Skip to main content
SpringerLink
Log in

Coupled oscillations of a protein microtubule immersed in cytoplasm: an orthotropic elastic shell modeling

  • Original Paper
  • Published:
Journal of Biological Physics Aims and scope Submit manuscript

Abstract

Revealing vibration characteristics of sub-cellular structural components such as membranes and microtubules has a principal role in obtaining a deeper understanding of their biological functions. Nevertheless, limitations and challenges in biological experiments at this scale necessitates the use of mathematical and computational models as an alternative solution. As one of the three major cytoskeletal filaments, microtubules are highly anisotropic structures built from tubulin heterodimers. They are hollow cylindrical shells with a ∼ 25 nm outer diameter and are tens of microns long. In this study, a mechanical model including the effects of the viscous cytosol and surrounding filaments is developed for predicting the coupled oscillations of a single microtubule immersed in cytoplasm. The first-order shear deformation shell theory for orthotropic materials is used to model the microtubule, whereas the motion of the cytosol is analyzed by considering the Stokes flow. The viscous cytosol and the microtubule are coupled through the continuity condition across the microtubule–cytosol interface. The stress and velocity fields in the cytosol induced by vibrating microtubule are analytically determined. Finally, the influences of the dynamic viscosity of the cytosol, filament network elasticity, microtubule shear modulus, and circumferential wave-number on longitudinal, radial, and torsional modes of microtubule vibration are elucidated.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Cifra, M., Pokorný, J., Havelka, D., Kučera, O.: Electric field generated by axial longitudinal vibration modes of microtubule. BioSystems 100, 122–131 (2010)

    Article  Google Scholar 

  2. Pokorný, J., Hašek, J., Jelínek, F.: Endogenous electric field and organization of living matter. Electromagnetic Biology and Medicine 24, 185–197 (2005)

    Article  Google Scholar 

  3. Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P.: Molecular Biology of the Cell. Garland Science, New York (2005)

    Google Scholar 

  4. Kasas, S., Cibert, C., Kis, A., Rios, P.D.L., Riederer, B.M., Forro, L., Dietler, G., Catsicas, S.: Oscillation modes of microtubules. Biol. Cell 96, 697–700 (2004)

    Article  Google Scholar 

  5. Jordan, M.A., Wilson, L.: Microtubules as a target for anticancer drugs. Nat. Rev. Cancer 4, 253–265 (2004)

    Article  Google Scholar 

  6. Yi, L.J., Chang, T.C., Ru, C.Q.: Buckling of microtubules under bending and torsion. J. Appl. Phys. 103, 103516–103521 (2008)

    Article  ADS  Google Scholar 

  7. Deriu, M.A., Soncini, M., Orsi, M., Patel, M., Essex, J.W., Montevecchi, F.M., Redaelli, A.: Anisotropic elastic network modeling of entire microtubules. Biophys. J. 99, 2190–2199 (2010)

    Article  ADS  Google Scholar 

  8. Sirenko, Y.M., Stroscio, M.A., Kim, K.W.: Elastic vibration of microtubules in a fluid. Phys. Rev. E 53, 1003–1010 (1996)

    Article  ADS  Google Scholar 

  9. Pokorný, J., Jelínek, F., Trkal, V., Lamprecht, I., Holzel, R.: Vibrations in microtubules. Astrophy. Space Sci. 23, 171–179 (1997)

    Google Scholar 

  10. Portet, S., Tuszynski, J.A., Hogue, C.W.V., Dixon, J.M.: Elastic vibrations in seamless microtubules. Eur. Biophys. J. 34, 912–920 (2005)

    Article  Google Scholar 

  11. Wang, C.Y., Ru, C.Q., Mioduchowski, A.: Vibration of microtubules as orthotropic elastic shells. Physica E 35, 48–56 (2006)

    Article  ADS  Google Scholar 

  12. Wang, C.Y., Zhang, L.C.: Circumferential vibration of microtubules with long axial wavelength. J. Biomech. 41, 1892–1896 (2008)

    Article  Google Scholar 

  13. Shi, Y.J., Guo, W.L., Ru, C.Q.: Relevance of Timoshenko-beam model to microtubules of low shear modulus. Physica E 41, 213–219 (2008)

    Article  ADS  Google Scholar 

  14. Civalek, Ö., Akgöz, B.: Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler-Bernoulli beam modeling. Sci. Iran. 17(6), 367–375 (2010)

    MATH  Google Scholar 

  15. Tounsi, A., Heireche, H., Benhassaini, Missouri, M.: Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory. J. Theor. Biol. 266, 250–255 (2010)

    Article  Google Scholar 

  16. Ghavanloo, E., Daneshmand, F., Amabili, M.: Vibration analysis of a single microtubule surrounded by cytoplasm. Physica E 43, 192–198 (2010)

    Article  ADS  Google Scholar 

  17. Samarbakhsh, A., Tuszynski J.A.: Vibrational dynamics of bio- and nano-filaments in viscous solution subjected to ultrasound: implications for microtubules. Eur. Biophys. J. 40(9), 937–946 (2011)

    Article  Google Scholar 

  18. Kasza, K.E., Rowat, A.C., Liu, J.Y., Angelini, T.E., Brangwynne, C.P., Koenderink, G.H., Weitz, D.A.: The cell as a material. Curr. Opin. Cell Biol. 19, 101–107 (2007)

    Article  Google Scholar 

  19. Tuszynski, J.A., Luchko, T., Portet, S., Dixon, J.M.: Anisotropic elastic properties of microtubules. Eur. Phys. J. E 17, 29–35 (2005)

    Article  Google Scholar 

  20. Pierson, G.B., Burton, P.R., Himes, R.H.: Alterations in number of protofilaments in microtubules assembled in vitro. J. Cell Biol. 76, 223–228 (1978)

    Article  Google Scholar 

  21. Chretien, D., Wade, R.H.: New data on the microtubule surface lattice. Biol. Cell 71, 161–174 (1991)

    Article  Google Scholar 

  22. Li, T.: A mechanics model of microtubule buckling in living cells. J. Biomech. 41, 1722–1729 (2008)

    Article  Google Scholar 

  23. Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, New York (2008)

    Book  MATH  Google Scholar 

  24. Selmane, A., Lakis, A.A.: Dynamic analysis of anisotropic open cylindrical shells. Comput. Struct. 62(1), 1–12 (1997)

    Article  MATH  Google Scholar 

  25. Pickard, W.F.: The role of cytoplasmic streaming in symplastic transport. Plant Cell Environ. 26, 1–15 (2003)

    Article  Google Scholar 

  26. Fu, Y., Zhang, J.: Modeling and analysis of microtubules based on a modified couple stress theory. Physica E 42, 1741–1745 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  27. Sadd, M.H.: Elasticity: Theory, Applications, and Numerics. Elsevier Butterworth–Heinemann, New York (2005)

    Google Scholar 

  28. Chen, S.S.: Dynamics of a rod-shell system conveying fluid. Nucl. Eng. Des. 30, 223–233 (1974)

    Article  Google Scholar 

  29. Yeh, T.T., Chen, S.S.: The effect of fluid viscosity on coupled tube/fluid vibrations. J. Sound Vib. 59(4), 453–467 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  30. de Pablo, P.J., Schaap, I.A.T., MacKintosh, F.C., Schmidt, C.F.: Deformation and collapse of microtubules on the nanometer scale. Phys. Rev. Lett. 91, 098101 (2003)

    Article  ADS  Google Scholar 

  31. Felgner, H., Frank, R., Schiwa, M.: Flexural rigidity of microtubules measured with the use of optical tweezers. J. Cell Sci., Suppl. 109, 509–516 (1996)

    Google Scholar 

  32. Fushimi, K., Verkman, A.S.: Low viscosity in the aqueous domain of cell cytoplasm measured by picosecond polarization microfluorimetry. J. Cell Biol. 112, 719–725 (1991)

    Article  Google Scholar 

  33. Kis, A., Kasas, S., Babić, B., Kulik, A., Benoît, W., Briggs, G.A.D., Schönenberger, C., Catsicas, S., Forró, L.: Nanomechanics of microtubules. Phys. Rev. Lett. 89, 248101 (2002)

    Article  ADS  Google Scholar 

  34. Huang, G.Y., Mai, Y.W., Ru, C.Q.: Surface deflection of a microtubule loaded by a concentrated radial force. Nanotechnology 19, 12501–12508 (2008)

    Google Scholar 

  35. Zheng, J., Pollack, G.: Long-range forces extending from polymer-gel surfaces. Phys. Rev. E 68, 031408 (2003)

    Article  ADS  Google Scholar 

  36. Zheng, J., Chin, W., Khijniak, E., Khijniak, E. Jr., Pollack, GH.: Surfaces and interfacial water: evidence that hydrophilic surfaces have long-range impact. Adv. Colloid Interface Sci. 127, 19–27 (2006)

    Article  Google Scholar 

  37. Chai, B., Yoo, H., Pollack, G.: Effect of radiant energy on near-surface water. J. Phys. Chem. B 113, 13953–13958 (2009)

    Article  Google Scholar 

  38. Chai, B., Zheng, J., Zhao, Q., Pollack, G.: Spectroscopic studies of solutes in aqueous solution. J. Phys. Chem. A 112, 2242–2247 (2008)

    Article  Google Scholar 

  39. Pokorný, J., Vedruccio, C., Cifra, M., Kučera, O.: Cancer physics: diagnostics based on damped cellular elastoelectrical vibrations in microtubules. Eur. Biophys. J. 40(7), 747–759 (2011)

    Article  Google Scholar 

  40. Havelka, D., Cifra, M., Kučera, O., Pokorný, J., Vrba, J.: High-frequency electric field and radiation characteristics of cellular microtubule network. J. Theor. Biol. 286, 31–40 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Shiraz University, Shiraz, 71348-51154, Iran

    Farhang Daneshmand

  2. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street W., Montreal, Québec, Canada, H3A 2K6

    Farhang Daneshmand & Marco Amabili

Authors
  1. Farhang Daneshmand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Amabili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farhang Daneshmand.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Daneshmand, F., Amabili, M. Coupled oscillations of a protein microtubule immersed in cytoplasm: an orthotropic elastic shell modeling. J Biol Phys 38, 429–448 (2012). https://doi.org/10.1007/s10867-012-9263-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10867-012-9263-y

Keywords

Search

Navigation