## 1. Introduction

Microtubules (MTs) are major cytoskeletal proteins. MTs are cytoskeletal biopolymers shaped as nanotubes. They are hollow cylinders formed by Proto-Filaments (PFs) representing a series of proteins known as tubulin dimers. Each dimer is an electric dipole. These dimers are in a straight position within the PFs or placed in radial positions pointing out of the cylindrical surface. MTs compriseaninteresting type of protein structure that may be a good candidate for designing and manufacturing electronic nano-devices. MTs dynamical behavior is modeled by nonlinear partial differential equations (NPDEs). These equations are mathematical models of physical circumstances that emerge in various fields of engineering, plasma physics, solid state physics, optical fibers, chemistry, hydrodynamics, biology, fluid mechanics and geochemistry. To date solving NPDEs exactly or approximately, a plethora of methods have been in use. These include, but are not limited to, (G′/G)-expansion [

1,

2,

3,

4,

5,

6], Frobenius decomposition [

7], local fractional variation iteration [

8], local fractional series expansion [

9], multiple exp-function algorithm [

10,

11], transformed rational function [

12], exp-function method [

13,

14], trigonometric series function [

15], inverse scattering [

16], homogeneous balance [

17,

18], first integral [

19,

20,

21,

22], F-expansion [

23,

24,

25], Jacobi function [

26,

27,

28,

29], Sumudu transform [

30,

31,

32], solitary wave ansatz [

33,

34,

35,

36], novel (G′/G) -expansion [

37,

38,

39,

40,

41,

42], modified direct algebraic method [

43,

44], and last but not least, the

$\mathrm{exp}(-\mathsf{\Phi}(\xi ))$-Expansion Method [

45,

46,

47,

48,

49,

50].

The objective of this paper is to apply the latter method, namely the

$\mathrm{exp}(-\mathsf{\Phi}(\xi ))$-Expansion Method, to construct the exact solutions for the following two NPDEs modeling MT dynamics, [

51,

52,

53,

54,

55,

56,

57,

58,

59]. In particular, in presenting the questions to be solved, for comparison purposes, we follow the initial set up established by Zayed and Alurrfi [

56], solving the extended Riccatti equations (see Equations (1) and (2)). We then depart generically from their development by using an entirely distinct method, albeit we compare our final results with theirs in [

56], keeping in focus the developments in [

57,

58,

59], as well.

- (i)
The model of nonlinear dynamics of microtubules assuming a single longitudinal degree of freedom per tubulin dimer is described by the nonlinear PDE (see [

59]),

where

$A,$ and

$B$ are positive parameters,

$m$, is the mass of the dimer,

$z(x,t)$, is the traveling wave,

$E$ is the magnitude of intrinsic electric field,

$l$, is the MT length,

$q>0$, is the excess charge within the dipole,

$\gamma $, is the viscosity coefficient and,

$k$, is a harmonic constant describing the nearest-neighbor interaction between the dimers belonging to the same PFs. In [

48], authors have used the Jacobi elliptic function method to find the exact solutions of Equation (1), the physical details and derivations of which were discussed there, although omitted here for obvious reasons.

- (ii)
The nonlinear PDE describing the nonlinear dynamics of radially dislocated MTs:

Here,

$z(x,t)$, is the corresponding angular displacement when the whole dimer rotates and,

$l$, is the MT length,

$p$ is the magnitude of intrinsic electric field,

$k$, stands for inter-dimer bonding interaction within the same PFs,

$I$, is the moment of inertia of the single dimer and

$\mathrm{\Gamma}$ is the viscosity coefficient. In [

57], authors have used the simple equation method to find the exact solutions of Equation (2), after relating physical aspects and equation derivation being omitted here.

This paper is organized as follows: In

Section 2, we give the description of the

$\mathrm{exp}(-\mathsf{\Phi}(\xi ))$-Expansion Method, while in

Section 3, we apply the said method to solve the given NPDEs, Equations (1) and (2). In

Section 4, physical explanations are given, followed by the conclusion in

Section 5. The paper ends with relevant acknowledgments, and a rich list of references for interested readers.

## 2. Description of the $\mathrm{exp}(-\mathsf{\Phi}(\xi ))$-Expansion Method

Following th initial setup in [

56], we consider the nonlinear evolution equation in the form,

where,

$F$, is a polynomial in,

$u(x,t)$, and its partial derivatives, involving nonlinear terms and highest order derivatives. The focal steps of the method are as follows:

**Step 1.** It is well known that, for a given wave equation, a travelling wave,

$u(\xi )$, is a solution which depends upon,

$x$, and,

$t$, only through a unified variable,

$\xi $, such that,

where,

${k}_{1}$ and

$\omega $, are constants. Based on this we have,

and so on, for other derivatives.

We reduce Equation (3) to the following ODE:

Here,

$Q$ is a polynomial in,

$u(\xi )$, and its total derivatives, such that

$\prime =\frac{d}{d\xi}$.

**Step 2.** We assume that Equation (6) has the formal solution:

where, the

${A}_{i}$’s are constants to be determined, such that

${A}_{N}\ne 0$ and

$\mathsf{\Phi}=\mathsf{\Phi}(\xi )$ satisfies the following ODE:

Consequently, we get the following possibilities for Equation (8):

**Cluster 1:** When

$\mu \ne 0,$ ${\lambda}^{2}-4\mu >0,$ we get,

**Cluster 2:** When

$\mu \ne 0,$ ${\lambda}^{2}-4\mu <0,$ we get,

**Cluster 3:** When

$\mu =0,$ $\lambda \ne 0,$ and

${\lambda}^{2}-4\mu >0,$ we obtain,

**Cluster 4:** When

$\mu \ne 0,$ $\lambda \ne 0,$ and

${\lambda}^{2}-4\mu =0,$ we obtain

**Cluster 5:** When

$\mu =0,$ $\lambda =0,$ and

${\lambda}^{2}-4\mu =0,$ we then have,

where

${A}_{N},\cdots \cdots ,\hspace{0.17em}V,\hspace{0.17em}\lambda ,\hspace{0.17em}\mu $, are constants to be determined, such that

${A}_{N}\ne 0.$ The positive integer, m, can be determined by considering the homogeneous balance between nonlinear terms and the highest order derivatives occurring in the ODE in Equation (6), after using Equation (7).

**Step 3.** We interchange Equation (7) into Equation (6) and then we expand the function $\mathrm{exp}(-\mathsf{\Phi}(\xi ))$. As a result of this interchange, we get a polynomial of $\mathrm{exp}(-\mathsf{\Phi}(\xi ))$. We equate all the coefficients of same power of $\mathrm{exp}(-\mathsf{\Phi}(\xi ))$ to zero. This procedure yields a system of algebraic equations which could be solved to obtain the values of ${A}_{N},\cdots \cdots ,\hspace{0.17em}V,\hspace{0.17em}\lambda ,\hspace{0.17em}\mu $ which after substitution into Equation (7) along with general solutions of Equation (8) completes the setup for getting the traveling wave solutions of the NPDE in Equation (3).

## 5. Physical Interpretations of Some Obtained Solutions

In this section, attempting to shed lights on the corresponding physical behavior, we to discuss nonlinear dynamics of MTs whether as nano-bioelectronics transmission lines like or radially dislocated MTs, based on the obtained traveling wave solutions, from Equations (24)–(28), and (41)–(50), respectively. We examine the nature of some obtained solutions of Equations (1) and (2) by selecting particular values of the parameters and graphing the resulting exact solutions using mathematical software Maple 13, represented in

Figure 1,

Figure 2,

Figure 3,

Figure 4,

Figure 5 and

Figure 6.

From our obtained solutions, we observe that Equations (24)–(28), and (41)–(50), exude kink type solitons, singular kink shape solitons, and periodic solutions. Equation (24) shows kink shaped soliton profile for,

${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =1,$ $\lambda =3,$ $E=1$, within the interval

$-10\le x,\hspace{0.17em}t\le 10$ which is represented in

Figure 1 and

Figure 2. Equation (25) provides a periodic solution profile for,

${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =3,$ $\lambda =1,$ $E=5$ within the interval

$-1\le x,\hspace{0.17em}t\le 1,$ which is represented in

Figure 3 and

Figure 4. Equation (26) provides a singular kink soliton profile for,

${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =0,$ $\lambda =2,$ $E=1$, within the interval

$-10\le x,\hspace{0.17em}t\le 10,$ which is represented in

Figure 5 and

Figure 6. Equations (27) and (28) also represent singular kink type wave solutions which are similar to

Figure 5 and

Figure 6. Equations (41) and (46) provide kink soliton profile, for

${A}_{0}=2,$ $T=\frac{3}{2},$ $\omega =-1,$ ${k}_{1}=1,$ $\mu =1,$ $\lambda =3,$ and

$E=1$, within the interval,

$-10\le x,\hspace{0.17em}t\le 10,$ as in

Figure 1 and

Figure 2. Equations (42) and (47) provide periodic solutions for,

${A}_{0}=2,$ $T=\frac{3}{2},$ $\omega =-1,$ ${k}_{1}=1,$ $\mu =3,$ $\lambda =1,$ $E=5$, within the interval,

$-1\le x,\hspace{0.17em}t\le 1,$ as in

Figure 3 and

Figure 4. Equations (43) and (48), provide singular kink soliton profiles for,

${A}_{0}=2,$ $T=\frac{3}{2},$ $\omega =-1,$ ${k}_{1}=1,$ $\mu =0,$ $\lambda =2,$ and,

$E=1$, within the interval

$-10\le x,\hspace{0.17em}t\le 10,$ as in

Figure 5 and

Figure 6. Equations (44) and (45), as well as Equations (49) and (50), also represent singular Kink type wave solutions which are similar to

Figure 5 and

Figure 6.

**Figure 1.**
The solitary wave 3D graphics of Equation (24) shows a kink shaped soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =1,$ $\lambda =3,$ $E=1$ within the interval $-10\le x,\hspace{0.17em}t\le 10$.

**Figure 1.**
The solitary wave 3D graphics of Equation (24) shows a kink shaped soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =1,$ $\lambda =3,$ $E=1$ within the interval $-10\le x,\hspace{0.17em}t\le 10$.

**Figure 2.**
The solitary wave 2D graphics of Equation (24) shows a kink shaped soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =1,$ $\lambda =3,$ $E=1$, $t=2$.

**Figure 2.**
The solitary wave 2D graphics of Equation (24) shows a kink shaped soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =1,$ $\lambda =3,$ $E=1$, $t=2$.

**Figure 3.**
The solitary wave 3D graphics of Equiation (25) provides a periodic solution profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =3,$ $\lambda =1,$ $E=5$ within the interval $-1\le x,\hspace{0.17em}t\le 1$.

**Figure 3.**
The solitary wave 3D graphics of Equiation (25) provides a periodic solution profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =3,$ $\lambda =1,$ $E=5$ within the interval $-1\le x,\hspace{0.17em}t\le 1$.

**Figure 4.**
The solitary wave 2D graphics of Equation (25) provides a periodic solution profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =3,$ $\lambda =1,$ $E=5$, $t=2$.

**Figure 4.**
The solitary wave 2D graphics of Equation (25) provides a periodic solution profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =3,$ $\lambda =1,$ $E=5$, $t=2$.

**Figure 5.**
The solitary wave 3D graphics of Equation (26) provides a singular kink soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =0,$ $\lambda =2,$ $E=1$ within the interval $-10\le x,\hspace{0.17em}t\le 10$.

**Figure 5.**
The solitary wave 3D graphics of Equation (26) provides a singular kink soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =0,$ $\lambda =2,$ $E=1$ within the interval $-10\le x,\hspace{0.17em}t\le 10$.

**Figure 6.**
The solitary wave 2D graphics of Equation (26) provides a singular kink soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =0,$ $\lambda =2,$ $E=1$, $t=2$.

**Figure 6.**
The solitary wave 2D graphics of Equation (26) provides a singular kink soliton profile for, ${A}_{0}=1,$ $m=1,$ $\omega =-1,$ ${k}_{1}=1,$ $k=2,$ $l=2,$ $A=2,$ $B=3,$ $\mu =0,$ $\lambda =2,$ $E=1$, $t=2$.