Relevance of Timoshenko-beam model to microtubules of low shear modulus

Microtubules are characterized by extremely low shear modulus that is a few orders of magnitude lower than longitudinal modulus. In this paper, the effects of transverse shearing due to low shear modulus of microtubules are investigated using a Timoshenko-beam model, with detailed comparison between the Timoshenko-beam model, classical isotropic Euler–Bernoulli beam model and a more accurate 2D orthotropic elastic shell model. It is confirmed that transverse shearing is mainly responsible for the length-dependent flexural rigidity of an isolated microtubule reported in the literature, which cannot be explained by the widely used Euler–Bernoulli beam model. Indeed, the length-dependent flexural rigidity predicted by the Timoshenko-beam model is found to be in good quantitative agreement with known experimental data. In particular, the present Timoshenko-beam model predicts that, because of the length dependence of flexural rigidity, microtubules of different lengths could sustain almost equal maximum axial compressive force against column buckling, a conclusion that could have some interesting consequences to the mechanical behavior of cells. These results recommend that the Timoshenko-beam model offers a unified simple 1D model, which can capture the length dependence of flexural rigidity and be applied to various static and dynamic problems of microtubule mechanics.