Skip to main content
SpringerLink
Log in

Link, Writhe, and Twist

  • Chapter
  • First Online:
Modeling Nonlinear Problems in the Mechanics of Strings and Rods

Part of the book series: Interaction of Mechanics and Mathematics ((IMM))

  • 2608 Accesses

Abstract

We review the concepts of writhe, twist, and linking as applied to space curves and ribbons. The application of these concepts to DNA is also discussed.

“In drawing the various closed curves which have a given number of double points, I found it desirable to have some simple mode of ascertaining whether a particular form was a new one, or only a deformation of one of those I had already obtained.”

P. G. Tait [337, Page 289].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We refer the reader to Bergou et al. [22] and Hanson [155, Chapter 20] for details on the numerical computation of the Bishop frame for space curves.

  2. 2.

    The total surface area of a unit sphere is 4π. In topology, the area A in Figure 3.2 is known as a solid angle.

  3. 3.

    Our historical comments in this section are based entirely on the (recent) insightful papers by Epple [95, 96] and Ricca and Nipoti [301]. The latter paper contains a translation of the page in Gauss’ notebook where Eqn. (3.28) is presented as well as copies of letters from Maxwell to Tait discussing the linking number.

  4. 4.

    Our convention for writing \(L_{\mbox{ k}}\left (\mathcal{S}_{1},\mathcal{S}_{2}\right )\) is taken from Spivak [329, Problem 8.28, Page 402] and differs from Gauss’ original prescription by a minus sign. As a result, our computations using Eqn. (3.28), such as the results shown in Figure 3.9, agree with those found by counting the signed crossings using Eqn. (3.34).

  5. 5.

    These methods are discussed in several places in the literature and summarized in Rolfsen’s text [303, Chapter 5, Section D].

  6. 6.

    Fuller’s version of Eqn. (3.38) differs from ours in that \(T_{\text{w}}\left (\mathcal{S},\mathbf{e}_{n}\right )\) is replaced by the more general case \(T_{\text{w}}\left (\mathcal{S},\mathbf{u}\right )\) in Eqn. (3.38). Alternative proofs of Fuller [111, Eqn. (6.3)] can be found in Aldinger et al. [7] and Kamien [176].

  7. 7.

    The parameters for this curve are discussed in Section 1.3.4.

  8. 8.

    The orientability condition on the ribbon is satisfied when u(s) = u(s + ℓ) where s ∈ [0, ℓ] on \(\mathcal{S}\). Thus, the ribbon is not a Möbius strip.

  9. 9.

    As emphasized in [7], the domain of integration excludes those points s 1 = s 2 where the integrand becomes unbounded.

  10. 10.

    The primary reference for our summary of the methods used to compute the writhing number is [7]. We also recommend the later works [20, 21, 85, 169, 176] for helpful perspectives and insights on this topic.

  11. 11.

    The curves considered in [360] are discussed in Exercise 3.9.

  12. 12.

    See [169, 360] and references therein.

  13. 13.

    We refer the reader to [239] for a discussion of Călugăreanu’s legacy and the roles played by Călugăreanu’s theorem and Eqn. (3.60).

  14. 14.

    For examples, see [18, 360, 361].

  15. 15.

    Underwound is also termed negatively supercoiled in contrast to the case σ > 0 which is termed positively supercoiled.

  16. 16.

    This result was first established by Maxwell [95, 301].

  17. 17.

    The curve \(\mathcal{S}_{2}\) is not identical to the curve \(\mathcal{S}_{2}\) that is defined in Eqn. (3.52)2.

References

  1. Agoston, M.K.: Computer Graphics and Geometric Modelling: Mathematics. Springer-Verlag, London (2005). URL http://dx.doi.org/10.1007/b138899

  2. Aldinger, J., Klapper, I., Tabor, M.: Formulae for the calculation and estimation of writhe. Journal of Knot Theory and its Ramifications 4 (3), 343–372 (1995). URL http://dx.doi.org/10.1142/S021821659500017X

  3. Bauer, W.R., Lund, R.A., White, J.H.: Twist and writhe of a DNA loop containing intrinsic bends. Proceedings of the National Academy of Sciences 90 (3), 833–837 (1993)

    Article  Google Scholar 

  4. Berger, M.A.: Topological quantities: Calculating winding, writhing, linking, and higher order invariants. In: R.L. Ricca (ed.) Lectures on Topological Fluid Mechanics, Lecture Notes in Mathematics, pp. 75–97. Springer-Verlag, Berlin, Heidelberg (2009). URL http://dx.doi.org/10.1007/978-3-642-00837-5_2

  5. Berger, M.A., Prior, C.: The writhe of open and closed curves. Journal of Physics A: Mathematical and General 39 (26), 8321 (2006). URL http://dx.doi.org/10.1088/0305-4470/39/26/005

  6. Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Transactions on Graphics (SIGGRAPH) 27 (3), 63:1–63:12 (2008)

    Google Scholar 

  7. Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly 82 (3), 246–251 (1975). URL http://dx.doi.org/10.2307/2319846

  8. Bustamante, C., Bryant, Z., Smith, S.B.: Ten years of tension: Single-molecule DNA mechanics. Nature 421, 423–427 (2003). URL http://dx.doi.org/10.1038/nature01405

  9. Călugăreanu, G.: L’intégrale de Gauss et l’analyse des nœuds tridimensionnels. Revue Roumaine Mathématique Pures et Appliqué 4, 5–20 (1959)

    Google Scholar 

  10. Călugăreanu, G.: Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Mathematical Journal 11 (86), 588–625 (1961)

    MathSciNet  MATH  Google Scholar 

  11. Călugăreanu, G.: Sur les enlacements tridimensionnels des courbes fermées. Comptes Rendus Academie R. P. Romîne 11, 829–832 (1961)

    MATH  Google Scholar 

  12. Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. Journal of Elasticity 60 (3), 173–221 (2000). URL http://dx.doi.org/10.1023/A:1010911113919

  13. Coleman, B.D., Swigon, D.: Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 362 (1820), 1281–1299 (2004). URL http://dx.doi.org/10.1098/rsta.2004.1393

  14. Coleman, B.D., Swigon, D., Tobias, I.: Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact. Physical Review E 61, 759–770 (2000). URL http://dx.doi.org/10.1103/PhysRevE.61.759

  15. Crick, F.H.: Linking numbers and nucleosomes. Proceedings of the National Academy of Sciences 73 (8), 2639–2643 (1976). URL http://www.pnas.org/content/73/8/2639.abstract

  16. Daune, M.: Molecular Biophysics: Structures in Motion. Oxford University Press, New York (1999). Translated from the French by W. J. Duffin.

    Google Scholar 

  17. Dennis, M.R., Hannay, J.H.: Geometry of Călugăreanu’s theorem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 461 (2062), 3245–3254 (2005). URL http://dx.doi.org/10.1098/rspa.2005.1527

  18. Epple, M.: Orbits of asteroids, a braid, and the first link invariant. The Mathematical Intelligencer 20 (1), 45–52 (1998). URL http://dx.doi.org/10.1007/BF03024400

  19. Epple, M.: Topology, matter, and space, I: Topological notions in 19th-century natural philosophy. Archive for History of Exact Sciences 52 (4), 297–392 (1998). URL http://www.jstor.org/stable/41134050

  20. Fuller, F.B.: The writhing number of a space curve. Proceedings of the National Academy of Sciences 68, 815–819 (1971). URL http://www.pnas.org/content/68/4/815.abstract

  21. Fuller, F.B.: Decomposition of the linking number of a closed ribbon: A problem from molecular biology. Proceedings of the National Academy of Sciences 75 (8), 3557–3561 (1978). URL http://dx.doi.org/10.1073/pnas.75.8.3557

  22. Goyal, S., Perkins, N.C.: Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness. International Journal of Non-Linear Mechanics 43 (10), 1121–1129 (2008). URL http://dx.doi.org/10.1016/j.ijnonlinmec.2008.06.013

  23. Guggenheimer, H.: Computing frames along a trajectory. Computer Aided Geometric Design 6 (1), 77–78 (1989). URL http://dx.doi.org/10.1016/0167-8396(89)90008-3

  24. Hanson, A.J.: Visualizing Quaternions. Morgan Kaufmann, San Francisco, CA; Amsterdam; Boston (2006)

    Google Scholar 

  25. Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy, and the stability of DNA minicircles. Biopolymers 70 (2), 145–157 (2003). URL http://dx.doi.org/10.1002/bip.10430

  26. Kamien, R.D.: Local writhing dynamics. The European Physical Journal B 1 (1), 1–4 (1998). URL http://dx.doi.org/10.1007/s100510050145

  27. Kreyszig, E.: Differential Geometry, Revised and reprinted edn. Toronto University Press, Toronto (1964)

    MATH  Google Scholar 

  28. Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38 (4), 605–618 (1996). URL http://dx.doi.org/10.1137/S0036144593253290

  29. Livingston, C.: Knot Theory. Mathematical Association of America, Washington, DC (1993)

    MATH  Google Scholar 

  30. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, fourth edn. Cambridge University Press, Cambridge (1927)

    Google Scholar 

  31. Maxwell, J.C.: A Treatise on Electricity and Magnetism, vol. 2, third edn. Clarendon Press, Oxford (1892)

    Google Scholar 

  32. McConnell, A.J.: Applications of the Absolute Differential Calculus. Blackie and Son, London (1947). Corrected reprinted edition

    Google Scholar 

  33. McMillen, T., Goriely, A.: Tendril perversion in intrinsically curved rods. Journal of Nonlinear Science 12 (3), 241–281 (2002). URL http://dx.doi.org/10.1007/s00332-002-0493-1

  34. Moffatt, H.K., Ricca, R.L.: Helicity and the Călugăreanu invariant. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 439 (1906), 411–429 (1992). URL http://www.jstor.org/stable/52228

  35. Murasugi, K.: Knot theory and its applications. Birkhäuser Boston Inc., Boston, MA (1996). Translated from the 1993 Japanese original by Bohdan Kurpita

    MATH  Google Scholar 

  36. Neukirch, S., Starostin, E.L.: Writhe formulas and antipodal points in plectonemic DNA configurations. Physical Review E 78 (4), 041,912, 9 (2008). URL http://dx.doi.org/10.1103/PhysRevE.78.041912

  37. Oprea, J.: Differential Geometry and its Applications, second edn. Pearson Prentice Hall, Upper Saddle River, NJ (2003)

    Google Scholar 

  38. Pohl, W.F.: The self-linking number of a closed space curve. Journal of Mathematics and Mechanics 17, 975–985 (1967/1968)

    Google Scholar 

  39. Pohl, W.F.: DNA and differential geometry. The Mathematical Intelligencer 3 (1), 20–27 (1980/81). URL http://dx.doi.org/10.1007/BF03023391

  40. Ricca, R.L., Nipoti, B.: Gauss’ linking number revisited. Journal of Knot Theory and its Ramifications 20 (10), 1325–1343 (2011). URL http://dx.doi.org/10.1142/S0218216511009261

  41. Rolfsen, D.: Knots and Links, Mathematics Lecture Series, vol. 7. Publish or Perish, Inc., Houston, Texas (1990). Corrected reprint of the 1976 original

    Google Scholar 

  42. Sarkar, A., Léger, J.F., Chatenay, D., Marko, J.F.: Structural transitions in DNA driven by external force and torque. Physical Review E 63 (5), 051,903 (2001). URL http://link.aps.org/doi/10.1103/PhysRevE.63.051903

  43. Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I, second edn. Publish or Perish, Inc., Wilmington, Delaware (1979)

    Google Scholar 

  44. Swigon, D., Coleman, B.D., Tobias, I.: The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophysical Journal 74 (5), 2515–2530 (1998). URL http://dx.doi.org/10.1016/S0006-3495(98)77960-3

  45. Tait, P.G.: Note on the measure of beknottedness. Proceedings of the Royal Society of Edinburgh 9, 289–298 (1877–1878). URL http://www.maths.ed.ac.uk/~aar/papers/beknot.pdf

  46. Thomson, W., Tait, P.G.: Treatise on Natural Philosophy. Oxford University Press, Oxford (1867)

    MATH  Google Scholar 

  47. White, J.H.: Self-linking and the Gauss integral in higher dimensions. American Journal of Mathematics 91, 693–728 (1969). URL http://www.jstor.org/stable/2373348

  48. White, J.H., Bauer, W.R.: Calculation of the twist and the writhe for representative models of DNA. Journal of Molecular Biology 189 (2), 329–341 (1986). URL http://dx.doi.org/10.1016/0022-2836(86)90513-9

  49. White, J.H., Bauer, W.R.: Applications of the twist difference to DNA structural analysis. Proceedings of the National Academy of Sciences 85 (3), 772–776 (1988). URL http://www.pnas.org/content/85/3/772.full.pdf

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA, USA

    Oliver M. O’Reilly

Authors
  1. Oliver M. O’Reilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

O’Reilly, O.M. (2017). Link, Writhe, and Twist. In: Modeling Nonlinear Problems in the Mechanics of Strings and Rods. Interaction of Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-50598-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50598-5_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50596-1

  • Online ISBN: 978-3-319-50598-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation