Bridge number and integral Dehn surgery

In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair (R,K), we consider the knot K^n gotten by twisting K n times along R and give a lower bound on the bridge number of K^n with respect to Heegaard splittings of M -- as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of K^n tends to infinity with n. In application, we look at a family of knots K^n found by Teragaito that live in a small Seifert fiber space M and where each K^n admits a Dehn surgery giving the 3-sphere. We show that the bridge number of K^n with respect to any genus 2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving the 3-sphere

Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.Comment: 76 page, 48 figures; referee comments incorporated and typos fixed; accepted at TAM

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 involving two or more DNA segments. 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.Comment: 60 pages, 74 figure

Knots with unknotting number 1 and essential Conway spheres

For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K) contains an EM-tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway's table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsvath-Szabo, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram. As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.Comment: This is the version published by Algebraic & Geometric Topology on 19 November 200

Crystal structure of the Anabaena sensory rhodopsin transducer.

We present crystal structures of the Anabaena sensory rhodopsin transducer (ASRT), a soluble cytoplasmic protein that interacts with the first structurally characterized eubacterial retinylidene photoreceptor Anabaena sensory rhodopsin (ASR). Four crystal structures of ASRT from three different spacegroups were obtained, in all of which ASRT is present as a planar (C4) tetramer, consistent with our characterization of ASRT as a tetramer in solution. The ASRT tetramer is tightly packed, with large interfaces where the well-structured beta-sandwich portion of the monomers provides the bulk of the tetramer-forming interactions, and forms a flat, stable surface on one side of the tetramer (the beta-face). Only one of our four different ASRT crystals reveals a C-terminal alpha-helix in the otherwise all-beta protein, together with a large loop from each monomer on the opposite face of the tetramer (the alpha-face), which is flexible and largely disordered in the other three crystal forms. Gel-filtration chromatography demonstrated that ASRT forms stable tetramers in solution and isothermal microcalorimetry showed that the ASRT tetramer binds to ASR with a stoichiometry of one ASRT tetramer per one ASR photoreceptor with a K(d) of 8 microM in the highest affinity measurements. Possible mechanisms for the interaction of this transducer tetramer with the ASR photoreceptor via its flexible alpha-face to mediate transduction of the light signal are discussed

Structural basis of human LRG1 recognition by Magacizumab, a humanized monoclonal antibody with therapeutic potential

The formation of new dysfunctional blood vessels is a crucial stage in the development of various conditions such as macular degeneration, diabetes, cardiovascular disease, neurological disease and inflammatory disorders, as well as during tumor growth, eventually contributing to metastasis. An important factor involved in pathogenic angiogenesis is leucine-rich α-2-glycoprotein 1 (LRG1), the antibody blockade of which has been shown to lead to a reduction in both choroidal neovascularization and tumor growth in mouse models. In this work, the structural interactions between the LRG1 epitope and the Fab fragment of Magacizumab, a humanized function-blocking IgG4 against LRG1, are analysed, determining its specific binding mode and the key residues involved in LRG1 recognition. Based on these structural findings, a series of mutations are suggested that could be introduced into Magacizumab to increase its affinity for LRG1, as well as a model of the entire Fab–LRG1 complex that could enlighten new strategies to enhance affinity, consequently leading towards an even more efficient therapeutic