Microtubules are the basic components of the eukaryotic cytoskeleton. We discuss a class of nonlinear waves traveling in microtubules. The waves are obtained on the basis of a kind of z-model. The model used is extended to account for (i) the possibility for nonlinear interaction between neighboring dimers and (ii) the possibility of asymmetry in the double-well potential connected to the external electric field caused by the interaction of a dimer with all the other dimers. The model equation obtained is solved by means of the specific case of the Simple Equations Method. This specific case is denoted by SEsM(1,1), and the equation of Riccati is used as a simple equation. We obtain three kinds of waves with respect to the relation of their velocity with the specific wave velocity $v_c$ determined by the parameters of the dimer: (i) waves with $v>v_c$, which occur when there is nonlinearity in the interaction between neighboring dimers; (ii) waves with $v<v_c$ (they occur when the interaction between neighboring dimers is described by Hooke’s law); and (iii) waves with $v=v_c$. We devote special attention to the last kind of waves. In addition, we discuss several waves which travel in the case of the absence of friction in a microtubule system.

The dynamics of microtubules is considered in the frameworks of the longitudinal model. By using the approximation of the slowly varying envelopes, we obtain the approximate solution of the model equation in the form of the oscillating kink. The evolution of this solution is compared with the results of the numerical simulation. Several types of the dynamics of the oscillating kinks, which are determined by the parameters of the approximate solution, are identified.