Research Papers

Localized Vibration of a Microtubule Surrounded by Randomly Distributed Cross Linkers

[+] Author and Article Information
M. Z. Jin

Department of Mechanical Engineering,
University of Alberta,
Edmonton AB T6G 2G8, Canada
e-mail: mingzhao@ualberta.ca

C. Q. Ru

Department of Mechanical Engineering,
University of Alberta,
Edmonton AB T6G 2G8, Canada
e-mail: cru@ualberta.ca

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received October 1, 2013; final manuscript received April 3, 2014; accepted manuscript posted April 11, 2014; published online May 12, 2014. Assoc. Editor: Mohammad Mofrad.

J Biomech Eng 136(7), 071002 (May 12, 2014) (7 pages) Paper No: BIO-13-1459; doi: 10.1115/1.4027413 History: Received October 01, 2013; Revised April 03, 2014; Accepted April 11, 2014

Based on finite element simulation, the present work studies free vibration of a microtubule surrounded by 3D randomly distributed cross linkers in living cells. A basic result of the present work is that transverse vibration modes associated with the lowest frequencies are highly localized, in sharp contrast to the through-length modes predicted by the commonly used classic elastic foundation model. Our simulations show that the deflected length of localized modes increases with increasing frequency and approaches the entire length of microtubule when frequency approaches the minimum classic frequency given by the elastic foundation model. In particular, unlike the length-sensitive classic frequencies predicted by the elastic foundation model, the lowest frequencies of localized modes predicted by the present model are insensitive to the length of microtubules and are at least 50% lower than the minimum classic frequency for infinitely long microtubules and could be one order of magnitude lower than the minimum classic frequency for shorter microtubules (only a few microns in length). These results suggest that the existing elastic foundation model may have overestimated the lowest frequencies of microtubules in vivo. Finally, based on our simulation results, some empirical relations are proposed for the critical (lowest) frequency of localized modes and the associated wave length. Compared to the classic elastic foundation model, the localized vibration modes and the associated wave lengths predicted by the present model are in better agreement with some known experimental observations.

Copyright © 2014 by ASME
Topics: Waves , Vibration
Your Session has timed out. Please sign back in to continue.


Boal, D., 2002, Mechanics of the Cell, Cambridge University Press, Cambridge, UK.
Stamenović, D., and Coughlin, M. F., 1999, “A Quantitative Model of Cellular Elasticity Based on Tensegrity,” ASME J. Biomech. Eng., 122(1), pp. 39–43. [CrossRef]
Molodtsov, M. I., Ermakova, E. A., Shnol, E. E., Grishchuk, E. L., Mcintosh, J. R., and Ataullakhanov, F. I., 2005, “A Molecular-Mechanical Model of the Microtubule,” Biophys. J., 88(5), pp. 3167–3179. [CrossRef]
Mandato, C. A., and Bement, W. M., 2003, “Actomyosin Transports Microtubules and Microtubules Control Actomyosin Recruitment During Xenopus Oocyte Wound Healing,” Curr. Biol., 13(13), pp. 1096–1105. [CrossRef]
Lee, A. A., Karlon, W. J., Graham, D. A., Dela Cruz, S., and Ratcliffe, A., 2001, “Fluid Shear Stress-Induced Alignment of Cultured Vascular Smooth Muscle Cells,” ASME J. Biomech. Eng., 124(1), pp. 37–43. [CrossRef]
Rai, A. k., Rai, A., Ramaiya, A., Jha, R., and Mallik, R., 2013, “Molecular Adaptations Allow Dynein to Generate Large Collective Forces inside Cells,” Cell, 152(1–2), pp. 172–182. [CrossRef]
Tounsi, A., Heireche, H., Benhassaini, H., and Missouri, M., 2010, “Vibration and Length-Dependent Flexural Rigidity of Protein Microtubules Using Higher Order Shear Deformation Theory,” J. Theor. Biol., 266(2), pp. 250–255. [CrossRef]
Allen, K. B., Sasoglu, F. M., and Layton, B. E., 2008, “Cytoskeleton-Membrane Interactions in Neuronal Growth Cones: A Finite Analysis Study,” ASME J. Biomech. Eng., 131(2), p. 021006. [CrossRef]
Pokorný, J., 2004, “Excitation of Vibrations in Microtubules in Living Cells,” Bioelectrochemistry, 63(1–2), pp. 321–326. [CrossRef]
Daneshmand, F., 2012, “Microtubule Circumferential Vibrations in Cytosol,” Proc. Inst. Mech. Eng.s, Part H: J. Eng. Med., 226(8), pp. 589–599. [CrossRef]
Behrens, S., Wu, J., Habicht, W., and Unger, E., 2004, “Silver Nanoparticle and Nanowire Formation by Microtubule Templates,” Chem. Mater., 16(16), pp. 3085–3090. [CrossRef]
Sirenko, Y. M., Stroscio, M. A., and Kim, K. W., 1996, “Elastic Vibrations of Microtubules in a Fluid,” Phys. Rev. E, 53(1), pp. 1003–1010. [CrossRef]
Wang, C. Y., Ru, C. Q., and Mioduchowski, A., 2006, “Orthotropic Elastic Shell Model for Buckling of Microtubules,” Phys. Rev. E, 74(5), p. 052901. [CrossRef]
Portet, S., Tuszyński, J. A., Hogue, C. W. V., and Dixon, J. M., 2005, “Elastic Vibrations in Seamless Microtubules,” Eur. Biophys. J., 34(7), pp. 912–920. [CrossRef]
Xiang, P., and Liew, K. M., 2012, “Free Vibration Analysis of Microtubules Based on an Atomistic-Continuum Model,” J. Sound Vib., 331(1), pp. 213–230. [CrossRef]
Shen, H.-S., 2011, “Nonlinear Vibration of Microtubules in Living Cells,” Curr. Appl Phys., 11(3), pp. 812–821. [CrossRef]
Ghavanloo, E., Daneshmand, F., and Amabili, M., 2010, “Vibration Analysis of a Single Microtubule Surrounded by Cytoplasm,” Physica E, 43(1), pp. 192–198. [CrossRef]
Zeverdejani, M. K., and Beni, Y. T., 2013, “The Nano Scale Vibration of Protein Microtubules Based on Modified Strain Gradient Theory,” Curr. Appl Phys., 13(8), pp. 1566–1576. [CrossRef]
Mehrbod, M., and Mofrad, M. R. K., 2011, “On the Significance of Microtubule Flexural Behavior in Cytoskeletal Mechanics,” PloS one, 6(10), p. e25627. [CrossRef]
Marrari, Y., Clarke, E. J., Rouvière, C., and Houliston, E., 2003, “Analysis of Microtubule Movement on Isolated Xenopus Egg Cortices Provides Evidence That the Cortical Rotation Involves Dynein as Well as Kinesin Related Proteins and Is Regulated by Local Microtubule Polymerisation,” Dev. Biol., 257(1), pp. 55–70. [CrossRef]
Jin, M. Z., and Ru, C. Q., 2013, “Localized Buckling of a Microtubule Surrounded by Randomly Distributed Cross Linkers,” Phys. Rev. E, 88(1), p. 012701. [CrossRef]
Brangwynne, C. P., Mackintosh, F. C., Kumar, S., Geisse, N. A., Talbot, J., Mahadevan, L., Parker, K. K., Ingber, D. E., and Weitz, D. A., 2006, “Microtubules Can Bear Enhanced Compressive Loads in Living Cells Because of Lateral Reinforcement,” J. Cell Biol., 173(5), pp. 733–741. [CrossRef]
Jin, M. Z., and Ru, C. Q., 2012, “Compressed Microtubules: Splitting or Buckling,” J. Appl. Phys., 111(6), p. 064701. [CrossRef]
Li, C., Ru, C. Q., and Mioduchowski, A., 2006, “Length-Dependence of Flexural Rigidity as a Result of Anisotropic Elastic Properties of Microtubules,” Biochem. Biophys. Res. Commun., 349(3), pp. 1145–1150. [CrossRef]
Enemark, S., Deriu, M. A., Soncini, M., and Redaelli, A., 2008, “Mechanical Model of the Tubulin Dimer Based on Molecular Dynamics Simulations,” ASME J. Biomech. Eng., 130(4), p. 041008. [CrossRef]
Hawkins, T., Mirigian, M., Selcuk Yasar, M., and Ross, J., 2010, “Mechanics of Microtubules,” J. Biomech., 43(1), pp. 23–30. [CrossRef]
Kikumoto, M., Kurachi, M., Tosa, V., and Tashiro, H., 2006, “Flexural Rigidity of Individual Microtubules Measured by a Buckling Force With Optical Traps,” Biophys. J., 90(5), pp. 1687–1696. [CrossRef]
Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., and Watson, J., 1994, Molecular Biology of the Cell, Garland Publishing, London, UK.
Peter, S. J., and Mofrad, M. R. K., 2012, “Computational Modeling of Axonal Microtubule Bundles under Tension,” Biophys. J., 102(4), pp. 749–757. [CrossRef]
Svitkina, T. M., Verkhovsky, A. B., and Borisy, G. G., 1996, “Plectin Sidearms Mediate Interaction of Intermediate Filaments With Microtubules and Other Components of the Cytoskeleton,” J. Cell Biol., 135(4), pp. 991–1007. [CrossRef]
Wang, N., Naruse, K., Stamenovic, D., Fredberg, J. J., Mijailovich, S. M., Toric-Norrelykke, I. M., Polte, T., Mannix, R., and Ingber, D. E., 2001, “Mechanical Behavior in Living Cells Consistent With the Tensegrity Model,” Proc. Natl. Acad. Sci. U.S.A., 98(14), pp. 7765–7770. [CrossRef]
Timoshenko, S., Young, D. H., and Weaver, W. J., 1974, Vibration Problems in Engineering, John Wiley & Sons, New York.
Felgner, H., Frank, R., Biernat, J., Mandelkow, E. M., Mandelkow, E., Ludin, B., Matus, A., and Schliwa, M., 1997, “Domains of Neuronal Microtubule-Associated Proteins and Flexural Rigidity of Microtubules,” J. Cell Biol., 138(5), pp. 1067–1075. [CrossRef]


Grahic Jump Location
Fig. 2

Three lowest transverse (mode number = 7, 11, 15) and longitudinal (mode number = 1, 2, 3) vibration frequencies predicted by Eqs. (1) and (2) are in good agreement with simulations given by the present equivalent 2D elastic foundation model shown in Fig. 1(a). Simply supported end conditions are applied for all cases.

Grahic Jump Location
Fig. 3

Illustration of vibration modes from the randomly distributed 3D cross linker model. The deflection patterns of these three typical cases are localized. The vibration mode with longer deflected length is associated with higher vibration frequency.

Grahic Jump Location
Fig. 4

Vibration frequencies (from the lowest) of a microtubule with length of 50 μm, k of 39 pN/nm and Ld of 25 nm predicted by the randomly distributed 3D cross linker model. All transverse vibration modes with lower frequencies shown here are localized.

Grahic Jump Location
Fig. 1

Finite element models: (a) equivalent elastic foundation model; (b) presently developed randomly distributed 3D cross linker model

Grahic Jump Location
Fig. 5

The frequencies of transverse vibration are distributed more densely for longer microtubules, where microtubule length varies from 240 Ld to 2800 Ld. Here the critical frequency (fL) marked by the dashed line is obtained by the mean value of the lowest 10 very close frequencies, above which densely distributed frequencies of localized modes are identified.

Grahic Jump Location
Fig. 6

The deflected length d (through which the deflection is larger than 5% of the maximum deflection) versus vibration frequencies obtained from four tests with the same Ld of 25 nm and k of 39 pN/nm but different random angles assigned to cross linkers. Each data point is obtained by the mean value of 10 frequencies and associated deflected lengths. The data point with lowest frequency and shortest wave length indicates the critical frequency fL and the associated wave length λL of localized mode.

Grahic Jump Location
Fig. 7

The total deflected length d versus vibration frequencies obtained from our randomly distributed 3D cross linker model for microtubules of length (L) of 5000, 6000, 8000, and 10,000 nm (with k of 39 pN/nm and Ld of 25 nm). All frequencies of localized modes are lower than the lower-bound minimum frequency fmin of 448 MHz given by Eq. (2), and beyond fmin the deflection spreads through the whole microtubule and vibration modes are no longer localized.

Grahic Jump Location
Fig. 8

The dependency of the critical frequency fL (a) and the associated wave length λL (b) of localized vibration on Ld and k. The lower-bound minimum frequency fmin given by Eq. (2) is always higher than the critical frequency fL of localized modes given by the present model.

Grahic Jump Location
Fig. 9

Comparison between the critical frequency fL predicted by empirical formula (5) and numerical simulation with variable k (a) and Ld (b)

Grahic Jump Location
Fig. 10

Comparison between the wave length λL predicted by empirical formula (6) and numerical simulation results with variable Ld (a) and k (b). The formula (6) is valid only when the predicted λL is longer than 13 Ld.

Grahic Jump Location
Fig. 11

Effect of the bending rigidity EI on the critical frequency f* (a) and the associated wave length λ* (b) of localized vibration

Grahic Jump Location
Fig. 12

Effect of ρA on the critical frequency f* of localized vibration



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In