Isabel K. Darcy Mathematics Department University of Iowa ©2008 I.K. Darcy. All rights reserved

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Rob Scharein’s KnotPlot.com

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

Statistics of knots, geometry of conformations, and evolution of proteins. Rhonald C. Lua, Alexander Y. Grosberg PLoS Comput Biol May;2(5) unknot Direct Center Random and more complicated knots for random closure

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

Solving tangle equations

A + B = C = = 0 Most tangles don’t have inverses

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

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

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

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

The tangle equations corresponding to the electron micrograph:

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), protein = three dimensional ball protein-bound DNA = strings.

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

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

A difference topology experiment: