×

Untangling DNA. (English) Zbl 0712.57002

“Knot theory - a new kind of applicable mathematics!” This starts an excellent survey article which provides the reader with a glimpse of a topological approach to enzymology, the geometry of DNA, site-specific recombination, tangles and 4-plats, Tn3 resolvase, and Phage \(\lambda\) Integrase. Resolvase recombinations lead to tangle equations and it is shown how the Cyclic Surgery Theorem of Culler, Gordon, Luecke-Shalen and the Determination of Knots by their Complements Theorem of Gordon-Luecke can be used to solve certain such tangle equations for the tangle factors. The author rightfully concludes the article with the hope that he has provided the reader with some reason to question a statement of Cipra [Science 241, 1291-1292 (1988)]: “Gordon and Luecke’s solution to Tietze’s 80 year-old problem has no immediate practical or even theoretical applications.”
Reviewer: W.Heil

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
92C40 Biochemistry, molecular biology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] W. R. Bauer, F. H. C. Crick, and J. H. White, Supercoiled DNA,Scientific American 243 (1980), 100–113.
[2] G. Burde and H. Zieschang,Knots, Berlin, de Gruyter (1985).
[3] B. A. Cipra, To have and have knot: when are knots alike?,Science 241 (1988), 1291–1292. · Zbl 1226.57009
[4] J. H. Conway, On enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra, Proc. Con). Oxford 1967, Pergamon (1970), 329–358.
[5] M. C. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen, Dehn surgery on knots,Ann. of Math. 125 (1987). 237–300. · Zbl 0633.57006
[6] E. E. David, Jr., Renewing U.S. mathematics: an agenda to begin the second century,Notices of the A.M.S. 35 (1988), 1119–1123.
[7] C. Ernst and D. W. Sumners,A calculus for rational tangles: applications to DNA recombination, preprint, Florida State University (1988). · Zbl 0727.57005
[8] D. Gabai, Foliations and surgery on knots,Bull. A.M.S. 15 (1986), 83–97. · Zbl 0606.57014
[9] C. M. Gordon and J. Luecke, Knots are determined by their complements, preprint, University of Texas (1988). · Zbl 0672.57009
[10] V. F. R. Jones, A polynomial invariant for knots and links via Von Neumann Algebras,Bull. A.M.S. 12 (1985), 103–111. · Zbl 0564.57006
[11] L. H. Kauffman, On knots,Ann. of Math. Studies 115, Princeton Univ. Press (1987). · Zbl 0627.57002
[12] M. A. Krasnow, A. Stasiak, S. J. Spengler, F. Dean, T. Koller, and N. R. Cozzarelli, Determination of the absolute handedness of knots and catenanes of DNA,Nature 304 (1983), 559–560.
[13] W. B. R. Lickorish, Polynomials for links,Bull. L.M.S. 20 (1988), 558–588. · Zbl 0685.57001
[14] W. B. R. Lickorish, Prime knots and tangles,Trans. A.M.S. 267 (1981), 321–332. · Zbl 0472.57004
[15] R. Lipkin, Unraveling more than just knots,Insight 4 (Oct. 24, 1988), 54–55.
[16] W. F. Pohl, DNA and differential geometry,The Mathematical Intelligencer 3 (1980), 20–27. · Zbl 0447.92013
[17] J. Simon, Topological chirality of certain molecules,Topology 25 (1986), 229–235. · Zbl 0615.57005
[18] S. J. Spengler, A. Stasiak, and N. R. Cozzarelli, The stereostructure of knots and catenanes produced by phage {\(\lambda\)} integrative recombination: implications for mechanism and DNA structure,Cell 42 (1985), 325–334.
[19] L. A. Steen, The science of patterns,Science 240 (1988), 611–616. · Zbl 1226.00006
[20] D. W. Sumners, The role of knot theory in DNA research,Geometry and Topology, Manifolds, Varieties and Knots (C. McCrory, T. Schifrin, eds.), New York: Marcel Dekker (1987), 297–318.
[21] D. W. Sumners, Knots, macromolecules and chemical dynamics,Graph Theory and Topology in Chemistry (R. B. King and D. H. Rouvray, eds.), Studies in Physical and Theoretical Chemistry 51, New York: Elsevier (1987), 3–22.
[22] W. Thompson, On vortex atoms,Philosophical Magazine 34 (July, 1867), 15–24.
[23] D. M. Walba, Topological stereochemistry,Tetrahedron 41 (1985), 3161–3212.
[24] J. C. Wang, DNA topoisomerases,Scientific American 247 (1982), 94–109.
[25] J. H. White, K. C. Millett, and N. R. Cozzarelli, Description of the topological entanglement of DNA catenanes and knots by a powerful method involving strand passage and recombination,J. Mol. Biology 197 (1987), 585–603.
[26] S. A. Wasserman and N. R. Cozzarelli, Biochemical topology: applications to DNA recombination and replication,Science 232 (1986), 951–960.
[27] S. A. Wasserman, J. M. Dungan, N. R. Cozzarelli, Discovery of a predicted DNA knot substantiates a model for site-specific recombination,Science 229 (1985), 171–174.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.