On a Class of Nonlinear Waves in Microtubules

Abstract

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.

1. Introduction

Microtubules are important structural elements of many biological systems [1,2,3,4,5,6]. Microtubules are among the three principal elements of the cytoskeleton, along with microfilaments and intermediate filaments. These three elements play an important role in cell division, movement, and maintaining cell shape. Microtubules are rigid hollow rods of approximately 25 nm in diameter and between 200 nm and 25 $\mathsf{\mu }$m in length. They consist of a cytoplasmic, globular protein subunit called tubulin. Each microtubule subunit contains two polypeptides, $\alpha$-tubulin and $\beta$-tubulin, which form heterodimers. Multiple units of these dimers polymerize to form a chain called the protofilament. Thirteen of these protofilaments are arranged into a cylindrical pattern to form a microtubule.

In a microtubule, the subunits are organized such that they all face the same direction to form 13 parallel protofilaments. As a consequence of this, the microtubule is polar. It has $\alpha$-tubulin exposed at one end and $\beta$-tubulin at the other. The positively charged end of the microtubule grows relatively fast. The negatively charged end grows relatively slowly. Each protofilament is arranged in a parallel orientation with respect to the other. The orientation is such that the positive end always has $\beta$ subunits exposed. The negative end has $\alpha$ subunits exposed. Microtubules have several important functions, as mentioned above. An additional function is the formation of an internal transport network for moving materials throughout the cell and between the exterior and interior of the cell.

Because of all the above, microtubules are widely studied. And numerous of these studies are devoted to the propagation of solitary nonlinear waves in microtubules [7,8,9,10,11]. Various models of the wave motion in microtubules are known. An overview of many important models can be found, for example, in [12]. There are three classes of models: u-models, z-models, and $\phi$-models. u-Models assume radial oscillations of the dimer. The coordinate u is the projection of the top of the dimer on the direction of the PF (the protofilament). For more details on this model, see [13]. In the z-model, the z-coordinate is the longitudinal displacement of the dimer [14]. The $\phi$-model is a radial model. The angle $\phi$ determines the radial displacement of the dimer representing the angle between the dimer and the direction of the PF.

The excitations in microtubules are of interest in connection with the energy transfer mechanism in microtubules. In [15], kink-like excitations are discussed on the basis of a longitudinal displacement model including double-well quartic potential. A model of the tangential oscillations of the dimer is discussed in [16]. In [17], a version of the u-model is discussed in presence of Morse potential. A u-model and a specific case of SEsM(1,1) called extended tanh method are discussed in [18]. Ref. [19] considers the $\phi$-model of microtubules and uses the nonlinear Schrödinger equation as the model equation. Analytical and numerical solutions of this model equation are discussed. Additional discussion of the $\phi$ model can be found in [20]. The $\phi$-model with quartic potential of the general form is discussed in [21]. Finally, solutions of a z-model are discussed in [14].

In this article, we apply version SEsM(1,1) of the methodology called Simple Equations Method (SEsM). The basis of the application is the simple equation of the Riccati kind. The text is organized as follows: In Section 2, we present the mathematical formulation of the problem and obtain the basic model equation. Section 3 contains several notes about the SEsM and its application. In Section 4, we present the obtained solutions and discuss their properties. Several concluding remarks are summarized in Section 5.

2. Mathematical Formulation of the Problem

We consider the longitudinal displacement of dimers in a single protofilament. The model assumes that $z_n$ is the longitudinal displacement of a dimer at position n in a direction of the protofilament. The equation of motion for the dimer equals $m{\displaystyle \frac{d{z}_n^2}{d{t}^2}}$ (m is the mass of the dimer), and we consider the following forces:

• The elastic force of the nearest-neighbor interaction among the dimers, given by Hooke’s law:

  $$F_h=k[(z_{n+1}-z_n)-(z_n-z_{n-1})],$$

  (1)

  where k is the intradimer stiffness parameter.
• The anharmonic part of the nearest-neighbor interaction, corresponding to the weak nonlinear interaction:

  $$F_{ah}=k^{*}[{(z_{n+1}-z_n)}^2-{(z_n-z_{n-1})}^2],$$

  (2)

  where $k^{*}$ is the coefficient of the nonlinear interaction among the dimers.
• The force connected with the dipole electric field:

  $$F_d=-qE,$$

  (3)

  where q is the excess charge within the dipole and E is the magnitude of the intrinsic electric field.
• The force connected with the influence of the external electric field caused by the interaction of the dimer with all other dimers (corresponding to an asymmetric double-well potential):

  $$F_e=[A{z}_n-B{z}_n^3]+P+Q{z}_n^2,A\ge 0,B\ge 0.$$

  (4)

  where the part $A{z}_n-B{z}_n^3$ is connected to the classic symmetric double-well potential $-{\displaystyle \frac{A}{2}}{z}_n^2+{\displaystyle \frac{B}{4}}{z}_n^4$. The part $P+Q{z}_n^2$ is associated with additional asymmetry in the potential of the interaction of the dimers.
• The force connected to friction:

  $$F_f=-g{\displaystyle \frac{d{z}_n}{dt}},$$

  (5)

  where g is the friction coefficient.

Thus, the differential equation for the determination of $z_n\left(t\right)$ becomes

$$\begin{array}{ccc}\hfill m{\displaystyle \frac{d^2{z}_n}{d{t}^2}}& =& F_h+F_{ah}+F_d+F_e+F_f=\hfill \\ & & k[(z_{n+1}-z_n)-(z_n-z_{n-1})]+F_{ah}+k^{*}[{(z_{n+1}-z_n)}^2-{(z_n-z_{n-1})}^2]-qE+\hfill \\ & & A{z}_n-B{z}_n^3+P+Q{z}_n^2-g{\displaystyle \frac{d{z}_n}{dt}}.\hfill \end{array}$$

(6)

Next, we apply the continuum limit:

$$\begin{array}{ccc}\hfill z_n\left(t\right)& \to & z(x,t),\hfill \\ \hfill z_{n+1}\left(t\right)& \to & z(x,t)+R_0{\displaystyle \frac{\partial z(x,t)}{\partial x}}+{\displaystyle \frac{1}{2}}{R}_0^2{\displaystyle \frac{{\partial }^{2}z(x,t)}{\partial x^2}}+\cdots ,\hfill \\ \hfill z_{n-1}\left(t\right)& \to & z(x,t)-R_0{\displaystyle \frac{\partial z(x,t)}{\partial x}}+{\displaystyle \frac{1}{2}}{R}_0^2{\displaystyle \frac{{\partial }^{2}z(x,t)}{\partial x^2}}+\cdots ,\hfill \end{array}$$

(7)

where $R_0$ is the equilibrium distance in the space between adjacent dimers. The result of differential Equation (6) is

$$m{\displaystyle \frac{{\partial }^{2}z}{\partial t^2}}-k{R}_0^2{\displaystyle \frac{{\partial }^{2}z}{\partial x^2}}-{\displaystyle \frac{k{R}_0^4}{12}}{\displaystyle \frac{{\partial }^{4}z}{\partial x^4}}-2k^{*}{R}_0^3{\displaystyle \frac{\partial z}{\partial x}}{\displaystyle \frac{{\partial }^{2}z}{\partial x^2}}-qE-Az+B{z}^3-P-Q{z}^2+g{\displaystyle \frac{\partial z}{\partial t}}=0$$

(8)

In order to treat the case of linear forces together with the case of nonlinear forces, we consider a slightly modified version of (8):

$$m{\displaystyle \frac{{\partial }^{2}z}{\partial t^2}}-k{R}_0^2{\displaystyle \frac{{\partial }^{2}z}{\partial x^2}}-b{\displaystyle \frac{k{R}_0^4}{12}}{\displaystyle \frac{{\partial }^{4}z}{\partial x^4}}-2b{k}^{*}{R}_0^3{\displaystyle \frac{\partial z}{\partial x}}{\displaystyle \frac{{\partial }^{2}z}{\partial x^2}}-qE-Az+B{z}^3-P-Q{z}^2+g{\displaystyle \frac{\partial z}{\partial t}}=0.$$

(9)

In (9), we introduce the parameter b. When $b=1$, we have the case in (8), which describes the interaction under the action of nonlinear nearest-neighbor interactions between the dimers. For $b=0$, we obtain the case of interaction according to the linear Hooke’s law.

Equation (9) is similar to the damped Boussinesq equation. Various aspects of several versions of the damped Boussinesq equations have been studied by many authors [22,23,24,25,26,27,28,29,30].

We assume the solutions of (9) of the kind of traveling wave

$$z(x,t)=z\left(\xi \right),\xi =x-vt,$$

(10)

where v is the velocity of the wave. We set

$$\begin{array}{c}\hfill \alpha =-{\displaystyle \frac{bk{R}_0^4}{12}};\beta =-2b{k}^{*}{R}_0;\gamma =m{v}^2-k{R}_0^2;\delta =-gv;B=ϵ;\\ \hfill -A=\nu ;-Q=\mu ;\eta =-(P+qE);\end{array}$$

(11)

Equation (9) is reduced to

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\gamma {\displaystyle \frac{d^{2}z}{d{\xi }^2}}+\delta {\displaystyle \frac{dz}{d\xi }}+ϵz^3+\mu z^2+\nu z+\eta =0$$

(12)

Thus, $\alpha$ is connected to the linear part of interaction between the nearest neighbors in the protofilament. $\beta$ is connected to the weakly nonlinear part of the interaction between the nearest neighbors. $\gamma$ is connected to the velocity of the wave. $ϵ$ and $\nu$ are connected to the two parts of the symmetric double-well potential of the interaction. $\mu$ is connected to a part of the asymmetric potential of interaction among the dimers. $\delta$ accounts for the friction. $\eta$ accounts for two effects: the influence of the dipole field and for the second part of the asymmetric potential of the interaction among the dimers.

We can consider $A,B,P,$ and Q to be independent variables. Thus, below we count $ϵ,\mu ,\nu ,$ and $\eta$ as such independent variables.

Finally, we note that $\gamma =0$ defines a characteristic velocity of the wave $v_c^2=k{R}_0/m$. Below, we obtain waves which travel with velocity that can be larger or smaller than $v_c$.

We compare exact solutions for the case $b=1$ of (12) to the corresponding solutions for the case $b=0$. We proceed as follows: First, we obtain a solution for the case $b=1$. Then, we obtain the corresponding solution for the case $b=0$. Finally, we compare characteristic quantities for these two solutions. After this, we proceed to the next solution for the case $b=1$ and repeat the procedure.

3. Notes About the Application of the SEsM

As methodology for the solution of (12), we use the Simple Equations Method (SEsM) [31,32,33,34]. A large review of the methodology can be seen in [31], and the connection between the methodology and other methods is discussed in [34]. The notation of the SEsM is SEsM(m,n). This means that in the general case, we have to solve m nonlinear differential equations by means of n simple equations. We use the specific case of the SEsM labelled SEsM(1,1). This specific case is known also as the Modified method of the Simplest Equation (MMSE). It has many applications to various problems; see, for example, [35].

We specify further the use of SEsM(1,1) as follows: We use solutions to the simple equation

$${\displaystyle \frac{df}{d\xi }}=\sum _{j=0}^M{b}_{j}f{\left(\xi \right)}^j,$$

(13)

where $b_j$ are parameters which have to be determined. The solution $z\left(\xi \right)$ is to be searched in the form

$$z\left(\xi \right)=\sum _{i=0}^N{a}_{i}f{\left(\xi \right)}^i,$$

(14)

where $a_i$ are parameters which are to be determined in the course of the application of the SEsM.

For the successful application of the SEsM, we need two items: the form of the balance equation occurring in SEsM(1,1) and the explicit form of the solutions to the simple equations of the kind in (13).

We have several possibilities for simple equations for SEsM(1,1). In this article, we use the equation of Riccati as a simple equation. This means that we have assumed that $f\left(\xi \right)$ is a solution to (13) for $M=2$. In other words,

$${\displaystyle \frac{df}{d\xi }}=b_0+b_{1}f+b_2{f}^2.$$

(15)

The general solution to the equation of Riccati is

$$f\left(\xi \right)=-{\displaystyle \frac{b_1}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+{\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}},$$

(16)

where ${\theta }^2=b_1^2-4b_0{b}_2$ and C, D, and E are constants.

In this article, we discuss the case of the presence of the term $ϵz^3$ (and $ϵ>0$). In this case, the balance equation for SEsM(1,1) is $M=N+1$, for $N=1,2,\dots$ In addition, we use the equation of Riccati as a simple equation. Then, $M=2$. From the balance equation, it follows that $N=1$. Thus, below, we discuss the case $N=1$, $M=2$. For more information about obtaining balance equations within the scope of the SEsM, see the detailed description of the methodology in [31].

We note that for the case $b=0$ and $ϵ\ne 0$, the balance equation is again $M=N+1$. Then, the solutions for this case, which correspond to the solutions for the case $b=1$, are constructed again by means of SEsM(1,1) on the basis of the Riccati equation as a simple equation. An exception to this is the case $b=0$, $\gamma =0$, which leads to $\alpha =\beta =\gamma =0$. In this case, the equation of Riccati cannot be used as a simple equation.

4. Studied Exact Solutions of (12)

We consider the possible specific cases with respect to the presence or absence of terms in (12). Specific cases lead to different systems of nonlinear algebraic equations after the application of SEsM(1,1). The results below were checked by MATLAB software (https://www.mathworks.com). The figures were prepared by xmgrace software (https://plasma-gate.weizmann.ac.il/Grace/).

4.1. All Parameters in (12) Are Different from 0

Lemma 1. 

Let us consider the case where all parameters in (12) are different from 0. The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) to the following system of nonlinear algebraic equations:

$$\begin{array}{ccc}\hfill 2a_1{b}_2^3(12\alpha b_2+\beta a_1)& =& 0,\hfill \\ \hfill 5a_1{b}_1{b}_2^2(12\alpha b_2+\beta a_1)& =& 0,\hfill \\ \hfill \alpha [(2(a_1{b}_1^2+2a_1{b}_2{b}_0)b_2+12a_1{b}_1^2{b}_2+12a_1{b}_2^2{b}_0)b_2+36a_1{b}_1^2{b}_2^2+& & \hfill \\ \hfill 24a_1{b}_2^3{b}_0]+\beta (a_1{b}_1^2+2a_1{b}_2{b}_0)a_1{b}_2+3\beta a_1^2{b}_1^2{b}_2+2\beta a_1^2{b}_2^2{b}_0+ϵa_1^3+2\gamma a_1{b}_2^2& =& 0,\hfill \\ \hfill \delta a_1{b}_2+3ϵa_0{a}_1^2+4\beta a_1^2{b}_1{b}_0{b}_2+\beta (a_1{b}_1^2+2a_1{b}_2{b}_0)a_1{b}_1+\mu a_1^2+\alpha \left[\right((a_1{b}_1^2+& & \hfill \\ \hfill 2a_1{b}_2{b}_0)b_1+6a_1{b}_1{b}_2{b}_0)b_2+(2(a_1{b}_1^2+2a_1{b}_2{b}_0)b_2+12a_1{b}_1^2{b}_2+12a_1{b}_2^2{b}_0)b_1+& & \hfill \\ \hfill 36a_1{b}_1{b}_2^2{b}_0]+3\gamma a_1{b}_1{b}_2& =& 0,\hfill \\ \hfill \alpha [((a_1{b}_1^2+2a_1{b}_2{b}_0)b_1+6a_1{b}_1{b}_2{b}_0)b_1+(2(a_1{b}_1^2+2a_1{b}_2{b}_0)b_2+& & \hfill \\ \hfill 12a_1{b}_1^2{b}_2+12a_1{b}_2^2{b}_0\left)b_0\right]+\delta a_1{b}_1+\beta a_1^2{b}_1^2{b}_0+\beta (a_1{b}_1^2+2a_1{b}_2{b}_0)a_1{b}_0+& & \hfill \\ \hfill 3ϵa_0^2{a}_1+\nu a_1+2\mu a_0{a}_1+\gamma (a_1{b}_1^2+2a_1{b}_2{b}_0)& =& 0,\hfill \\ \hfill \alpha [(a_1{b}_1^2+2a_1{b}_2{b}_0)b_1+6a_1{b}_1{b}_2{b}_0]b_0+\nu a_0+\eta +\delta a_1{b}_0+\gamma a_1{b}_1{b}_0+& & \hfill \\ \hfill \beta a_1^2{b}_1{b}_0^2+ϵa_0^3+\mu a_0^2& =& 0.\hfill \end{array}$$

(17)

Proof. 

The application of SEsM(1,1) to (12) means that we have to substitute (14) and (13) into (12) for the case $M=2$, $N=1$. Then, we have to equate to 0 the coefficients of the obtained polynomial of $f\left(\xi \right)$. This leads us to the system in (17). □

Theorem 1. 

An exact solution to (12) is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0-12{\displaystyle \frac{\alpha b_2}{\beta }}\{{\displaystyle \frac{\beta (-\delta \beta +36ϵa_0\alpha +12\mu \alpha )}{512b_2ϵ{\alpha }^2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\hfill \end{array}$$

(18)

where

$$\begin{array}{c}\hfill {\theta }^2=b_1^2-4b_0{b}_2=-{\displaystyle \frac{\gamma {\beta }^2+72ϵ{\alpha }^2}{\alpha {\beta }^2}}>0.\end{array}$$

(19)

Proof. 

Let us consider the case where all parameters in (12) are different from 0. We apply SEsM(1,1) to (12) on the basis of Equations (13) and (14) in the case $M=2$, $N=1$. As a result, we obtain the system of nonlinear algebraic equations in (17). A solution of this system is

$$\begin{array}{ccc}\hfill a_1& =& -12{\displaystyle \frac{\alpha b_2}{\beta }},\hfill \\ \hfill b_0& =& {\displaystyle \frac{\frac{{\beta }^4{(-\delta \beta +36ϵa_0\alpha +12\mu \alpha )}^2}{46656{\alpha }^3{ϵ}^2}+\gamma {\beta }^2+72ϵ{\alpha }^2}{4\alpha b_2{\beta }^2}},\hfill \\ \hfill b_1& =& -{\displaystyle \frac{\beta (-\delta \beta +36ϵa_0\alpha +12\mu \alpha )}{216ϵ{\alpha }^2}}\hfill \\ \hfill \nu & =& {\displaystyle \frac{-{\beta }^6{\delta }^2+144{\beta }^4{\mu }^2{\alpha }^2+15552\gamma {\beta }^2{\alpha }^3{ϵ}^2+1119744{\alpha }^5{ϵ}^3}{432{\alpha }^2ϵ{\beta }^4}}\hfill \\ \hfill \eta & =& {\displaystyle \frac{1}{23328{\beta }^4{ϵ}^2{\alpha }^3}}(20155392\mu {\alpha }^6{ϵ}^3+{\delta }^3{\beta }^7+46656\delta {\beta }^3\gamma {\alpha }^3{ϵ}^2+3359232\delta \beta {ϵ}^3{\alpha }^5+\hfill \\ & & 279936\mu {\alpha }^4\gamma {\beta }^2{ϵ}^2+864{\mu }^3{\alpha }^3{\beta }^4-18{\beta }^2{\delta }^6\mu \alpha ).\hfill \end{array}$$

(20)

This solution leads to the exact solution (18) to (12). □

The following comments are in order:

• We see that the parameters $\nu$ and $\eta$ are not independent anymore. The parameter $\nu$ is connected to the parameter A of the double-well potential. This means that when we fix the other parameters in the relationship of $\nu$, A also becomes fixed. As the parameter B is not fixed, there are many profiles of the double-well potential which allow for the propagation of this wave. The relationship for $\eta$ means that there is a relation between the parameters P and E. Thus, the fixation of the electric field E determines the value of the parameter P of the asymmetric part of the double-well potential. We assume that $\nu >0$ and $\eta >0$. For the above relationships of the parameters, the solution to (12) is (18).
• We note that this solution is obtained under the condition that $\nu \ne 0$ and $\eta \ne 0$. When one of these quantities or both of them are 0, we have to search for another solution to the system of nonlinear algebraic equations.
• In (19), we have $\alpha <0$. We assume that $ϵ>0$. Then, we obtain $\gamma >-{\displaystyle \frac{72ϵ{\alpha }^2}{{\beta }^2}}$, which leads to the condition on the wave velocity of $v^2>{\displaystyle \frac{k{R}_0}{m}}-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}=v_c-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}$. As $ϵ>0$, then $B>0$, and the velocity of the wave can be smaller than $v_c$. But this wave can have a velocity which is equal to the characteristic velocity, and even the wave velocity can be larger than $v_c$.

Let us now consider the solution for the case $b=0$ (the case of linear interaction—Hooke’s law).

Lemma 2. 

Let us consider the case where all parameters in (12) are different from 0 except for $b=0$. The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) to the following system of nonlinear algebraic equations:

$$\begin{array}{ccc}\hfill a_1(ϵa_1^2+2\gamma b_2^2)& =& 0\hfill \\ \hfill a_1(3ϵa_0{a}_1+\delta b_2+\mu a_1+3\gamma b_1{b}_2)& =& 0\hfill \\ \hfill a_1[\nu +\delta b_1+\gamma (b_1^2+2b_2{b}_0)+3ϵa_0^2+2\mu a_0]& =& 0\hfill \\ \hfill \gamma a_1{b}_1{b}_0+\nu a_0+\eta +\delta a_1{b}_0+ϵa_0^3+\mu a_0^2& =& 0.\hfill \end{array}$$

(21)

Proof. 

The application of SEsM(1,1) to (12) with $b=0$ means that we have to substitute (14) and (13) into (12) for the case $M=2$, $N=1$. Then, we have to equate to 0 the coefficients of the obtained polynomial of $f\left(\xi \right)$. This leads us to the system in (21). □

Theorem 2. 

An exact solution to (12) with $b=0$ is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0+{\left({\displaystyle \frac{-2\gamma }{ϵ}}\right)}^{1/2}\{-{\displaystyle \frac{2^{1/2}[6a_0\gamma ϵ-2^{1/2}\delta {(-\gamma ϵ)}^{1/2}+2\mu \gamma ]}{12\gamma {(-ϵ\gamma )}^{1/2}}}-{\displaystyle \frac{\theta }{2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\\ \hfill {\displaystyle \frac{b_{2}D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\end{array}$$

(22)

where

$$\begin{array}{c}\hfill {\theta }^2={\left\{{\displaystyle \frac{2^{1/2}[6a_0\gamma ϵ-2^{1/2}\delta {(-\gamma ϵ)}^{1/2}+2\mu \gamma ]}{6\gamma {(-ϵ\gamma )}^{1/2}}}\right\}}^2+\\ \hfill 2{\displaystyle \frac{2^{1/2}[9\eta \gamma ϵ+2{\delta }^2ϵa_0+2^{1/2}\delta a_0\mu {(-ϵ\gamma )}^{1/2}+3·2^{1/2}{a}_0^2ϵ\delta {(-ϵ\gamma )}^{1/2}+3a_0^2ϵ\gamma \mu +2{\mu }^2{a}_0\gamma ]}{3\gamma [2\delta -\mu {(-2\gamma )}^{1/2}]{(-\gamma )}^{1/2}}}.\end{array}$$

(23)

Proof. 

Let us consider the case where all parameters in (12) are different from 0 except for $b=0$. We apply SEsM(1,1) to (12) with $b=0$ on the basis of Equations (13) and (14) for the case $M=2$, $N=1$. As a result, we obtain the system of nonlinear algebraic equations in (21). A solution to (21) is

$$\begin{array}{ccc}\hfill a_1& =& b_2{\left({\displaystyle \frac{-2\gamma }{ϵ}}\right)}^{1/2},\hfill \\ \hfill b_0& =& -{\displaystyle \frac{2^{1/2}[9\eta \gamma ϵ+2{\delta }^2ϵa_0+2^{1/2}\delta a_0\mu {(-ϵ\gamma )}^{1/2}+3·2^{1/2}{a}_0^2ϵ\delta {(-ϵ\gamma )}^{1/2}+3a_0^2ϵ\gamma \mu +2{\mu }^2{a}_0\gamma ]}{6\gamma b_2[2\delta -\mu {(-2\gamma )}^{1/2}]{(-\gamma )}^{1/2}}},\hfill \\ \hfill b_1& =& {\displaystyle \frac{2^{1/2}[6a_0\gamma ϵ-2^{1/2}\delta {(-\gamma ϵ)}^{1/2}+2\mu \gamma ]}{6\gamma {(-ϵ\gamma )}^{1/2}}},\hfill \\ \hfill \nu & =& -{\displaystyle \frac{2^{3/2}{\mu }^3\gamma {(-ϵ\gamma )}^{1/2}-4{\delta }^3{ϵ}^2+27·2^{1/2}\eta \gamma {ϵ}^2{(-ϵ\gamma )}^{1/2}-6{\mu }^2\gamma \delta ϵ}{9\gamma ϵ[2\delta ϵ-2^{1/2}\mu {(-ϵ\gamma )}^{1/2}]}}.\hfill \end{array}$$

(24)

This solution corresponds to solution (22) to (12) with $b=0$. □

We note the following:

• In the case $b=0$, $\eta$ is an independent variable. This means that the asymmetric part of the double-well potential is not restricted in the form.
• In addition, it is necessary that $\gamma /ϵ<0$ and $\gamma <0$ (in order to have $\theta$ real and ${\theta }^2>0$). But the requirement of $\gamma <0$ leads to $v^2<k{R}_0/m$. This means that in the case $b=0$, the velocity of the wave has to be smaller than $v_c$, which is a difference with respect to the case $b=1$. In order to have $v>v_c$, we must have $ϵ<0$, and we consider the nonnegative values of $ϵ$.

Figure 1 shows a comparison of the two waves in (18) and (22) (Figure (a)) and the corresponding potentials (Figure (b)). For the wave presented with a solid line, we have $b=1$, and the wave presented by a dashed line corresponds to $b=0$. The wave presented with a solid line travels with $v>v_c$, and the wave presented by a dashed line travels with $v<v_c$. Figure (b) shows the asymmetric double-well potentials connected to the waves from Figure (a). We observe that the switch from $b=1$ to $b=0$ changes the profile of the waves as well as the form of the corresponding potential.

4.2. $\gamma =0$

Below, we discuss several waves which travel with velocity $v=v_c$. The most general case of such a wave corresponds to $\gamma =0$. In this case, we have to solve

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\delta {\displaystyle \frac{dz}{d\xi }}+ϵz^3+\mu z^2+\nu z+\eta =0$$

(25)

Theorem 3. 

Let us consider the case where all parameters in (12) are different from 0 except for $\gamma =0$. An exact solution to (12) for the specific case $\gamma =0$ is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0-12{\displaystyle \frac{\alpha }{\beta }}\{{\displaystyle \frac{\beta (-\delta \beta +36ϵa_0\alpha +12\mu \alpha )}{512ϵ{\alpha }^2}}-{\displaystyle \frac{\theta }{2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\}\hfill \end{array}$$

(26)

where

$$\begin{array}{c}\hfill {\theta }^2=b_1^2-4b_0{b}_2=-{\displaystyle \frac{72ϵ\alpha }{{\beta }^2}}>0.\end{array}$$

(27)

Proof. 

We apply SEsM(1,1) to (12) for the specific case $\gamma =0$ on the basis of Equations (13) and (14) for the case $M=2$, $N=1$. As a result, we obtain the system of nonlinear algebraic equations in (17) for the specific case $\gamma =0$. The solution of this system of algebraic equations leads us to solution (26) to (12) for the specific case $\gamma =0$. □

As $\alpha <0$, the relationship in (27) is satisfied when $ϵ>0$, which is the case indeed.

Figure 2 shows the wave profiles and the corresponding potentials for the waves in (18) and (26). The waves travel at different velocities. We note the following:

• The wave marked by a solid line in Figure 2a travels with velocity $v>v_c$. The wave marked by a dashed line travels with velocity $v=v_c$. We observe that the wave which travels at higher velocity has a larger size. The double-well potential for the wave with larger velocity has smaller minimal values in comparison with the potential connected to the wave which travels with $v=v_c$.
• Solution to kind (26) (based on the Riccati equation as the simple equation) is not present in the case $b=0$. For the case $b=0$, the simple equation has to be different from the Riccati equation. Because of this, these cases are not discussed in the text below. Thus, all the cases discussed below for $\gamma =0$ require $b=1$ in order to have a solution based on the Riccati equation as a simple equation.

4.3. $\gamma =\mu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\delta {\displaystyle \frac{dz}{d\xi }}+ϵz^3+\nu z+\eta =0$$

(28)

A solution to this equation is described by the following.

Theorem 4. 

Let us consider the case where all parameters in (12) are different from 0 except for $\gamma =\mu =0$. An exact solution to (12) for the specific case $\gamma \mu =0$ is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0-12{\displaystyle \frac{\alpha b_2}{\beta }}\{-{\displaystyle \frac{-\frac{\beta (-\delta \beta +36ϵa_0\alpha )}{216ϵ{\alpha }^2}}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\hfill \end{array}$$

(29)

where

$$\begin{array}{c}\hfill {\theta }^2=b_1^2-4b_0{b}_2=-{\displaystyle \frac{72ϵ\alpha }{{\beta }^2}}>0.\end{array}$$

(30)

Proof. 

We apply SEsM(1,1) to (12) for the specific case $\gamma =\mu =0$ on the basis of Equations (13) and (14) in the case $M=2$, $N=1$. As a result, we obtain the system of nonlinear algebraic equations in (17) for the specific case $\gamma =\mu =0$. The solution of this system of algebraic equations leads us to solution (29) to (12) for the specific case $\gamma =\mu =0$. □

We note that solution (29) describes waves with fixed velocity $v=\pm R_0{(k/m)}^{1/2}$.

Figure 3 compares the profiles and potentials of to the two waves in (18) and (29). We note the following:

• The wave marked by a solid line in Figure 3a travels with velocity $v>v_c$.
• The wave marked by a dashed line travels with velocity $v=v_c$. In addition, a part of the double-well potential is not present, as $Q=0$.
• We observe that in this case, the differences in the wave profiles are larger in comparison with the differences in the profiles of the waves in Figure 2.
• The same large difference is observed in the case of potentials shown in Figure 3b.

4.4. $\gamma =\nu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\delta {\displaystyle \frac{dz}{d\xi }}+ϵz^3+\mu z^2+\eta =0.$$

(31)

One solution to this equation is given by the following.

Theorem 5. 

Let us consider the case where all parameters in (12) are different from 0 except for $\gamma =\nu =0$. An exact solution to (12) for the specific case $\gamma =\nu =0$ is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0-{\displaystyle \frac{12\alpha b_2}{\beta }}\{-{\displaystyle \frac{\beta \delta -12\mu \alpha -\frac{\beta a_0{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}{6\alpha }}{2b_2{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\\ \hfill {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\end{array}$$

(32)

where

$$\begin{array}{ccc}\hfill {\theta }^2& =& {\displaystyle \frac{{\left[(144{\mu }^2{\alpha }^2-{\beta }^2{\delta }^2)\right]}^{1/3}}{3^{1/3}{\left(\beta \alpha \right)}^{2/3}}}.\hfill \end{array}$$

Proof. 

We apply SEsM(1,1) to (12) in the specific case $\gamma =\nu =0$ on the basis of Equations (13) and (14) in the case $M=2$, $N=1$. As a result, we obtain a system of nonlinear algebraic equations. A solution to this system is

$$\begin{array}{ccc}\hfill a_1& =& -{\displaystyle \frac{12\alpha b_2}{\beta }},\hfill \\ \hfill b_0& =& {\displaystyle \frac{1}{4b_2{\beta }^2}}\left[{\displaystyle \frac{{\beta }^{4/3}{\left[({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}{3^{1/3}{\alpha }^{2/3}}}+{\displaystyle \frac{{\left(\frac{\beta a_0{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}{6\alpha }-\delta \beta +12\mu \alpha \right)}^2}{9^{2/3}{\left[\alpha ({\delta }^2{\beta }^2-144{\mu }^2{\alpha }^2)\right]}^{2/3}}}\right],\hfill \\ \hfill b_1& =& {\displaystyle \frac{\beta \delta -12\mu \alpha -\frac{\beta a_0{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}{6\alpha }}{{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}},\hfill \\ \hfill ϵ& =& {\displaystyle \frac{\beta {\left[9\beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}}{216{\alpha }^{5/3}}},\hfill \\ \hfill \eta & =& {\displaystyle \frac{1}{18{\beta }^2\alpha {\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{2/3}}}\{a_0{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{4/3}-62208{\mu }^3{\alpha }^5+\hfill \\ & & 144{\delta }^3{\beta }^3{\alpha }^2-15552\delta \beta {\alpha }^4{\mu }^2-9{\delta }^2{\beta }^3\alpha a_0{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}+\hfill \\ & & 1296\beta a_0{\mu }^2{\alpha }^3{\left[9\alpha \beta ({\beta }^2{\delta }^2-144{\mu }^2{\alpha }^2)\right]}^{1/3}\}.\hfill \end{array}$$

(33)

The solution of this system of algebraic equations leads us to solution (29) to (12) for the specific case $\gamma =\nu =0$. □

Several notes about solution (32) are as follows:

• The velocity of the wave in (32) is $v=\pm R_0{(k/m)}^{1/2}$.
• For this solution, we have $\nu =-A=0$, which means that a part of the symmetric part of the double-well potential is absent.
• The relationships of $ϵ$ and of $\eta$ mean that for the existence of the wave, there are restrictions on the values of the parameters B of the symmetric part of the double-well potential and the parameter P of the asymmetric part of the double-well potential.

4.5. $\gamma =\mu =\nu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\delta {\displaystyle \frac{dz}{d\xi }}+ϵz^3+\eta =0.$$

(34)

A solution to this equation is given by

Theorem 6. 

Let us consider the case where all parameters in (12) are different from 0 except for $\gamma =\mu =\nu =0$. An exact solution to (12) for the specific case $\gamma =\mu =\nu =0$ is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0-{\displaystyle \frac{12\alpha b_2}{\beta }}\left\{-{\displaystyle \frac{b_1}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+{\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\right\},\end{array}$$

(35)

where

$${\theta }^2=-{\displaystyle \frac{1}{3^{1/3}}}{\left({\displaystyle \frac{\delta }{\alpha }}\right)}^{2/3}.$$

Proof. 

We apply SEsM(1,1) to (12) for the specific case $\gamma =\mu =\nu =0$ on the basis of Equations (13) and (14) in the case $M=2$, $N=1$. As a result, we obtain a system of nonlinear algebraic equations. A solution to this system is

$$\begin{array}{ccc}\hfill a_1& =& -{\displaystyle \frac{12\alpha b_2}{\beta }},\hfill \\ \hfill b_0& =& {\displaystyle \frac{1}{4b_2{\beta }^2}}\left[{\displaystyle \frac{{\delta }^{1/3}{\beta }^2}{3^{1/3}{\alpha }^{2/3}}}+{\displaystyle \frac{{\left(\frac{{\delta }^{2/3}{\beta }^2{a}_0}{2·3^{1/3}{\alpha }^{2/3}}-\delta \beta \right)}^2}{9^{2/3}{\left({\delta }^2\alpha \right)}^{2/3}}}\right],\hfill \\ \hfill b_1& =& -{\displaystyle \frac{\left(\frac{{\delta }^{2/3}{\beta }^2{a}_0}{2·3^{1/3}{\alpha }^{2/3}}-\delta \beta \right)}{9^{1/3}\beta {\left({\delta }^2\alpha \right)}^{1/3}}},\hfill \\ \hfill ϵ& =& {\displaystyle \frac{9^{1/3}{\delta }^{2/3}{\beta }^2}{216{\alpha }^{5/3}}}\hfill \\ \hfill \eta & =& -{\displaystyle \frac{9^{1/3}{\left({\delta }^2\alpha \right)}^{1/3}\left(9^{1/3}{\left({\delta }^2\alpha \right)}^{1/3}\beta a_0-9^{1/3}{\left({\delta }^2\alpha \right)}^{1/3}\beta a_0-16\delta \alpha \right)}{18\alpha \beta }}.\hfill \end{array}$$

(36)

The solution of this system of algebraic equations leads us to solution (35) to (12) for the specific case $\gamma =\mu =\nu =0$. □

Several notes about the obtained solution are as follows:

• The velocity of the wave in (35) is $v=v_c=\pm R_0{(k/m)}^{1/2}$.
• $\nu =-A=0$, which means that a part of the symmetry component of the double-well potential is missing.
• In addition, $\mu =-D=0$, which means that a part of the asymmetric components of the double-well potential are also missing.
• Moreover, the relationships of $ϵ$ and of $\eta$ mean that for the existence of the wave, there are restrictions on the values of the parameters B of the symmetric part of the double-well potential and the parameter P of the asymmetric part of the double-well potential.
• We note that the right-hand side of the relationship of $\theta$ is negative (as $\delta /\alpha >0$). Then, this solution cannot be associated with a wave in the tubule.

4.6. $\delta =0$

This case corresponds to the absence of friction in the system. In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\gamma {\displaystyle \frac{d^{2}z}{d{\xi }^2}}+ϵz^3+\mu z^2+\nu z+\eta =0$$

(37)

A solution to this equation is given by the following.

Theorem 7. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =0$. An exact solution to (12) for the specific case $\delta =0$ is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0-12{\displaystyle \frac{\alpha b_2}{\beta }}\{-{\displaystyle \frac{-\frac{\beta (36ϵa_0\alpha +12\mu \alpha )}{216ϵ{\alpha }^2}}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\}\hfill \end{array}$$

(38)

where

$$\begin{array}{c}\hfill {\theta }^2=b_1^2-4b_0{b}_2=-{\displaystyle \frac{\gamma {\beta }^2+72ϵ{\alpha }^2}{\alpha {\beta }^2}}.\end{array}$$

(39)

Proof. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =0$. The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) to the system of nonlinear algebraic equations in (17) for the specific case $\delta =0$. A solution to this system of algebraic equations is (38). □

We note the following:

• We remember that $\delta =-gv$ was connected with friction. If we assume that $v\ne 0$, then we have $g=0$. This means that this wave can propagate when the friction force is absent.
• We assume that $\alpha <0$ and $ϵ\ge 0$. Because of this, we obtain $\gamma \ge -{\displaystyle \frac{72ϵ{\alpha }^2}{{\beta }^2}}$, which leads to the condition on the wave velocity of $v^2\ge {\displaystyle \frac{k{R}_0}{m}}-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}=v_c-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}$.
• As $ϵ>0$, then $B>0$, and the velocity of the wave can be smaller than $v_c$. But, in principle, this wave can have a velocity which is equal to the characteristic velocity, and even the wave velocity can be larger than $v_c$. This can happen when $B<0$.

Let us now consider the case of linear interaction ($b=0$). The solution for this case is given by the following.

Theorem 8. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =b=0$. An exact solution to (12) for the specific case $\delta =b=0$ is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0+b_2{\left({\displaystyle \frac{-2\gamma }{ϵ}}\right)}^{1/2}\{-{\displaystyle \frac{2^{1/2}(3a_0ϵ+\mu )}{6b_2{(-ϵ\gamma )}^{1/2}}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\\ \hfill {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\end{array}$$

(40)

where

$$\begin{array}{c}\hfill {\theta }^2={\displaystyle \frac{2}{9\gamma ϵ}}[9\nu ϵ-{(3a_0ϵ+\mu )}^2].\end{array}$$

(41)

In addition,

$$\nu ={\displaystyle \frac{2{\mu }^3+27\eta {ϵ}^2}{9ϵ\mu }}.$$

(42)

Proof. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =b=0$. The application of SEsM(1,1) a with simple equation of kind (15) reduces (12) to the system of nonlinear algebraic equations in (17) for the specific case $\delta =b=0$. A solution to this system of algebraic equations is (40). □

Several notes about this solution are as follows:

• The condition $ϵ\gamma <0$ is present. This influences the velocity of the wave. The wave travels with velocity $v<v_c$. In addition, we must have $9\nu ϵ<{(3a_0ϵ+\mu )}^2$.
• Figure 4 shows two waves which can travel in the tubule in the case of the absence of friction. Figure 4a shows the profiles of the waves. The wave marked with a solid line travels with velocity $v>v_c$. The wave marked with a dashed line travels with velocity $v<v_c$.
• The differences are significant not only for the profile of the waves but also for the corresponding potentials, which are shown in Figure 4b (for the wave traveling with $v>v_c$ and corresponding to the case $b=1$) and in Figure 4c (for the wave which travels with $v<v_c$ and corresponding to the case $b=0$, which describes the case of linear interaction between the neighboring dimers).

4.7. $\delta =\mu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\gamma {\displaystyle \frac{d^{2}z}{d{\xi }^2}}+ϵz^3+\nu z+\eta =0.$$

(43)

A solution to this equation is given by the following.

Theorem 9. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =\mu =0$. An exact solution of (12) for the specific case $\delta =\mu =0$ is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0-12{\displaystyle \frac{\alpha b_2}{\beta }}\{{\displaystyle \frac{\beta (-\delta \beta +36ϵa_0\alpha +12\mu \alpha )}{512b_2ϵ{\alpha }^2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\hfill \end{array}$$

(44)

where

$$\begin{array}{c}\hfill {\theta }^2=b_1^2-4b_0{b}_2=-{\displaystyle \frac{\gamma {\beta }^2+72ϵ{\alpha }^2}{\alpha {\beta }^2}}.\end{array}$$

(45)

Proof. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =\mu =0$. The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) to the system of nonlinear algebraic equations in (17) for the specific case $\delta =\mu =0$. A solution to this system of algebraic equations is (44). □

Several notes about the obtained solution are as follows.:

• $\delta =-gv$ was connected with friction. If we assume that $v\ne 0$, then we have $g=0$. This means that this wave can propagate when the friction force is absent.
• In addition, $\mu =-D=0$, which means that also a part of the anharmonic part of the double-well potential is missing.
• From (45), it follows that $\gamma >-{\displaystyle \frac{72ϵ{\alpha }^2}{{\beta }^2}}$, which leads to the condition on the wave velocity of $v^2>{\displaystyle \frac{k{R}_0}{m}}-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}=v_c-{\displaystyle \frac{k^{2}B}{8m{k^{*}}^2}}$.
• As $ϵ>0$, then $B>0$, and the velocity of the wave can be smaller than $v_c$.
• In principle, this wave can have a velocity which is equal to the characteristic velocity, and even the wave velocity can be larger than $v_c$. This happens when $B<0$.
• Figure 5 shows a comparison of the two waves in (38) and (44) with similar parameters for the following cases. The wave marked with a solid line in Figure 5a travels with $v>v_c$. The wave marked with a dashed line is the same wave but for the case $Q=0$.
• We observe that in the case of the absence of friction, the differences between the profiles of the two waves are not very large (these differences can be observed in the case of relatively large values of $\mid \xi \mid$). The same conclusion holds for the potentials connected to the two waves.

Let us now consider the case of linear interaction ($b=0$). A solution for this case is given by the following.

Theorem 10. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =\mu =b=0$. An exact solution to (12) for the specific case $\delta =\mu =0$ is

$$\begin{array}{ccc}\hfill z\left(\xi \right)& =& a_0+b_2{\left({\displaystyle \frac{-2\gamma }{ϵ}}\right)}^{1/2}\{-{\displaystyle \frac{\frac{2^{1/2}{a}_0ϵ}{{(-ϵ\gamma )}^{1/2}}}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\hfill \\ & & {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\hfill \end{array}$$

(46)

where

$$\begin{array}{c}\hfill {\theta }^2={\displaystyle \frac{2}{\gamma }}(\nu -a_0^2ϵ).\end{array}$$

(47)

Proof. 

Let us consider the case where all parameters in (12) are different from 0 except for $\delta =\mu =0$. The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) to the system of nonlinear algebraic equations in (17) for the specific case $\delta =\mu =0$. A solution to this system of algebraic equations is (46). □

Several notes about this solution are as follows:

• We have the requirement $ϵ\gamma <0$, which leads to $\gamma <0$ and to $v<v_c$.
• An additional requirement is $\nu <a_0^2ϵ$. Moreover, we have $\eta =0$.
• Figure 6 shows a comparison of the wave profiles and the potential for two waves which travel in the case of the absence of friction and in the case of $b=0$ (linear interaction among the dimers). The wave marked with a dashed line in Figure 6a travels when $\mu =0$, in other words, when a part of the double-well potential is not present.
• We observe differences in the sizes of the two waves and the differences between potentials around $z=0$, as shown in Figure 6b.

4.8. $\delta =\nu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\gamma {\displaystyle \frac{d^{2}z}{d{\xi }^2}}+ϵz^3+\mu z^2+\eta =0.$$

(48)

As $\nu =0$, we have to search for new solutions to the general system of nonlinear algebraic equations. One solution is given by the following.

Theorem 11. 

Let us consider the case where all parameters in (12) are different from 0 except for $\nu =0$. A solution for this specific case is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0-{\displaystyle \frac{\alpha -\frac{\mu +\frac{\beta a_0{Q}_2}{2}}{3\alpha Q_2}}{\beta }}\{-{\displaystyle \frac{-\frac{\mu +\frac{\beta a_0{Q}_2}{2}}{3\alpha Q_2}}{2b_2}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\\ \hfill {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\end{array}$$

(49)

where

$$\begin{array}{c}\hfill {\theta }^2={\left(-{\displaystyle \frac{\mu +\frac{\beta a_0{Q}_2}{2}}{3\alpha Q_2}}\right)}^2-{\displaystyle \frac{\gamma {\beta }^2+12{\alpha }^2\beta Q_2+\frac{{\beta }^2{\left[\mu +\beta a_0\frac{Q_2}{2}\right]}^2}{9\alpha Q_2^2}}{4\alpha {\beta }^2}}.\end{array}$$

(50)

Proof. 

The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) with $\nu =0$ to a system of nonlinear algebraic equations. This system has a solution

$$\begin{array}{ccc}\hfill a_1& =& -12{\displaystyle \frac{\alpha b_2}{\beta }},\hfill \\ \hfill b_0& =& {\displaystyle \frac{\gamma {\beta }^2+12{\alpha }^2\beta Q_2+\frac{{\beta }^2{\left[\mu +\beta a_0\frac{Q_2}{2}\right]}^2}{9\alpha Q_2^2}}{4\alpha b_2{\beta }^2}},\hfill \\ \hfill b_1& =& -{\displaystyle \frac{\mu +\frac{\beta a_0{Q}_2}{2}}{3\alpha Q_2}},\hfill \\ \hfill Q_2& =& {\displaystyle \frac{1}{36{\alpha }^2}}\left\{Q_1^{1/3}+{\displaystyle \frac{{\beta }^2{\gamma }^2}{Q_1^{1/3}}}-\beta \gamma \right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill Q_1& =& -\beta \left[648{\mu }^2{\alpha }^3+{\gamma }^3{\beta }^2-36\mu {\alpha }^2{\left({\displaystyle \frac{324{\mu }^2{\alpha }^3+{\gamma }^3{\beta }^2}{\alpha }}\right)}^{1/2}\right],\hfill \\ \hfill ϵ& =& {\displaystyle \frac{\beta }{216{\alpha }^2}}\left(Q_1^{1/3}+{\displaystyle \frac{{\gamma }^2{\beta }^2}{Q_1^{1/3}}}-\gamma \beta \right),\hfill \\ \hfill \eta & =& {\displaystyle \frac{1}{18}}\{Q_1^{1/3}(40310784{\mu }^5{\alpha }^8\beta +62208{\mu }^3{\alpha }^5{\beta }^3{\gamma }^3-2239488{\mu }^4{\alpha }^7\beta Q_3^{1/2}+\hfill \\ & & 1296\beta {\mu }^2{\alpha }^3{a}_0{Q}_1^{4/3}+{\beta }^6{a}_0{\gamma }^6{Q}_1^{1/3}+2{\beta }^3{a}_0{\gamma }^3{Q}_1^{4/3}+a_0{Q}_1^{7/3}\left)\right\}/\left\{\alpha {\beta }^3\right(3\beta {\gamma }^2{Q}_1^{4/3}-\hfill \\ & & 2\gamma Q_1^{5/3}+{\beta }^3{\gamma }^4{Q}_1^{2/3}-144{\alpha }^2{\beta }^3{\gamma }^3\mu Q_3^{1/2}+3888{\alpha }^3\beta 63{\gamma }^3{\mu }^2+3{\beta }^5{\gamma }^6+\hfill \\ & & 839808{\alpha }^6\beta {\mu }^4-46656{\alpha }^5\beta {\mu }^3{Q}_3^{1/2}\left)\right\},\hfill \\ \hfill Q_3& =& {\displaystyle \frac{324{\mu }^2{\alpha }^3+{\gamma }^2{\beta }^2}{\alpha }}.\hfill \end{array}$$

(51)

This solution leads to solution (51) to (12) for the specific case $\nu =0$. □

We make the following considerations:

• We remember that $\delta =-gv$ was connected with friction. If we assume that $v\ne 0$, then we have $g=0$. This means that this wave can propagate when the friction force is absent.
• In addition, we have $\nu =-A=0$, which means that a part of the double-well potential is missing.

Let us now consider the case $b=0$. The solution for this case is given by the following.

Theorem 12. 

Let us consider the case where all parameters in (12) are different from 0 except for $\nu =0$ and $b=0$. A solution for this specific case is

$$\begin{array}{c}\hfill z\left(\xi \right)=a_0+2^{1/2}{b}_2{\left(-{\displaystyle \frac{\gamma }{ϵ}}\right)}^{1/2}\{-{\displaystyle \frac{2a_0ϵ+2^{2/3}{(-\eta {ϵ}^2)}^{1/3}}{2^{3/2}{b}_2{(-ϵ\gamma )}^{1/2}}}-{\displaystyle \frac{\theta }{2b_2}}tanh\left[{\displaystyle \frac{\theta (\xi +C)}{2}}\right]+\\ \hfill {\displaystyle \frac{D}{{cosh}^2\left[\frac{\theta (\xi +C)}{2}\right]\left\{E-\frac{2b_{2}D}{\theta }tanh\left[\frac{\theta (\xi +C)}{2}\right]\right\}}}\},\end{array}$$

(52)

where

$$\begin{array}{c}\hfill {\theta }^2=-{\displaystyle \frac{3·2^{1/3}{(-\eta {ϵ}^2)}^{2/3}}{ϵ\gamma }}.\end{array}$$

(53)

In addition,

$$\mu ={\displaystyle \frac{3}{2^{1/3}}}{(-\eta {ϵ}^2)}^{1/3}.$$

(54)

Proof. 

The application of SEsM(1,1) with a simple equation of kind (15) reduces (12) with $\nu =0$ and $b=0$ to a system of nonlinear algebraic equations. This system has a solution

$$\begin{array}{ccc}\hfill a_1& =& 2^{1/2}{b}_2{\left(-{\displaystyle \frac{\gamma }{ϵ}}\right)}^{1/2},\hfill \\ \hfill b_0& =& {\displaystyle \frac{2^{4/3}{(-\eta {ϵ}^2)}^{2/3}-2a_0^2{ϵ}^2}{ϵ\gamma }},\hfill \\ \hfill b_1& =& -{\displaystyle \frac{2a_0ϵ+2^{2/3}{(-\eta {ϵ}^2)}^{1/3}}{2^{1/2}{(-ϵ\gamma )}^{1/2}}},\hfill \\ \hfill \mu & =& {\displaystyle \frac{3}{2^{1/3}}}{(-\eta {ϵ}^2)}^{1/3},\hfill \end{array}$$

(55)

which leads to solution (52). □

Here, we again have the requirement $\gamma <0$, which means that the wave velocity is $v<v_c$.

4.9. $\delta =\mu =\nu =0$

In this case, the model equation becomes

$$\alpha {\displaystyle \frac{d^{4}z}{d{\xi }^4}}+\beta {\displaystyle \frac{d^{2}z}{d{\xi }^2}}{\displaystyle \frac{dz}{d\xi }}+\gamma {\displaystyle \frac{d^{2}z}{d{\xi }^2}}+ϵz^3+\eta =0.$$

(56)

Here, we have to search for new solutions as for all solutions in the general case, we have $\nu \ne 0$. The application of SEsM(1,1) with a simple equation of kind (15) leads to $\eta =0$ and to solutions for which $\theta =0$. Such solutions are constants and are not of interest to us.

5. Concluding Remarks

In this article, we discuss exact analytical solutions to Equation (12), which is a model equation of wave dynamics connected to biological microtubules. The exact solutions are obtained by means of the Simple Equations Method (SEsM). We apply version SEsM(1,1) of the methodology based on the use of the equation of Riccati as a simple equation. The results of the study are exact solutions to (12) for different values of the model parameters. We can compare the case $b=1$ (case of nonlinear interaction force between dimers) to the case $b=0$ (case of linear interaction force between dimers—Hooke’s law). The first interesting result is obtained for the case where the parameters in (12) are different from 0. In this case, a specific value of the wave velocity exists. This value is $v_c^2=k{R}_0/m$, and it depends on the parameters of the dimer and on a parameter of the interaction between the dimers. For the case of the linear force of the interaction between dimers, the resulting waves have $v<v_c$. For the case of the possibility of nonlinear interaction, the wave velocity can become $v>v_c$.

Next, the interesting case $\gamma =0$ is discussed. As $ϵ\ne 0$ in the case of linear interaction between the dimers, the solution is connected to a simple equation which is more complicated than the equation of Riccati. Such solutions will be discussed elsewhere. The interesting result is that in the case of nonlinear interaction $b=1$, there are solutions to model Equation (12) which can be constructed on the basis of the Riccati equation as a simple equation in SEsM(1,1).

In the case $\gamma =\mu =0$, we have the absence of the part of the asymmetric part of the double-well potential. The solution for the case $\gamma =0$ continues to exist. The solution of (12) becomes more complicated when $\gamma =\nu =0$ (in comparison to the case $\gamma =\mu =0)$. In this case, a part of the harmonic part of the potential is missing ($A=0$). Finally, the solution for the case $b=1$ exists when $\gamma =\mu =\nu =0$.

The second-largest class of solutions considered in this article are solutions to (12) for $\delta =0$ (absence of friction in the system). When the other parameters have values which are different from 0, the solution based on the use of Riccati equation as a simple equation exists in both the cases $b=1$ and $b=0$. The solution continues to exist for $\delta =\mu =0$ and for $\delta =\nu =0$. For the case $\delta =\mu =\nu =0$, the system of nonlinear algebraic equations of SEsM leads to $\theta =0$. Then, the solution obtained on the basis of the use of Riccati equation as a simple equation is constant.

We have obtained kink solutions of the studied equation, connected to the $z-$model of the wave propagation in the microtubules. According to the model, the kink soliton is responsible for the energy transfer along the microtubule. Microtubules are used as road networks for motor proteins. Their use is as follows: The cell’s compartment which needs energy or substance can launch a soliton of the above kind. This soliton is sent as a signal along the closest microtubule. This soliton activates molecular motors which are close to microtubule used for the propagation of the soliton.

The modeling of waves in microtubules has many perspectives. One of them is to consider the microtubule as an elastic tube and to study waves in such a system [36,37]. Finally, we note that we can apply the SEsM to (12) also on the basis of other equations as simple equations. Results from this research will be reported elsewhere.