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Inferring scale-dependent non-equilibrium activity using carbon nanotubes


In living systems, irreversible, yet stochastic, molecular interactions form multiscale structures (such as cytoskeletal networks), which mediate processes (such as cytokinesis and cellular motility) in a close relationship between the structure and function. However, owing to a lack of methods to quantify non-equilibrium activity, their dynamics remain poorly characterized. Here, by measuring the time-reversal asymmetry encoded in the conformational dynamics of filamentous single-walled carbon nanotubes embedded in the actomyosin network of Xenopus egg extract, we characterize the multiscale dynamics of non-equilibrium activity encoded in bending-mode amplitudes. Our method is sensitive to distinct perturbations to the actomyosin network and the concentration ratio of adenosine triphosphate to adenosine diphosphate. Thus, our method can dissect the functional coupling of microscopic dynamics to the emergence of larger scale non-equilibrium activity. We relate the spatiotemporal scales of non-equilibrium activity to the key physical parameters of a semiflexible filament embedded in a non-equilibrium viscoelastic environment. Our analysis provides a general tool to characterize steady-state non-equilibrium activity in high-dimensional spaces.

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Fig. 1: Shape fluctuations of a filament encode the non-equilibrium dynamics of the environment.
Fig. 2: Angular momentum tensor indicates scale-dependent non-equilibrium activity.
Fig. 3: The tensor represents a phase space of mode couplings violating detailed balance.
Fig. 4: Distinct perturbations at the microscopic level produce distinguishable effects on larger scale non-equilibrium activity.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The computational methods that support the plots within this paper are described in the Supplementary Information and the underlying code is available from the corresponding author upon reasonable request.


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We thank J.M. Horowitz and T.R. Gingrich for discussions. N.F. acknowledges support from a Sloan Research Fellowship, a Human Frontier Science Program Career Development Award (HFSP CDA-00053/2016C) and a Gordon and Betty Moore Foundation Grant (Grant No 7729). A.B. acknowledges support from a National Defense Science and Engineering Graduate Fellowship. J.F.P. acknowledges support from a Fannie and John Hertz Foundation Fellowship and an International Human Frontier Science Program Organization Cross-Disciplinary Fellowship (LT000901/2021-C).

Author information

Authors and Affiliations



N.F. conceived and supervised the project. A.B. and J.F.P. designed and performed experiments. A.B. developed the theory, performed simulations and analysed data. Y.J. built the imaging setup. All authors discussed the results and co-wrote the paper.

Corresponding author

Correspondence to Nikta Fakhri.

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The authors declare no competing interests.

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Nature Nanotechnology thanks Jacinta Conrad and Sarah Loos for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Tables 1 and 2, Figs. 1–20 and Theory Sections 2.1–2.10.

Supplementary Video 1

Temporal dynamics of active actomyosin network in Xenopus egg extract.

Supplementary Video 2

Temporal dynamics of carbon nanotube filament in passive viscous medium (water).

Supplementary Video 3

Temporal dynamics of carbon nanotube filament in passive viscoelastic medium (actin gel).

Supplementary Video 4

Temporal dynamics of carbon nanotube filament in active actomyosin network (Xenopus egg extract).

Supplementary Video 5

Temporal evolution of ensemble-averaged mode correlation tensor for active wild-type/passive gels.

Supplementary Video 6

Temporal evolution of ensemble-averaged mode correlation tensor for active wild-type/active gels treated with cytochalasin D.

Supplementary Video 7

Temporal evolution of ensemble-averaged mode correlation tensor for active wild-type/active gels treated with α-actinin.

Supplementary Video 8

Temporal evolution of ensemble-averaged mode correlation tensor for active wild-type/active gels treated with apyrase.

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Bacanu, A., Pelletier, J.F., Jung, Y. et al. Inferring scale-dependent non-equilibrium activity using carbon nanotubes. Nat. Nanotechnol. 18, 905–911 (2023).

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