English

M E Zayed Elsayed,A E Alurrfi Khaled

doi:10.5897/ijps2015.4289

Abstract

Microtubules (MTs) are major cytoskeletal proteins. They are hollow cylinders formed by protofilaments (PFs) representing series of proteins known as tubulin dimers. Each dimer is an electric dipole. These diamers are in a straight position within PFs or in radially displaced positions pointing out of cylindrical surface. In this paper, the authors demonstrate how the generalized projective Riccati equations method can be used in the study of the nonlinear dynamics of MTs. To this end, the authors apply this method to construct the exact solutions with parameters for two nonlinear PDEs describing MTs. The first equation describes the model of microtubules as nanobioelectronics transmission lines. The second equation describes the dynamics of radial dislocations in microtubules. As a result, hyperbolic, trigonometric and rational function solutions are obtained. When these parameters are taken as special values, solitary wave solutions are derived from the exact solutions. Comparison between our recent results and the well-known results is given. Key words: Generalized projective Riccati equations method, models of microtubules (MTs), exact solutions, solitary solutions, trigonometric solutions rational solutions.

Highlights

In the recent years, investigations of exact solutions to nonlinear partial differential equations (NPDEs) play an important role in the study of nonlinear physical phenomena
The authors demonstrate how the generalized projective Riccati equations method can be used in the study of the nonlinear dynamics of MTs
The authors apply this method to construct the exact solutions with parameters for two nonlinear PDEs describing MTs

Summary

INTRODUCTION

Investigations of exact solutions to nonlinear partial differential equations (NPDEs) play an important role in the study of nonlinear physical phenomena. Presented an indirect method to seek more solitary wave solutions of some NPDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation (Bountis et al 1986). The objective of this paper is to apply the generalized projective Riccati equations method to construct the exact solutions for the following two nonlinear PDEs of microtubules (MTs):. The authors give the main steps (Conte and Musette, 1992; Zayed and Alurrfi, 2014d; Zhang et al, 2001; Yan, 2003; Yomba, 2005) of this method. The authors determine the positive integer N in (6) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in Equation (5). From (12), (13), (19), (20) and (22), the authors deduce that if r 1 , the exact wave solution was realized:

Result

The authors have

Conclusions