y. Xu, T. Mori, C. H. Johnson, P. L. Stewart, and M. Egli, Analysis of KaiA–KaiC protein interactions in the cyano‐bacterial circadian clock using hybrid structural methods, EMBO J 25, 2017 (2006).
  41. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 1998).
  42. A. Dechant and S.-I. Sasa, Current fluctuations and transport efficiency for general Langevin systems, J. Stat. Mech. (2018) 063209.
  43. W. Hwang and C. Hyeon, Energetic costs, precision, and transport efficiency of molecular motors, J. Phys. Chem. Lett. 9, 513 (2018).
  44. J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nat. Phys. 16, 15 (2020).
  45. T. R. Gingrich and J. M. Horowitz, Fundamental bounds on first passage time fluctuations for currents, Phys. Rev. Lett. 119, 170601 (2017).
  46. P. Kim and C. Hyeon, Thermodynamic optimality of glycolytic oscillations, J. Phys. Chem. B 125, 5740 (2021).
  47. P. Pietzonka, A. C. Barato, and U. Seifert, Universal bound on the efficiency of molecular motors, J. Stat. Mech. (2016) 124004.
  48. V. Rodríguez-Franco, M. Spiering, P. Bianco, F. Ritort, and M. Mañosas, Continuous-time random walk model for the diffusive motion of helicases, QRB Discovery 6, e26 (2025).
  49. A. C. Barato and U. Seifert, Cost and precision of Brownian clocks, Phys. Rev. X 6, 041053 (2016).
  50. T. Koyuk and U. Seifert, Operationally accessible bounds on fluctuations and entropy production in periodically driven systems, Phys. Rev. Lett. 122, 230601 (2019).
  51. D. A. Beard and H. Qian, Chemical Biophysics: Quantitative Analysis of Cellular Systems (Cambridge University Press, Cambridge, 2008).
  52. Q. Li and Y. Wang, Coupling and internal noise sustain synchronized oscillation in calcium system, Biophys. Chem. 129, 23 (2007).
  53. M. J. Rust, S. S. Golden, and E. K. O’Shea, Light-driven changes in energy metabolism directly entrain the cyanobacterial circadian oscillator, Science 331, 220 (2011).
  54. J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing, Proc. Natl. Acad. Sci. USA 101, 10955 (2004).
  55. J. K. Kim and D. B. Forger, A mechanism for robust circadian timekeeping via stoichiometric balance, Mol. Syst. Biol. 8, 630 (2012).
  56. M. Fang, A. G. Chavan, A. LiWang, and S. S. Golden, Synchronization of the circadian clock to the environment tracked in real time, Proc. Natl. Acad. Sci. USA 120, e2221453120 (2023).
  57. A. Pikovsky, M. Rosenblum, and J. Kurths, in Synchronization: A Universal Concept in Nonlinear Science, edited by B. Chirikov, C. Predrag, F. Moss, and H. Swinney (Cambridge University Press, Cambridge, 2002).
  58. M. L. Heltberg, S. Krishna, L. P. Kadanoff, and M. H. Jensen, A tale of two rhythms: Locked clocks and chaos in biology, Cell Syst. 12, 291 (2021).
  59. P. Kim and C. Hyeon, Glycolytic oscillations under periodic drivings, J. R. Soc. Interface 21, 20230588 (2024).
  60. C. Maggi, F. Saglimbeni, V. C. Sosa, R. Di Leonardo, B. Nath, and A. Puglisi, Thermodynamic limits of sperm swimming precision, PRX Life 1, 013003 (2023).
  61. A. Murugan, D. A. Huse, and S. Leibler, Speed, dissipation, and error in kinetic proofreading, Proc. Natl. Acad. Sci. USA 109, 12034 (2012).
  62. Y. Song and C. Hyeon, Thermodynamic cost, speed, fluctuations, and error reduction of biological copy machines, J. Phys. Chem. Lett. 11, 3136 (2020).
  63. J. D. Mallory, O. A. Igoshin, and A. B. Kolomeisky, Do we understand the mechanisms used by biological systems to correct their errors? J. Phys. Chem. B 124, 9289 (2020).
  64. I. Jarmoskaite, P. Tijerina, and R. Russell, ATP utilization by a DEAD-box protein during refolding of a misfolded group I intron ribozyme, J. Biol. Chem. 296, 100132 (2021).
  65. Y. Song, D. Thirumalai, and C. Hyeon, Moderate activity of RNA chaperone maximizes the yield of self-spliced pre-RNA in vivo, Proc. Natl. Acad. Sci. USA 119, e2209422119 (2022).
  66. Y. Song and C. Hyeon, Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation, eLife 10, e70034 (2021).
  67. D. Zhang, C. Zhang, Q. Ouyang, and Y. Tu, Free energy dissipation enhances spatial accuracy and robustness of self-positioned turing pattern in small biochemical systems, J. R. Soc. Interface 20, 20230276 (2023).
  68. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  69. A. T. Winfree, The Geometry of Biological Time (Springer, New York, 2001), Vol. 12.
  70. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, Heidelberg, 2003).
  71. Y. Song, J. O. Park, L. Tanner, Y. Nagano, J. D. Rabinowitz, and S. Y. Shvartsman, Energy budget of Drosophila embryogenesis, Curr. Biol. 29, R566 (2019).
  72. X. Yang, M. Heinemann, J. Howard, G. Huber, S. Iyer-Biswas, G. Le Treut, M. Lynch, K. L. Montooth, D. J. Needleman, S. Pigolotti, et al., Physical bioenergetics: Energy fluxes, budgets, and constraints in cells, Proc. Natl. Acad. Sci. USA 118, e2026786118 (2021).
  73. I. Di Terlizzi, M. Gironella, D. Herraez-Aguilar, T. Betz, F. Monroy, M. Baiesi, and F. Ritort, Variance sum rule for entropy production, Science 383, 971 (2024).
  74. Z. Padamsey and N. L. Rochefort, Paying the Brain's energy bill, Curr. Opin. Neurobiol. 78, 102668 (2023).
  75. S. D. Jamadar, A. Behler, H. Deery, and M. Breakspear, The metabolic costs of cognition, Trends Cognit. Sci. 29, 541 (2025).
  76. M. B. Elowitz and S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature (London) 403, 335 (2000).
  77. J. Stricker, S. Cookson, M. R. Bennett, W. H. Mather, L. S. Tsimring, and J. Hasty, A fast, robust and tunable synthetic gene oscillator, Nature (London) 456, 516 (2008).
  78. L. Potvin-Trottier, N. D. Lord, G. Vinnicombe, and J. Paulsson, Synchronous long-term oscillations in a synthetic gene circuit, Nature (London) 538, 514 (2016).

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Phys. Rev. Research 8, 023033– Published 9 April, 2026
  • Received 29 July 2025
  • Accepted 13 March 2026
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