Polygonal motion and adaptable phototaxis via flagellar beat switching in the microswimmer Euglena gracilis

An Author Correction to this article was published on 09 October 2018

This article has been updated


Biological microswimmers exhibit versatile strategies for sensing and navigating their environment, such as run-and-tumble and curvature modulation. Here, we report a striking phototactic behaviour of the microswimmer Euglena gracilis, where these eukaryotic cells swim in polygonal trajectories due to a sudden increase in light intensity. While smoothly curved trajectories are common for microswimmers, such quantized ones have not been reported previously. We find that this polygonal behaviour emerges from periodic switching between the flagellar beating patterns of helical swimming and spinning behaviours. We develop and experimentally validate a biophysical model that describes the phase relationship between the eyespot, cell orientation, light detection and cellular reorientation, accounting for all three behavioural states. Coordinated switching between these behaviours selects for ballistic, superdiffusive, diffusive or subdiffusive motion (including tuning the effective diffusion constant over several orders of magnitude), thereby enabling navigation in spatially structured light fields, such as edge avoidance and gradient descent. This feedback control links multiple system scales (flagellar beats, cellular behaviours and phototaxis strategies), with implications for other natural and synthetic microswimmers.

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Fig. 1: Euglena gracilis cells swim in striking polygonal patterns following a step-up in light intensity.
Fig. 2: Euglena switches between two flagellar beating patterns to achieve the three behavioural states of helical swimming, polygonal swimming and spinning.
Fig. 3: The number of spinning-type beats determines the turning angle and hence the order of the polygon.
Fig. 4: A biophysical model accounts for the reorientation feedback of Euglena in response to the detected light, capturing the transitions between the three behavioural states.
Fig. 5: Model simulations and experiments reveal distinct phase relations between eyespot and cell orientation for the three different behavioural states.
Fig. 6: Euglena accomplish versatile phototaxis strategies including edge avoidance and gradient descent, through behavioural state switching and selection of different anomalous diffusion types.

Data availability

The data sets and computer codes generated during and analysed during the current study are available from the corresponding author on reasonable request.

Change history

  • 09 October 2018

    In the version of this Article originally published, the angular oscillation of amplitude in Fig. 4a was incorrectly labelled ζ; it should have been ξ. Also, the blue line in the top-right corner of Fig. 6d should not have been dash-dotted but solid. These have now been corrected in all versions of the Article.


  1. 1.

    Miller, M. B. & Bassler, B. L. Quorum sensing in bacteria. Annu. Rev. Microbiol. 55, 165–199 (2001).

    Article  Google Scholar 

  2. 2.

    Berg, H. C. & Brown, D. A chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239, 500–504 (1972).

    ADS  Article  Google Scholar 

  3. 3.

    Friedrich, B. M. & Jülicher, F. Chemotaxis of sperm cells. Proc. Natl Acad. Sci. USA 104, 13256–13261 (2007).

    ADS  Article  Google Scholar 

  4. 4.

    Drescher, K., Goldstein, R. E. & Tuval, I. Fidelity of adaptive phototaxis. Proc. Natl Acad. Sci. USA 107, 11171–11176 (2010).

    ADS  Article  Google Scholar 

  5. 5.

    Diehn, B. Phototaxis and sensory transduction in Euglena. Science 181, 1009–1015 (1973).

    ADS  Article  Google Scholar 

  6. 6.

    Häder, D. Simulation of phototaxis in the flagellate Euglena gracilis. J. Biol. Phys. 19, 95–108 (1993).

    Article  Google Scholar 

  7. 7.

    Machemer, H. Swimming sensory cells: electrical membrane parameters, receptor properties and motor control in ciliated protozoa. Verh. Dtsch. Zool. Ges. 1977, 86–110 (1977).

    Google Scholar 

  8. 8.

    Ogawa, N., Oku, H., Hashimoto, K. & Ishikawa, M. A physical model for galvanotaxis of Paramecium cell. J. Theor. Biol. 242, 314–328 (2006).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Riedel-Kruse, I. H., Chung, A. M., Dura, B., Hamilton, A. L. & Lee, B. C. Design, engineering and utility of biotic games. Lab Chip 11, 14–22 (2011).

    Article  Google Scholar 

  10. 10.

    Kantsler, V., Dunkel, J., Blayney, M. & Goldstein, R. E. Rheotaxis facilitates upstream navigation of mammalian sperm cells. eLife 3, e02403 (2014).

    Article  Google Scholar 

  11. 11.

    Arrieta, J., Barreira, A., Chioccioli, M., Polin, M. & Tuval, I. Phototaxis beyond turning: persistent accumulation and response acclimation of the microalga Chlamydomonas reinhardtii. Sci. Rep. 7, 3447 (2017).

    ADS  Article  Google Scholar 

  12. 12.

    Schaller, K., David, R. & Uhl, R. How Chlamydomonas keeps track of the light once it has reached the right phototactic orientation. Biophys. J. 73, 1562–1572 (1997).

    Article  Google Scholar 

  13. 13.

    Bennett, R. R. & Golestanian, R. A steering mechanism for phototaxis in Chlamydomonas. J. R. Soc. Interface 12, 20141164 (2015).

    Article  Google Scholar 

  14. 14.

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

    Article  Google Scholar 

  15. 15.

    Leptos, K. C., Chioccioli, M., Furlan, S., Pesci, A. I. & Goldstein, R. E. An adaptive flagellar photoresponse determines the dynamics of accurate phototactic steering in Chlamydomonas. Preprint at https://www.biorxiv.org/content/early/2018/02/17/254714 (2018).

  16. 16.

    Hill, N. A. & Vincent, R. V. A simple model and strategies for orientation in phototactic microorganisms. J. Theor. Biol. 163, 223–235 (1993).

    Article  Google Scholar 

  17. 17.

    Giometto, A., Altermatt, F., Maritan, A., Stocker, R. & Rinaldo, A. Generalized receptor law governs phototaxis in the phytoplankton Euglena gracilis. Proc. Natl Acad. Sci. USA 112, 7045–7050 (2015).

    ADS  Article  Google Scholar 

  18. 18.

    Rossi, M., Cicconofri, G., Beran, A., Noselli, G. & DeSimone, A. Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes. Proc. Natl Acad. Sci. USA 114, 13085–13090 (2017).

    ADS  Article  Google Scholar 

  19. 19.

    Wan, K. Y. & Goldstein, R. E. Time irreversibility and criticality in the motility of a flagellate microorganism. Phys. Rev. Lett. 121, 058103 (2018).

    ADS  Article  Google Scholar 

  20. 20.

    Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325, 487–490 (2009).

    ADS  Article  Google Scholar 

  21. 21.

    Lauga, E., DiLuzio, W. R., Whitesides, G. M. & Stone, H. A. Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90, 400–412 (2006).

    ADS  Article  Google Scholar 

  22. 22.

    Ozasa, K. et al. Temporal change of photophobic step-up responses of Euglena gracilis investigated through motion analysis. PLoS One 12, e0172813 (2017).

    Article  Google Scholar 

  23. 23.

    Nichols, K. M., Jacklet, A. & Rikmenspoel, R. Effects of Mg2+ and Ca2+ on photoinduced Euglena flagellar responses. J. Cell. Biol. 84, 355–363 (1980).

    Article  Google Scholar 

  24. 24.

    Arroyo, M., Heltai, L., Millán, D. & DeSimone, A. Reverse engineering the euglenoid movement. Proc. Natl Acad. Sci. USA 109, 17874–17879 (2012).

    ADS  Article  Google Scholar 

  25. 25.

    Lee, S. A. et al. Trap it!: a playful human–biology interaction for a museum installation. In Proc. 33rd Annual ACM Conference on Human Factors in Computing Systems 2593–2602 (ACM, 2015).

  26. 26.

    Cira, N. J. et al. A biotic game design project for integrated life science and engineering education. PLoS Biol. 13, e1002110 (2015).

    Article  Google Scholar 

  27. 27.

    Hossain, Z. et al. Interactive and scalable biology cloud experimentation for scientific inquiry and education. Nat. Biotechnol. 34, 1293–1298 (2016).

    Article  Google Scholar 

  28. 28.

    Hossain, Z. et al. Design guidelines and empirical case study for scaling authentic inquiry-based science learning via open online courses and interactive biology cloud labs. Int. J. Artif. Intell. Educ. https://doi.org/10.1007/s40593-017-0150-3 (2017).

    Article  Google Scholar 

  29. 29.

    Kim, H. et al. LudusScope: accessible interactive smartphone microscopy for life-science education. PLoS One 11, e0162602 (2016).

    Article  Google Scholar 

  30. 30.

    Morimoto, K. & Takemura, A. Developing teaching materials using LED for a phototaxis in Euglena. J. Res. Sci. Educ. 45, 73–77 (2005).

    Google Scholar 

  31. 31.

    Ozasa, K., Lee, J., Song, S., Hara, M. & Maeda, M. Gas/liquid sensing via chemotaxis of Euglena cells confined in an isolated micro-aquarium. Lab Chip 13, 4033–4039 (2013).

    Article  Google Scholar 

  32. 32.

    Ozasa, K., Lee, J., Song, S., Hara, M. & Maeda, M. Two-dimensional optical feedback control of Euglena confined in closed-type microfluidic channels. Lab Chip 11, 1933–1940 (2011).

    Article  Google Scholar 

  33. 33.

    Lam, A. T. et al. Device and programming abstractions for spatiotemporal control of active micro-particle swarms. Lab Chip 17, 1442–1451 (2017).

    Article  Google Scholar 

  34. 34.

    Krajvovic, J., Vesteg, M. & Schwartzbach, S. D. Euglenoid flagellates: A multifaceted biotechnology platform. J. Biotechnol. 202, 135–145 (2015).

    Article  Google Scholar 

  35. 35.

    Leander, B. S., Lax, G., Karnkowska, A. & Simpson, A. G. B. Handbook of the Protists 1047–1088 (Springer, Cham, 2017).

  36. 36.

    Ascoli, C., Barbi, M., Frediani, C. & Mure, A. Measurements of Euglena motion parameters by laser light scattering. Biophys. J. 24, 585–599 (1978).

    Article  Google Scholar 

  37. 37.

    Ogawa, T. et al. The flux of Euglena gracilis cells depends on the gradient of light intensity. PLoS One 11, e0168114 (2016).

    Article  Google Scholar 

  38. 38.

    Metzler, R. & Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000).

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Metzler, R., Jeon, J.-H., Cherstvy, A. G. & Barkai, E. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014).

    Article  Google Scholar 

  40. 40.

    Häder, D.-P. & Griebenow, K. Orientation of the green flagellate, Euglena gracilis, in a vertical column of water. FEMS Microbiol. Lett. 53, 159–167 (1988).

    Article  Google Scholar 

  41. 41.

    Shoji, E., Nishimori, H., Awazu, A., Izumi, S. & Iima, M. Localized bioconvection patterns and their initial state dependency in Euglena gracilis suspensions in an annular container. J. Phys. Soc. Jpn 83, 043001 (2014).

    ADS  Article  Google Scholar 

  42. 42.

    Wioland, H., Woodhouse, F. G., Dunkel, J. & Goldstein, R. E. Ferromagnetic and antiferromagnetic order in bacterial vortex lattices. Nat. Phys. 12, 341–345 (2016).

    Article  Google Scholar 

  43. 43.

    Riedel, I. H., Kruse, K. & Howard, J. A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309, 300–303 (2005).

    ADS  Article  Google Scholar 

  44. 44.

    Dai, B. et al. Programmable artificial phototactic microswimmer. Nat. Nanotech. 11, 1087–1092 (2016).

    ADS  Article  Google Scholar 

  45. 45.

    Lozano, C., Ten Hagen, B., Löwen, H. & Bechinger, C. Phototaxis of synthetic microswimmers in optical landscapes. Nat. Commun. 7, 12828 (2016).

    ADS  Article  Google Scholar 

  46. 46.

    Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. Living crystals of light-activated colloidal surfers. Science 339, 936–940 (2013).

    ADS  Article  Google Scholar 

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We thank members of the Riedel-Kruse laboratory, B. Friedrich, J. Dunkel, N. Ouellette and A. Macdonald. This work was supported by NSF grant no. 1324753, the Stanford Discovery Innovation Fund and the Croucher Foundation (through a postdoctoral fellowship to A.C.H.T.).

Author information




A.C.H.T. and I.H.R.-K. were responsible for the project idea, the theory and manuscript preparation; A.C.H.T. was responsible for the modelling, the experiments and data analysis (except Fig. 6: A.C.H.T. and A.T.L.).

Corresponding author

Correspondence to Ingmar H. Riedel-Kruse.

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Supplementary information

Supplementary Information

Supplementary Methods, Supplementary Figures 1–14, Supplementary Table 1, Supplementary References 1–4

Reporting Summary

Supplementary Video 1

Euglena swimming in a polygonal trajectory at intermediate light intensity. The images were sampled at 200 fps and the video is replayed at 10 × slower than the real time

Supplementary Video 2

Euglena swimming in a helical trajectory at low light intensity, recorded at 400 fps. The images were sampled at 400 fps and the video is replayed at 20 × slower than the real time

Supplementary Video 3

Euglena spinning around locally at high light intensity, recorded at 400 fps. The images were sampled at 400 fps and the video is replayed at 20 × slower than the real time

Supplementary Video 4

Various Euglena flagellar beat patterns for helical swimming, spinning, and polygonal swimming. The images are sampled at 200 fps

Supplementary Video 5

Comparison of simulation and experiment for helical swimming

Supplementary Video 6

Comparison of simulation and experiment for spinning

Supplementary Video 7

Comparison of simulation and experiment for polygonal swimming

Supplementary Video 8

Experimental tracking of Euglena exhibiting light avoidance from a light barrier

Supplementary Video 9

Experimental tracking of Euglena exhibiting spinning behaviour initially, followed by swimming in polygonal paths of increasing order, thereby expanding the search radius to navigate the light edge

Supplementary Video 10

Experimental tracking of Euglena exhibiting biased run-and-tumble down the light gradient

Supplementary Video 11

Simulations of Euglena exhibiting light avoidance from a light barrier

Supplementary Video 12

Simulations of Euglena exhibiting polygonal paths of increasing order and navigating the light edge

Supplementary Video 13

Simulations of Euglena exhibiting biased run-and-tumble down the light gradient

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Tsang, A.C.H., Lam, A.T. & Riedel-Kruse, I.H. Polygonal motion and adaptable phototaxis via flagellar beat switching in the microswimmer Euglena gracilis. Nature Phys 14, 1216–1222 (2018). https://doi.org/10.1038/s41567-018-0277-7

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