Isabel K. Darcy

Mathematics Department Applied Mathematical and Computational Sciences (AMCS)University of Iowahttp://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).

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Rob Scharein’s

KnotPlot.com

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

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.

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

Mathematical Model

Protein =

DNA =

=

==

http://www.ch.cam.ac.uk/magnus/molecules/nucleic/dna1.jpg

http://www.accessexcellence.org/RC/VL/GG/images/dna_replicating.gif

(J. Mann) http://www.sbs.utexas.edu/herrin/bio344/

Postow L. et.al. PNAS;2001;98:8219-8226

Cellular 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:

Mathematical Model

Protein =

DNA =

=

==

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.

Slide (modified) from Soojeong Kim

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

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

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:

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

A difference topology experiment:

x2 + x3 = 4

x2 + x3 = 4

x1 + x3 = 3

x1 + x2 = 3

implies

x1 = 1,

x2 = 2,

x3 = 2

Thus T =

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

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)

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.

How to solve tangle equations

1.) Brute forceEx: solve x + 3 = 5

0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5

How to solve tangle equations

1.) Brute forceEx: solve x + 3 = 5

0 + 3 = 5, 1 + 3 = 5, 2 + 3 = 5

2.) Make simplifying assumptions

How to solve tangle equations

1.) Brute forceEx: solve x + 3 = 5

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

2.) Make simplifying assumptions

3.) Use mathematics

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.

Which tangles are rational?

Which tangles are rational?

This one is not rational.The others are all rational

Rob Scharein’s

KnotPlot.com

Cellular 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:

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

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).

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.

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.