# TOPOLOGICAL ANALYSIS OF MU-TRANSPOSITION

• Accepted : 2013.02.22
• Published : 2013.06.25

#### Abstract

An n-string tangle is a three dimensional ball with n-strings which are properly embedded in the ball. In early 90's, C. Ernst and D. Sumners first used a tangle to describe a DNA-protein complex. In this model, DNA is represented by a string and protein is represented by a ball. Mu is a protein which binds to DNA at three sites and a DNA-Mu complex is called Mu-transpososome. Knowing the DNA topology within Mu-transpososome is very important to understand DNA transposition by Mu protein. In 2002, Pathania et al. determined that the DNA configuration within the Mu transpososome is three branched and five noded [12]. In 2007, Darcy et al. analyzed this by using mathematical tangle and concluded that the three branched and five noded DNA configuration is the only biologically reasonable solution [4]. In this paper, based on the result of Pathania et al. and Darcy et al., the author determines the DNA topology within the DNA-Mu complex after the whole Mu transposition process. Furthermore, a new experiment is designed which can support the Pathania et al.'s result. The result of this new experiment is predicted through mathematical knot thory.

#### References

1. C. Adams, The knot book: an elementary introduction to the mathematical theory of knots, American mathematical society, 1st ed., Providence, Rhode Island, 2004.
2. A.D. Bates and A. Maxwell, DNA Topology, IRL Press Oxford, 1993.
3. I. K. Darcy, A. Bhutra, J. Chang, N. Druivenga, C. McKinney, R. K. Medikonduri, S. Mills, J. Navarra Madsen, A. Ponnusamy, J. Sweet, and T. Thompson, Coloring the Mu transpososome, BMC Bioinformatics, 7:Art. No. 435 (2006).
4. I.K. Darcy, J. Luecke, and M. Vazquez, Tangle analysis of difference topology experiments: Applications to a Mu protein-dna complex, Algebraic and Geometric Topology, 9 (2009), 2247-2309. https://doi.org/10.2140/agt.2009.9.2247
5. F.B. Dean, A. Stasiak, T. Koller, and N.R. Cozzarelli, Duplex DNA knots produced by Escherichia Coli topoisomerase I, J. Biol. Chem., 260 (1985), 4795-4983.
6. C. Ernst and D.W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc., 108 (1990), 489-515. https://doi.org/10.1017/S0305004100069383
7. C. Ernst and D. W. Sumners, Solving tangle equations arising in a DNA recombination model, Math. Proc. Camb. Phil. Soc., 126 (1990), 23-36.
8. International Human Genome Sequencing Consortium Initial sequencing and analysis of the human genome, Nature, 409 (2001), 860-921. https://doi.org/10.1038/35057062
9. S. Kim and I. Darcy, Topological Analysis of DNA-protein complexes, Mathematics of DNA Structure, function and interacctions, The IMA volumes in Mathematics and its applications, Springer Science + Business Media, LLC, New York, 2009
10. S. Kim, A 4-string tangle analysis of DNA-protein complexes based on difference topology, Ph.D thesis (2009)
11. B. Lewin, Genes, Oxford University Press, 5th ed., New York, 1994.
12. S. Pathania, M. Jayaram, and R. M. Harshey, Path of DNA within the Mu transpososome: transposase interactions bridging two Mu ends and the enhancer trap five DNA supercoils, Cell, 109(4) (2002), 425-436. https://doi.org/10.1016/S0092-8674(02)00728-6
13. S. Pathania, M. Jayaram, and R. M. Harshey, The Mu Transpososome Through a Topological Lens, Critical Reviews in Biochemistry and Molecular Biology, 41 (2006), 387-405. https://doi.org/10.1080/10409230600946015
14. D. Rolfsen, Knots And Links, AMS Chelsea Publishing, Providence, Rhode Island, 1976.
15. Y. Saka and M. Vazquez, Tanglesolve: topological analysis of site-specific recombination, Bioinformatics, 18 (2002), 1011-1012. https://doi.org/10.1093/bioinformatics/18.7.1011
16. 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. https://doi.org/10.1016/S0092-8674(85)80128-8
17. D. W. Sumners, C. Ernst, N.R. Cozzarelli, and S.J. Spengler, Mathematical analysis of the mechanisms of DNA recombination using tangles, Quarterly Reviews of Biophysics, 28 (1995).
18. Alexandre A. Vetcher, Alexander Y. Lushnikov, Junalyn Navarra-Madsen, Robert G, Scharein, Yuri L. Lyubchenko, Isabel K. Darcy, and Stephen D. Levene, DNA topology and geometry in Flp and Cre recombination, J. Mol. Biol., 357 (2006), 1089-1104. https://doi.org/10.1016/j.jmb.2006.01.037
19. A. V. Vologodskii Circular DNA. Annual Review of Biochemistry, John Wiley and Sons Inc, 40 (1971), 899- 942. https://doi.org/10.1146/annurev.bi.40.070171.004343
20. S.A. Wasserman and N.R. Cozzarelli, Determination of the stereostructure of the product of Tn3 resolvase by a general method, Proc. Nat. Acad. Sci. U.S.A., 82 (1985), 1079-1083. https://doi.org/10.1073/pnas.82.4.1079