A 4-string tangle analysis of DNA-protein complexes based on difference topology

Abstract

An n-string tangle is a three-dimensional ball with n-strings properly embedded in it. In the late 1980s, Ernst and Sumners introduced a tangle model for protein-DNA complexes. The protein is modeled by a three-dimensional ball and the protein-bound DNA is modeled by strings embedded inside the ball. Originally the tangle model was applied to proteins such as Tn3 resolvase which binds two DNA segments. This protein breaks and rejoins two DNA segments and can create knotted DNA. A 2-string tangle model can be used for this complex. More recently, Pathania, Jayaram and Harshey determined that the topological structure of DNA within a Mu protein complex consists of three DNA segments containing five crossings. Since Mu binds DNA sequences at three sites, this Mu protein-DNA complex can be modeled by a 3-string tangle. Darcy, Leucke and Vazquez analyzed Pathania et al.'s experimental results by using 3-string tangle analysis. There are protein-DNA complexes that involve four or more DNA sites. When a protein binds circular DNA at four sites, a protein-DNA complex can be modeled by a 4-string tangle with four loops outside of the tangle. We determine a biologically relevant 4-string tangle model. We also develop mathematics for solving 4-string tangle equations to determine the topology of DNA within a protein complex.