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Tangle analysis of difference topology experiments: Applications to a Mu protein-DNA complex

Isabel K Darcy, John Luecke and Mariel Vazquez

Algebraic & Geometric Topology 9 (2009) 2247–2309

We develop topological methods for analyzing difference topology experiments involving 3–string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number.

$3$–string tangle, DNA topology, tangle method, difference topology, Mu transpososome, graph planarity, Dehn surgery, handle addition lemma
Mathematical Subject Classification 2000
Primary: 57M25, 92C40
Received: 31 October 2007
Revised: 14 July 2009
Accepted: 20 July 2009
Published: 30 October 2009
Isabel K Darcy
Department of Mathematics
University of Iowa
Iowa City, IA 52242
John Luecke
Department of Mathematics
The University of Texas at Austin
1 University Station C1200
Austin, TX 78712-0257
Mariel Vazquez
Department of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132