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Geometry and Mechanics of Uniform n-Plies: from Engineering Ropes to Biological Filaments

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Abstract

We study the mechanics of uniform n-plies, correcting and extending previous work in the literature. An n-ply is the structure formed when n pretwisted strands coil around one another in helical fashion. Such structures are encountered widely in engineering (mooring ropes, power lines) and biology (DNA, proteins). We first show that the well-known lock-up phenomenon for n=2, described by a pitchfork bifurcation, gets unfolded for higher n. Geometrically, n-plies with n>2 are all found to behave qualitatively the same. Next, using elastic rod theory, we consider the mechanics of n-plies, allowing for axial end forces and end moments while ignoring friction. An exact expression for the interstrand pressure force is derived, which is used to investigate the onset of strand separation in plied structures. After defining suitable displacements we also give an alternative variational formulation and derive (nonlinear) constitutive relationships for torsion and extension (including their coupling) of the overall ply. For a realistic loading problem in which the ends are not free to rotate one needs to consider the topological conservation law, and we show how the concepts of link and writhe can be extended to n-plies.

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References

  1. J.F. Allemand, D. Bensimon, R. Lavery and V. Croquette, Stretched and overwound DNA forms a Pauling-like structure with exposed bases. Proc. National Acad. Sci. USA 95 (1998) 14152–14157.

    Article  ADS  Google Scholar 

  2. S.S. Antman, Nonlinear Problems of Elasticity. Springer, New York (1995).

    MATH  Google Scholar 

  3. U. Bockelmann, B. Essevaz-Roulet and F. Heslot, DNA strand separation studied by single molecule force measurements. Phys. Rev. E 58 (1998) 2386–2394.

    Article  ADS  Google Scholar 

  4. G. Călugăreanu, Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11 (1961) 588–625.

    MathSciNet  Google Scholar 

  5. A. Cardou and C. Jolicoeur, Mechanical models of helical strands. ASME Appl. Mech. Rev. 50 (1997) 1–14.

    Google Scholar 

  6. C.R. Chaplin, Torsional failure of a wire rope mooring line during installation in deep water. Eng. Failure Anal. 6 (1998) 67–82.

    Article  Google Scholar 

  7. B.D. Coleman and D. Swigon, Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elasticity 60 (2000) 173–221.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.A. Costello, Theory of Wire Rope, 2nd edn. Springer, New York (1997).

    Google Scholar 

  9. E.H. Dill, Kirchhoff's theory of rods. Arch. Hist. Exact Sci. 44 (1992) 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Essevaz-Roulet, U. Bockelmann and F. Heslot,Mechanical separation of the complementary strands of DNA. Proc. National Acad. Sci. USA 94 (1997) 11935–11940.

    Article  ADS  Google Scholar 

  11. B. Fain, J. Rudnick and S. Östlund, Conformations of linear DNA. Phys. Rev. E 55 (1997) 7364–7368.

    Article  ADS  Google Scholar 

  12. W.B. Fraser and D.M. Stump, The equilibrium of the convergence point in two-strand yarn plying. Internat. J. Solids Struct. 35 (1998) 285–298.

    Article  MATH  Google Scholar 

  13. W.B. Fraser and D.M. Stump, Twist in balanced-ply structures. J. Text. Inst. 89 (1998) 485–497.

    Article  Google Scholar 

  14. F.B. Fuller, The writhing number of a space curve. Proc. National Acad. Sci. USA 68 (1971) 815–819.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. F.B. Fuller, Decomposition of the linking of a closed ribbon: a problem from molecular biology. Proc. National Acad. Sci. USA 75 (1978) 3557–3561.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. D.E. Gilbert and J. Feigon, Multistranded DNA structures. Curr. Opin. Struct. Biol. 9 (1999) 305–314.

    Article  Google Scholar 

  17. O. Gonzalez and J.H. Maddocks, Global curvature, thickness, and the ideal shapes of knots. Proc. National Acad. Sci. USA 96 (1999) 4769–4773.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. O. Gonzalez, J.H. Maddocks and J. Smutny, Curves, circles, and spheres. Contemporary Math. 304 (2002) 195–215.

    MATH  MathSciNet  Google Scholar 

  19. J.W.S. Hearle and A.E. Yegin, The snarling of highly twisted monofilaments. Part I: The loadelongation behaviour with normal snarling. J. Text. Inst. 63 (1972) 477–489; Part II: Cylindrical snarling. J. Text. Inst. 63 (1972) 490-501.

    Google Scholar 

  20. W. Jiang, A general formulation of the theory of wire ropes. ASME J. Appl. Mech. 62 (1995) 747–755.

    Article  MATH  Google Scholar 

  21. W.G. Jiang, J.L. Henshall and J.M. Walton, A concise finite element model for three-layered straight wire rope strand. Internat. J. Mech. Sci. 42 (2000) 63–86.

    Article  MATH  Google Scholar 

  22. C. Kang, X. Zhang, R. Ratliff, R. Moyzis and A. Rich, Crystal structure of four-stranded Oxytricha telomeric DNA. Nature 356 (1992) 126–131.

    Article  ADS  Google Scholar 

  23. V. Katritch, W.K. Olson, P. Pierański, J. Dubochet and A. Stasiak, Properties of ideal composite knots. Nature 388 (1997) 148–151.

    Article  ADS  Google Scholar 

  24. T. Kunoh and C.M. Leech, Curvature effects on contact position of wire strands. Internat. J. Mech. Sci. 27 (1985) 465–470.

    Article  Google Scholar 

  25. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Cambridge Univ. Press, Cambridge (1927).

    MATH  Google Scholar 

  26. A. Maritan, C. Micheletti, A. Trovato and J.R. Banavar, Optimal shapes of compact strings. Nature 406 (2000) 287–290.

    Article  ADS  Google Scholar 

  27. M. Moakher and J.H. Maddocks, A double-strand elastic rod theory. Preprint Institut Bernoulli, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland (2002).

  28. D.A.D. Parry and J.M. Squire (eds), Fibrous Proteins (Special issue). J. Struct. Biol. 122 (1998).

  29. M. Peyrard and A.R. Bishop, Statistical mechanics of a nonlinear model for DNA denaturation. Phys. Rev. Lett. 62 (1989) 2755–2758.

    Article  ADS  Google Scholar 

  30. P. Pierański, In search of ideal knots. In: [35], pp. 20–41.

  31. S. Przybyl and P. Pierański, Helical close packings of ideal ropes. European Phys. J. E 4 (2001) 445–449.

    Article  ADS  Google Scholar 

  32. I. Rouzina and V.A. Bloomfield, Force-induced melting of the DNA double helix 1: Thermodynamic analysis. Biophys. J. 80 (2001) 882–893; 2: Effect of solution conditions. Biophys. J. 80 (2001) 894-900.

    Article  Google Scholar 

  33. A. Sarkar, J.-F. Léger, D. Chatenay and J.F. Marko, Structural transitions in DNA driven by external force and torque. Phys. Rev. E 63 (2001) 051903.

    Article  ADS  Google Scholar 

  34. T. Simonsson, G-quadruplex DNA structures - variations on a theme. Biol. Chem. 382 (2001) 621–628.

    Article  Google Scholar 

  35. A. Stasiak, V. Katritch and L.H. Kauffman (eds), Ideal Knots, Series on Knots and Everything, Vol. 19. World Scientific, Singapore (1998).

    MATH  Google Scholar 

  36. A. Stasiak and J.H. Maddocks, Best packing in proteins and DNA. Nature 406 (2000) 251–253.

    Article  ADS  Google Scholar 

  37. E.L. Starostin, On the writhe of non-closed curves. Preprint, Institut Bernoulli, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland (2002); available at arXiv: physics/0212095.

  38. T.R. Strick, J.-F. Allemand, D. Bensimon, A. Bensimon and V. Croquette, The elasticity of a single supercoiled DNA molecule. Science 271 (1996) 1835–1837.

    ADS  Google Scholar 

  39. D.M. Stump, W.B. Fraser and K.E. Gates, The writhing of circular cross-section rods: undersea cables to DNA supercoils. Proc. Roy. Soc. London A 454 (1998) 2123–2156.

    MATH  ADS  Google Scholar 

  40. J.M.T. Thompson, G.H.M. van der Heijden and S. Neukirch, Super-coiling of DNA plasmids: The mechanics of the generalised ply. Proc. Roy. Soc. London A 458 (2002) 959–985.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. L.R.G. Treloar, The geometry of multy-ply yarns. J. Text. Inst. 47 (1956) T348–T368.

    Google Scholar 

  42. P.N.T. Unwin and P.D. Ennis, Two configurations of a channel-forming membrane protein. Nature 307 (1984) 609–613.

    Article  ADS  Google Scholar 

  43. G.H.M. van der Heijden, The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc. Roy. Soc. London A 457 (2001) 695–715.

    MATH  MathSciNet  ADS  Google Scholar 

  44. G.H.M. van der Heijden and J.M.T. Thompson, Helical and localised buckling in twisted rods: a uniform analysis of the symmetric case. Nonlinear Dynamics 21 (2000) 71–99.

    Article  MATH  MathSciNet  Google Scholar 

  45. G.H.M. van der Heijden, J.M.T. Thompson and S. Neukirch, A variational approach to loaded ply structures. J. Vibration Control 9 (2003) 175–185.

    Article  MATH  MathSciNet  Google Scholar 

  46. K.M. Vasquez, L. Narayanan and P.M. Glazer, Specific mutations induced by triplex-forming oligonucleotides in mice. Science 290 (2000) 530–533.

    Article  ADS  Google Scholar 

  47. J.H. White, Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91 (1969) 693–728.

    MATH  MathSciNet  Google Scholar 

  48. M. Yeager and B.J. Nicholson, Structure of gap junction intercellular channels. Curr. Opin. Struct. Biol. 6 (1996) 183–192.

    Article  Google Scholar 

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Authors and Affiliations

  1. Bernoulli Mathematics Institute, Swiss Federal Institute of Technology, Lausanne, CH-1015, Switzerland

    S. Neukirch

  2. Centre for Nonlinear Dynamics, University College London, London, WC1E 6BT, U.K.

    G.H.M. van der Heijden

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  1. S. Neukirch
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  2. G.H.M. van der Heijden
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Neukirch, S., van der Heijden, G. Geometry and Mechanics of Uniform n-Plies: from Engineering Ropes to Biological Filaments. Journal of Elasticity 69, 41–72 (2002). https://doi.org/10.1023/A:1027390700610

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