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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