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Low-dimensional dynamics of two coupled biological oscillators


The circadian clock and the cell cycle are two biological oscillatory processes that coexist within individual cells. These two oscillators were found to interact, which can lead to their synchronization. Here, we develop a method to identify a low-dimensional stochastic model of the coupled system directly from time-lapse imaging in single cells. In particular, we infer the coupling and nonlinear dynamics of the two oscillators from thousands of mouse and human single-cell fluorescence microscopy traces. This coupling predicts multiple phase-locked states showing different degrees of robustness against molecular fluctuations inherent to cellular-scale biological oscillators. For the 1:1 state, the predicted phase-shifts following period perturbations were validated experimentally. Moreover, the phase-locked states are temperature-independent and evolutionarily conserved from mouse to human, hinting at a common underlying dynamical mechanism. Finally, we detect a signature of the coupled dynamics in a physiological context, explaining why tissues with different proliferation states exhibited shifted circadian clock phases.

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Fig. 1: Reconstructing the phase dynamics and coupling of two biological oscillators.
Fig. 2: Influence of the cell cycle on the circadian phase enables 1:1 phase-locking.
Fig. 3: The coupling between the cell cycle and the circadian oscillator predicts phase-shifts and phase-locking attractors in perturbation experiments.
Fig. 4: Single-cell data and stochastic simulations reveal robust 1:1 and 1:2 phase-locked states.
Fig. 5: Conserved influence of the cell cycle on the circadian clock in human U2OS osteosarcoma cells.
Fig. 6: Temperature cycles do not entrain circadian oscillators in dividing cells and proliferation genes are associated with tissue-specific circadian phases.

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Data availability

The data supporting the figures and other findings of this study are available from the corresponding author on request.

Code availability

The code is available at


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We thank R. Cannavo for engineering the U2OS-Dual cell line and J. Bieler for initial analyses. We also thank F. Kuttler from the EPFL Biomolecular Screening Facility and L. Bozzo and J. Artacho from the EPFL Bioimaging and Optics Core Facility for assistance with the imaging. Fluorescence-activated cell sorting was performed at the EPFL Flow Cytometry Core Facility. This work was supported by the Swiss National Science Foundation grant 310030_173079 and the EPFL. E.R.P. was supported by a Canadian Institute of Health Research (CIHR 358808) and a Transition Postdoc Fellowship (51FSP0163584).

Author information

Authors and Affiliations



C.D., E.R.P. and F.N. designed and participated in the study concept. C.D. and E.R.P. developed computational analysis tools. E.R.P. performed the experiments. C.D. and E.R.P. processed and analysed the experimental data. C.D., E.R.P. and F.N. interpreted the results. E.R.P. and F.N. acquired the funding. F.N. supervised the study. C.D., E.R.P. and F.N. wrote the manuscript.

Corresponding author

Correspondence to Felix Naef.

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The authors declare no competing interests.

Additional information

Peer review information: Nature Physics thanks Mogens Jensen, Joris Paijmans and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–5, methods and refs. 1–9.

Reporting Summary

Supplementary Table 1

Correlation analysis of tissue-specific phases and amplitudes with the mean transcript levels in mouse for every gene. Gene Ontology analyses for the top 200 positively and top 200 negatively correlated genes are provided for both phase and amplitude correlations.

Supplementary Video 1

Single-cell trajectories of circadian and cell-cycle phases cluster around a 1:1 phase-locked state.

Supplementary Video 2

Simulations of the deterministic model show phase-locking and quasiperiodicity.

Supplementary Video 3

Spectral analysis of the deterministic and stochastic simulations shows qualitative differences as a function of the cell-cycle period.

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Droin, C., Paquet, E.R. & Naef, F. Low-dimensional dynamics of two coupled biological oscillators. Nat. Phys. 15, 1086–1094 (2019).

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