Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids

  • Bernard D. Coleman
  • David Swigon


Methods are presented for obtaining exact analytical representations of supercoiled equilibrium configurations of impenetrable elastic rods of circular cross-section that have been pretwisted and closed to form rings, and a discussion is given of applications in the theory of the elastic rod model for DNA. When, as here, self-contact is taken into account, and the rod is assumed to be inextensible, intrinsically straight, transversely isotropic, and homogeneous, the important parameters in the theory are the excess link Δℒ (a measure of the amount the rod was twisted before its ends were joined), the ratio ω of the coefficients of torsional and flexural rigidity, and the ratio d of cross-sectional diameter to the length of the axial curve C. Solutions of the equations of equilibrium are given for cases in which self-contact occurs at isolated points and along intervals. Bifurcation diagrams are presented as graphs of Δℒ versus the writhe of C and are employed for analysis of the stability of equilibrium configurations. It is shown that, in addition to primary, secondary, and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the axial curve is a circle, there are families of equilibrium configurations that are isolas in the sense that they are not connected to bifurcation branches by paths of equilibrium configurations compatible with the assumed impenetrability of the rod. Each of the isolas found to date is connected to a bifurcation branch by a path which, although made up of solutions of the governing equations, contains regions on which the condition of impenetrability does not hold.

contact problems for elastic rods DNA topology 


  1. 1.
    I. Tobias, D. Swigon, and B.D. Coleman, Elastic stability of DNA configurations: I. General theory. Phys. Rev. E 61 (2000) 747–758.MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    B.D. Coleman, D. Swigon, and I. Tobias, Elastic stability of DNA configurations: II. Supercoiling of miniplasmids. Phys. Rev. E 61 (2000) 759–770.MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    D. Swigon, Configurations with self-contact in the theory of the elastic rod model for DNA, Doctoral Dissertation, Rutgers University, New Brunswick (1999).Google Scholar
  4. 4.
    G. Kirchhoff, Ñber das Gleichgewicht und die Bewegung eines unendlich dünen elastischen Stabes. J. Reine angew. Math. (Crelle) 56 (1859) 285–313.MATHCrossRefGoogle Scholar
  5. 5.
    A. Clebsch, Theorie der Elasticität Fester Körper, Teubner, Leipzig (1862).Google Scholar
  6. 6.
    G. Kirchhoff, Vorlesungen über mathematische Physik, Mechanik, Vol. 28, Teubner, Leipzig (1876).Google Scholar
  7. 7.
    E.H. Dill, Kirchhoff's theory of rods. Arch. Hist. Exact Sci. 44 (1992) 1–23.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Courant, Differential and Integral Calculus, Vol. II, Blackie, London (1936).MATHGoogle Scholar
  9. 9.
    F.B. Fuller, The writhing number of a space curve. Proc. Natl. Acad. Sci. USA. 68 (1971) 815–819.MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    J.H. White, An introduction to the geometry and topology of DNA structure. In: Mathematical Methods for DNA Sequences, CRC, Boca Raton, Florida (1989), pp. 225–253.Google Scholar
  11. 11.
    G. Calugareanu, Sur les classes d'isotopie des noeuds tridimensionnels et leurs invariants. Czechoslovak Math. J. 11 (1961) 588–625.MATHMathSciNetGoogle Scholar
  12. 12.
    J.H. White, Self-linking and the Gauss integral in higher dimensions. Amer. J. Math. 91 (1969) 693–728.MATHMathSciNetGoogle Scholar
  13. 13.
    R.A. Litherland, J. Simon, O. Durumeric, and E. Rawdon, Thickness of knots. Topology Appl. 91 (1999) 233–244.MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Le Bret, Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers 23 (1984) 1835–1867.CrossRefGoogle Scholar
  15. 15.
    L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Pergamon Press, Oxford (1959).Google Scholar
  16. 16.
    I. Tobias, B.D. Coleman, and W. Olson, The dependence of DNA tertiary structure on end conditions: Theory and implications for topological transitions. J. Chem. Phys. 101 (1994) 10990–10996.CrossRefADSGoogle Scholar
  17. 17.
    B.D. Coleman, I. Tobias, and D. Swigon, Theory of the influence of end conditions on self-contact in DNA loops. J. Chem. Phys. 103 (1995) 9101–9109.CrossRefADSGoogle Scholar
  18. 18.
    D. Swigon, B.D. Coleman, and I. Tobias, The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophys. J. 74 (1998) 2515–2530.CrossRefADSGoogle Scholar
  19. 19.
    B.D. Coleman, E.H. Dill, M. Lembo, Z. Lu, and I. Tobias, On the dynamics of rods in the theory of Kirchhoff and Clebsch. Arch. Rational Mech. Anal. 121 (1993) 339–359.MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    J.W. Milnor, Topology from the Differentiable Viewpoint, 2nd edn, The University Press of Virginia, Charlotesville (1969).Google Scholar
  21. 21.
    P.J. Holmes, G. Domokos, J. Schmitt, and I. Szeberényi, Constrained Euler buckling: An interplay of computation an analysis. Comp. Meth. in Appl. Mech. and Eng. 170 (1999) 175–207.MATHCrossRefGoogle Scholar
  22. 22.
    W.F. Pohl, The self-linking number of a closed space curve. J. Math. Mech. 17 (1968) 975–985.MATHMathSciNetGoogle Scholar
  23. 23.
    P.J. Hagerman, Flexibility of DNA. Annu. Rev. Biophys. Biophys. Chem. 17 (1988) 265–286.CrossRefGoogle Scholar
  24. 24.
    J.M. Schurr, B.S. Fujimoto, P. Wu, and L. Song, Fluorescence studies of nucleic acids: Dynamics, rigidities, and structures. In: J.R. Lakowicz (ed.), Topics in Fluorescence Spectroscopy, Vol. 3: Biochemical Applications, Plenum Press, New York (1992).Google Scholar
  25. 25.
    D.S. Horowitz and J.C. Wang, Torsional rigidity of DNA and length dependence of the free energy of DNA supercoiling. J. Mol. Biol. 173 (1984) 75–91.CrossRefGoogle Scholar
  26. 26.
    C. Bouchiat and M. Mezard, Elasticity model of supercoiled DNA molecule. Phys. Rev. Lett. 80 (1998) 1556–1559.CrossRefADSGoogle Scholar
  27. 27.
    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.ADSGoogle Scholar
  28. 28.
    P.J. Heath, J.B. Clendenning, B.S. Fujimoto, and J.M. Schurr, Effect of bending strain on the torsion elastic constant of DNA. J. Mol. Biol. 260 (1996) 718–730.CrossRefGoogle Scholar
  29. 29.
    E.E. Zajac, Stability of two planar loop elasticas. J. Appl. Mech. 29 (1962) 136–142.MATHMathSciNetGoogle Scholar
  30. 30.
    G. Domokos, A group-theoretic approach to the geometry of elastic rings. J. Nonlinear Sci. 5 (1995) 453–478.MATHMathSciNetCrossRefADSGoogle Scholar
  31. 31.
    G. Domokos and T.J. Healey, Hidden symmetry of global solutions in twisted elastic rings. J. Nonlin. Sci. (October 2000), accepted.Google Scholar
  32. 32.
    D.M. Stump, W.B. Fraser, and K.E. Gates, The writhing of circular cross-section rods: From undersea cables to DNA supercoils. Proc. Roy. Soc. London A 454 (1998) 2123–2156.MATHADSCrossRefGoogle Scholar
  33. 33.
    B. Fain and J. Rudnick, Conformations of closed DNA. Phys. Rev. E 60 (1999) 7239–7252.MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Bernard D. Coleman
    • 1
  • David Swigon
    • 1
  1. 1.Department of Mechanics and Materials Science, RutgersThe State University of New JerseyPiscatawayU.S.A.

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