In the present work, we focus on the longitudinal model of microtubules (MTs) proposed by Satar i c ̵́ et al. [Phy. Rev. E 48, 89 (1993)], that consider cell MTs to have ferroelectric properties, i.e., a displacive ferro-distortive system of dimers and usually referred to as u-model of MTs. It has been shown that during the hydrolysis of GTP into GDP, the energy released is transferred along the MTs trough kink-like solitons. Substantially, we propose to theoretically investigate the dynamic of MTs by intrinsically taking into account the effect of the oriented molecules of polarized cytoplasmic water and enzymes surrounding the MT. In this regards, we introduce a cubic nonlinear term in the electric potential characterizing the polyelectrolyte features of MTs and show that in addition to the kink and antikink solitons, asymmetrical bright and dark solitons, and discrete modes can also propagate along the MTs. Theses results are supported by numerical analysis. The investigation shows us that the nonlinear dynamics of MTs is strongly impacted by the intrinsic electric field, the polyelectrolyte and the viscosity effects. Moreover, new solitons and discrete solitary modes may help to find new phenomena occurring in the microtubulin systems.