The generalized projective Riccati equations method and its applications for solving two nonlinear PDEs describing microtubules

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.

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): (i) The nonlinear PDE describing the nonlinear dynamics of MTs as nanobioelectronics transmission lines: where ( , ) z x t is the traveling wave, m is the mass of the dimer, k is a harmonic constant describing the nearest-neighbor interaction between the dimers belonging to the same protofilaments (PFs), l is the MT length, E is the magnitude of intrinsic electric field, q>0 is the excess charge within the dipole,  is the viscosity coefficient and , ABare positive parameters.The physical details of the derivation of Equation (1) has been discussed in Zekovic et al. (2014) which are omitted here for simplicity.The authors (Zekovic et al., 2014) have used the Jacobi elliptic function method to find the exact solutions of Equation ( 1).
(ii) The nonlinear PDE describing the nonlinear dynamics of radial dislocations in MTs: (2) where ( , ) xt  is the corresponding angular displacement when the whole dimer rotates with the angular displacement ( , ) xt of the single dimer, c stands for inter-dimer bonding interaction within the same protofilaments (PFs), h is the MT length, p is the electric dipole moment, H is the magnitude of intrinsic electric field and  is the viscosity coefficient.The physical details of the derivation of Equation ( 2) has been discussed in Zdravkovic et al. (2014) which are omitted here for simplicity.The authors (Zdravkovic et al., 2014) have used the simplest equation method to find the exact solutions of Equation (2).

Description of the generalized projective Riccati equations method
Considering the following NPDE: ( , , , , , ,...) 0, where F is a polynomial in ( , ) u x t and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved.In the following, 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.
Step 1.The authors use the wave transformation where 1 , k and  are constants, to reduce Equation (3)   to the following ODE: ( , ', '',...) 0, where Q is a polynomial in () u  and its total derivatives, such that ' d d   .
Step 2. The authors assume that Equation (5) has the formal solution: where 0 , i AA and i B are constants to be determined later.The functions ()  and ()  satisfy the ODEs: '( ) ( ) ( ) Where where 1 r  and , R  are nonzero constants.
If , 0, R   Equation ( 5) has the formal solution: 0 ( ) ( ), where () Step 3. 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).
Step 4. Substitute (6) along with Equations ( 7) -(9) into Equation ( 5) or ((10) along with Equation (11) into Equation ( 5)).Collecting all terms of the same order of ). Setting each coefficient to zero, yields a set of algebraic equations which can be solved to find the values of 01 , , , , , ii A A B k  and R .
Step 4. It is well known (Yomba, 2005) that Equations ( 7) and ( 8) admit the following solutions: where C is nonzero constant.

APPLICATIONS
In this part, the authors will apply the proposed method described in description of the generalized projective Riccati equations method, to find the exact solutions of the two nonlinear PDEs (1) and (2).

Example 1. Exact solutions of the nonlinear PDE (1) describing the nonlinear dynamics of MTS as nanobioelectronics transmission lines
The authors find the exact wave solutions of Equation (1).
To this end, the authors use the transformation (4) to reduce Equation (1) into the following ODE: where and  in Equation ( 17), the authors get 1.N  Consequently, the authors have the formal solution of Equation ( 17) as follows: where 01 , AA and 1 B are constants to be determined later.Substituting ( 20) into (17) and using ( 7) -( 9), the lefthand side of Equation ( 17) becomes a polynomial in ()  and ()  .Setting the coefficients of this polynomial to be zero, yields the following system of algebraic equations: Case 1.If authors substitute 1   into the algebraic equations ( 21) and solve them by Maple 14, the following results were realized: Result 1.The authors have (2 9 ) 2 , 27 where 0   , 2 60   .
In this case, the authors deduce that if 1 r  , then the exact wave solution was realized: Result 3. The authors have (2 9 ) 2 , 27 In this case, the authors deduce that if 1 r  , then the exact wave solution was realized: while if 1 r  , then the authors have the exact wave solution (2 9 ) 2 , 27   .From ( 14), ( 15), ( 19), ( 20) and ( 30), the authors deduce the following exact wave solutions Result 2. The authors have (2 9 ) 2 , 27 In this case, the authors deduce the exact wave solutions Result 3. The authors have (2 9 ) 2 , 27   and 32 (1 ) 0   .In this case, the authors deduce the exact wave solutions Zayed and Alurrfi 395 or 17) and using ( 11), the left-hand side of Equation ( 17) becomes a polynomial in () Setting the coefficients of this polynomial to be zero, yields the following system of algebraic equations: On solving the above, the algebraic equations using the Maple 14, the authors have the following result: From ( 10), ( 16), ( 19) and ( 39), the authors deduce the following rational solution Example 2. Exact solutions of the nonlinear PDE (2) describing the nonlinear dynamics of radial dislocations in MTs In this subsection, the authors find the exact solutions of Equation (2).To this end, the authors use the transformation (4) to reduce Equation (2) into the following ODE: where  41), the authors get 1.N  Consequently, the authors have the formal solution of Equation ( 41) as follows: where 01 , AA and 1 B are constants to be determined later.Substituting ( 44) into (41) and using ( 7) -( 9), the lefthand side of Equation ( 41) becomes a polynomial in ()  and ()  .Setting the coefficients of this polynomial to be zero, yields the following system of algebraic equations: If the authors substitute 1   into the algebraic Equations ( 45) and solve them by Maple 14, the authors have the following results: Result 1.The authors have From ( 12), ( 13), ( 43), ( 44) and ( 46), the authors deduce that if 1 r  , then the authors have the exact wave solution 63 ( ) 1 tanh , 24 Note that our solution ( 47) is in agreement with the solution (43) obtained in Z d r a v k o v i c e t a l .
Result 2. The authors have where 0 r  .
In this case, the authors deduce that if 1 r  , then the authors have the exact wave solution where 0 r  .
In this case, the authors deduce that if 1 r  , then the authors have the exact wave solution Result 4. The authors have where 2 0 r  .
In this case, the authors deduce that if 1 r  , then the authors have the exact wave solution while if 1 r  , then the authors have the exact wave solution.
Finally, note that the case 1, 1, 0, rR      is rejected for example 2, because the authors have complex solutions for Equation (2).

PHYSICAL EXPLANATIONS OF SOME OBTAINED SOLUTIONS
Solitary waves can be obtained from each traveling wave solution by setting particular values to its unknown parameters.In this section, the authors have presented some graphs of solitary waves constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation.Using mathematical software Maple 14, three dimensional plots of some obtained exact traveling wave solutions have been shown in Figures 1 to 6.

The nonlinear PDE (1) describing the nonlinear dynamics of MTs as nanobioelectronics transmission lines
The obtained solutions for the nonlinear PDE (1) incorporate three types of explicit solutions namely, hyperbolic, trigonometric and rational.From these explicit results, it is easy to say that the solution ( 23) is a kink shaped soliton solution; the solution ( 24) is a singular kink shaped soliton solution; the solutions ( 26), ( 28) are bell-kink shaped soliton solution; the solution ( 29) is a singular bell-kink shaped soliton solution, the solutions (31), ( 32), ( 34), ( 35), ( 37), ( 38) are periodic solutions and the solution ( 40) is rational solution.The graphical representation of the solutions ( 23), ( 26), ( 34) and ( 38) can be plotted as shown in Figures 1 to 4.

The nonlinear PDE (2) describing the nonlinear dynamics of radial dislocations in MTs
The obtained solutions for the nonlinear PDE (2) are hyperbolic.From the obtained solutions for this equation, the authors observe that the solution (47) is a kink shaped soliton solution, the solution (48) is a singular kink shaped soliton solution, the solution (50), (54) are bell-kink shaped soliton solutions and the solutions (52), (55) are singular bell-kink shaped soliton solutions.The graphical representation of the solutions ( 52) and ( 54) can be plotted as shown in Figure 5 and 6.

Remark:
The authors have checked all our solutions with Maple 14 by putting them back into the original Equations ( 1) and (2).
. The plot of (52) when . The plot of (54) when

Conclusions
The generalized projective Riccati equations method was used in this paper to obtain some new exact solutions of the two nonlinear evolution Equations (1) and ( 2) which describe the model of MTs as nano-bioelectronics transmission lines and the dynamics of radial dislocations in MTs, respectively.On comparing our results in this paper with the well-known results obtained in Zekovic et al. (2014) and Zdravkovic et al. (2014), the authors deduce that their results are new and not published elsewhere except the result (47) which is in agreement with the result of (43) obtained in Zdravkovic et al. (2014).It is to be noted here that the obtained solutions are of type kink, soliton with singularities and periodic.Solitons are the solutions in the form sec h and 2 sec h , the graph of soliton is a wave that goes up only.It is not like periodic solutions sine, cosine, etc, as in trigonometric function, that goes above and below the horizontal.Kink is also called a soliton; it is in the form tanh not 2 tanh .