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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Madras and G. Slade,The Self-Avoiding walk (Birkhäuser, Boston, 1993).
W. R. Bauer, F. H. C. Crick, and J. H. White,Sci. Amer. 243:118 (1980).
N. S. Anderson, J. W. Campbell, M. M. Harding, D. A. Rees, and J. W. B. Samuel,J. Mol. Biol. 45:85 (1969).
D. A. Rees,Polysaccharide Conformation, inMTP International Review of Science, Organic Chemistry, Series One, Vol. 7, G. O. Aspinall, ed. (Butterworths 1973).
F. B. Fuller,Proc. Natl. Acad. Sci. USA 91:513 (1971).
E. J. Janse van Rensburg, E. Orlandini, D. W. Summers, M. C. Tesi, and S. G. Whittington,Phys. Rev. E 50:R4279 (1994).
A. V. Vologodskii and N. R. Cozzarelli,Annu. Rev. Biophys. Biomol. Struct. 23:609 (1994).
E. Orlandini, E. J. Janse van Rensburg, and S. G. Whittington,J. Stat. Phys. 82:1159, (1996).
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).
J. B. Wilker and S. G. Whittington,J. Phys. A: Math. Gen. 12:L245 (1979).
J. M. Hammersley and D. J. A. Welsh,Q. J. Math. Oxford 13:108 (1962).
H. Kesten,J. Math. Phys. 4:960 (1963).
J. M. Hammersley, Private communication.
G. Burde and H. Zieschang,Knots (de Gruyter, Berlin, 1985).
D. Rolfsen,Knots and Links (Publish or Perish, Wilmington, 1976).
D. W. Sumners, and S. G. Whittington,J. Phys. A: Math. Gen. 21:1689 (1988).
C. E. Soteros, D. W. Sumners and S. G. Whittington,Math. Proc. Camb. Phil. Soc. 111:75 (1992).
H. Schubert,Acta Math. 90:131 (1953).
M. Thistlethwaite,Unpublished.
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).
J. H. White,Am. J. Math. 91:693 (1969).
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.)
G. Calugareano,Czech. Math. J. 11:588 (1961).
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.
K. V. Klenin, A. V. Vologodskii, V. V. Anshelevich, A. M. Dykhne, and M. D. Frank-Kamenetskii,J. Biomol. Struct. 5:1173 (1988).
M. O. Fenley, W. K. Olson, I. Tobias, and G. S. Manning,Biophys. Chem. 50:255 (1994).
M. C. Tesi, E. J. Janse van Rensburg, E. Orlandini, D. W. Sumners, and S. G. Whittington,Phys. Rev. E 49:868 (1994).
M.-H. Hao and W. K. Olson,Macromolecules 22:3292 (1989).
A. V. Vologodskii, S. D. Levene, K. V. Klenin, M. Frank-Kamenetskii, and N. R. Cozzarelli,J. Mol. Biol. 227:1224 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Janse van Rensburg, E.J., Orlandini, E., Sumners, D.W. et al. Entanglement complexity of lattice ribbons. J Stat Phys 85, 103–130 (1996). https://doi.org/10.1007/BF02175557
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02175557