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.

  • Article
  • Published:

Active hydraulics and odd elasticity of muscle fibres

Nature Physics volume 20, pages 1501–1508 (2024)Cite this article

Subjects

Abstract

Muscle is a complex, hierarchically organized, soft contractile engine. To understand the limits on the rate of contraction and muscle energetics, we construct a coarse-grained multiscale model that describes muscle as an active sponge. Our analysis of existing experiments across species and muscle types highlights the importance of spatially heterogeneous strains and local volumetric deformations. Our minimal theoretical model shows how contractions induce intracellular fluid flow and power active hydraulic oscillations, yielding the limits of ultrafast muscular contractions. We further demonstrate that the viscoelastic response of muscle is naturally non-reciprocal—or odd—owing to its active and anisotropic nature. This enables an alternate mode of muscular power generation from periodic cycles in spatial strain alone, contrasting with previous descriptions based on temporal cycles. Our work suggests a revised view of muscle dynamics that emphasizes the multiscale spatiotemporal origins of soft hydraulic power, with potential implications for physiology, biomechanics and locomotion.

Access through your institution
Buy or subscribe

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

Access options

Access through your institution

Buy this article

  • Purchase on SpringerLink
  • Instant access to full article PDF
Buy now

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

Fig. 1: Muscle fibres are multiscale soft, wet, active engines.
Fig. 2: Volume changes and strain cycles are generic in self-oscillating muscle.
Fig. 3: Active hydraulics limits muscular contraction rates.
Fig. 4: Odd elasticity and strain cycle engine in muscle.
Fig. 5: Active and non-reciprocal viscoelasticity across muscle types and species.

Similar content being viewed by others

Data availability

All original data supporting the findings of this work were obtained from published literature as indicated in Supplementary Section V. Source data are provided with this paper.

References

  1. Huxley, A. Reflections on Muscle Sherrington Lectures (Liverpool Univ. Press, 1980).

  2. Needham, D. M. Machina Carnis: The Biochemistry of Muscular Contraction in its Historical Development (Cambridge Univ. Press, 1971).

  3. Nyitrai, M. et al. What limits the velocity of fast-skeletal muscle contraction in mammals? J. Mol. Biol. 355, 432–442 (2006).

    Article  Google Scholar 

  4. Swank, D. M., Vishnudas, V. K. & Maughan, D. W. An exceptionally fast actomyosin reaction powers insect flight muscle. Proc. Natl Acad. Sci. USA 103, 17543–17547 (2006).

    Article  ADS  Google Scholar 

  5. Mead, A. F. et al. Fundamental constraints in synchronous muscle limit superfast motor control in vertebrates. eLife 6, e29425 (2017).

    Article  Google Scholar 

  6. Gordon, A. M., Homsher, E. & Regnier, M. Regulation of contraction in striated muscle. Physiol. Rev. 80, 853–924 (2000).

    Article  Google Scholar 

  7. Powers, J. D., Malingen, S. A., Regnier, M. & Daniel, T. L. The sliding filament theory since Andrew Huxley: multiscale and multidisciplinary muscle research. Annu. Rev. Biophys. 50, 373–400 (2021).

    Article  Google Scholar 

  8. Millman, B. M. The filament lattice of striated muscle. Physiol. Rev. 78, 359–391 (1998).

    Article  Google Scholar 

  9. Mogilner, A. & Manhart, A. Intracellular fluid mechanics: coupling cytoplasmic flow with active cytoskeletal gel. Annu. Rev. Fluid Mech. 50, 347–370 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  10. Moeendarbary, E. et al. The cytoplasm of living cells behaves as a poroelastic material. Nat. Mater. 12, 253–261 (2013).

    Article  ADS  Google Scholar 

  11. Skotheim, J. M. & Mahadevan, L. Physical limits and design principles for plant and fungal movements. Science 308, 1308–1310 (2005).

    Article  ADS  Google Scholar 

  12. Rome, L. C. & Lindstedt, S. L. The quest for speed: muscles built for high-frequency contractions. Physiology 13, 261–268 (1998).

    Article  Google Scholar 

  13. Syme, D. A. & Josephson, R. K. How to build fast muscles: synchronous and asynchronous designs. Integr. Comp. Biol. 42, 762–770 (2002).

    Article  Google Scholar 

  14. Josephson, R. Contraction dynamics and power output of skeletal muscle. Annu. Rev. Physiol. 55, 527–546 (1993).

    Article  Google Scholar 

  15. Szent-Györgyi, A. The contraction of myosin threads. Stud. Inst. Med. Chem. Univ. Szeged. 1, 17–26 (1942).

    Google Scholar 

  16. Bugyi, B. & Kellermayer, M. The discovery of actin: “to see what everyone else has seen, and to think what nobody has thought”. J. Muscle Res. Cell Motil. 41, 3–9 (2019).

    Article  Google Scholar 

  17. Kaminer, B. Water loss during contracture of muscle. J. Gen. Physiol. 46, 131–142 (1962).

    Article  Google Scholar 

  18. Trombitás, K., Baatsen, P., Schreuder, J. & Pollack, G. H. Contraction-induced movements of water in single fibres of frog skeletal muscle. J. Muscle Res. Cell Motil. 14, 573–584 (1993).

    Article  Google Scholar 

  19. Cecchi, G., Bagni, M., Griffiths, P., Ashley, C. & Maeda, Y. Detection of radial crossbridge force by lattice spacing changes in intact single muscle fibers. Science 250, 1409–1411 (1990).

    Article  ADS  Google Scholar 

  20. Pinto, J. & Win, R. Non-uniform strain distribution in papillary muscles. Am. J. Physiol. 233, H410–H416 (1977).

    Google Scholar 

  21. Neering, I., Quesenberry, L., Morris, V. & Taylor, S. Nonuniform volume changes during muscle contraction. Biophys. J. 59, 926–933 (1991).

    Article  ADS  Google Scholar 

  22. Ghosh, S. et al. Deformation microscopy for dynamic intracellular and intranuclear mapping of mechanics with high spatiotemporal resolution. Cell Rep. 27, 1607–1620 (2019).

    Article  Google Scholar 

  23. Washio, T., Shintani, S. A., Higuchi, H., Sugiura, S. & Hisada, T. Effect of myofibril passive elastic properties on the mechanical communication between motor proteins on adjacent sarcomeres. Sci. Rep. 9, 9355 (2019).

    Article  ADS  Google Scholar 

  24. Kono, F., Kawai, S., Shimamoto, Y. & Ishiwata, S. Nanoscopic changes in the lattice structure of striated muscle sarcomeres involved in the mechanism of spontaneous oscillatory contraction (SPOC). Sci. Rep. 10, 16372 (2020).

    Article  Google Scholar 

  25. Chan, W. P. & Dickinson, M. H. In vivo length oscillations of indirect flight muscles in the fruit fly Drosophila virilis. J. Exp. Biol. 199, 2767–2774 (1996).

    Article  Google Scholar 

  26. Irving, T. & Maughan, D. In vivo X-ray diffraction of indirect flight muscle from Drosophila melanogaster. Biophys. J. 78, 2511–2515 (2000).

    Article  Google Scholar 

  27. Cass, J. A. et al. A mechanism for sarcomere breathing: volume change and advective flow within the myofilament lattice. Biophys. J. 120, 4079–4090 (2021).

    Article  ADS  Google Scholar 

  28. Malingen, S. A. et al. In vivo X-ray diffraction and simultaneous EMG reveal the time course of myofilament lattice dilation and filament stretch. J. Exp. Biol. 223, jeb224188 (2020).

    Article  Google Scholar 

  29. Mijailovich, S. M. et al. Three-dimensional stochastic model of actin–myosin binding in the sarcomere lattice. J. Gen. Physiol. 148, 459–488 (2016).

    Article  Google Scholar 

  30. Sleboda, D. A. & Roberts, T. J. Internal fluid pressure influences muscle contractile force. Proc. Natl Acad. Sci. USA 117, 1772–1778 (2020).

    Article  ADS  Google Scholar 

  31. Malingen, S. A., Hood, K., Lauga, E., Hosoi, A. & Daniel, T. L. Fluid flow in the sarcomere. Arch. Biochem. Biophys. 706, 108923 (2021).

    Article  Google Scholar 

  32. Wang, H. F. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology (Princeton Univ. Press, 2017).

  33. Schoenberg, M. Geometrical factors influencing muscle force development. I. The effect of filament spacing upon axial forces. Biophys. J. 30, 51–67 (1980).

    Article  ADS  Google Scholar 

  34. Schoenberg, M. Geometrical factors influencing muscle force development. II. Radial forces. Biophys. J. 30, 69–77 (1980).

    Article  ADS  Google Scholar 

  35. Guo, B. & Guilford, W. H. Mechanics of actomyosin bonds in different nucleotide states are tuned to muscle contraction. Proc. Natl Acad. Sci. USA 103, 9844–9849 (2006).

    Article  ADS  Google Scholar 

  36. Pringle, J. W. S. The Croonian Lecture, 1977. Stretch activation of muscle: function and mechanism. Proc. R. Soc. Lond. B 201, 107–130 (1978).

    Article  ADS  Google Scholar 

  37. Ait-Mou, Y. et al. Titin strain contributes to the frank–starling law of the heart by structural rearrangements of both thin-and thick-filament proteins. Proc. Natl Acad. Sci. USA 113, 2306–2311 (2016).

    Article  ADS  Google Scholar 

  38. Guérin, T., Prost, J. & Joanny, J.-F. Dynamical behavior of molecular motor assemblies in the rigid and crossbridge models. Eur. Phys. J. E 34, 60 (2011).

    Article  Google Scholar 

  39. Josephson, R. K., Malamud, J. G. & Stokes, D. R. Asynchronous muscle: a primer. J. Exp. Biol. 203, 2713–2722 (2000).

    Article  Google Scholar 

  40. Biot, M. A. Mechanics of Incremental Deformations (John Wiley & Sons, 1965).

  41. Fruchart, M., Scheibner, C. & Vitelli, V. Odd viscosity and odd elasticity. Annu. Rev. Condens. Matter Phys. 14, 471–510 (2023).

    Article  ADS  Google Scholar 

  42. Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020).

    Article  Google Scholar 

  43. Zahalak, G. I. Non-axial muscle stress and stiffness. J. Theor. Biol. 182, 59–84 (1996).

    Article  ADS  Google Scholar 

  44. Josephson, R. K. Mechanical power output from striated muscle during cyclic contraction. J. Exp. Biol. 114, 493–512 (1985).

    Article  Google Scholar 

  45. Tanner, B. C. et al. Thick-to-thin filament surface distance modulates cross-bridge kinetics in drosophila flight muscle. Biophys. J. 103, 1275–1284 (2012).

    Article  ADS  Google Scholar 

  46. Palmer, B. M. et al. Two-state model of acto-myosin attachment-detachment predicts c-process of sinusoidal analysis. Biophys. J. 93, 760–769 (2007).

    Article  ADS  Google Scholar 

  47. Kawai, M. & Brandt, P. W. Sinusoidal analysis: a high resolution method for correlating biochemical reactions with physiological processes in activated skeletal muscles of rabbit, frog and crayfish. J. Muscle Res. Cell Motil. 1, 279–303 (1980).

    Article  Google Scholar 

  48. Roca-Cusachs, P., Conte, V. & Trepat, X. Quantifying forces in cell biology. Nat. Cell Biol. 19, 742–751 (2017).

    Article  Google Scholar 

  49. Marden, J. H. & Allen, L. R. Molecules, muscles, and machines: universal performance characteristics of motors. Proc. Natl Acad. Sci. USA 99, 4161–4166 (2002).

    Article  ADS  Google Scholar 

  50. Ilton, M. et al. The principles of cascading power limits in small, fast biological and engineered systems. Science 360, eaao1082 (2018).

    Article  Google Scholar 

  51. Labonte, D. A theory of physiological similarity in muscle-driven motion. Proc. Natl Acad. Sci. USA 120, e2221217120 (2023).

    Article  MathSciNet  Google Scholar 

  52. Mirvakili, S. M. & Hunter, I. W. Artificial muscles: mechanisms, applications, and challenges. Adv. Mater. 30, 1704407 (2018).

    Article  Google Scholar 

  53. de Gennes, P.-G. A semi-fast artificial muscle. C. R. Acad. Sci. IIB 5, 343–348 (1997).

    Google Scholar 

Download references

Acknowledgements

S.S. acknowledges support from the Harvard Society of Fellows and L.M. acknowledges partial support from the NSF-Simons Center for Mathematical and Statistical Analysis of Biology under grant number 1764269, the Simons Foundation and the Henri Seydoux Fund. We thank S. Srinivasan for useful discussions.

Author information

Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, MA, USA

    Suraj Shankar & L. Mahadevan

  2. Department of Physics, University of Michigan, Ann Arbor, MI, USA

    Suraj Shankar

  3. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    L. Mahadevan

  4. Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, USA

    L. Mahadevan

Authors
  1. Suraj Shankar
    View author publications

    Search author on:PubMed Google Scholar

  2. L. Mahadevan
    View author publications

    Search author on:PubMed Google Scholar

Contributions

L.M. conceived the research topic and approach. S.S. and L.M. formulated the theoretical model. S.S. performed the analytical calculations, and compiled and analysed the data. S.S. and L.M. wrote the paper.

Corresponding authors

Correspondence to Suraj Shankar or L. Mahadevan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Kenneth Campbell, and the other, anonymous, reviewer 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 Figs, 1–3, Tables 1 and 2 and Discussion.

Source data

Source Data Fig. 2

Data for strain in oscillating muscle extracted from references cited in the main text.

Source Data Fig. 3

Parameters for muscle dynamics estimated from references provided in Supplementary Table 1.

Source Data Fig. 5

Muscle rheology data extracted from references cited in the main text.

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

Shankar, S., Mahadevan, L. Active hydraulics and odd elasticity of muscle fibres. Nat. Phys. 20, 1501–1508 (2024). https://doi.org/10.1038/s41567-024-02540-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue date:

  • DOI: https://doi.org/10.1038/s41567-024-02540-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing