Isabel K. Darcy

BIOLOGICAL DISTANCES ON DNA KNOTS AND LINKS: Applications to XER recombination

The mathematics of tangles has been very useful in studying recombinases which act processively and which require DNA to be in a certain configuration in order for the enzyme to act. Electron micrographs of the enzyme-DNA complex show the enzyme as a blob with DNA looping out of it. The configuration of the DNA within the blob cannot be determined form the electron micrographs. However, mathematics can in some cases determine the configuration of the DNA within the enzyme blob as well as the enzyme action. In this paper, several theorems used to analyze recombinase experiments are summarized. In particular Xer recombinase, an enzyme which does not act processively is analyzed. Unfortunately, for enzymes which do not act processively, infinitely many possibilities exist. Several experiments are proposed to reduce this number and to emphasize both the usefulness and limitations of tangle analysis. Although the local action cannot be mathematically determined without more biological assumptions, it is possible to determine the topology of the synaptic complex through additional biolgical experiments.

SOLVING UNORIENTED TANGLE EQUATIONS INVOLVING 4-PLATS

The system of unoriented tangle equations and is completely solved for the tangles U and as a function of where K1 and K2 are 4-plats, and and rational tangles such that |f1g2 - g1f2| > 1. As an application, it is completely determined when one 4-plat can be obtained from another 4-plat via a signed crossing change.

TopoICE-R: 3D visualization modeling the topology of DNA recombination

UNLABELLED TopoICE-R is a three-dimensional visualization and manipulation software for solving 2-string tangle equations and can be used to model the topology of DNA bound by proteins such as recombinases and topoisomerases. AVAILABILITY This software, manual and example files are available at www.knotplot.com/download for Linux, Windows and Mac.

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

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

Rational tangle surgery and Xer recombination on catenanes

The protein recombinase can change the knot type of circular DNA. The action of a recombinase converting one knot into another knot is normally mathematically modeled by band surgery. Band surgeries on a 2-bridge knot N((4mn-1)/(2m)) yielding a (2,2k)-torus link are characterized. We apply this and other rational tangle surgery results to analyze Xer recombination on DNA catenanes using the tangle model for protein-bound DNA.

Coloring the Mu transpososome.

BackgroundTangle analysis has been applied successfully to study proteins which bind two segments of DNA and can knot and link circular DNA. We show how tangle analysis can be extended to model any stable protein-DNA complex.ResultsWe discuss a computational method for finding the topological conformation of DNA bound within a protein complex. We use an elementary invariant from knot theory called colorability to encode and search for possible DNA conformations. We apply this method to analyze the experimental results of Pathania, Jayaram, and Harshey (Cell 2002). We show that the only topological DNA conformation bound by Mu transposase which is biologically likely is the five crossing solution found by Pathania et al (although other possibilities are discussed).ConclusionOur algorithm can be used to analyze the results of the experimental technique described in Pathania et al in order to determine the topological conformation of DNA bound within a stable protein-DNA complex.

Determining the topology of stable protein-DNA complexes.

Difference topology is an experimental technique that can be used to unveil the topological structure adopted by two or more DNA segments in a stable protein-DNA complex. Difference topology has also been used to detect intermediates in a reaction pathway and to investigate the role of DNA supercoiling. In the present article, we review difference topology as applied to the Mu transpososome. The tools discussed can be applied to any stable nucleoprotein complex.

Modeling protein-DNA complexes with tangles

The mathematics of tangles has been applied to model protein-DNA binding. A tangle consists of strings properly embedded in a 3-dimensional ball. The protein complex can be thought of as a 3D ball while the DNA segments bound by the protein complex can be thought of as strings embedded within the ball. This simple model can be used to determine the topology of protein-bound DNA. We review some tangle models and related software for modeling recombinase and topoisomerase action.

Applications of topology to DNA

0. Some preliminaries. Many biological processes affect topological properties of DNA. Duplex DNA consists of two backbone strands wound about each other in a righthanded helical fashion. Each strand consists of a sugar phosphate backbone with a nitrogenous base attached to each sugar. The four possible bases are adenine (A), guanine (G), cytosine (C), and thymine (T). The two strands are held together by hydrogen bonding between the bases with A always paired with T, and G always paired with C. The DNA of most bacteria and viruses is circular. Although human DNA is linear, it is extremely long and tacked down to a protein scaffold at various points on the DNA. This periodic attachment endows human DNA with topological constraints similar to those for circular DNA. These topological constraints can interfere with vital metabolic cellular processes such as replication and transcription. Most mathematicians have, at some point, taken a strip of paper, put an even number of twists in it before taping the ends together, and cut the strip down the middle. The result is two linked strips of paper. This is what occurs when DNA replicates if one thinks of the two edges of the strip as being the sugar phosphate backbones of the two strands of DNA. Hence, enzymes are required to solve these topological entanglement problems which arise through cellular metabolism and replication. In this case topoisomerases, which are enzymes that mediate the passage of one segment of DNA through an enzyme-bridged transient break in the backbone strands of another DNA segment, are responsible for unlinking the DNA. Other enzymes

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.