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An n-string tangle is a three dimensional ball with n-strings properly embedded in it. In late the 80's, C. Ernst and D. Sumners introduced a tangle model of protein-DNA complexes. This model assumes that the protein is a 3-dimensional ball and the protein-bound DNA are strings embedded inside the ball.
Originally the tangle model was applied to proteins such as Cre recombinate which binds two DNA segments. The protein breaks and rejoins the DNA segments and then creatss knotted DNA. When this kind of protein complex bounds circular DNA, there will be two DNA loops outside of the DNA-protein complex. Hence we can use a 2-string tangle model for this complex. More recently, Pathania, Jayaram and Harshey predicted that the topological structure within the Mu protein complex consists of three DNA segments containing five crossigs. Since Mu binds DNA sequences at 3 sites, the 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-tangle analysis.
Based on the 3-string tangle analysis of Mu protein-DNA complex, we addressed the possibility that a protein binds DNA sequences at four sites. Such a protein complex bound to a circular DNA molecule is modeled by a 4-string tangle with four loops outside of the tangle. In this thesis, we decided a biologically relevant 4-string tangle model. We also developed mathematics for solving tangle equations to predict the topology of DNA within the protein complex.