Localized discrete and asymmetric dark-bright soliton-like modes as nonlinear dynamics in microtubules

In the present work, we focus on the longitudinal model of microtubules (MTs) proposed by Satarić et al. (1993) [12], and that considers MT cells to have ferroelectric properties (behaviors) due to dipolar oscillations of dimers within MTs, i.e., a displacive ferrodistortive system of heterodimers in MTs and usually referred to as <i>u</i>-model of MTs. It has been shown that during the hydrolysis of guanosine 5'- triphosphate into guanosine 5'-diphosphate, the energy released is transferred along the MTs through kink-like solitons. Substantially, we propose to theoretically investigate the dynamic of MTs by intrinsically taking into account the effect of 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 anti-kink dynamics, asymmetrical bright and dark solitons, and discrete modes can also propagate along the MTs. These 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 viscous effects. Moreover, new solitonic dynamics and discrete solitary modes might aid in the discovery of novel microtubulin system phenomena.