Nonlinear Dynamics

, Volume 92, Issue 2, pp 479–486 | Cite as

General model of microtubules

  • Slobodan Zdravković
  • Miljko V. Satarić
  • Vladimir Sivčević
Original Paper

Abstract

In the present work, we deal with nonlinear dynamics of microtubules. A new model, describing nonlinear dynamics of microtubules, is introduced. Its advantages over two existing models are demonstrated. We show that dynamics of microtubules can be explained in terms of kink solitons. Also, circumstances yielding to either subsonic or supersonic solitons are discussed.

Keywords

Microtubules Kink solitons General model 

Notes

Acknowledgements

This work was supported by funds from Serbian Ministry of Education, Sciences and Technological Development (Grants No. III45010), Project of AP Vojvodina No. 114-451-2708/2016-03 and Serbian Academy of Sciences and Arts.

References

  1. 1.
    Amos, L.A.: Focusing—in on microtubules. Curr. Opin. Struct. Biol. 10, 236–241 (2000)CrossRefGoogle Scholar
  2. 2.
    Satarić, M.V., Tuszyński, J.A., Žakula, R.B.: Kinklike excitations as an energy-transfer mechanism in microtubules. Phys. Rev. E 48, 589–597 (1993)CrossRefGoogle Scholar
  3. 3.
    Zdravković, S., Satarić, M.V., Zeković, S.: Nonlinear dynamics of microtubules—a longitudinal model. Europhys. Lett. 102, 38002 (2013)CrossRefGoogle Scholar
  4. 4.
    Satarić, M.V., Tuszyński, J.A.: Relationship between the nonlinear ferroelectric and liquid crystal models for microtubules. Phys. Rev. E 67, 011901 (2003)CrossRefGoogle Scholar
  5. 5.
    Krebs, A., Goldie, K.N., Hoenger, A.: Complex formation with kinesin motor domains affects the structure of microtubules. J. Mol. Biol. 335, 139–153 (2004)CrossRefGoogle Scholar
  6. 6.
    Zdravković, S., Kavitha, L., Satarić, M.V., Zeković, S., et al.: Modified extended tanh-function method and nonlinear dynamics of microtubules. Chaos Solitons Fract. 45, 1378–1386 (2012)CrossRefMATHGoogle Scholar
  7. 7.
    Zdravković, S., Zeković, S., Bugay, A.N., Satarić, M.V.: Localized modulated waves and longitudinal model of microtubules. Appl. Math. Comput. 285, 248–259 (2016)MathSciNetGoogle Scholar
  8. 8.
    Zdravković, S., Satarić, M.V., Maluckov, A., Balaž, A.: A nonlinear model of the dynamics of radial dislocations in microtubules. Appl. Math. Comput. 237, 227–237 (2014)MathSciNetMATHGoogle Scholar
  9. 9.
    Drabik, P., Gusarov, S., Kovalenko, A.: Microtubule stability studied by three-dimensional molecular theory of solvation. Biophys. J. 92, 394–403 (2007)CrossRefGoogle Scholar
  10. 10.
    Nogales, E., Whittaker, M., Milligan, R.A., Downing, K.H.: High-resolution model of the microtubule. Cell 96, 79–88 (1999)CrossRefGoogle Scholar
  11. 11.
    Ali, A.H.A.: The modified extended tanh-function method for solving coupled MKdV and coupled Hirota–Satsuma coupled KdV equations. Phys. Lett. A 363, 420–425 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    El-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons Fract. 31, 840–852 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kavitha, L., Akila, N., Prabhu, A., Kuzmanovska-Barandovska, O., et al.: Exact solitary solutions of an inhomogeneous modified nonlinear Schrödinger equation with competing nonlinearities. Math. Comput. Model. 53, 1095–1110 (2011)CrossRefMATHGoogle Scholar
  14. 14.
    Sekulić, D.L., Satarić, M.V., Živanov, M.B.: Symbolic computation of some new nonlinear partial differential equations of nanobiosciences using modified extended tanh-function method. Appl. Math. Comput. 218, 3499–3506 (2011)MathSciNetMATHGoogle Scholar
  15. 15.
    Pokorný, J., Jelínek, J., Trkal, V., Lamprecht, I., et al.: Vibrations in microtubules. J. Biol. Phys. 23, 171–179 (1997)CrossRefGoogle Scholar
  16. 16.
    Havelka, D., Cifra, M., Kučera, O., Pokorný, J., et al.: High-frequency electric field and radiation characteristics of cellular microtubule network. Theor. Biol. 286, 31–40 (2011)CrossRefGoogle Scholar
  17. 17.
    Schoutens, J.E.: Dipole–dipole interactions in microtubules. J. Biol. Phys. 31, 35–55 (2005)CrossRefGoogle Scholar
  18. 18.
    Tuszyński, J.A., Brown, J.A., Crawford, E., Carpenter, E.J., et al.: Molecular dynamics simulations of tubulin structure and calculations of electrostatic properties of microtubules. Math. Comput. Model. 41, 1055–1070 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Cifra, M., Pokorný, J., Havelka, D., Kučera, O.: Electric field generated by axial longitudinal vibration modes of microtubule. BioSystems 100, 122–131 (2010)CrossRefGoogle Scholar
  20. 20.
    Hunyadi, V., Chretien, D., Flyvbjerg, H., Janosi, I.M.: Why is the microtubule lattice helical? Biol. Cell. 99, 117–128 (2007)CrossRefGoogle Scholar
  21. 21.
    Sekulić, D.L., Satarić, B.M., Zdravković, S., Bugay, A.N., et al.: Nonlinear dynamics of C-terminal tails in cellular microtubules. Chaos 26, 073119 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Guigas, G., Kalla, C., Weiss, M.: Probing the nanoscale viscoelasticity of intracellular fluids in living cells. Biophys. J. 93, 316–323 (2007)CrossRefGoogle Scholar
  23. 23.
    Goychuk, I.: Fractional-time random walk subdiffusion and anomalous transport with finite mean residence times; faster, not slower. Phys. Rev. E 86, 021113 (2012)CrossRefGoogle Scholar
  24. 24.
    Sekulić, D.L., Satarić, B.M., Tuszyński, J.A., Satarić, M.V.: Nonlinear ionic pulses along microtubules. Eur. Phys. J. E 34, 49–59 (2011)CrossRefGoogle Scholar
  25. 25.
    Zdravković, S., Maluckov, A., Đekić, M., Kuzmanović, S., et al.: Are microtubules discrete or continuum systems? Appl. Math. Comput. 242, 353–360 (2014)MathSciNetMATHGoogle Scholar
  26. 26.
    Zdravković, S., Bugay, A.N., Aru, G.F., Maluckov, A.: Localized modulated waves in microtubules. Chaos 24, 023139 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zdravković, S., Bugay, A.N., Parkhomenko, AYu.: Application of Morse potential in nonlinear dynamics of microtubules. Nonlinear Dyn. 90, 2841–2849 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kudryashov, N.A.: Exact solitary waves of the Fisher equation. Phys. Lett. A 342, 99–106 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fract. 24, 1217 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kudryashov, N.A., Loguinova, N.B.: Extended simplest equation method for nonlinear differential equations. Appl. Math. Comput. 205, 396–402 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut za Nuklearne Nauke VinčaUniverzitet u BeograduBeogradSerbia
  2. 2.Fakultet Tehničkih NaukaUniverzitet u Novom SaduNovi SadSerbia
  3. 3.Fizički FakultetUniverzitet u BeograduBeogradSerbia

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