1. Isabel K. Darcy Mathematics Department University of Iowa http://www.math.uiowa.edu/~idarcy ©2008 I.K. Darcy. All rights reserved

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5. Rob Scharein’s KnotPlot.com

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

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

10. 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

11. 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

12. Solving tangle equations

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

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

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

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

20. 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

21. The tangle equations corresponding to the electron micrograph:

22. 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.

23. 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

24. 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

26. A difference topology experiment: