DNA Topology Review

  • Garrett Jones
  • Candice Reneé PriceEmail author
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 15)


DNA holds the instructions for an organism’s development, reproduction, and, ultimately, death. It encodes much of the information a cell needs to survive and reproduce. It is important for inheritance and coding for proteins, and contains the genetic instruction guide for life and its processes. But also, DNA of an organism has a complex and interesting topology. For information retrieval and cell viability, some geometric and topological features of DNA must be introduced, and others quickly removed. Proteins perform these amazing feats of topology at the molecular level; thus, the description and quantization of these protein actions require the language and computational machinery of topology. The use of tangle algebra to model the biological processes that give rise to knotting in DNA provides an excellent example of the application of topological algebra to biology. The tangle algebra approach to knotting in DNA began with the study of the site-specific recombinase Tn3 resolvase. This chapter is a summary of some basic knot theory and biology. We then describe the tangle model developed by Ernst and Sumners using the Tn3 resolvase as an example. We conclude with applications of the tangle model to other biological problems.

Subject Classification 2010



  1. 1.
    C.C. Adams, The Knot Book (American Mathematical Society, Providence, 2004), An elementary introduction to the mathematical theory of knots, Revised reprint of the 1994 original. MR MR2079925 (2005b:57009)Google Scholar
  2. 2.
    B. Alberts, D. Bray, K. Hopkins, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, Essential Cell Biology, 2nd edn. (Garland Science/Taylor & Francis Group, New York, 2003)Google Scholar
  3. 3.
    J.W. Alexander, Topological invariants of knots and links. Trans. Am. Math. Soc. 30(2), 275–306 (1928)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A.D. Bates, A. Maxwell, Dna Topology (Oxford University Press, Oxford, 2005)Google Scholar
  5. 5.
    J.M. Berger, S.J. Gamblin, S.C. Harrison, J.C. Wang, Structure and mechanism of DNA topoisomerase ii. Nature 379(6562), 225–232 (1996)CrossRefGoogle Scholar
  6. 6.
    D. Buck, E. Flapan, A topological characterization of knots and links arising from site-specific recombination. J. Phys. A 40(41), 12377–12395 (2007). MR 2394909 (2010h:92064)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D. Buck, C.V. Marcotte, Tangle solutions for a family of DNA-rearranging proteins. Math. Proc. Camb. Philos. Soc. 139(1), 59–80 (2005). MR 2155505 (2006j:57010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Buck, K. Valencia, Characterization of knots and links arising from site-specific recombination on twist knots. J. Phys. A Math. Theor. 44(4), 045002 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    C.R. Calladine, H.R. Drew, B.F. Luisi, A.A. Travers, Understanding DNA, 3rd edn. (Elsevier Academic Press, Amsterdam, 2004)Google Scholar
  10. 10.
    J.J. Champoux, DNA topoisomerases: structure, function, and mechanism. Annu. Rev. Biochem. 70, 369–413 (2001)CrossRefGoogle Scholar
  11. 11.
    J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) (Pergamon, Oxford, 1970), pp. 329–358. MR 0258014 (41 #2661)Google Scholar
  12. 12.
    P.R. Cromwell, Knots and Links (Cambridge University Press, Cambridge, 2004). MR MR2107964 (2005k:57011)Google Scholar
  13. 13.
    I.K. Darcy, Biological distances on DNA knots and links: applications to XER recombination. J. Knot Theory Ramifications 10(2), 269–294 (2001), Knots in Hellas ‘98, Vol. 2 (Delphi). MR 1822492 (2002m:57008)Google Scholar
  14. 14.
    I. Darcy, J. Chang, N. Druivenga, C. McKinney, R. Medikonduri, S. Mills, J. Navarra-Madsen, A. Ponnusamy, J. Sweet, T. Thompson, Coloring the Mu transpososome. BMC Bioinf. 7(1), 435 (2006)Google Scholar
  15. 15.
    I. Darcy, J. Luecke, M. Vazquez, Tangle analysis of difference topology experiments: applications to a mu protein-DNA complex. Algebr. Geom. Topol. 9(4), 2247–2309 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    P. Dröge, N.R Cozzarelli, Recombination of knotted substrates by tn3 resolvase. Proc. Natl. Acad. Sci. 86(16), 6062–6066 (1989)CrossRefGoogle Scholar
  17. 17.
    E. Eftekhary, Heegaard floer homologies of pretzel knots. arXiv preprint math/0311419.Google Scholar
  18. 18.
    C. Ernst, D. Sumners, A calculus for rational tangles: applications to DNA recombination. Math. Proc. Camb. Philos. Soc. 108, 489–515 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12(2), 239–246 (1985)MathSciNetCrossRefGoogle Scholar
  20. 20.
    F.B. Fuller, Decomposition of the linking number of a closed ribbon: a problem from molecular biology. Proc. Natl. Acad. Sci. U. S. A. 75(8), 3557–3561 (1978). MR 0490004 (58 #9367)MathSciNetCrossRefGoogle Scholar
  21. 21.
    N.D.F. Grindley, K.L. Whiteson, P.A. Rice, Mechanisms of site-specific recombination. Annu. Rev. Biochem. 75(1), 567–605 (2006), PMID: 16756503CrossRefGoogle Scholar
  22. 22.
    J. Hardin, G.P. Bertoni, L.J. Kleinsmith, Becker’s World of the Cell, 8th edn. (Benjamin Cummings, San Francisco, 2010)Google Scholar
  23. 23.
    H.C. Ibarra, D.A.L. Navarro, An algorithm based on 3-braids to solve tangle equations arising in the action of gin {DNA} invertase. Appl. Math. Comput. 216(1), 95–106 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    M. Jayaram, R.M. Harshey, The Mu transpososome through a topological lens. Crit. Rev. Biochem. Mol. Biol. 41(6), 387–405 (2006)CrossRefGoogle Scholar
  25. 25.
    V.F.R. Jones, A polynomial invariant for knots via von Neumann algebra. Bull. Am. Math. Soc. (N.S.) 12, 103–111 (1985)MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Khovanov, A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    S. Kim, I.K. Darcy, Topological Analysis of DNA-Protein Complexes. Mathematics of DNA structure, function and interactions (Springer, Berlin, 2009), pp. 177–194CrossRefGoogle Scholar
  28. 28.
    K. Murasugi, Knot Theory and Its Applications (Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, 2008), Translated from the 1993 Japanese original by Bohdan Kurpita, Reprint of the 1996 translation [MR1391727]. MR 2347576Google Scholar
  29. 29.
    F.J. Olorunniji, D.E. Buck, S.D. Colloms, A.R. McEwan, M.C.M. Smith, W.M. Stark, S.J. Rosser, Gated rotation mechanism of site-specific recombination by ϕc31 integrase. Proc. Natl. Acad. Sci. 109(48), 19661–19666 (2012)CrossRefGoogle Scholar
  30. 30.
    P. Ozsváth, Knot Floer Homology. Advanced Summer School in Knot Theory, May 2009, International Center for Theoretical PhysicsGoogle Scholar
  31. 31.
    S. Pathania, M. Jayaram, R.M. Harshey, Path of DNA within the Mu transpososome. Cell 109(4), 425–436 (2002)CrossRefGoogle Scholar
  32. 32.
    C.R. Price, A Biological Application for the Oriented Skein Relation (ProQuest LLC, Ann Arbor, 2012), Thesis (Ph.D.)–The University of Iowa. MR 3078590Google Scholar
  33. 33.
    K. Reidemeister, Knotentheorie (Springer, Berlin, 1974), Reprint. MR MR0345089 (49 #9828)CrossRefGoogle Scholar
  34. 34.
    D. Rolfsen, Knots and Links. Mathematics Lecture Series, vol. 7 (Publish or Perish Inc., Houston, 1990), Corrected reprint of the 1976 original. MR MR1277811 (95c:57018)Google Scholar
  35. 35.
    R.G. Scharein, Interactive topological drawing, Ph.D. thesis, Department of Computer Science, The University of British Columbia, 1998Google Scholar
  36. 36.
    M.C.M. Smith, H.M. Thorpe, Diversity in the serine recombinases. Mol. Microbiol. 44(2), 299–307 (2002)CrossRefGoogle Scholar
  37. 37.
    D.W. Sumners, Lifting the curtain: using topology to probe the hidden action of enzymes. Not. Am. Math. Soc. 42 (1995), 528–537.MathSciNetzbMATHGoogle Scholar
  38. 38.
    D.W. Sumners, C. Ernst, S.J. Spengler, N.R. Cozzarelli, Analysis of the mechanism of DNA recombination using tangles. Q. Rev. Biophys. 28, 253–313 (1995)CrossRefGoogle Scholar
  39. 39.
    L.-P. Tan, G.Y.J. Chen, S.Q. Yao, Expanding the scope of site-specific protein biotinylation strategies using small molecules. Bioorg. Med. Chem. Lett. 14(23), 5735–5738 (2004)CrossRefGoogle Scholar
  40. 40.
    M. Vazquez, D.W. Sumners, Tangle analysis of Gin site-specific recombination. Math. Proc. Camb. Philos. Soc. 136(3), 565–582 (2004). MR 2055047 (2005d:57013)MathSciNetCrossRefGoogle Scholar
  41. 41.
    J.C. Wang, Untangling the Double Helix (Cold Spring Harbor Laboratory Press, 2009), DNA Entanglement and the Action of the DNA TopoisomerasesGoogle Scholar
  42. 42.
    S.A. Wasserman, N.R. Cozzarelli, Determination of the stereostructure of the product of tn3 resolvase by a general method. Proc. Natl. Acad. Sci. U. S. A. 82(4), 1079–1083 (1985). (English)CrossRefGoogle Scholar
  43. 43.
    S.A. Wasserman, N.R. Cozzarelli, Biochemical topology: applications to DNA recombination and replication. Science 232(4753), 951–960 (1986). (English)CrossRefGoogle Scholar
  44. 44.
    S.A. Wasserman, J.M. Dungan, N.R. Cozzarelli, Discovery of a predicted DNA knot substantiates a model for site-specific recombination. Science 229(4709), 171–174 (1985)CrossRefGoogle Scholar
  45. 45.
    Wikipedia, Deoxyribose—Wikipedia, the free encyclopedia, 2012. Accessed 20 May 2012Google Scholar
  46. 46.
    Wikipedia, DNA—Wikipedia, the free encyclopedia, 2012. Accessed 20 May 2012Google Scholar
  47. 47.
    Wikipedia, Nucleic acid—Wikipedia, the free encyclopedia, 2012. Accessed 20 May 2012Google Scholar
  48. 48.
    G. Witz, A. Stasiak, DNA supercoiling and its role in DNA decatenation and unknotting. Nucl. Acids Res. 38(7), 2119–2133 (2010)CrossRefGoogle Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Wisconsin-Stevens PointStevens PointUSA
  2. 2.Department of MathematicsUniversity of San DiegoSan DiegoUSA

Personalised recommendations