Modulated wave dynamics in microtubules under long range interactions

In this work, we investigate the dynamics of modulated waves in a long range extension of the damped two-component model of microtubules (MTs) nonlinear dynamics. The global dynamical equation of motion of dimers oscillations is found by including in the inter-dimer coupling and hydrodynamical frictional forces a linear term that decays as a power-law, with exponent(s). By considering a slowly varying envelope, we apply the powerful multiple scale expansion in the semi-discrete approximation and complex Ginzburg–Landau equation (CGL) that admits breather wave solution is derived. The CGL coefficients as well as the dispersion relation and group velocity of envelope are found to be dependent on the long-range interactions (LRI) parameters. By performing a linear stability analysis of modulational instability (MI) of plane waves, the instability criterion is derived and plotted as a function of the LRI parameters. It is observed that strong LRI reduces unstable zone of modulational instability meanwhile weak LRI widens. Our analytical predictions are in good agreements with numerical experiments and confirm the essential role of LRI couplings in MTs structural stability in relation to polymerization and depolymerization. It could be thought that weak LRI couplings promotes dynamic instability with the increased propensity for microtubule depolymerization.