Vibrational dynamics of bio- and nano-filaments in viscous solution subjected to ultrasound: implications for microtubules.

In this paper, using a new analytical method, we solved the beam equation for a uniform bio- and nano-filament in a viscous solution. The filament is assumed to be attached at its two ends and driven by ultrasound plane waves. To obtain analytical solutions, we converted the beam equation to an equation that allows us the use of the method of separation of variables. We then reconstructed the solution of the original beam equation from the solution of the converted equation. Subsequently, we have used the parametric equations derived in this paper to investigate the resonance condition for a microtubule (MT) in an aqueous solution. We show that by using ultrasound plane waves, one cannot satisfy a resonance condition for MTs treated as rigid rods. In order to achieve resonance, a single mode of the MT vibration must be excited with a harmonic number larger than a threshold value found here. Single mode excitation not only helps to transfer a minimum amount of energy to the surrounding medium compared with multi-mode excitation, but it also allows for a simultaneous high amplitude and high mode quality that is impossible using plane waves. In order to overcome this difficulty, we propose to use an ultrasound generation device as a potential technical solution characterized by both frequency control and optimized energy transfer to the MT. Finally, the minimum required intensity of the ultrasound at the location of the MT in order to break it is shown to be on the order of 10(5) W/m(2), which corresponds to 170 dB.