Abstract
Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. In this paper, an orthotropic elastic shell model with transverse shearing is developed to study the effects of transverse shearing on shell-like mechanics of microtubules. The study is based on a detailed comparison between four elastic beam and shell models with and without transverse shearing. It is shown that the length-dependent flexural rigidity of microtubules predicted by the present orthotropic shell model with transverse shearing is in good agreement with known experimental data and is consistently close to that given by the Timoshenko-beam model. Our results show that transverse shearing is essential for shell-like deformation of microtubules when the axial wave-length is not extremely long (compared to the diameter of microtubules which is ~25 nm) or the circumferential wave-number is larger than unity. In particular, transverse shearing is found to significantly lower the critical pressure for buckling of a long microtubule under radial pressure and leads to an even better agreement with recently observed experimental data. These results suggest that the 2D orthotropic shell model with transverse shearing is suitable to study the shell-like mechanics of microtubules for short axial wave-length and circumferential wave-number exceeding unity.
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References
Nogales E.: Structural insights into microtubule function. Annu. Rev. Biochem. 69, 277–302 (2000)
Howard J.: Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates Inc, Sunderland (2001)
Cotterill R.: Biophysics-An Introduction. Wiley, New York (2002)
Boal D.: Mechanics of the Cell. Cambridge University Press, Cambridge (2002)
Scholey J.M., Mascher I.B., Mogilner A.: Cell division. Nature 422, 746–752 (2003)
Schliwa M., Woehlke G.: Molecular motors. Nature 422, 759–765 (2003)
Carter N.J., Cross R.A.: Mechanics of the kinesin step. Nature 435, 308–312 (2005)
Kurachi M., Hoshi M., Tashiro H.: Buckling of a single microtubule by optical trapping forces: direct measurement of microtubule rigidity. Cell Motil. Cytoskeleton 30, 221–228 (1995)
Dogterom M., Yurke B.: Measurement of the force-velocity relation for growing microtubules. Science 278, 856–860 (1997)
Takasone T., Juodkazis S., Kawagishi Y., Yamaguchi A., Matsuo S., Sakakibara H., Nakayama H., Misawa H.: Flexural rigidity of a single microtubule. Japanese J. Appl. Phys. 41, 3015–3019 (2002)
Kikumoto M., Kurachi M., Tosa V., Tashiro H.: Flexural rigidity of individual microtubules measured by a buckling force with optical traps. Biophys. J. 90, 1687–1696 (2006)
Brangwynne C.P., Mackintosh F.C., Kumar S., Geisse N.A., Talbot J., Mahadevevan L., Parker K.K., Ingber D.E., Weitz D.E.: Microtubules can bear enhanced compressive loads in living cells because of lateral reinforcement. J. Cell Biol. 173, 733–741 (2006)
Venier P., Maggs A.C., Carlier M.F., Pantaloni D.: Analysis of microtubule rigidity using hydrodynamic flow and thermal fluctuations. J. Biol. Chem. 269, 13353–13360 (1994)
Gittes F., Mickey B., Nettleton J., Howard J.: Flexural rigidity of microtubules and actin filaments measured from thermal fluctuation in shape. J. Cell Biol. 120, 923–934 (1995)
Vinckier A., Dumortier C., Engelborghs Y., Hellemans L.: Dynamical and mechanical study of immobilized microtubules with atomic force microscopy. J. Vac. Sci. Tech. B 14, 1427–1431 (1996)
Cassimeris L., Gard D., Tran P.T., Erickson H.P.: XMAP215 is a long thin molecule that does not increase microtubule stiffness. J. Cell Sci. 114, 3025–3033 (2001)
Janson M.E., Dogterom M.A.: Bending mode analysis for growing microtubules: evidence for a velocity-dependent rigidity. Biophys. J. 87, 2723–2736 (2004)
Pampaloni F., Lattanzi G., Jonas A., Surrey T., Frey E., Florin E.: Thermal fluctuation of grafted microtubules provides evidence of a length-dependent persistent length. PNAS 103, 10248–10253 (2006)
Kis A., Kasas S., Babić B., Kulik A.J., Benoît W., Briggs G.A.D., Schönenberger C., Catsicas S., Forró L.: Nanomechanics of microtubules. Phys. Rev. Lett. 89, 248101-1-4 (2002)
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)
Pablo P.J., Schaap L.A.T., Mackintosh F.C., Schmit C.F.: Deformation and collapse of microtubules on the nanometer scale. Phys. Rev. Lett. 91, 098101-1-4 (2003)
Needleman D.J., Ojeda-Lopez M.A., Raviv U., Ewert K., Jayna B., Jones J.B., Miller H.P., Wilson L., Safinya C.R.: Synchrotron X-ray diffraction study of microtubules buckling and bundling under osmotic stress: a probe of interprotofilament interactions. Phys. Rev. Lett. 93, 198104-1-4 (2004)
Sirenko Y.M., Stroscio M.A., Kim K.W.: Elastic vibration of microtubules in a fluid. Phys. Rev. E 53, 1003–1010 (1996)
Kasas S., Kis A., Riederer B.M., Forro L., Dietler G., Catsicas S.: Mechanical properties of microtubules explored using the finite elements method. Chem. Phys. Chem. 5, 252–257 (2004)
Nogales E., Whittaker M., Milligan R.A., Downing K.H.: High-resolution model of the microtubule. Cell 96, 79–88 (1999)
VanBuren V., Odde D.J., Cassimeris L.: Estimates of lateral and longitudinal bond energies within the microtubule lattice. Proc. Nat. Acad. Sci. USA 99, 6035–6040 (2002)
Tuszynski J.A., Luchko T., Portet S., Dixon J.M.: Anisotropic elastic properties of microtubules. Eur. Phys. J. E 17, 29–35 (2005)
Timoshenko S.P., Young D.H., Weave W.: Vibration Problems in Engineering. Wiley, New York (1974)
Saito T., Parbery R.D., Okuno S., Kawand S.: Parameter identification for aluminum honeycomb sandwich panels based on orthotropic Timoshenko beam theory. J. Sound Vib. 208, 271–287 (1997)
Shi, Y.J., Guo, W.L., Ru, C.Q.: Relevance of Timoshenko-beam model for microtubules of low shear modulus. Physica E (2008). doi:10.1016/j.physe.2008.06.025
Flügge W.: Stresses in Shells. Springer, Berlin (1960)
Bert C.W., Birman V.: Parametric instability of thick, orthotropic, circular cylindrical shells. Acta Mech. 71, 61–76 (1988)
Christoforou A.P., Swanson S.R.: Analysis of simply-supported orthotropic cylindrical shells subject to lateral impact loads. J. Appl. Mech. (ASME) 57, 376–382 (1990)
Li C., Ru C.Q., Mioduchowski A.: Length-dependence of flexural rigidity as a result of anisotropic elastic properties of microtubules. Biochem. Biophys. Res. Commun. 349, 1145–1150 (2006)
Wang C.Y., Ru C.Q., Mioduchowski A.: Orthotropic elastic shell model for buckling of microtubules. Phys. Rev. E 74(052901), 1–4 (2006)
Wang C.Y., Ru C.Q., Mioduchowski A.: Vibration of microtubules as orthotropic elastic shells. Physica E 35, 48–56 (2006)
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C. Q. Ru is on leave from the University of Alberta, Canada.
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Gu, B., Mai, Y.W. & Ru, C.Q. Mechanics of microtubules modeled as orthotropic elastic shells with transverse shearing. Acta Mech 207, 195–209 (2009). https://doi.org/10.1007/s00707-008-0121-8
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DOI: https://doi.org/10.1007/s00707-008-0121-8