Solving tangle equations arising in a DNA recombination model

Ernst, Claus; Sumners, D. W.

doi:10.1017/s0305004198002989

Cited by 42 publications

(39 citation statements)

References 7 publications

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“…Theorem 1·5 ( [13,22] These results will be used in Section 4 to prove the theorems for Gin action. In the next section we introduce the Gin recombination system.…”

Section: Given Tanglesmentioning

confidence: 94%

“…Determining when the intervening tangles are rational, or sums of rational tangles is not easy. Ernst and Sumners have extensively studied this problem [10][11][12][13]. The next few concepts and results will be used in Section 4 to prove that the tangles O and R arising in the system corresponding to Gin recombination are rational.…”

Section: Fig 2 Definitionmentioning

confidence: 99%

“…The work of Ernst and Sumners allows to find solutions for O and R that are rational or sums of rational tangles, when substrate and products of recombination are known 4-plats [10][11][12][13]. If only two equations are provided, it is sometimes impossible to find a unique solution pair (O, R) for the equations.…”

Section: Applying the Tangle Model To Site-specific Recombinationmentioning

confidence: 99%

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Tangle analysis of Gin site-specific recombination

Vázquez¹,

Sumners²

2004

Math. Proc. Camb. Phil. Soc.

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We use the tangle model to study the action of the site-specific recombinase Gin, an enzyme that can introduce topological changes on circular DNA molecules. Gin and its bound DNA are modelled as a 2-string tangle which undergoes changes during recombination, thereby changing the topology of the DNA substrate. We show that the tangles involved in the analysis are all rational tangles. This technique allows us to prove that, under the model's assumptions, there is a unique topological description of the enzymatic action. The Gin system is one of the few to date where tangle analysis can be carried out systematically and rigorously, yielding a single, biologically reasonable solution. Preview and motivationSite-specific recombination alters the genome of an organism by moving, inserting or inverting DNA segments. When acting on circular DNA molecules, site-specific recombinases can change the topology and geometry of the molecules [25,31]. The tangle model provides mathematical tools to analyze such changes [26,27,28]. The detailed tangle formalism and an application of the model to the Tn3 resolvase system were first presented in [10].Here we analyze the enzymatic action of Gin, a site-specific recombinase encoded by bacteriophage Mu (a virus that infects bacteria). Gin inverts a DNA segment within the phage genome to extend the phage's range of infection. The phage genome contains two recombination sites that Gin recognizes, cleaves and rearranges to complete one round of recombination (Figure 1). Gin recombination is processive, i.e. it can carry out more than one recombination round while binding only once to its substrate. We study the case where, prior to recombination, the phage genome is a single circular DNA molecule, effectively the unknot in S 3 . Recombination can then be analyzed by using tangles to model an enzyme such as Gin together with the DNA bound to the enzyme (see: [29]). Tangles, defined formally below, are

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“…Theorem 1·5 ( [13,22] These results will be used in Section 4 to prove the theorems for Gin action. In the next section we introduce the Gin recombination system.…”

Section: Given Tanglesmentioning

confidence: 94%

Section: Fig 2 Definitionmentioning

confidence: 99%

Section: Applying the Tangle Model To Site-specific Recombinationmentioning

confidence: 99%

See 1 more Smart Citation

Tangle analysis of Gin site-specific recombination

Vázquez¹,

Sumners²

2004

Math. Proc. Camb. Phil. Soc.

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“…other prime or locally knotted) can be more difficult or even impossible. 16,28 Results and Discussion…”

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confidence: 99%

Tangle Analysis of Xer Recombination Reveals only Three Solutions, all Consistent with a Single Three-dimensional Topological Pathway

Vázquez

Colloms

Sumners

2005

Journal of Molecular Biology

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The product of Xer recombination at directly repeated psi sites on a circular unknotted DNA molecule is a right-hand four-noded catenane. Here, we use tangle equations to analyze the topological changes associated with Xer recombination at psi. This mathematical method allows computation of all possible topological pathways consistent with the experimental data. We give a rigorous mathematical proof that, under reasonable biological assumptions, there are only three solutions to the tangle equations. One of the solutions corresponds to a synaptic complex with antiparallel alignment of recombination core sites, the other two correspond to parallel alignment of cores. We show that all three solutions can be unified into a single three-dimensional model for Xer recombination. Thus the three distinct mathematical solutions do not necessarily represent distinct three-dimensional pathways, and in this case the three distinct tangle solutions are different planar projections of the same three-dimensional configuration. Keywords: site-specific recombination; Xer recombination; topological mechanism; tangle equations; DNA knots *Corresponding author IntroductionSite-specific recombinases catalyze the exchange of genetic material between specific sites on a DNA molecule. This recombination mediates a number of biologically important processes including: integration and excision of viral DNA into and out of its host genome; inversion of DNA segments to regulate gene expression; resolution of multimeric DNA molecules to allow proper segregation at cell division; and plasmid copy number regulation. 1,2 Recombination reactions can be divided into two stages, synapsis and strand-exchange. In the first stage, the two recombination sites are brought together by the recombinase and accessory proteins to form a specific synaptic complex. In the second stage, the DNA strands are cleaved, exchanged and religated within this synapse, to form recombinant products. When the DNA substrate is circular, topological changes associated with recombination can be observed and quantified. These topological changes can be used to infer topological details of the synaptic complex and the strand-exchange mechanism, 3,4 and can be analyzed mathematically using the tangle method (reviewed in the next section). Site-specific recombinases are divided into two families, based on sequence similarity and reaction mechanism: the tyrosine recombinases, which include Flp, Cre, Int and XerC/XerD and recombine through a Holliday junction (HJ) intermediate; 6,7 and the serine recombinases, which include Tn3 and gd resolvases, as well as DNA invertases such as Gin, and recombine via double-strand cleaved intermediates. 8Many serine recombinases display topological selectivity, i.e. they distinguish between sites in different orientations, and between inter-and intramolecular sites. These systems also display topological specificity, i.e. the topology of the substrate uniquely determines the topology of the reaction product. Topological specificity implies that...

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“…2. In DNA, strand switching using topoisomerase of types I and II is vital for the structure of DNA replication [50]. The mixed interaction of topological change and physical evolution of the molecules in vitro may benefit from a mixed state summation that averages quantities respecting the hierarchy of interactions.…”

Section: Reconnection (In Vortices)mentioning

confidence: 99%

Skein Invariants of Links and Their State Sum Models

Kauffman

Lambropoulou

2017

Symmetry

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We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally.

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Solving tangle equations arising in a DNA recombination model

Cited by 42 publications

References 7 publications

Tangle analysis of Gin site-specific recombination

Tangle analysis of Gin site-specific recombination

Tangle Analysis of Xer Recombination Reveals only Three Solutions, all Consistent with a Single Three-dimensional Topological Pathway

Skein Invariants of Links and Their State Sum Models

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