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

Abstract

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. 57M25, 92C40 IntroductionBacteriophage Mu is a virus that infects bacteria. Mu transposase is involved in transposing the Mu genome within the DNA of the bacterial host. In [38], Pathania et al determined the shape of DNA bound within the Mu transposase protein complex using an experimental technique called difference topology; see Harshey and Jayaram [26;30], Kilbride et al [31], Grainge et al [24], Pathania et al [38;39] and Yin et al [59;60]. Their conclusion was based on the assumption that the DNA is in a branched supercoiled form as described near the end of Section 1 (see Figures 9 and 25). We show that this restrictive assumption 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 found in [38] (shown in Figure 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.The topological structure of the Mu transpososome will be a key ingredient in determining its more refined molecular structure and in understanding the basic mechanism crystallographic data is combined with the known Tn3 resolvase topological mechanisms of binding and strand-exchange to piece together a more detailed geometrical picture of two resolvase-DNA complexes (Sin and ı ).In Section 1 we provide some biological background and describe eight difference topology experiments from [38]. In Section 2, we translate the biological problem of determining the shape of DNA bound by Mu into a mathematical model. The mathematical model consists of a system of ten 3-string tangle equations ( Figure 11). Using 2-string tangle analysis, we simplify this to a system of four tangle equations ( Figure 24). In Section 3 we characterize solutions to these tangle equations in terms of knotted graphs. This allows us to exhibit infinitely many different 3-string tangle solutions. The existence of solutions different from the PJH solution raises the possibility of alternate acceptable models. In Sections 3-5, we show that all solutions to the mathematical problem other than the PJH solution are too complex to be biologically reasonable, where the complexity is measured either by the rationality or by the minimal crossing number of the 3-string tangle solution.In Section 3, we show that the only rational solution i...

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

“…In section 3.3, we explain the methodology of difference topology and its application to a Mu protein-DNA complex. In section 3.4, we summarize the 3-string tangle analysis of the Mu protein-DNA complex in [6]. Finally, in chapter 4, we introduce a 4-string tangle model for a protein which binds four DNA segments.…”

“…[6] Let T be an n-string tangle. T is split if there is a properly embedded disk separating some of its strands from other strands ( In particular, when n = 2, much theory was developed which was initiated by John H. Conway [3].…”

“…In the next section we will discuss the biological model for a Mu protein-DNA complex given in [15], while in section 5 we will summarize the mathematical tangle analysis given in [6].…”

“…One can prove that the solution set for T to the system of three equations in Figure 3.8a is the same as the solution set for T if all six experiments are considered [5,6].…”

“…3-string tangle analysis is much more complicated than 2-string tangle analysis. The experimental results in [15] were mathematically [6] and computationally [5] analyzed by using a 3-string tangle model.…”