Results on the Classification of Rational 3-Tangles

Cabrera-Ibarra, Hugo

doi:10.1142/s021821650400307x

Cited by 8 publications

(3 citation statements)

References 6 publications

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“…better explained in terms of 3-string tangles. Some efforts to classifying rational 3-string tangles and solving 3-string tangle equations are underway [C1,C2,EE,D1]. In this paper we find tangle solutions for the relevant 3-string tangle equations; we characterize 2 The chirality of the products was only determined when the loxP sites were placed on both sides of the enhancer sequence.…”

Section: Figure 14mentioning

confidence: 89%

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

Darcy

Luecke

Vázquez

2009

Algebr. Geom. Topol.

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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...

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Section: Figure 14mentioning

confidence: 89%

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

Darcy

Luecke

Vázquez

2009

Algebr. Geom. Topol.

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“…Similar mathematics does not yet exist for solving n -string tangle equations for n > 2. Some work has been done on 3-string tangles [ 39 ] and solving 3-string tangles equations involving the class of 3-string tangles called 3-braids [ 40 ]. There is also some work on classifying n -string tangles (for example, [ 41 ]).…”

Section: Discussionmentioning

confidence: 99%

Coloring the Mu transpososome

et al. 2006

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“…Especially, if we use two proper generators of them then we obtain a family of all the rational 3-tangle diagrams presented by the braids group with three strings, B 3 , such as the first diagram in Figure 1. Recently, Cabrera-Ibarra ( [2], [3]) found a pair of invariants which is defined for all rational 3-tangles. Each invariant is a 3×3 matrix with complex number entries.…”

Section: Introductionmentioning

confidence: 99%

On the classification of a large set of rational 3-tangle diagrams

Kwon

2017

J. Knot Theory Ramifications

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Abstract. We note that a rational 3-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational 3-tangle diagrams up to isotopy. ([5]) However, there is no perfect classification about rational 3-tangle diagrams such as the classification of rational 2-tangle diagrams corresponding to rational numbers. In this paper, we classify a large set of rational 3-tangles which are generated by only three generators.

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Results on the Classification of Rational 3-Tangles

Cited by 8 publications

References 6 publications

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

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

Coloring the Mu transpososome

On the classification of a large set of rational 3-tangle diagrams

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