Skip to main content
SpringerLink
Log in

Micro-swimming without flagella: Propulsion by internal structures

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

Since a first proof-of-concept for an autonomous micro-swimming device appeared in 2005 a strong interest on the subject ensued. The most common configuration consists of a cell driven by an external propeller, bio-inspired by bacteria such as E.coli. It is natural to investigate whether micro-robots powered by internal mechanisms could be competitive. We compute the translational and rotational velocity of a spheroid that produces a helical wave on its surface, as has been suggested for the rod-shaped cyanobacterium Synechococcus. This organisms swims up to ten body lengths per second without external flagella. For the mathematical analysis we employ the tangent plane approximation method, which is adequate for amplitudes, frequencies and wave lengths considered here. We also present a qualitative discussion about the efficiency of a device driven by an internal rotating structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berg, H.C., How Spirochetes May Swim, J. Theoret. Biol., 1976, vol. 56, no. 2, pp. 269–273.

    Article  Google Scholar 

  2. Blake, J. R., A Spherical Envelope Approach to Ciliary Propulsion, J. Fluid Mech., 1971, vol. 46, pp. 199–208.

    Article  MATH  Google Scholar 

  3. Blake, J. R., Infinite Models for Ciliary Propulsion, J. Fluid Mech., 1971, vol. 49, pp. 209–222.

    Article  MATH  Google Scholar 

  4. Brahamsha, B., Non-Flagellar Swimming in Marine Synechococcus, J. Mol. Microbiol. Biotechnol., 1999, vol. 1, pp. 59–62.

    Google Scholar 

  5. Brennen, C., An Oscillating-Boundary-Layer Theory for Ciliary Propulsion, J. Fluid Mech., 1974, vol. 65, pp. 799–824.

    Article  MATH  Google Scholar 

  6. http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture2.pdf

  7. Chan, B., Balmforth, N., and Hosoi, A., Building a Better Snail: Lubrication and Adhesive Locomotion, Phys. Fluids, 2005, 113101.

  8. Childress, S., Mechanics of Swimming and Flying, Cambridge: Cambridge Univ. Press, 1978.

    Google Scholar 

  9. Childress, S., Inertial Swimming as a Singular Perturbation, in ASME 2008 Dynamic Systems and Control Conference, Paper DSCC2008-2294, 1413–1420 (2008).

  10. Childress, S., Spagnolie, S., and Tokieda, T., A Bug on a Raft: Recoil Locomotion in a Viscous Fluid, J. Fluid Mech., 2011, vol. 669, pp. 527–556.

    Article  MathSciNet  MATH  Google Scholar 

  11. Chwang, A.T., Winet, H., and Wu, T.Y., A Theoretical Mechanism of Spirochete Locomotion, J. Mechanochem. Cell Motil., 1974, vol. 3, no. 1, pp. 69–76.

    Google Scholar 

  12. Cortez, R., The Method of Regularized Stokeslets, SIAM J. Sci. Comput., 2001, vol. 23, no. 4, pp. 1204–1225.

    Article  MathSciNet  MATH  Google Scholar 

  13. Cortez, R., Fauci, L., and Medovikov, A., The Method of Regularized Stokeslets in Three Dimensions: Analysis, Validation and Application to Helical Swimming, Phys. Fluids, 2005, vol. 17, pp.1–14.

    Article  MathSciNet  Google Scholar 

  14. Cortez, R., Fauci, L., and Cowen, N., Simulation of Swimming Organisms: Coupling Internal Mechanics with External Fluid Dynamics, Computing in Science & Engineering (IEEE/AIP), May–June 2004, pp. 38–45.

  15. Dreyfus, R., Baudry, J., Roper, M., Fermigier, M., Stone, H., and Bibette, J., Microscopic Artificial Swimmers, Nature, 2005, vol. 437, pp. 862–865.

    Article  Google Scholar 

  16. Ebbens, S. and Howse, J., In Pursuit of Propulsion at the Nanoscale, Soft Matter, 2010, vol. 6, pp. 726–738.

    Article  Google Scholar 

  17. Ehlers, K.M., Samuel, A.D., Berg, H.C., and Montgomery, R., Do Cyanobacteria Swim Using Traveling Surface Waves?, Proc. Natl. Acad. Sci. USA, 1996, vol. 93, no. 16, pp. 8340–8343.

    Article  Google Scholar 

  18. Ehlers, K. and Koiller, J., Could Cell Membranes Produce Acoustic Streaming? Making the Case for Synechococcus Self-Propulsion, Mathl. Computer Modelling, 2011, vol. 53, nos. 7–8, pp. 1489–1504.

    Article  Google Scholar 

  19. Ehlers, K. and Oster, G., The Mysterious Swimming of Synechococcus, PLoS Comp. Biol., submitted.

  20. Fair, M.C. and Anderson, J.L. Electrophoresis of Nonuniformly Charged Ellipsoidal Particles, J. Colloid Interface Sci., 1989, vol. 127, no. 2, pp. 388–400.

    Article  Google Scholar 

  21. Faraday, M., On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles Upon Vibrating Elastic Surfaces, Philos. Trans. R. Soc. Lond., 1831, vol. 121, pp. 299–340.

    Article  Google Scholar 

  22. Happel, J. and Brenner, H., Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, New York: Springer, 1983.

    Google Scholar 

  23. Jahn, T. L. and Landman, M.D., Locomotion of Spirochetes, Trans. Amer. Microscopical Soc., 1965, vol. 84, no. 3, pp. 95–406.

    Google Scholar 

  24. Jung, S., Mareck, K., Fauci, L., and Shelley, M. J., Rotational Dynamics of a Superhelix Towed in a Stokes Fluid, Phys. Fluids, 2007, vol. 19, 103105.

    Article  Google Scholar 

  25. Kanso, E., Swimming in an Inviscid Fluid, Theoret. Comput. Fluid Dynamics, 2010, vol. 24, nos. 1–4, pp. 201–207.

    Article  MATH  Google Scholar 

  26. Koiller, J., Ehlers, K., and Montgomery, R., Problems and Progress in Microswimming, J. Nonlinear Science, 1996, vol. 6, no. 6, pp. 507–541.

    Article  MathSciNet  MATH  Google Scholar 

  27. Koiller, J., Ehlers, K., and Chalub, F., Acoustic Streaming: The Small Invention of Cyanobacteria? Arbor/CSIC, 2010, vol. 746, pp. 1089–1115.

    Article  Google Scholar 

  28. Kozlov, V.V. and Ramodanov, S. M., On the Motion of a Body with a Rigid Shell and Variable Mass Geometry in a Perfect Fluid, Dokl. Ross. Akad. Nauk, 2002, vol. 382, no. 4, pp. 478–481 [Dokl. Phys., 2002, vol. 47, no. 2, pp. 132–135].

    MathSciNet  Google Scholar 

  29. Kulić, I.M., Thaokar, R., and Schiessel, H., Twirling DNA Rings — Swimming Nanomotors Ready for a Kickstart, Europhys. Lett., 2005, vol. 72, no. 4, pp. 527–533.

    Article  Google Scholar 

  30. Lac, E., Barthés-Biesel, D., Pelekasis, N.A., and Tsamopoulos, J., Spherical Capsules in Three-Dimensional Unbounded Stokes Flows: Effect of the Membrane Constitutive Law and Onset of Buckling, J. Fluid Mech., 2004, vol. 516, pp. 303–334.

    Article  MathSciNet  MATH  Google Scholar 

  31. Lighthill, J., On the Squirming Motion of Nearly Spherical Deformable Bodies through Liquids at Very Small Reynolds Numbers, Comm. Pure Appl. Math., 1952, vol. 5, pp. 109–118.

    Article  MathSciNet  MATH  Google Scholar 

  32. Lighthill, J., Acoustic Streaming, J. Sound Vibration, 1978, vol. 61, no. 3, pp. 391–418.

    Article  MATH  Google Scholar 

  33. Lighthill, J., Helical Distributions of Stokeslets, J. Engrg. Math., 1996, vol. 30, pp. 35–78.

    Article  MathSciNet  MATH  Google Scholar 

  34. Ludwig, W., Zur Theorie der Flimmerbewegung (Dynamik, Nutzeffekt, Energiebilanz), Z. Vergl. Physiol., 1930, vol. 13, pp. 397–504.

    Google Scholar 

  35. McCarren, J., Microscopic, Genetic, and Biochemical Characterization of Non-Flagellar Swimming Motility in Marine Cyanobacteria, Ph.D. Thesis, Scripps Institution of Oceanography, Univ. of California, San Diego, 2005 (download from http://openwetware.orgwikiJay McCarren)

    Google Scholar 

  36. Miloh, T., Galper, A., Self-Propulsion of General Deformable Shapes in a Perfect Fluid, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1993, vol. 442, pp. 273–299.

    Article  MATH  Google Scholar 

  37. Nan, B., Chen, J., Neu, J.C., Berry, R.M., Oster, G., and Zusman, D.R., Myxobacteria Gliding Motility Requires Cytoskeleton Rotation Powered by Proton Motive Force, www.pnas.org/cgi/doi/10.1073/pnas.1018556108 (early edition, 2011).

  38. Nyborg, W., Acoustic Streaming due to Attenuated PlaneWaves, J. Acoust. Soc. America, 1953, vol. 25, no. 1, pp. 68–75.

    Article  MathSciNet  Google Scholar 

  39. Purcell, E. M., Life at Low Reynolds Number, Amer. J. Phys., 1977, vol. 45, no. 1, pp. 3–10.

    Article  MathSciNet  Google Scholar 

  40. Purcell, E.M., The Efficiency of Propulsion by a Rotating Flagellum, Proc. Natl. Acad. Sci. USA, 1997, vol. 94, pp. 11307–11311.

    Article  Google Scholar 

  41. Nelson, B. J., Kaliakatsos, I.K., and Abbott, J. J., Microrobots for Minimally Invasive Medicine, Annu. Rev. Biomed. Eng., 2010, vol. 12, pp.55–85.

    Article  Google Scholar 

  42. Shapere, A. and Wilczek, F., Self Propulsion at Low Reynolds Numbers, Phys. Rev. Lett., 1987, vol. 58:20, pp. 2051–2054.

    Article  Google Scholar 

  43. Shapere, A. and Wilczek, F., Geometry of Self-Propulsion at Low Reynolds Number, J. Fluid Mech., 1989, vol. 198, pp. 557–585.

    Article  MathSciNet  MATH  Google Scholar 

  44. Shapere, A., A Theory of Something, Ph.D. Thesis, Princeton University, 1989.

  45. Sitti, M., Miniature Devices: Voyage of the Microrobots, Nature, 2009, vol. 458, pp. 1121–1122.

    Article  Google Scholar 

  46. Stone, H.A. and Samuel, A.D.T., Propulsion of Microorganisms by Surface Distortions, Phys. Rev. Lett., 1990, vol. 77, pp. 4102–4104.

    Article  Google Scholar 

  47. Taylor, G. I., Analysis of the Swimming Microscopic Organisms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1951, vol. 209, pp. 447–461.

    Article  MATH  Google Scholar 

  48. Taylor, G. I., The Action of Waving Cylindrical Tails in Propelling Microscopic Organisms, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 1952, vol. 211, pp. 225–239.

    Article  MATH  Google Scholar 

  49. Tortora, G., Valdastri, P., Ekawahyu Susilo, E., Menciassi, A., Dario, P., Rieber, F., and Schurr, M., Propeller-Based Wireless Device for Active Capsular Endoscopy in the Gastric District, Minimally Invasive Therapy & Allied Technologies, 2009, vol. 18, no. 5, pp. 280–290.

    Article  Google Scholar 

  50. Takayama, T. and Hirose, S., Amphibious 3D Active Cord Mechanism “HELIX” with Helical Swimming Motion, IEEE/RSJ International Conference on Intelligent Robots and Systems (2002).

  51. Yu, T. S., Lauga, E., and Hosoi, A. E., Experimental Investigations of Elastic Tail Propulsion at Low Reynolds Number, Phys. Fluids, 2006, vol. 18, 091701.

    Article  Google Scholar 

  52. Wang, C.-Y. and Jahn, T. L., A Theory for the Locomotion of Spirochetes, J. Theoret. Biol., 1972, vol. 36, no. 1, pp. 53–60.

    Article  Google Scholar 

  53. Waterbury, J. B., Wiley, J. M., Franks, D. G., Valois, F. W., and Watson, S.W., A Cyanobacterium Capable of Swimming Motility, Science, 1985, vol. 230, no. 4721, pp. 74–76.

    Article  Google Scholar 

  54. Williams, W.E., Boundary Effects in Stokes Flows, J. Fluid Mech., 1966, vol. 24, no. 2, pp. 285–291.

    Article  MathSciNet  Google Scholar 

  55. Wolgemuth, C. W., Charon, N. W., and Goldstein, S. F., The Flagellar Cytoskeleton of the Spirochetes, J. Mol. Microbiol. Biotechnol., 2006, vol. 11, pp. 221–227.

    Article  Google Scholar 

  56. Wu, T.Y., Swimming of a Waving Plate, J. Fluid Mech., 1961, vol. 10, pp. 321–344.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Truckee Meadows Community College, 7000 Dandini Blvd, Reno, NV, 89512, USA

    Kurt M. Ehlers

  2. Escola de Matemática Aplicada, Fundação Getulio Vargas Praia de Botafogo 190, Rio de Janeiro, RJ, 22250-040, Brazil

    Jair Koiller

Authors
  1. Kurt M. Ehlers
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jair Koiller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kurt M. Ehlers.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ehlers, K.M., Koiller, J. Micro-swimming without flagella: Propulsion by internal structures. Regul. Chaot. Dyn. 16, 623–652 (2011). https://doi.org/10.1134/S1560354711060050

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354711060050

MSC2010 numbers

Keywords

Search

Navigation