We study the nonlinear dynamics of a microtubule (MT) assuming that its structure is known [1,2]. It suffices now to mention that MT is a long whole cylinder whose surface is formed of, usually, 13 long structures called protofilaments (PFs), representing a series of 8nm long dimers. A key point is that the dimer is an electric dipole, which means that MT behaves as ferroelectric [3]. We rely on the so-called general model of MTs and start with Hamiltonian [4] \begin{equation*} H=\sum_n \left[\frac{I}{2}\dot{\varphi}_n^2+\frac{k}{2}(\varphi_{n+1}-\varphi_n)^2+W(\varphi_n)-pE\cos{\varphi_n}\right]. \end{equation*} The angle $\varphi_n$ describes the dimer`s oscillation and $n$ is its position. The term $W(\varphi)$, called $W$-potential, represents the interaction of a single dimer with all other ones that do not belong to the same PF. We study both symmetrical and non-symmetrical $W$-potentials through the following three cases:
Case 1:
$W_1=-\frac{A}{2}\varphi_n^2+\frac{B}{4}\varphi_n^4$, $A>0$, $B>0$
Case 2:
$W_2=-\frac{A}{2}\varphi_n^2+\frac{B}{4}\varphi_n^4-C\varphi_n $, $A>0$, $B>0$, $C>0$
Case 3:
$W_3=-\frac{A}{2}\varphi_n^2+\frac{B}{4}\varphi_n^4-D\varphi_n^3$, $A>0$, $B>0$, $D>0$
We use a continuum approximation $\varphi_n(t)\Rightarrow\varphi(x,t)$, look for the travelling wave solutions, and obtain appropriate dynamical equations of motion whose solutions are kink and antikink solitons, according to the tangent hyperbolic function method [5]. An advantage of the cases 2 and 3 is demonstrated. Also, we suggest that this issue should be studied within a new two-component model of MT [6,7].
References
[1] S. Zdravković, J. Serb. Chem. Soc. 82 (5) (2017) 469.
[2] S. Zdravković, Mechanical Models of Microtubules. In Complexity in Biological and Physical Systems, Ed. by Ricardo Lopez-Ruiz (IntechOpen, 2018) Chapter 1.
[3] M.V. Satarić, J.A. Tuszynski, R.B. Žakula, Phys. Rev. E 48 (1993) 589.
[4] S. Zdravković, M.V. Satarić, V. Sivčević, Nonlinear Dyn. 92 (2018) 479.
[5] S.A. El-Wakil, M.A. Abdou, Chaos Soliton Fract. 31 (2007) 840.
[6] S. Zdravković, Proceedings of the 1st Conference on Nonlinearity, Chapter 5, Editors: B. Dragovich, Z. Cupic (Serbian Academy of Nonlinear Sciences, Belgrade, 2020, Serbia, ISBN: 978-86-905633-4-0).
[7] S. Zdravković, S. Zeković, A.N. Bugay, J. Petrović, Chaos Soliton Fract. 152 (2021) 111352.