The microtubule-associated protein (MAP) tau plays a key role in the regulation of microtubule assembly and spatial organization. Tau hyperphosphorylation affects its binding on the tubulin surface and has been shown to be involved in several pathologies such as Alzheimer's disease. As the tau binding site on the microtubule lays close to the disordered and highly flexible tubulin C-terminal tails (CTTs), these are likely to impact the tau-tubulin interaction. Since the disordered tubulin CTTs are missing from the available experimental structures, we used homology modeling to build two complete models of tubulin heterotrimers with different isotypes for the β-tubulin subunit (βI/αI/βI and βIII/αI/βIII). We then performed long timescale classical Molecular Dynamics simulations for the tau-R2/tubulin assembly (in systems with and without CTTs) and analyzed the resulting trajectories to obtain a detailed view of the protein interface in the complex and the impact of the CTTs on the stability of this assembly. Additional analyses of the CTT mobility in the presence, or in the absence, of tau also highlight how tau might modulate the CTT activity as hooks that are involved in the recruitment of several MAPs. In particular, we observe a wrapping phenomenon, where the β-tubulin CTTs form a loop over tau-R2, thus stabilizing its interaction with the tubulin surface and simultaneously reducing the CTT availability for interactions with other MAPs.