An innovative analytical simulation for scale-dependent transverse vibration of isolated protein microtubules

Abstract

Microtubules, one of the cytoskeleton's main components, perform various important tasks and vibrate in many of them. Therefore, it is crucial to understand the vibration of microtubules. In this paper, it is aimed to present an innovative analytical simulation for the scale-dependent transverse vibration of isolated protein microtubules. For this purpose, the isolated protein microtubules are considered with scale effect under the assumptions of modified strain gradient theory and arbitrary boundary conditions. The model that satisfies the arbitrary boundary conditions is constructed by supporting the isolated protein microtubules at both ends with springs that can be deformed in the transverse direction. A hollow circular beam structure for isolated protein microtubules supported by transverse deformable springs at both ends is used based on the modified strain gradient theory and Rayleigh beam theory, which includes rotary inertia. The transverse displacement function of isolated protein microtubules is represented by the Fourier sine series. Using Stokes' transformation and boundary conditions that depend on the material length scale parameters contained in the modified strain gradient theory, a coefficient matrix including the isolated protein microtubules' properties is constructed. It is clear from the literature that no work analyzes the vibration of isolated protein microtubules with the mentioned procedures and under arbitrary boundary conditions. So, in this study, for the first time, the transverse vibration of isolated protein microtubules in a size-dependent mechanical model is considered under deformable boundary conditions, which is considered a more realistic support condition.