Skip to main content

Thank you for visiting 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.

Flagella-like beating of actin bundles driven by self-organized myosin waves


Wave-like beating of eukaryotic cilia and flagella—threadlike protrusions found in many cells and microorganisms—is a classic example of spontaneous mechanical oscillations in biology. This type of self-organized active matter raises the question of the coordination mechanism between molecular motor activity and cytoskeletal filament bending. Here we show that in the presence of myosin motors, polymerizing actin filaments self-assemble into polar bundles that exhibit wave-like beating. Importantly, filament beating is associated with myosin density waves initiated at twice the frequency of the actin-bending waves. A theoretical description based on curvature control of motor binding to the filaments and of motor activity explains our observations in a regime of high internal friction. Overall, our results indicate that the binding of myosin to actin depends on the actin bundle shape, providing a feedback mechanism between the myosin activity and filament deformations for the self-organization of large motor filament assemblies.

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

Access options

Rent or buy this article

Get just this article for as long as you need it


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

Fig. 1: Beating of an actin filament bundle with myosin II.
Fig. 2: Beating of an actin filament bundle with myosin V.
Fig. 3: Beating properties as a function of actin bundle length.
Fig. 4: Interplay between myosin V density and actin curvature waves.
Fig. 5: Wave-like beating in a model of an active filament bundle.

Data availability

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


  1. Nicastro, D. The molecular architecture of axonemes revealed by cryoelectron tomography. Science 313, 944–948 (2006).

    Article  ADS  Google Scholar 

  2. Lindemann, C. B. & Lesich, K. A. Flagellar and ciliary beating: the proven and the possible. J. Cell Sci. 123, 519–528 (2010).

    Article  Google Scholar 

  3. Pazour, G. J., Agrin, N., Leszyk, J. & Witman, G. B. Proteomic analysis of a eukaryotic cilium. J. Cell Biol. 170, 103–113 (2005).

    Article  Google Scholar 

  4. Brokaw, C. J. Molecular mechanism for oscillation in flagella and muscle. Proc. Natl Acad. Sci. USA 72, 3102–3106 (1975).

    Article  ADS  Google Scholar 

  5. Riedel-Kruse, I. H., Hilfinger, A., Howard, J. & Julicher, F. How molecular motors shape the flagellar beat. HFSP J. 1, 192–208 (2007).

    Article  Google Scholar 

  6. Machin, K. E. Wave propagation along flagella. J. Exp. Biol. 35, 796–806 (1958).

    Article  Google Scholar 

  7. Brokaw, C. J. Computer simulation of flagellar movement VIII: coordination of dynein by local curvature control can generate helical bending waves. Cell Motil. Cytoskeleton 53, 103–124 (2002).

    Article  Google Scholar 

  8. Brokaw, C. & Rintala, D. Computer simulation of flagellar movement. III. Models incorporating cross-bridge kinetics. J. Mechanochem. Cell 3, 77–86 (1975).

    Google Scholar 

  9. Hines, M. & Blum, J. J. Bend propagation in flagella. I. Derivation of equations of motion and their simulation. Biophys. J. 23, 41–57 (1978).

    Article  ADS  Google Scholar 

  10. Lindemann, C. B. A ‘geometric clutch’ hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168, 175–189 (1994).

    Article  ADS  Google Scholar 

  11. Camalet, S. & Jülicher, F. Generic aspects of axonemal beating. New J. Phys. 2, 24 (2000).

    Article  ADS  Google Scholar 

  12. Sartori, P., Geyer, V. F., Scholich, A., Julicher, F. & Howard, J. Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella. eLife 5, e13258 (2016).

    Article  Google Scholar 

  13. Oriola, D., Gadêlha, H. & Casademunt, J. Nonlinear amplitude dynamics in flagellar beating. R. Soc. Open Sci. 4, 160698 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  14. Mondal, D., Adhikari, R. & Sharma, P. Internal friction controls active ciliary oscillations near the instability threshold. Sci. Adv. 6, eabb0503 (2020).

    Article  ADS  Google Scholar 

  15. Nandagiri, A. et al. Flagellar energetics from high-resolution imaging of beating patterns in tethered mouse sperm. eLife 10, e62524 (2021).

    Article  Google Scholar 

  16. Geyer, V. F., Howard, J. & Sartori, P. Ciliary beating patterns map onto a low-dimensional behavioural space. Nat. Phys. 18, 332–337 (2022).

    Article  Google Scholar 

  17. Bourdieu, L. et al. Spiral defects in motility assays: a measure of motor protein force. Phys. Rev. Lett. 75, 176–179 (1995).

    Article  ADS  Google Scholar 

  18. Placais, P. Y., Balland, M., Guerin, T., Joanny, J. F. & Martin, P. Spontaneous oscillations of a minimal actomyosin system under elastic loading. Phys. Rev. Lett. 103, 158102 (2009).

    Article  ADS  Google Scholar 

  19. Sanchez, T., Welch, D., Nicastro, D. & Dogic, Z. Cilia-like beating of active microtubule bundles. Science 333, 456–459 (2011).

    Article  ADS  Google Scholar 

  20. Jülicher, F. & Prost, J. Spontaneous oscillations of collective molecular motors. Phys. Rev. Lett. 78, 4510–4513 (1997).

    Article  ADS  Google Scholar 

  21. Guerin, T., Prost, J., Martin, P. & Joanny, J. F. Coordination and collective properties of molecular motors: theory. Curr. Opin. Cell Biol. 22, 14–20 (2010).

    Article  Google Scholar 

  22. Reymann, A. C. et al. Nucleation geometry governs ordered actin networks structures. Nat. Mater. 9, 827–832 (2010).

    Article  ADS  Google Scholar 

  23. Reymann, A. C., Guerin, C., Thery, M., Blanchoin, L. & Boujemaa-Paterski, R. Geometrical control of actin assembly and contractility. Methods Cell. Biol. 120, 19–38 (2014).

    Article  Google Scholar 

  24. Richard, M. et al. Active cargo positioning in antiparallel transport networks. Proc. Natl Acad. Sci. USA 116, 14835–14842 (2019).

    Article  ADS  Google Scholar 

  25. Letort, G. et al. Geometrical and mechanical properties control actin filament organization. PLoS Comput. Biol. 11, e1004245 (2015).

    Article  Google Scholar 

  26. Köhler, S., Lieleg, O. & Bausch, A. R. Rheological characterization of the bundling transition in F-actin solutions induced by methylcellulose. PLoS ONE 3, e2736 (2008).

    Article  ADS  Google Scholar 

  27. Gray, J. The movement of the spermatozoa of the bull. J. Exp. Biol. 35, 96–108 (1958).

    Article  Google Scholar 

  28. Rikmenspoel, R. Movements and active moments of bull sperm flagella as a function of temperature and viscosity. J. Exp. Biol. 108, 205–230 (1984).

    Article  Google Scholar 

  29. De La Cruz, E. M. & Ostap, E. M. Relating biochemistry and function in the myosin superfamily. Curr. Opin. Cell Biol. 16, 61–67 (2004).

    Article  Google Scholar 

  30. Howard, J. Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, 2001).

  31. Mehta, A. D. et al. Myosin-V is a processive actin-based motor. Nature 400, 590–593 (1999).

    Article  ADS  Google Scholar 

  32. Rief, M. et al. Myosin-V stepping kinetics: a molecular model for processivity. Proc. Natl Acad. Sci. USA 97, 9482–9486 (2000).

    Article  ADS  Google Scholar 

  33. Clemen, A. E. et al. Force-dependent stepping kinetics of myosin-V. Biophys. J. 88, 4402–4410 (2005).

    Article  ADS  Google Scholar 

  34. Sakamoto, T. et al. Neck length and processivity of myosin V. J. Biol. Chem. 278, 29201–29207 (2003).

    Article  Google Scholar 

  35. Pierobon, P. et al. Velocity, processivity, and individual steps of single myosin V molecules in live cells. Biophys. J. 96, 4268–4275 (2009).

    Article  ADS  Google Scholar 

  36. Howard, J. Mechanical signaling in networks of motor and cytoskeletal proteins. Annu. Rev. Biophys. 38, 217–234 (2009).

    Article  Google Scholar 

  37. Ma, R., Klindt, G. S., Riedel-Kruse, I. H., Jülicher, F. & Friedrich, B. M. Active phase and amplitude fluctuations of flagellar beating. Phys. Rev. Lett. 113, 048101 (2014).

    Article  ADS  Google Scholar 

  38. Egelman, E. H., Francis, N. & DeRosier, D. J. F-actin is a helix with a random variable twist. Nature 298, 131–135 (1982).

    Article  ADS  Google Scholar 

  39. Galkin, V. E., Orlova, A., Schroder, G. F. & Egelman, E. H. Structural polymorphism in F-actin. Nat. Struct. Mol. Biol. 17, 1318–1323 (2010).

    Article  Google Scholar 

  40. Galkin, V. E., Orlova, A. & Egelman, E. H. Actin filaments as tension sensors. Curr. Biol. 22, R96–R101 (2012).

    Article  Google Scholar 

  41. Reynolds, M. J., Hachicho, C., Carl, A. G., Gong, R. & Alushin, G. M. Actin nucleotide state modulates the F-actin structural landscape evoked by bending forces. (2022).

  42. Kozuka, J., Yokota, H., Arai, Y., Ishii, Y. & Yanagida, T. Dynamic polymorphism of single actin molecules in the actin filament. Nat. Chem. Biol. 2, 83–86 (2006).

    Article  Google Scholar 

  43. McGough, A., Pope, B., Chiu, W. & Weeds, A. Cofilin changes the twist of F-actin: implications for actin filament dynamics and cellular function. J. Cell Biol. 138, 771–781 (1997).

    Article  Google Scholar 

  44. Risca, V. I. et al. Actin filament curvature biases branching direction. Proc. Natl Acad. Sci. USA 109, 2913–2918 (2012).

    Article  ADS  Google Scholar 

  45. Tsaturyan, A. K. et al. Strong binding of myosin heads stretches and twists the actin helix. Biophys. J. 88, 1902–1910 (2005).

    Article  ADS  Google Scholar 

  46. Jegou, A. & Romet-Lemonne, G. The many implications of actin filament helicity. Semin. Cell Dev. Biol. 102, 65–72 (2020).

    Article  Google Scholar 

  47. Mei, L. et al. Molecular mechanism for direct actin force-sensing by α-catenin. eLife 9, e62514 (2020).

    Article  Google Scholar 

  48. Winkelman, J. D., Anderson, C. A., Suarez, C., Kovar, D. R. & Gardel, M. L. Evolutionarily diverse LIM domain-containing proteins bind stressed actin filaments through a conserved mechanism. Proc. Natl Acad. Sci. USA 117, 25532–25542 (2020).

  49. Sun, X. et al. Mechanosensing through direct binding of tensed F-actin by LIM domains. Dev. Cell 55, 468–482 (2020).

    Article  Google Scholar 

  50. Shimozawa, T. & Ishiwata, S. Mechanical distortion of single actin filaments induced by external force: detection by fluorescence imaging. Biophys. J. 96, 1036–1044 (2009).

    Article  ADS  Google Scholar 

  51. Carvalho, K. et al. Actin polymerization or myosin contraction: two ways to build up cortical tension for symmetry breaking. Philos. Trans. R. Soc. B 368, 20130005 (2013).

    Article  Google Scholar 

  52. Margossian, S. S. & Lowey, S. Preparation of myosin and its subfragments from rabbit skeletal muscle. Methods Enzym. 85, 55–71 (1982).

    Article  Google Scholar 

  53. Snyder, G. E., Sakamoto, T., Hammer, J. A., Sellers, J. R. & Selvin, P. R. Nanometer localization of single green fluorescent proteins: evidence that myosin V walks hand-over-hand via telemark configuration. Biophys. J. 87, 1776–1783 (2004).

    Article  ADS  Google Scholar 

  54. Steger, C. An unbiased detector of curvilinear structures. IEEE Trans. Pattern Anal. Mach. Intell. 20, 113–125 (1998).

    Article  Google Scholar 

Download references


We are indebted to M. Rief for providing the heavy meromyosin II molecules. We thank the Cell and Tissue Imaging core facility (PICT-IBiSA) of the Institut Curie, a member of the French National Research Infrastructure France-BioImaging (ANR10-INBS-04). We thank J. Manzi for protein purification and characterization; F. Di Federico for the genetic construct of His–pWA–streptavidin; J.-Y. Tinevez, A. Allard and I. Bonnet for help with MATLAB programming; H. Ennomani and C. Guérin for help with actin micropatterning; and J. Plastino and C. Sykes for fruitful discussions. This work was supported by the French National Agency for Research (ANR-12-BSV5 0014; P.M. and L.B. and ANR-21-CE30-0057; PM and FJ), Labex Cell(n)Scale ANR-11-LABX-0038 and ANR-10-IDEX-001-02, United States National Institutes of Health Grant R35-GM135656 (E.M.D.L.C.) and European Research Council (741773 (AAA); L.B.).

Author information

Authors and Affiliations



M.P. and M.M. contributed equally to the work. M.P., M.M., M.R., A.C., J.-F.J., F.J., L.B. and P.M. designed and performed the research and analysed the data. Y.T., W.C., E.M.D.L.C. and J.R.S. provided the new reagents. M.P., M.M., J.-F.J, F.J., L.B. and P.M. wrote the article.

Corresponding author

Correspondence to Pascal Martin.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Hermes Bloomfield-Gadêlha and the other, anonymous, reviewer(s) 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.

Extended data

Extended Data Fig. 1 Actin micropatterns with or without myosin motors.

a, Radial network of actin filaments that have grown away from the border of a single 60-µm nucleation disk of actin polymerization (no motors). There is a branched filament network on the disk surface. b, With the same disk of actin nucleation as in a, polymerization in the presence of myosin II motors added in bulk results in the formation of actin filament bundles that are curved near their tips. c and d, Same as in a and b, respectively, but with a smaller, 9-µm nucleation disk. e, Similar actin pattern as that shown in d but bundling and beating are here driven by myosin V. f, A line of contiguous 9-µm disks results in a more complex arrangement of actin filament bundles. Although the bundles are static in their basal half, myosin-V driven beating is observed in a region that spans about 20 µm from the bundles’ tips. Beating of the bundles seen in d, e and f can be visualized in Supplementary Videos 1, 2, and 3, respectively. The beating properties of the bundle in the box of d and f are analysed in Figs. 1 and 2 of the main text, respectively.

Extended Data Fig. 2 Tangent-angle maximal amplitude as a function of bundle length.

The beating bundles are driven by myosin II (black disks) or by myosin V (white disks). The inset shows the same relation for a single growing bundle driven by myosin II. In both cases, the maximal amplitude \(\psi _{MAX}\) of tangent-angle oscillation saturates beyond a bundle length of about 11 µm.

Extended Data Fig. 3 Actin fluorescence blinking betrays out-of-the-plane beating.

a, Oscillation of actin-fluorescence intensity (arbitrary units) as a function of time at the arc length s= 6 µm along the centre line of an actin-filament bundle. The period of oscillation is T = 3.3 s. b, Oscillation of the tangent angle ψ to the bundle’s centre line as a function of time at the same arc length as in a. The period of oscillation is twice that of the fluorescence oscillation shown in a. c, Phase ϕ of the actin fluorescence oscillation (solid line) and of the tangent angle oscillation (dashed line) as a function of arc length s. The observed phase accumulation of fluorescence oscillation reveals propagation of a travelling wave with half the wavelength but with the same velocity, here 2.3 µm s−1, as those of the corresponding actin-bending waves. d, Oscillation of the apparent bundle length L as a function of time, with the same period as that of fluorescence oscillations shown in a. The apparent bundle length is defined as the arc length of the bundle’s tip that is detected in the actin-fluorescence image. This behaviour is interpreted as the consequence of a small out-of-the-plane component to the beat of the actin bundle, which modulates the measured intensity of actin fluorescence within the field depth of the spinning-disk microscope. The beating actin bundle was here driven by myosin II. See Supplementary Video 5 for an animated representation of the same data.

Extended Data Fig. 4 Actin- and myosin-fluorescence profiles.

a, Actin-fluorescence profiles along the centre line of a beating actin-filament bundle. b, Ratio of myosin-fluorescence and actin-fluorescence profiles. The order of the line colours, from dark blue to red, indicates time progression (time interval of 2 s); the profiles are plotted over one period of the myosin-density waves. Same beating actin bundle as in Figs. 2 and 4 in the main text.

Extended Data Fig. 5 Myosin localization within a beating actin bundle.

a, Fluorescence images for actin (top row, in red) and myosin V (middle row, in cyan), as well as merged signals (bottom row), with an 8-s time interval over one period of the beat. b, Position of the myosin-density peak (disks) along the bundle’s centreline (grey lines) as a function of time over 4.5 periods of the beat, with a sampling time of 2 s. Same bundle in ab as in Figs. 2 and 4 in the main text. c, Same representation as in b for an actin-bundle that grows in time. The panels correspond to consecutive sections of a 137-s long recording, each lasting about 45 s. d, Mean position of the myosin-density peak for the data shown in each panel of c, with the corresponding colour code. Error bars are s.d. from the mean. The motors move vertically as the bundle grows, remaining near the centre of the figure-of-eight pattern of the actin beat. The data shown in c–d are associated with the Supplementary Video 9.

Extended Data Fig. 6 Myosin peak position with respect to actin bundle curvature.

a, At the initiation of a myosin-density wave, corresponding to the sudden appearance of a peak in the myosin-density profile along the bundle’s centre line (Fig. 4a and d in the main text and Supplementary Video 7), the myosin-density peak (disks) is localized near an arc length of maximal positive curvature (Fig. 4e in the main text). b, A quarter period of tangent-angle oscillation later, here 8 s, the myosin-density peak is localized farther toward the bundle’s tip, leaving the positively curved region of the actin-filament bundle behind. c, Half a period of tangent-angle oscillation after the initiation of the first myosin density wave, a second myosin-density wave is initiated at about the same arc length as in a, where curvature of the centre line is negative and near a local maximum of absolute value (see Fig. 4e in the main text). d, Eight seconds later, the myosin-density peak is localized farther toward the bundle’s tip, leaving the negatively curved region behind. Same data as in Figs. 2 and 4 of the main text, for 14 successive periods of tangent-angle oscillations. Red (blue) lines correspond to the production of myosin-density waves associated with positive (negative) curvature of the actin-filament bundle.

Extended Data Fig. 7 Motor peak position (disks) on filament beating pattern (grey lines) in the model.

Same data as in Fig. 5 of the Main Text and Supplementary Video 11; model parameter values in Supplementary Tables 1 and 2.

Supplementary information

Supplementary Information

Supplementary text and Tables 1 and 2.

Reporting Summary

Supplementary Video 1

Self-organized beating driven by myosin II motors. Video associated with the data shown in Fig. 1 (Extended Data Fig. 1d). Video accelerated by ten times; time interval between successive video frames, 0.225 s. Scale bar, 5 µm.

Supplementary Video 2

Self-organized beating driven by myosin V (example 1). Video accelerated by 100 times; time interval between successive video frames, 2 s. Scale bar, 10 µm.

Supplementary Video 3

Self-organized beating driven by myosin V (example 2). Video associated with the data shown in Fig. 3 (Extended Data Fig. 1f). Video accelerated by 12 times; time interval between successive video frames, 2.02 s. Scale bar, 10 µm.

Supplementary Video 4

Automatic tracking of a beating bundle’s centre line (red). Video associated with the data shown in Fig. 1 (Extended Data Fig. 1d and Supplementary Video 1). Video accelerated by ten times; time interval between successive video frames, 0.225 s. Scale bar, 3 µm.

Supplementary Video 5

Actin fluorescence profile as a function of time. Video associated with the data shown in Extended Data Fig. 3. In this example of a beating actin filament bundle driven by myosin II, the actin fluorescence at arc length s = 6 µm (indicated by a black disk) oscillates as a function of time with a period of 3.6 s, as does the apparent length of the actin bundle, which can be monitored by tracking the arc length of the white disk. The mean actin-fluorescence profile over the duration of the recording is shown as a red dashed line. Video accelerated by 2.5 times.

Supplementary Video 6

Growth of a beating actin bundle. Video associated with the data shown in Fig. 3b,c; analysis of the beating properties started at time t = 8 min. Beating driven by myosin II. Video accelerated by 180 times; time interval between successive video frames, 0.56 s. Scale bar, 5 µm.

Supplementary Video 7

Actin (left) and myosin V fluorescence (right). Video associated with the data shown in Figs. 2 and 4. Video accelerated by 12 times; time interval between successive video frames, 2.02 s. Scale bar, 3 µm.

Supplementary Video 8

Actin (top) and myosin V fluorescence (bottom) for three beating bundles. Video accelerated by 26.5 times; time interval between successive video frames, 1.13 s. Scale bar, 10 µm.

Supplementary Video 9

Actin (left) and myosin (right) fluorescence during growth of a beating bundle. Video associated with the data in Extended Data Fig. 5c,d. Video accelerated by 56.5 times; time interval between successive video frames, 1.13 s. Scale bar, 3 µm.

Supplementary Video 10

Actin bundle curvature and myosin V density profiles as a function of time. Video associated with the data shown in Figs. 2 and 4 and Supplementary Video 7. Video accelerated by six times; time interval between successive video frames, 2.02 s.

Supplementary Video 11

Filament-bending waves (left) and motor density waves (right) in the model. Video associated with the data shown in Fig. 5 and parameter values in the Supplementary Table 2. Video accelerated by three times; time interval between successive video frames, 0.2 s.

Source data

Source Data Fig. 2d

Statistical source data for Fig. 2d.

Source Data Fig. 3a

Statistical source data for Fig. 3a.

Source Data Fig. 3c

Statistical source data for Fig. 3c.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pochitaloff, M., Miranda, M., Richard, M. et al. Flagella-like beating of actin bundles driven by self-organized myosin waves. Nat. Phys. 18, 1240–1247 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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