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

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 involving two or more DNA segments. 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. In [PJH], Pathania et al determined the shape of DNA bound within the Mu transposase protein complex using an experimental technique called difference topology [HJ, KBS, GBJ, PJH, PJH2, YJPH, YJH] and by making certain assumptions regarding the DNA shape. We show that their most restrictive assumption (the plectonemic form described near the end of section 1) is not needed, and in doing so, conclude that the only biologically reasonable solution for the shape of DNA bound by Mu transposase is the one they found [PJH] (Figure 0.1). We will call this 3-string tangle the PJH solution. The 3-dimensional ball represents the protein complex, and the arcs represent the bound DNA. The Mu-DNA complex modeled by this tangle is called the Mu transpososome