Long-wavelength deformations and vibrational modes in empty and liquid-filled microtubules and nanotubes: A theoretical study

We propose a continuum model to predict long-wavelength vibrational modes of empty and liquid-filled tubules that are very hard to reproduce using the conventional force-constant matrix approach based on atomistic ab initio calculation. We derive simple quantitative expressions for long-wavelength longitudinal and torsional acoustic modes, flexural acoustic modes, as well as the radial breathing mode of empty or liquid-filled tubular structures that are based on continuum elasticity theory expressions for a thin elastic plate. We furthermore show that longitudinal and flexural acoustic modes of tubules are well described by those of an elastic beam resembling a nanowire. Our numerical results for biological microtubules and carbon nanotubes agree with available experimental data.