A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm

The buckling behaviors of microtubules (MTs) in a living cell have been studied based on the nonlocal anisotropic shell theory and Stokes flow theory. The analytical expressions for the buckling load and the growth rate of the buckling are obtained and discussed. In addition, the pressure on MTs, resulting from cytosol motion, is derived on the basis of the Stokes flow theory. The influences of filament network elasticity and the shear modulus of MTs, especially the cytosol viscosity and MT small scale effects on MT buckling behaviors, are investigated. The analytical results show that the MT buckling growth rate increases with the MT small scale parameter, while decreases as the filament network elastic modulus, the MT shear modulus and cytoplasm viscosity increase. Although the cytosol viscosity has a significant influence on the value of the growth rate, it shows little effects on the range of the axial wave number of buckling as well as the critical axial wave number that corresponds to the maximal growth rate. Finally, the MT buckling growth rates have been calculated using the beam model, the isotropic shell model, and the anisotropic shell model. The results indicate that using the anisotropy shell theory to model the buckling behavior of MTs is necessary.