Isabel K. Darcy

Mathematics Department University of Iowahttp://www.math.uiowa.edu/~idarcy

©2008 I.K. Darcy. All rights reserved

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Rob Scharein’s

KnotPlot.com

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

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

Statistics of knots, geometry of conformations, and evolution of proteins.Rhonald C. Lua, Alexander Y. Grosberg PLoS Comput Biol. 2006 May;2(5)

unknot 3.1 4.1 5.2

Direct 4516 164 9 3

Center 4692 20 3 1

Random 4697 15 0 1

and more complicated knots for random closure

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

Solving tangle equations

A + B = C

2 + 0 = 2

2 + -2 = 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

1 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5

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

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 complexHeichman 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.

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: