Microtubules are filamentous intracellular structures that are responsible for various kinds of movements in all eukaryotic cells. The dynamic assembly and disassembly of microtubules and the mechanical properties of these polymers are essential for many key cellular processes such as spermatogenesis and the processes of neurons. Mathematical and computational modeling, especially coupled mechanochemical modeling, has contributed a lot to understand their dynamics. However, it has remained a great challenge to reduce the critical discrepancies, which exist between the experimental observations and modeling results. During this research, the small scaling parameter of the nonlocal Euler-Bernoulli beam theory is analyzed to demonstrate the free vibration problem of microtubules.