Topological Analysis of Dna-protein Complexes

A tangle consists of strings properly embedded in a 3-dimensional ball. Tangles have been used to model protein-bound DNA. The protein is represented by the 3D ball and the protein-bound DNA is represented by the strings embedded in the 3D ball. We review tangle analysis of protein-DNA complexes involving three or four segments of DNA. 1. Introduction. An n-string tangle is a three dimensional ball with n-strings properly embedded in it. Tangles were studied by Conway in the 1960's [3]. In the 1980's, Ernst and Sumners introduced a mathematical tangle model for protein-bound DNA complexes [8]. In this model, the protein is modeled by a three dimensional ball and the protein-bound DNA is modeled by strings. They used a 2-string tangle model to analyze experimental results for Tn3 resolvase and phage λ integrase. Some proteins can break and rejoin DNA segments and will knot circular DNA molecules. The knot types of the products can be used to determine information regarding how these proteins act. Nick Cozzarelli also used such proteins to study other protein-DNA complexes [13]. Type II topoiso-merases will knot circular DNA by cutting the DNA, allowing a segment of DNA to pass through the break before resealing the DNA. In order to study the protein 13S condensin, DNA was first incubated with 13S con-densin allowing the condensin to bind the DNA. Topoisomerase was then added. A spectrum of knots resulted which was different than that when topoisomerase acts on DNA in the absence of condensin. The difference in the knot spectrum in the presence versus absence of condensin was used to determine the manner in which 13S condensin is bound to DNA. Pathania, Jayaram and Harshey extended these methods to derive the number of DNA crossings trapped in an unknown protein-DNA complex involving multiple DNA segments [15]. This methodology, called difference