Energy localization in microtubules with radial dislocation

In this study, we investigate the dynamics of microtubules in the presence of the cytosol viscosity using a discrete radial dislocations model. Applying the semi-discrete approximation, the discrete model is first converted into its continuous counterpart which is nothing else but the cubic Complex Ginzburg–Landau equation. Performing a linear stability analysis of plane waves, it is shown that the cytosol viscosity modifies the modulation instability of the system by enlarging the width of unstable zones but softens the instability as it shrinks the growth rate of instability. Furthermore, motivated by the existence of a range of biological processes during which microtubules exhibit a stationary behavior, we look for stationary state solutions and first apply a direct method to the cubic Complex Ginzburg–Landau equation. Coming back to the original discrete model, we show that both bright and anti-dark profiles solitary waves are good candidates likely to explain biological mechanisms that necessitate stationary microtubules. Our analytical predictions are corroborated by intensive numerical simulations with a pretty high accuracy. By providing system parameters related to obtaining stationary states, our work may find applications in targeted microtubules drugs that aim to stabilize cells growth like cancers.