Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Inferring scale-dependent non-equilibrium activity using carbon nanotubes

Nature Nanotechnology volume 18, pages 905–911 (2023)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

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.

References

  1. Needleman, D. & Dogic, Z. Active matter at the interface between materials science and cell biology. Nat. Rev. Mater. 2, 17048 (2017).

    Article  CAS  Google Scholar 

  2. Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323–345 (2010).

    Article  Google Scholar 

  3. Marchetti, M. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013).

    Article  CAS  Google Scholar 

  4. Bowick, M., Fakhri, N., Marchetti, M. & Ramaswamy, S. Symmetry, thermodynamics, and topology in active matter. Phys. Rev. X 12, 010501 (2022).

    CAS  Google Scholar 

  5. Yang, X. et al. Physical bioenergetics: energy fluxes, budgets, and constraints in cells. Proc. Natl Acad. Sci. USA 118, e2026786118 (2021).

    Article  CAS  Google Scholar 

  6. Tan, T. et al. Self-organized stress patterns drive state transitions in actin cortices. Sci. Adv. 4, eaar2847 (2018).

    Article  Google Scholar 

  7. Gladrow, J., Fakhri, N., MacKintosh, F. C., Schmidt, C. F. & Broedersz, C. P. Broken detailed balance of filament dynamics in active networks. Phys. Rev. Lett. 116, 248301 (2016).

    Article  CAS  Google Scholar 

  8. Landau, L., Lifshitz, E., Sykes, J., & Reid, W. Theory of Elasticity (Addison-Wesley 1959).

  9. Brangwynne, C. P., Koenderink, G. H., MacKintosh, F. C. & Weitz, D. A. Nonequilibrium microtubule fluctuations in a model cytoskeleton. Phys. Rev. Lett. 100, 118104 (2008).

    Article  Google Scholar 

  10. England, J. L. Dissipative adaptation in driven self-assembly. Nat. Nanotechnol. 10, 919–923 (2015).

    Article  CAS  Google Scholar 

  11. Mizuno, D., Tardin, C., Schmidt, C. F. & MacKintosh, F. C. Nonequilibrium mechanics of active cytoskeletal networks. Science 315, 370–373 (2007).

    Article  CAS  Google Scholar 

  12. Battle, C. et al. Broken detailed balance at mesoscopic scales in active biological systems. Science 352, 604–607 (2016).

    Article  CAS  Google Scholar 

  13. Egolf, D. A. Equilibrium regained: from nonequilibrium chaos to statistical mechanics. Science 287, 101–104 (2000).

    Article  CAS  Google Scholar 

  14. Prost, J., Jülicher, F. & Joanny, J-F. Active gel physics. Nat. Phys. 11, 111–117 (2015).

    Article  CAS  Google Scholar 

  15. O’Byrne, J., Kafri, Y., Tailleur, J. & van Wijland, F. Time irreversibility in active matter, from micro to macro. Nat. Rev. Phys. 4, 167–183 (2022).

    Article  Google Scholar 

  16. Gnesotto, F. S., Mura, F., Gladrow, J. & Broedersz, C. P. Broken detailed balance and non-equilibrium dynamics in living systems: a review. Rep. Prog. Phys. 81, 066601 (2018).

    Article  CAS  Google Scholar 

  17. Fakhri, N. et al. High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science 344, 1031–1035 (2014).

    Article  CAS  Google Scholar 

  18. Fakhri, N., Tsyboulski, D. A., Cognet, L., Weisman, R. B. & Pasquali, M. Diameter-dependent bending dynamics of single-walled carbon nanotubes in liquids. Proc. Natl Acad. Sci. USA 106, 14219–14223 (2009).

    Article  CAS  Google Scholar 

  19. Fakhri, N., MacKintosh, F. C., Lounis, B., Cognet, L. & Pasquali, M. Brownian motion of stiff filaments in a crowded environment. Science 330, 1804–1807 (2010).

    Article  CAS  Google Scholar 

  20. Murrell, M. P. & Gardel, M. L. F-actin buckling coordinates contractility and severing in a biomimetic actomyosin cortex. Proc. Natl Acad. Sci. USA 109, 20820–20825 (2012).

    Article  CAS  Google Scholar 

  21. Weiss, J. B. Coordinate invariance in stochastic dynamical systems. Tellus A 55, 208–218 (2003).

    Article  Google Scholar 

  22. Crooks, G. E. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721–2726 (1999).

    Article  CAS  Google Scholar 

  23. Ro, S. et al. Model-free measurement of local entropy production and extractable work in active matter. Phys. Rev. Lett. 129, 220601 (2022).

    Article  CAS  Google Scholar 

  24. Harunari, P. E., Dutta, A., Polettini, M. & Roldán, É. What to learn from a few visible transitions' statistics? Phys. Rev. X 12, 041026 (2022).

    CAS  Google Scholar 

  25. van der Meer, J., Ertel, B. & Seifert, U. Thermodynamic inference in partially accessible Markov networks: a unifying perspective from transition-based waiting time distributions. Preprint at https://arxiv.org/abs/2203.07427 (2022).

  26. van der Meer, J., Degünther, J. & Seifert, U. Time-resolved statistics of snippets as general framework for model-free entropy estimators. Preprint at https://arxiv.org/abs/2211.17032 (2022).

  27. Roldán, E., Barral, J., Martin, P., Parrondo, J. M. R. & Jülicher, F. Quantifying entropy production in active fluctuations of the hair-cell bundle from time irreversibility and uncertainty relations. New J. Phys. 23, 083013 (2021).

    Article  Google Scholar 

  28. Tucci, G. et al. Modeling active non-Markovian oscillations. Phys. Rev. Lett. 129, 030603 (2022).

    Article  CAS  Google Scholar 

  29. Skinner, D. J. & Dunkel, J. Improved bounds on entropy production in living systems. Proc. Natl Acad. Sci. USA 118, e2024300118 (2021).

    Article  CAS  Google Scholar 

  30. Weiss, J. B., Fox-Kemper, B., Mandal, D., Nelson, A. D. & Zia, R. K. P. Nonequilibrium oscillations, probability angular momentum, and the climate system. J. Stat. Phys. 179, 1010–1027 (2020).

    Article  Google Scholar 

  31. Gonzalez, J. P., Neu, J. C. & Teitsworth, S. W. Experimental metrics for detection of detailed balance violation. Phys. Rev. E 99, 022143 (2019).

    Article  CAS  Google Scholar 

  32. Zia, R. K. P., Weiss, J. B., Mandal, D. & Fox-Kemper, B. Manifest and subtle cyclic behavior in nonequilibrium steady states. J. Phys. Conf. Ser. 750, 012003 (2016).

    Article  Google Scholar 

  33. Li, J., Horowitz, J. M., Gingrich, T. R. & Fakhri, N. Quantifying dissipation using fluctuating currents. Nat. Commun. 10, 1666 (2019).

    Article  Google Scholar 

  34. Guo, M. et al. Probing the stochastic, motor-driven properties of the cytoplasm using force spectrum microscopy. Cell 158, 822–832 (2014).

    Article  CAS  Google Scholar 

  35. MacKintosh, F. C. & Levine, A. J. Nonequilibrium mechanics and dynamics of motor-activated gels. Phys. Rev. Lett. 100, 018104 (2008).

    Article  CAS  Google Scholar 

  36. Malik-Garbi, M. et al. Scaling behaviour in steady-state contracting actomyosin networks. Nat. Phys. 15, 509–516 (2019).

    Article  CAS  Google Scholar 

  37. MacKintosh, F. C., Käs, J. & Janmey, P. A. Elasticity of semiflexible biopolymer networks. Phys. Rev. Lett. 75, 4425–4428 (1995).

    Article  CAS  Google Scholar 

  38. Valentine, M. T., Perlman, Z. E., Mitchison, T. J. & Weitz, D. A. Mechanical properties of Xenopus egg cytoplasmic extracts. Biophys. J. 88, 680–689 (2005).

    Article  CAS  Google Scholar 

  39. Field, C. M., Pelletier, J. F. & Mitchison, T. J. in Methods in Cell Biology: Cytokinesis Vol. 137 (ed. Echard, A.) 395–435 (Academic, 2017).

  40. Riedl, J. et al. Lifeact: a versatile marker to visualize F-actin. Nat. Methods 5, 605–607 (2008).

    Article  CAS  Google Scholar 

  41. Chen, D. T., Heymann, M., Fraden, S., Nicastro, D. & Dogic, Z. ATP consumption of eukaryotic flagella measured at a single-cell level. Biophys. J. 109, 2562–2573 (2015).

    Article  CAS  Google Scholar 

  42. Ruhnow, F., Zwicker, D. & Diez, S. Tracking single particles and elongated filaments with nanometer precision. Biophys. J. 100, 2820–2828 (2011).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

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

  1. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Alexandru Bacanu, James F. Pelletier, Yoon Jung & Nikta Fakhri

  2. Department of Molecular and Cellular Biology, Harvard University, Cambridge, MA, USA

    Alexandru Bacanu

  3. Department of Stem Cell and Regenerative Biology, Harvard University, Cambridge, MA, USA

    Alexandru Bacanu

  4. Centro Nacional de Biotecnología (CNB), CSIC, Madrid, Spain

    James F. Pelletier

  5. Grupo Interdisciplinar de Sistemas Complejos (GISC), Madrid, Spain

    James F. Pelletier

Authors
  1. Alexandru Bacanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James F. Pelletier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yoon Jung
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nikta Fakhri
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

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

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bacanu, A., Pelletier, J.F., Jung, Y. et al. Inferring scale-dependent non-equilibrium activity using carbon nanotubes. Nat. Nanotechnol. 18, 905–911 (2023). https://doi.org/10.1038/s41565-023-01395-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41565-023-01395-2

This article is cited by

Search

Advanced search

Quick links

Find nanotechnology articles, nanomaterial data and patents all in one place. Visit Nano by Nature Research