Circumferential vibration of microtubules with long axial wavelength

This paper uses an orthotropic shell model to investigate in detail the long axial wavelength circumferential vibration of microtubules (MTs). The deformation patterns in the vibrations were explored and their phonon dispersion relations were presented for MTs with increasing radius. It was shown that with the growth of the axial wavelength, the associated frequency of MTs would finally approach a nonzero asymptotic value, rising considerably with the increase of circumferential wave number but dropping linearly with the growing radius. This study corrects the previous misunderstanding drawn by an oversimplified model, and points out that a parabolic dispersion law does not apply to the circumferential modes when the MT bending stiffness is properly considered.