Advertisement

Journal of Nonlinear Science

, Volume 22, Issue 6, pp 1013–1040 | Cite as

Traveling Waves in Elastic Rods with Arbitrary Curvature and Torsion

  • M. J. Ablowitz
  • V. Barone
  • S. De Lillo
  • M. SommacalEmail author
Article
  • 365 Downloads

Abstract

The dynamic Kirchhoff equations, describing a thin elastic rod of infinite length, are considered in connection with the study of the conformations of polymeric chains. A novel special traveling wave solution that can be interpreted as a conformational soliton propagating at constant speed is obtained, featuring arbitrary non-constant curvature and torsion of the rod, in the simple case of constant cross-section, homogeneous density and elastic isotropy. This traveling wave corresponds to a specific constraint on the twist-to-bend ratio of the constant stiffness parameters, which in turn appears to be compatible with the experimental evidence for the mechanical properties of real polymeric chains. Due to such a constraint, the square of the velocity of the solitary wave is directly proportional to the bending stiffness and inversely proportional to the density and to the principal momentum of inertia of the rod. Several applications to the study of conformational changes in polymeric chains are given.

Keywords

Continuum mechanics Elastic rod Dynamic Kirchhoff equations Polymeric chains Conformational soliton Protein folding 

Mathematics Subject Classification (2010)

74Axx 74Bxx 35C08 82D60 92D20 

Notes

Acknowledgements

This research was partially supported by the National Science Foundation under grant DMS-0905779.

References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. National Bureau of Standards, Washington (1968) Google Scholar
  2. Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995) zbMATHGoogle Scholar
  3. Argeri, M., Barone, V., De Lillo, S., Lupo, G., Sommacal, M.: Elastic rods in life- and material-sciences: a general integrable model. Physica D 238(13), 1031–1049 (2009a) MathSciNetzbMATHCrossRefGoogle Scholar
  4. Argeri, M., Barone, V., De Lillo, S., Lupo, G., Sommacal, M.: Existence of energy minimums for thin elastic rods in static helical configurations. Theor. Math. Phys. 159(3), 698–711 (2009b) zbMATHCrossRefGoogle Scholar
  5. Balakrishnan, R., Dandoloff, R.: Effect of conformations on charge transport along a thin elastic tube. Nonlinearity 21, 1–7 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  6. Berloff, N.G.: Nonlinear dynamics of secondary protein folding. Phys. Lett. A 337(4–6), 391–396 (2005) zbMATHCrossRefGoogle Scholar
  7. Careri, G.: Search for cooperative phenomena in hydrogen-bonded amide structures. In: Haken, H., Wagner, M. (eds.) Cooperative Phenomena, pp. 391–394. Springer, New York (1973) CrossRefGoogle Scholar
  8. Caspi, S., Ben-Jacob, E.: Toy model studies of soliton-mediated protein folding and conformation changes. Europhys. Lett. 47, 522–527 (1999) CrossRefGoogle Scholar
  9. Caspi, S., Ben-Jacob, E.: Conformation changes and folding of proteins mediated by Davydov’s soliton. Phys. Lett. A 272, 124–129 (2000) CrossRefGoogle Scholar
  10. Chouaieb, N., Goriely, A., Maddocks, J.H.: Helices. Proc. Natl. Acad. Sci. USA 103(25), 9398–9403 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  11. Cruzeiro, L.: The Davydov/Scott model for energy storage and transport in proteins. J. Biol. Phys. 35, 43–55 (2009) CrossRefGoogle Scholar
  12. D’Amore, M., Talarico, G., Barone, V.: Periodic and high-temperature disordered conformations of polytetrafluoroethylene chains: an ab initio modeling. J. Am. Chem. Soc. 128, 1099–1108 (2006) CrossRefGoogle Scholar
  13. Davydov, A.S.: The theory of contraction of proteins under their excitation. J. Theor. Biol. 38, 559–569 (1973) CrossRefGoogle Scholar
  14. De Lillo, S., Lupo, G., Sommacal, M.: Helical configurations of elastic rods in the presence of a long-range interaction potential. J. Phys. A 43, 085214 (2010) (19 pp.) MathSciNetCrossRefGoogle Scholar
  15. Dichmann, D., Li, Y., Maddocks, J.H.: Hamiltonian formulations and symmetries in rod mechanics. In: Mathematical Approaches to Biomolecular Structure and Dynamics. IMA Vol. Math. Appl., vol. 82, pp. 71–113. Springer, New York (1996) CrossRefGoogle Scholar
  16. Ding, F., Dokholyan, N.V., Buldyrev, S.V., Stanley, H.E., Shakhnovich, E.I.: Molecular dynamics simulation of the SH3 domain aggregation suggests a generic amyloidogenesis mechanism. J. Mol. Biol. 324, 851–857 (2002) CrossRefGoogle Scholar
  17. Dupuis, D.E., Guilford, W.H., Wu, J., Warshaw, D.M.: Actin filament mechanics in the laser trap. J. Muscle Res. Cell Motil. 18, 17–30 (1997) CrossRefGoogle Scholar
  18. Edler, J., Hamm, P., Scott, A.C.: Femtosecond study of self-trapped vibrational excitons in crystalline acetanilide. Phys. Rev. Lett. 88, 067403 (2002) CrossRefGoogle Scholar
  19. Goriely, A., Nizette, M.: Kovalevskaya rods and Kovalevskaya waves. Regul. Chaotic Dyn. 5(1), 95–106 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  20. Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. J. Nonlinear Sci. 11, 3–45 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  21. Gu, W., Wang, T., Zhu, J., Shi, Y., Liu, H.: Molecular dynamics simulation of the unfolding of the human prion protein domain under low pH and high temperature conditions. Biophys. Chem. 104, 79–94 (2003) CrossRefGoogle Scholar
  22. Heimburg, T., Jackson, A.D.: On soliton propagation in biomembranes and nerves. Proc. Natl. Acad. Sci. USA 102, 9790–9795 (2005) CrossRefGoogle Scholar
  23. Isambert, H., Maggs, A.: Dynamics and rheology of actin solutions. Macromolecules 29, 1036–1040 (1996) CrossRefGoogle Scholar
  24. Janmey, P.A., Tang, J.X., Schmidt, C.F.: Actin filaments. Biophys. Textb. Online (BTOL) (1999). https://www.biophysics.org/Portals/1/PDFs/Education/janmey.pdf
  25. Kim, J.L., Burley, S.K.: 1.9 Å resolution refined structure of TBP recognizing the minor groove of TATAAAAG. Nat. Struct. Biol. 1, 638–651 (1994) CrossRefGoogle Scholar
  26. Kim, Y., Geiger, J.H., Hahn, S., Sigler, P.B.: Crystal structure of a yeast TBP/TATA-box complex. Nature 365, 512–527 (1993) CrossRefGoogle Scholar
  27. Kirchhoff, G.: Vorlesungen über Mathematische Physik: Mechanik, 3rd edn. Teubner, Leipzig (1883) Google Scholar
  28. Klug, A.: Transcription. Opening the gateway. Nature 365, 486–487 (1993) CrossRefGoogle Scholar
  29. Landau, L.D., Lifshitz, E.M., Kosevich, A.M., Pitaevskii, L.P.: Theory of Elasticity. Pergamon Press, Oxford (1986) Google Scholar
  30. Langella, E., Improta, R., Barone, V.: Checking the pH-induced conformational transition of prion protein by molecular dynamics simulations: effect of protonation of histidine residues. Biophys. J. 87, 3623–3632 (2004) CrossRefGoogle Scholar
  31. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New York (1926) Google Scholar
  32. Malolepsza, E., Boniecki, M., Kolinski, A., Piela, L.: Theoretical model of prion propagation: a misfolded protein induces misfolding. Proc. Natl. Acad. Sci. USA 102(22), 7835–7840 (2005) CrossRefGoogle Scholar
  33. Peyrard, M. (ed.): Nonlinear Excitations in Biomolecules. In: Proceedings of Les Houches Workshop. Springer, New York (1995) Google Scholar
  34. Sanghani, S.R., Zakrzewska, K., Harvey, S.C., Lavery, R.: Molecular modelling of (A4T4NN)n and (T4A4NN)n: sequence elements responsible for curvature. Nucleic Acids Res. 24, 1632–1637 (1996) CrossRefGoogle Scholar
  35. Sekijima, M., Motono, C., Yamasaki, S., Kaneko, K., Akiyama, Y.: Molecular dynamics simulation of dimeric and monomeric forms of human prion protein: insight into dynamics and properties. Biophys. J. 85, 1176–1185 (2003) CrossRefGoogle Scholar
  36. Stember, J.N., Wriggers, W.: Bend-twist-stretch model for coarse elastic network simulation of biomolecular motion. J. Chem. Phys. 131, 074112 (2009) CrossRefGoogle Scholar
  37. Tretiak, S., Saxena, A., Martin, R.L., Bishop, A.R.: Photoexcited breathers in conjugated polyenes: an excited-state molecular dynamics study. Proc. Natl. Acad. Sci. USA 100, 2185–2190 (2003) CrossRefGoogle Scholar
  38. Urbanc, B., Cruz, L., Ding, F., Sammond, D., Khare, S., Buldyrev, S.V., Stanley, H.E., Dokholyan, N.V.: Molecular dynamics simulation of amyloid beta dimer formation. Biophys. J. 87, 2310–2321 (2004) CrossRefGoogle Scholar
  39. Volkov, S.N.: Propagation of local conformation transitions in molecular chains. Phys. Lett. A 136, 41–44 (1989) CrossRefGoogle Scholar
  40. Volkov, S.N., Kosevich, A.M.: Conformation oscillations of DNA. Mol. Biol. (Mosk.) 21(3), 797–806 (1987) Google Scholar
  41. Yasuda, R., Miyata, H., Kinosita, K. Jr.: Direct measurement of the torsional rigidity of single actin filaments. J. Mol. Biol. 263, 227–236 (1996) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • M. J. Ablowitz
    • 1
  • V. Barone
    • 2
    • 3
  • S. De Lillo
    • 4
    • 5
  • M. Sommacal
    • 6
    Email author
  1. 1.Applied Mathematics DepartmentUniversity of ColoradoBoulderUSA
  2. 2.Scuola Normale SuperiorePisaItaly
  3. 3.Istituto Nazionale di Fisica NucleareSezione di PisaPisaItaly
  4. 4.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  5. 5.Istituto Nazionale di Fisica NucleareSezione di PerugiaPerugiaItaly
  6. 6.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance

Personalised recommendations