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On a Class of Nonlinear Waves in Microtubules

Author

Listed:
  • Nikolay K. Vitanov

    (Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria)

  • Alexandr Bugay

    (Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia)

  • Nikolay Ustinov

    (Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia)

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.

Suggested Citation

  • Nikolay K. Vitanov & Alexandr Bugay & Nikolay Ustinov, 2024. "On a Class of Nonlinear Waves in Microtubules," Mathematics, MDPI, vol. 12(22), pages 1-23, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3578-:d:1521995
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    References listed on IDEAS

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    1. Zdravković, S. & Zeković, S. & Bugay, A.N. & Petrović, J., 2021. "Two component model of microtubules and continuum approximation," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Zdravković, Slobodan & Kavitha, Louis & Satarić, Miljko V. & Zeković, Slobodan & Petrović, Jovana, 2012. "Modified extended tanh-function method and nonlinear dynamics of microtubules," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1378-1386.
    3. Vitanov, Nikolay K. & Dimitrova, Zlatinka I. & Vitanov, Kaloyan N., 2015. "Modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations: further development of the methodology with applications," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 363-378.
    4. Vladimir V. Varlamov, 1999. "Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 22, pages 1-15, January.
    5. Zdravković, Slobodan & Zeković, Slobodan & Bugay, Aleksandr N. & Satarić, Miljko V., 2016. "Localized modulated waves and longitudinal model of microtubules," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 248-259.
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