Two-Dimensional Shell Vibration of Microtubule in Living Cell

The mechanical behavior of a eukaryotic cell is mainly determined by its cytoskeleton. Microtubules immersed in cytosol are a central part of the cytoskeleton. Cytosol is the viscous fluid in living cells. The microtubules permanently oscillate in the cytosol. In this study, two-dimensional vibration of a single microtubule in living cell is investigated. The Donnell’s shell theory equations for orthotropic materials is used to model the microtubule whereas the motion of the cytosol is modeled as Stokes flow characterized by a small Reynolds number with no-slip condition at microtubule-cytosol interface. The stress field in the cytosol induced by vibrating microtubule is determined analytically and the coupled vibrations of the microtubule-cytoplasm system are investigated. A coupled polynomial eigenvalue problem is developed in the present study and the variations of eigenvalues of coupled system with cytosol dynamic viscosity, microtubule circumferential Young’s modulus and circumferential wave number are examined.