Modified extended tanh-function method and nonlinear dynamics of microtubules

Abstract

A B S T R A C TWe here present a model of nonlinear dynamics of microtubules (MT) in the context of modified extended tanh-function (METHF) method. We rely on the ferroelectric model of MTs published earlier by Satarić et al [1] where the motion of MT subunits is reduced to a single longitudinal degree of freedom per dimer. It is shown that such nonlinear model can lead to existence of kink solitons moving along the MTs. An analytical solution of the basic equation, describing MT dynamics, was compared with the numerical one and a perfect agreement was demonstrated. It is now clearer how the values of the basic parameters of the model, proportional to viscosity and internal electric field, impact MT dynamics. Finally, we offer a possible scenario of how living cells utilize these kinks as signaling tools for regulation of cellular traffic as well as MT depolymerisation.

Mechanical Models of Microtubules

“…Of course, these solutions reproduce previously known results [25]. Figure 2 shows that the solutions of Eq.…”

“…z should be introduced as well as new transformations Z ¼ εz and T ¼ ε t. This allows a series expansion of F ξ ðÞ , that is F ε n AE 1 ðÞ l; εt ðÞ ! FZ ; T ðÞ AE F Z Z; T ðÞ εl þ 1 2 F ZZ Z; T ðÞ ε 2 l 2 (25) where indexes Z and ZZ denote the first and the second derivative with respect to Z. Hence, the function Φ n t ðÞbecomes…”

“…Somewhat improved and more general version of the first nonlinear model is what we call as U-model [25] and this will be explained in the following paragraphs. Both models are based on the fact mentioned above that the dimers are electric dipoles and that the whole MT can be regarded as ferroelectric [23,26], which means that the interaction between a single dimer and its surrounding can be modeled by W-potential [23,27].…”

“…Both models are based on the fact mentioned above that the dimers are electric dipoles and that the whole MT can be regarded as ferroelectric [23,26], which means that the interaction between a single dimer and its surrounding can be modeled by W-potential [23,27]. This yields to the following Hamiltonian for MT [23,25,28] …”

“…A common example for Φ is a solution of Riccati equation, which is either tangent or tangent hyperbolic. As only the latter function may have physical meaning, we call the method as tangent hyperbolic function method (THFM) [25,[33][34][35] and extended or modified extended THFM [36]. The function Φ can also be one of Jacobian elliptic functions [37] and, even, unknown [38].…”