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The energy cost and optimal design for synchronization of coupled molecular oscillators


A model of coupled molecular biochemical oscillators is proposed to study non-equilibrium thermodynamics of synchronization. Under general considerations, we find that chemical interactions within an ensemble of autonomous oscillators break detailed balance and thus cost energy. This extra energy cost, in addition to the energy dissipated for driving each individual oscillator, is necessary to power the coupling interactions such as oscillator–oscillator exchange reactions, which are responsible for correcting the phase error in each individual noisy oscillator with respect to the collective oscillation of the whole ensemble. By solving the steady-state distribution of the many-oscillator system analytically and numerically, we show that the system reaches its synchronized state through a non-equilibrium phase transition as energy dissipation increases. The critical energy dissipation per period depends on both the frequency and strength of the exchange reaction, which reveals an optimal (efficient) design for achieving maximum synchronization with a fixed energy budget. We apply our general theory to the Kai system in the cyanobacterial circadian clock and predict a relationship between the KaiC ATPase activity and synchronization of the KaiC hexamers. The theoretical framework established here can be extended to study thermodynamics of collective behaviours in other non-equilibrium active systems.

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Fig. 1: Non-equilibrium cycle dynamics of Poisson clock(s).
Fig. 2: Phase diagram and optimal design for synchronization.
Fig. 3: The cost of monomer shuffling for synchronization in the Kai system.

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Data availability

The data represented in Figs. 2 and 3 are available with the online version of this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author on request.

Code availability

Computer codes used in this work are available on request.


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We thank T. Theis for stimulating discussions and critical reading of the manuscript. This work is partially supported by NSFC (11434001,11774011). The work by Y.T. is partially supported by an NIH grant (R01-GM081747).

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Authors and Affiliations



D.Z. did the simulations and analysed the data; Y.C. did the simulations and analysed the data; Q.O. analysed the data; Y.T. initiated the project, developed the model, found the analytical solution and analysed the data; all wrote the paper.

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Correspondence to Yuhai Tu.

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The authors declare no competing interests.

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Peer review information Nature Physics thanks Andre Cardoso Barato and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Zhang, D., Cao, Y., Ouyang, Q. et al. The energy cost and optimal design for synchronization of coupled molecular oscillators. Nat. Phys. 16, 95–100 (2020).

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