1. Isabel K. Darcy Mathematics Department Applied Mathematical and Computational Sciences (AMCS) University of Iowa http://www.math.uiowa.edu/~idarcy ©2008 I.K. Darcy. All rights reserved This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285).

2. ==

3. ==

4. =

5. Rob Scharein’s KnotPlot.com

8. http://updatecenter.britannica.com/art?assemblyId=91&type=A

9. Duplex DNA knots produced by Escherichia coli topoisomerase I. Dean FB, Stasiak A, Koller T, Cozzarelli NR., J Biol Chem. 1985 Apr 25;260(8):4975-83.Dean FBStasiak A Koller TCozzarelli NR

10. Intricate Knots in Proteins: Function and Evolution Peter Virnau, Leonid A. Mirny, and Mehran Kardar, PLoS Comput Biol. 2006 September; 2(9): e122.

11. Statistics of knots, geometry of conformations, and evolution of proteins. Rhonald C. Lua, Alexander Y. Grosberg PLoS Comput Biol. 2006 May;2(5) unknot3.14.1 5.2 Direct451616493 Center 46922031 Random4697150 1 and more complicated knots for random closure

12. Mathematical Model Protein = DNA = = ==

13. http://www.ch.cam.ac.uk/magnus/ molecules/nucleic/dna1.jpg http://www.accessexcellence.org/R C/VL/GG/images/dna_replicating.g if

14. (J. Mann) http://www.sbs.utexas.edu/herrin/bio344/ Postow L. et.al. PNAS;2001;98:8219-8226

15. Cellular roles of DNA topoisomerases: a molecular perspectiveCellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, 430- 440 (June 2002) Topoisomerase II performing a crossing change on DNA:

16. Mathematical Model Protein = DNA = = ==

17. Protein-DNA complex Heichman and Johnson C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489-515. protein = three dimensional ball protein-bound DNA = strings. Slide (modified) from Soojeong Kim

18. Many mathematicians solve equations x + 3 = 5 x = 2 is a solution: 2 + 3 = 5 x = 1 is not a solution: 1 + 3 = 4 = 5

19. Solving tangle equations Equation A is from: Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils, Cell 2002 Shailja Pathania, Makkuni Jayaram and Rasika Harshey

20. New Pathway (post tangle analysis): Interactions of Phage Mu Enhancer and Termini that Specify the Assembly of a Topologically Unique Interwrapped Transpososome. Zhiqi Yin, Asaka Suzuki, Zheng Lou, Makkuni Jayaram and Rasika M. Harshey Path of DNA within the Mu Transpososome: Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils. Shailja Pathania, Makkuni Jayaram and Rasika M Harshey Cell 2002 Old Pathway :

21. Solving tangle equations Equation A is from: Path of DNA within the Mu Transpososome Transposase Interactions Bridging Two Mu Ends and the Enhancer Trap Five DNA Supercoils, Cell 2002 Shailja Pathania, Makkuni Jayaram and Rasika Harshey

22. A difference topology experiment:

27. x 2 + x 3 = 4

28. x 1 + x 3 = 3 x 1 + x 2 = 3 implies x 1 = 1, x 2 = 2, x 3 = 2 Thus T =

31. Theorem [ Darcy, I. K., Luecke, J., Vazquez, M., Tangle analysis of difference topology experiments: applications to a Mu protein-DNA complex ]: If T is a solution to the Mu-Cre equations, and if either Darcy, I. K., Luecke, J., Vazquez, M., Tangle analysis of difference topology experiments: applications to a Mu protein-DNA complex 1.) T is rational or split or has parallel strands or 2.) If T has fewer than 9 crossings, Then T = (assuming a particular handedness of the products)

32. Also solved up to 8 crossings computationally in Darcy, I. K., Bhutra, A., Chang, J., Druivenga, N., McKinney, C., Medikonduri, R. K., Mills, S., Navarra Madsen, J., Ponnusamy, A., Sweet, J., Thompson, T., Coloring the Mu Transpososome, BMC Bioinformatics. 2006 Oct 5;7:435.Coloring the Mu Transpososome

33. How to solve tangle equations 1.) Brute force Ex: solve x + 3 = 5 0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5

34. How to solve tangle equations 1.) Brute force Ex: solve x + 3 = 5 0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5 2.) Make simplifying assumptions

35. How to solve tangle equations 1.) Brute force Ex: solve x + 3 = 5 1 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5 2.) Make simplifying assumptions 3.) Use mathematics

36. A + B = C 2 + 0 = 2 2 + -2 = 0 Most tangles don’t have inverses

37. Some tangles (but not all) can be classified using fractions.

38. Which tangles are rational?

39. This one is not rational. The others are all rational

44. Rob Scharein’s KnotPlot.com

45. Cellular roles of DNA topoisomerases: a molecular perspectiveCellular roles of DNA topoisomerases: a molecular perspective, James C. Wang, Nature Reviews Molecular Cell Biology 3, 430- 440 (June 2002) Topoisomerase II performing a crossing change on DNA:

46. 3D visualization software to analyze topological outcomes of topoisomerase reactions I. K. Darcy, R. G. Scharein and A. Stasiak Nucleic Acids Research 2008 36(11):3515-3521

47. Cover: Visual presentation of knot distance metric created using the software TopoICE-X within KnotPlot. A pair of knots in this graph is connected by an edge if they can be converted into one another via a single intersegmental passage. This graph shows all mathematically possible topoisomerase reaction pathways involving small crossing knots. Darcy et al. (Nucleic Acids Res., 2008; 36: 3515–3521).3515–3521

48. Acknowledgements: Rob Scharein Andrzej Stasiak John Luecke Mariel Vazquez Steve Levene Bhutra, A., Chang, J., Druivenga, N., McKinney, C., Medikonduri, R. K., Mills, S., Navarra Madsen, J., Ponnusamy, A., Sweet, J., Thompson, T.

49. A protein bound to two segments of DNA can be modeled by a tangle. An electron micrograph of the Flp DNA complex is shown below: Electron micrograph courtesy of Kenneth Human and Steve Levene

50. The tangle equations corresponding to the electron micrograph:

51. Protein-DNA complex Heichman and Johnson C. Ernst, D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Camb. Phil. Soc. 108 (1990), 489-515. protein = three dimensional ball protein-bound DNA = strings.