Vibration and length-dependent flexural rigidity of protein microtubules using higher order shear deformation theory

Microtubules are hollow cylindrical filaments of the eukaryotic cytoskeleton characterized by extremely low shear modulus. A remarkable controversy has occurred in the literature, regarding the length dependence of flexural rigidity of microtubules predicted by the classical elastic beam model. In this study, a higher order shear deformable beam model for microtubules is employed to study unexplained length-dependent flexural rigidity and Young's modulus of microtubules reported in the literature. The formulation allows for warping of the cross-section of the microtubule and eliminates the need for using arbitrary shear correction coefficients as in other theories. It is showed that vibration frequencies predicted by the present parabolic shear deformation theory (PSDT) are much lower than that given by the approximate isotropic beam model for shorter microtubules, although the two models give almost identical results for sufficiently long microtubules. It is confirmed that transverse shearing and the warping of the cross-section of microtubules are 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 present model is found to be in qualitative agreement with the existing experimental data (Kurachi et al., 1995; Pampaloni et al., 2006). These results recommend that the parabolic shear deformation-beam theory 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.