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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.

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Topics: Waves , Vibration
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Figures

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Fig. 1

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

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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.

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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.

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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.

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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.

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Fig. 12

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

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Fig. 11

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

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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.

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Fig. 9

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

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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.

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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.

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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.

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