Advertisement

Journal of Statistical Physics

, Volume 85, Issue 1–2, pp 103–130 | Cite as

Entanglement complexity of lattice ribbons

  • E. J. Janse van Rensburg
  • E. Orlandini
  • D. W. Sumners
  • M. C. Tesi
  • S. G. Whittington
Articles

Abstract

We consider a discrete ribbon model for double-stranded polymers where the ribbon is constrained to lie in a three-dimensional lattice. The ribbon can be open or closed, and closed ribbons can be orientable or nonorientable. We prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes, and a theorem about the frequency of occurrence of certain patterns in these ribbons. We use this to derive results about the frequency of knots in closed ribbons, the linking of the boundary curves of orientable closed ribbons, and the twist and writhe of ribbons. We show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite-n limit. For an orientable ribbon, the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as √n, and similar results hold for the twist and writhe.

Key Words

Ribbon topological entanglement knot link satellite knot writhe double-stranded polymer 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Madras and G. Slade,The Self-Avoiding walk (Birkhäuser, Boston, 1993).Google Scholar
  2. 2.
    W. R. Bauer, F. H. C. Crick, and J. H. White,Sci. Amer. 243:118 (1980).Google Scholar
  3. 3.
    N. S. Anderson, J. W. Campbell, M. M. Harding, D. A. Rees, and J. W. B. Samuel,J. Mol. Biol. 45:85 (1969).Google Scholar
  4. 4.
    D. A. Rees,Polysaccharide Conformation, inMTP International Review of Science, Organic Chemistry, Series One, Vol. 7, G. O. Aspinall, ed. (Butterworths 1973).Google Scholar
  5. 5.
    F. B. Fuller,Proc. Natl. Acad. Sci. USA 91:513 (1971).Google Scholar
  6. 6.
    E. J. Janse van Rensburg, E. Orlandini, D. W. Summers, M. C. Tesi, and S. G. Whittington,Phys. Rev. E 50:R4279 (1994).Google Scholar
  7. 7.
    A. V. Vologodskii and N. R. Cozzarelli,Annu. Rev. Biophys. Biomol. Struct. 23:609 (1994).Google Scholar
  8. 8.
    E. Orlandini, E. J. Janse van Rensburg, and S. G. Whittington,J. Stat. Phys. 82:1159, (1996).Google Scholar
  9. 9.
    E. J. Janse van Rensburg, E. Orlandini, D. W. Sumners, M. C. Tesi, and S. G. Whittington,Topology and geometry of biopolymers inMathematical Approaches to Biomolecular Structure and Dynamics, J. Mesirov, K. Schulten, and D. W. Sumners eds. (Springer-Verlag, Berlin, 1995).Google Scholar
  10. 10.
    J. B. Wilker and S. G. Whittington,J. Phys. A: Math. Gen. 12:L245 (1979).Google Scholar
  11. 11.
    J. M. Hammersley and D. J. A. Welsh,Q. J. Math. Oxford 13:108 (1962).Google Scholar
  12. 12.
    H. Kesten,J. Math. Phys. 4:960 (1963).Google Scholar
  13. 13.
    J. M. Hammersley, Private communication.Google Scholar
  14. 14.
    G. Burde and H. Zieschang,Knots (de Gruyter, Berlin, 1985).Google Scholar
  15. 15.
    D. Rolfsen,Knots and Links (Publish or Perish, Wilmington, 1976).Google Scholar
  16. 16.
    D. W. Sumners, and S. G. Whittington,J. Phys. A: Math. Gen. 21:1689 (1988).Google Scholar
  17. 17.
    C. E. Soteros, D. W. Sumners and S. G. Whittington,Math. Proc. Camb. Phil. Soc. 111:75 (1992).Google Scholar
  18. 18.
    H. Schubert,Acta Math. 90:131 (1953).Google Scholar
  19. 19.
    M. Thistlethwaite,Unpublished.Google Scholar
  20. 20.
    E. J. Janse van Rensburg, E. Orlandini, D. W. Sumners, M. C. Tesi, and S. G. Whittington,J. Phys. A: Math. Gen. 26:L981 (1993).Google Scholar
  21. 21.
    J. H. White,Am. J. Math. 91:693 (1969).Google Scholar
  22. 22.
    J. H. White,Geometry and topology of DNA and DNA-protein interactions, inNew Scientific Applications of Geometry and Topology, D. W. Sumners, ed. (American Mathematical Society, Providence, Rhode Island, 1991, p. 17.)Google Scholar
  23. 23.
    G. Calugareano,Czech. Math. J. 11:588 (1961).Google Scholar
  24. 24.
    R. C. Lacher and D. W. SumnersData structures and algorithms for the computation of topological invariants of entanglements: Link, twist and writhe, inComputer Simulations of Polymers, R. J. Roe. ed. (Prentice-Hall, Englewood Cliffs, New Jersey, 1991), p. 365.Google Scholar
  25. 25.
    K. V. Klenin, A. V. Vologodskii, V. V. Anshelevich, A. M. Dykhne, and M. D. Frank-Kamenetskii,J. Biomol. Struct. 5:1173 (1988).Google Scholar
  26. 26.
    M. O. Fenley, W. K. Olson, I. Tobias, and G. S. Manning,Biophys. Chem. 50:255 (1994).Google Scholar
  27. 27.
    M. C. Tesi, E. J. Janse van Rensburg, E. Orlandini, D. W. Sumners, and S. G. Whittington,Phys. Rev. E 49:868 (1994).Google Scholar
  28. 28.
    M.-H. Hao and W. K. Olson,Macromolecules 22:3292 (1989).Google Scholar
  29. 29.
    A. V. Vologodskii, S. D. Levene, K. V. Klenin, M. Frank-Kamenetskii, and N. R. Cozzarelli,J. Mol. Biol. 227:1224 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • E. J. Janse van Rensburg
    • 1
  • E. Orlandini
    • 2
  • D. W. Sumners
    • 3
  • M. C. Tesi
    • 4
  • S. G. Whittington
    • 5
  1. 1.Department of MathematicsYork UniversityNorth YorkCanada
  2. 2.Theoretical PhysicsUniversity of OxfordOxfordU.K.
  3. 3.Department of MathematicsFlorida State UniversityTallahassee
  4. 4.Mathematical InstituteUniversity of OxfordOxfordU.K.
  5. 5.Department of ChemistryUniversity of TorontoTorontoCanada

Personalised recommendations