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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.

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.

A generic integrase protein, when acting on circular DNA, often changes the DNA topology by transforming unknotted circles into torus knots and links. Two systems of tangle equations — corresponding to two possible orientations of two DNA subsequences — arise when modelling this transformation.

With no a priori assumptions on the constituent tangles, we utilize Dehn surgery arguments to completely classify the tangle solutions for each of the two systems. A key step is to combine work of our previous paper [10] with recent results of Kronheimer, Mrowka, Ozsváth and Szabó [39] and work of Ernst [23] to show a certain prime tangle must in fact be a Montesinos tangle.

These tangle solutions are divided into three classes, common to both systems, plus a fourth class for one system that contains the sole Montesinos tangle. We discuss the possible biological implications of our classification, and of this novel solution.

Difference topology is a technique used to study any protein that can stably bind to DNA. This technique is used to determine the conformation of DNA bound by protein. Motivated by difference topology experiments, we use the skein relation tangle model as a novel technique to study experiments using topoisomerase to study SMC proteins, a family of proteins that stably bind to DNA. The oriented skein relation involves an oriented knot, $K_{+}$, with a distinguished positive crossing; a knot $K_{-}$, obtained by changing the distinguished positive crossing of $K_{+}$ to a negative crossing; a knot, $K_I$, resulting from the non-orientation persevering resolution of the distinguished crossing; and a link $K_D$, the orientation preserving resolution of the distinguished crossing. We refer to $(K_{+},K_{-},K_D,K_I)$ as the skein quadruple. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. They can act on direct repeat or inverted repeat sites, resulting in a link or knot, respectively. Thus, the skein quadruple is now viewed as $K_{\pm }=$ circular DNA substrate, $K_{\mp }=$ product of topoisomerase action, $K_D=$ product of recombinase action on directed repeat sites, and $K_I=$ product of recombinase action of inverted repeat sites.

Generalized Montesinos tangles are classified, and the system of unoriented tangle equations $N(U+P)=K_1$ and $N(U+R)=K_2$ is solved for a generalized Montesinos tangle $U$ where $P$ and $R$ are rational tangles and $K_1$ and $K_2$ are Montesinos knots/links.

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2\#b_3$, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$ nontrivial. We solve this system of equations under the assumption $\tilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e. $O$ is not $\pi$-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle.

Protein–DNA loops are essential for efficient transcriptional repression and activation. The geometry and stability of the archetypal Lac repressor tetramer (LacI)–DNA loop were investigated using designed hyperstable loops containing lac operators bracketing a sequence-directed bend. Electrophoretic mobility shift assays, DNA cyclization, and bulk and single-molecule fluorescence resonance energy transfer (FRET) demonstrate that the DNA sequence controls whether the LacI–DNA loop forms a compact loop with positive writhe or an open loop with little writhe. Monte Carlo methods for simulation of DNA ring closure were extended to DNA loops, including treatment of variable protein hinge angles. The observed distribution of topoisomer products upon cyclization provides a strong constraint on possible models. The experiments and modeling imply that LacI–DNA can adopt a wide range of geometries but has a strong intrinsic preference for an open form. The flexibility of LacI helps explain in vivo observations that DNA looping is less sensitive to DNA length and shape than that expected from the physical properties of DNA. While DNA cyclization suggests two pools of precursor loops for the 9C14 construct, single-molecule FRET demonstrates a single population. This discrepancy suggests that the LacI–DNA structure is strongly influenced by flanking DNA.

The expression of genes coding for determinants of DNA topology in the facultative intracellular pathogen Salmonella typhimurium was studied during adaptation by the bacteria to the intracellular environment of J774A.1 macrophage-like cells. A reporter plasmid was used to monitor changes in DNA supercoiling during intracellular growth. Induction of the dps and spv genes, previously shown to be induced in the macrophage, was detected, as was expression of genes coding for DNA gyrase, integration host factor and the nucleoid-associated protein H-NS. The topA gene, coding for the DNA relaxing enzyme topoisomerase I, was not induced. Reporter plasmid data showed that bacterial DNA became relaxed following uptake of S. typhimurium cells by the macrophage. These data indicate that DNA topology in S. typhimurium undergoes significant changes during adaptation to the intracellular environment. A model describing how this process may operate is discussed.