Nonlinear Dynamics

, Volume 88, Issue 3, pp 2013–2033 | Cite as

Localized discrete breather modes in neuronal microtubules

  • L. KavithaEmail author
  • E. Parasuraman
  • A. Muniyappan
  • D. Gopi
  • S. Zdravković
Original Paper


We made an attempt to provide a realistic picture of the localization of energy in microtubules (MTs), and we intend to model the nonlinear dynamics of MTs using the “double-well” \(\phi ^4\) form of the potential describing the dipole–dipole interactions. We investigate the modulational instability (MI) of the nonlinear plane wave solutions by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. A set of explicit criteria of MI is derived, and under the plane-wave perturbation, the constant amplitude solution becomes unstable and localized discrete breathers (DBs) solutions appear. We show numerically that MI is also an indicator of the presence of discrete breathers. We suggest that an electric field favourably leads the DB excitations towards the properly aligned end triggering a dissembly of the protofilament due to the energy release. These DBs could catalyse MT-associated proteins attachment/detachment and promote or inhibit the kinesin walk. We establish that the electromechanical vibrations in MTs can generate an electromagnetic field in the form of an electric pulse (breathers) which propagates along MT serving as signalling pathway in neuronal cells. The DBs in MT can be viewed as a bit of information whose propagation can be controlled by an electric filed. They might perform the role of elementary logic gates, thus implementing a subneuronal mode of computation. The generated DBs present us with novel possibilities for the direct interaction between the local electromagnetic field and the cytoskeletal structures in neurons. Thus, we emphasize that the effect of discreteness and electric field plays a significant role in MTs.


Microtubules Cytoskeleton Soliton Symbolic computation (computer algebra) 



L.K gratefully acknowledges the financial support by NBHM (2/48(9)/2011/-R and DII/1223), India, in the form of a major research project; DAE-BRNS (2009/20/37/7/BRNS/1819), India, in the form of Young Scientist Research Award, and ICTP, Italy, in the form of Regular Associateship. E.P. gratefully acknowledges Periyar University for providing the University Research Fellowship. A. M. gratefully acknowledges UGC for the Rajiv Gandhi National Fellowship. This research work was partially supported by Serbian Ministry of Education and Sciences (Grant III45010).


  1. 1.
    Hyams, J.S., Lloyd, C.W.: Microtubules. Wiley, New York (1994)Google Scholar
  2. 2.
    Kirschner, M., Mitchison, T.: Beyond self-assembly: from microtubules to morphogenesis. Cell 45, 329–342 (1986)CrossRefGoogle Scholar
  3. 3.
    Mitchison, T., Kirschner, M.: Dynamic instability of microtubule growth. Nature 312, 237–242 (1984)CrossRefGoogle Scholar
  4. 4.
    Lodish, H., Berk, A., Zipuski, S.L., Matsudaira, P., Baltimore, D., Darnell, J.: Molecular Cell Biology, 4th edn. Freeman, San Fransisco (2000)Google Scholar
  5. 5.
    Ouakad, H.M., Younis, M.I.: Dynamic response of slacked single-walled carbon nanotube resonators. Nonlinear Dyn. 67, 1419–1436 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Oosawa, F., Asakura, S.: Thermodynamic of the Polymerization of Protein. Academic Press, London (1975)Google Scholar
  7. 7.
    Johnson, K.A., Borisy, G.G.: Thermodynamic analysis of microtubule self-assembly in vitro. Mol. Biol. 133, 199–216 (1979)CrossRefGoogle Scholar
  8. 8.
    Desai, A., Mitchison, T.J.: Microtubule polymerization dynamics. Annu. Rev. Cell Dev. Biol. 13, 83–117 (1997)CrossRefGoogle Scholar
  9. 9.
    Haghshenas-Jaryani, M., Black, B., Ghaffari, S., Drake, J., Bowling, A., Mohanty, S.: Dynamics of microscopic objects in optical tweezers: experimental determination of underdamped regime and numerical simulation using multiscale analysis. Nonlinear Dyn. 76, 1013–1030 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Holy, T.E., Leibler, S.: Dynamic instability of microtubules as an efficient way to search in space. Proc. Natl. Acad. Sci. USA 91, 5682–5685 (1994)CrossRefGoogle Scholar
  11. 11.
    Wittmann, T., Waterman-Storer, C.M.: Cell motility: can Rho GTPases and microtubules point the way? J. Cell Sci. 114, 3795–3803 (2001)Google Scholar
  12. 12.
    Horio, T., Hotani, H.: Visualization of the dynamic instability of individual microtubules by dark-field microscopy. Nature 321, 605–607 (1986)CrossRefGoogle Scholar
  13. 13.
    Komarova, Y.A., Vorobjev, I.A., Borisy, G.G.: Life cycle of MTs: persistent growth in the cell interior, asymmetric transition frequencies and effects of the cell boundary. J. Cell Sci. 115, 3527–3539 (2002)Google Scholar
  14. 14.
    Drummond, D.R., Cross, R.A.: Dynamics of interphase microtubules in \(<\) i \(>\) Schizosaccharomyces pombe \(<\) /i \(>\). Curr. Biol. 10, 766–775 (2000)CrossRefGoogle Scholar
  15. 15.
    Straube, A.: How to measure microtubule dynamics? Mol. Biol. 777, 1–14 (2011)Google Scholar
  16. 16.
    van der Vaart, B., Akhmanova, A., Straube, A.: Regulation of microtubule dynamic instability. Biochem. Soc. Trans. 37, 1007–1013 (2009)CrossRefGoogle Scholar
  17. 17.
    Pokorneý, J.: Excitation of vibrations in microtubules in living cells. Bioelectrochemistry 63, 321–326 (2004)CrossRefGoogle Scholar
  18. 18.
    Pokorný, J.: Conditions for coherent vibrations in the cytoskeleton. Bioelectrochem. Bioenerg. 48, 267–271 (1999)CrossRefGoogle Scholar
  19. 19.
    Whitham, G.B.: Non-linear dispersive waves. Proc. R. Soc. Lond. A 283, 238–261 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, San Diego (1989)zbMATHGoogle Scholar
  21. 21.
    Battelli, F., Diblík, J., Fe\(\breve{{\rm c}}\)kan, M., Pickton, J., Pospí\(\breve{{\rm s}}\)il, M., Susanto, H.: Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss. Nonlinear Dyn. 81, 353–371 (2015)Google Scholar
  22. 22.
    Hendricks, A.G., Epureanu, B.I., Meyhfer, E.: Mechanistic mathematical model of kinesin under time and space fluctuating loads. Nonlinear Dyn. 53, 303–320 (2008)CrossRefzbMATHGoogle Scholar
  23. 23.
    Ansari, R., Ramezannezhad, H., Gholami, R.: Nonlocal beam theory for nonlinear vibrations of embedded multiwalled carbon nanotubes in thermal environment. Nonlinear Dyn. 67, 2241–2254 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chretien, D., Fuller, S.D., Karsenti, E.: Structure of growing microtubule ends: two-dimensional sheets close into tubes at variable rates. J. Cell. Biol. 129, 1311–1328 (1995)CrossRefGoogle Scholar
  25. 25.
    Glauber, R.: Coherent and incoherent states of the radiation field. J Phys. Rev. 131, 2766 (1963)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Daniel, M., Kavitha, L., Amuda, R.: Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole–dipole interaction. Phys. Rev. B 59, 13774 (1999)CrossRefGoogle Scholar
  27. 27.
    Cuevas, J., Archilla, J.F.R., Romero, F.R.: Stability of non-time-reversible phonobreathers. J. Phys. A Math. Theor. 44, 035102 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Campbell, D.K., Flach, S., Kivshar, Y.S.: Localizing energy through nonlinearity and discreteness. Phys. Today 57, 43 (2004)CrossRefGoogle Scholar
  29. 29.
    Yi, X., Wattis, J.A.D., Susanto, H., Cummings, L.J.: Discrete breathers in a two-dimensional spring-mass lattice. J. Phys. A Math. Theor. 42, 355207 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kosevich, A.M., Kovalev, A.S.: Self-localization of vibrations in a one-dimensional anharmonic chain. Zh. Eksp. Teor. Fiz. 67, 1793–1804 (1974)Google Scholar
  31. 31.
    Sievers, A.J., Takeno, S.: Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970 (1988)CrossRefGoogle Scholar
  32. 32.
    MacKay, R.S., Aubry, S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Daumont, I., Dauxois, T., Peyrard, M.: Modulational instability: first step towards energy localization in nonlinear lattices. Nonlinearity 10, 617–630 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Dauxois, T., Peyrard, M.: Energy localization in nonlinear lattices. Phys. Rev. Lett. 70, 3935 (1993)CrossRefGoogle Scholar
  35. 35.
    Kavitha, L., Muniyappan, A., Prabhu, A., Zdravković, S., Jayanthi, S., Gopi, D.: Nano breathers and molecular dynamics simulations in hydrogen-bonded chains. J. Biol. Phys. 39, 15–35 (2013)CrossRefGoogle Scholar
  36. 36.
    Kivshar, Y.S., Peyrard, M.: Modulational instabilities in discrete lattices. Phys. Rev. A 46, 3198 (1992)CrossRefGoogle Scholar
  37. 37.
    Ablowitz, M.J., Clarkson, P.A.: Solitons: nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  38. 38.
    Miura, M.R.: B\({\ddot{{\rm a}}}\)cklund Transformation. Springer, Berlin (1978)Google Scholar
  39. 39.
    Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Soliton Theory and Its Geometric Applications. Shanghai Scientific and Technical Publishers, Shanghai (1999)Google Scholar
  40. 40.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095 (1967)CrossRefzbMATHGoogle Scholar
  41. 41.
    Kavitha, L., Saravanan, M., Akila, N., Bhuvaneswari, S., Gopi, D.: Solitonic transport of energy–momentum in a deformed magnetic medium. Phys. Scr. 85, 035007 (2012)CrossRefzbMATHGoogle Scholar
  42. 42.
    Kavitha, L., Sathishkumar, P., Gopi, D.: Soliton-based logic gates using spin ladder. Commun. Nonlinear Sci. Numer. Simul. 15, 3900–3912 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192 (1971)CrossRefzbMATHGoogle Scholar
  44. 44.
    Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Inc, M., Fan, E.G.: Extended tanh-function method for finding travelling wave solutions of some nonlinear partial differential equations. Z. Naturforsch. 60, 7 (2005)Google Scholar
  46. 46.
    Kavitha, L., Akila, N., Prabhu, A., Kuzmanovska-Barandovska, O., Gopi, D.: Exact solitary solutions of an inhomogeneous modified nonlinear Schr\({\ddot{o}}\)dingerequation with competing nonlinearities. Math. Comput. Model 53, 1095–1110 (2011)CrossRefzbMATHGoogle Scholar
  47. 47.
    Kavitha, L., Srividya, B., Gopi, D.: Effect of nonlinear inhomogeneity on the creation and annihilation of magnetic soliton. J. Magn. Magn. Mater 322, 1793–1810 (2010)CrossRefGoogle Scholar
  48. 48.
    Kavitha, L., Srividya, B., Akila, N., Gopi, D.: Shape changing solitary solutions of a nonlocally damped nonlinear Schr\({\ddot{o}}\)dinger equation using symbolic computation. Nonlinear Sci. Lett. A 1, 95–107 (2010)Google Scholar
  49. 49.
    Kavitha, L., Sathishkumar, P., Gopi, D.: Energy-momentum transport through soliton in a site-dependent ferromagnet. Commun. Nonlinear Sci. Numer. Simul. 16, 1787–1803 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Kavitha, L., Venkatesh, M., Jayanthi, S., Gopi, D.: Propagation of proton solitons in hydrogen-bonded chains with an asymmetric double-well potential. Phys. Scr. 86, 025403 (2012)CrossRefGoogle Scholar
  51. 51.
    Zdravković, S., Kavitha, L., Satarić, M.V., Zeković, S., Petrović, J.: Modified extended tanh-function method and nonlinear dynamics of microtubules. Chaos Solitons Fractals 45, 1378–1386 (2012)CrossRefzbMATHGoogle Scholar
  52. 52.
    Kavitha, L., Prabhu, A., Gopi, D.: New exact shape changing solitary solutions of a generalized Hirota equation with nonlinear inhomogeneities. Chaos Solitons Fractals 42, 2322 (2009)CrossRefzbMATHGoogle Scholar
  53. 53.
    Kavitha, L., Sathishkumar, P., Gopi, D.: Shape changing soliton in a site-dependent ferromagnet using tanh-function method. Phys. Scr. 79, 015402 (2009)CrossRefzbMATHGoogle Scholar
  54. 54.
    Dai, C., Ni, Y.: The application of extended tanh-function approach in Toda lattice equations. Int. J. Theor. Phys. 46, 1455–1465 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Yan, Z.Y.: Abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional integrable Davey–Stewartson-type equation via a new method. Chaos Solitons Fractals 18, 299–309 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Kavitha, L., Saravanan, M., Srividya, B., Gopi, D.: Breatherlike electromagnetic wave propagation in an antiferromagnetic medium with Dzyaloshinsky–Moriya interaction. Phys. Rev. E 84, 066608 (2011)CrossRefGoogle Scholar
  57. 57.
    Kavitha, L., Parasuraman, E., Venkatesh, M., Mohamadou, A., Gopi, D.: Breather-like protonic tunneling in a discrete hydrogen bonded chain with heavy-ionic interactions. Phys. Scr. 87, 035007 (2013)CrossRefzbMATHGoogle Scholar
  58. 58.
    Zeković, S., Muniyappan, A., Zdravković, S., Kavitha, L.: Employment of Jacobian elliptic functions for solving problems in nonlinear dynamics of microtubules. Chin. Phys. B 23, 020504 (2014)CrossRefGoogle Scholar
  59. 59.
    He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    He, J.H., Abdou, M.A.: New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos Solitons Fractals 34, 1421–1429 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wu, X.H., He, J.H.: Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method. Comput. Math. Appl. 54, 966–986 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Bekir, A., Boz, A.: Exact solutions for a class of nonlinear partial differential equations using Exp-function method. Int. J. Nonlinear Sci. Numer. Simul. 8, 505 (2007)CrossRefGoogle Scholar
  63. 63.
    Kavitha, L., Srividya, B., Gopi, D.: Exact propagating dromion-like localized wave solutions of generalized (2 + 1)-dimensional Davey–Stewartson equations. Comput. Math. Appl. 62, 4691 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Nonlinear Mech. 31, 329–338 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Gao, Y.T., Tian, B.: Generalized tanh method with symbolic computation and generalized shallow water wave equation. Comput. Math. Appl. 33, 115–118 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Kuang, J.L., Leung, A.Y.T.: Chaotic flexural oscillations of a spinning nanoresonator. Nonlinear Dyn. 51, 9–29 (2008)CrossRefzbMATHGoogle Scholar
  68. 68.
    Elwakil, S.A., El-labany, S.K., Zahran, M.A., Sabry, R.: Modified extended tanh-function method for solving nonlinear partial differential equations. Phys. Lett. A 299, 179–188 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    L\({\ddot{u}}\), Z.S., Zhang, H.Q, : On a new modified extended tan\(h\)-function method. Commun. Theor. Phys. 39, 405–408 (2003)Google Scholar
  70. 70.
    Dai, C.Q., Zhang, J.F.: Exact solutions of discrete complex cubicquintic Ginzburg–Landau equation with non-local quintic term. Commun. Theor. Phys. 46, 23 (2006)CrossRefGoogle Scholar
  71. 71.
    Wang, Z., Zhang, H.Q.: Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation. Chaos Solitons Fractals 31, 197–204 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Wang, Z.: Discrete tanh method for nonlinear difference-differential equations. Comput. Phys. Commun. 180, 1104–1108 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Dai, Z., Liu, J., Liu, Z.: Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 15, 2331–2336 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Khawaja, U.Al, Bahlouli, H., Asad-uz-zaman, M., Al-Marzoug, S.M.: Modulational instability analysis of the Peregrine soliton. Commun. Nonlinear Sci. Numer. Simul. 19, 2706–2714 (2014)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Biswas, A., Milovic, D.: Bright and dark solitons of the generalized nonlinear Schr\({\ddot{o}}\)dinger equation. Commun. Nonlinear Sci. Numer. Simul. 15, 1473–1484 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Purich, D.L.: Enzyme catalysis: a new definition accounting for noncovalent substrate and product-like states. Trends Biochem. Sci. 26, 417–421 (2001)CrossRefGoogle Scholar
  77. 77.
    Witte, H., Neukirchen, D., Bradke, F.: Microtubule stabilization specifies initial neuronal polarization. J. Cell Biol. 180, 619–632 (2008)CrossRefGoogle Scholar
  78. 78.
    Wang, P., Zhang, Y., Lü, J., Yu, X.: Functional characteristics of additional positive feedback in genetic circuits. Nonlinear Dyn. 79, 397–408 (2015)CrossRefGoogle Scholar
  79. 79.
    Hirokawa, N., Takemura, R.: Molecular motors and mechanisms of directional transport in neurons. Nat. Rev. Neurosci. 6, 201–214 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  • L. Kavitha
    • 1
    • 2
    Email author
  • E. Parasuraman
    • 3
  • A. Muniyappan
    • 3
  • D. Gopi
    • 4
  • S. Zdravković
    • 5
  1. 1.Department of Physics, School of Basic and Applied SciencesCentral University of Tamilnadu (CUTN)ThiruvarurIndia
  2. 2.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Department of PhysicsPeriyar UniversitySalemIndia
  4. 4.Department of ChemistryPeriyar UniversitySalemIndia
  5. 5.Institut za nuklearne nauke VinčaUniverzitet u BeograduBeogradSerbia

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