142 research outputs found

    On the multiple Borsuk numbers of sets

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    The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

    The Fermat-Torricelli problem in normed planes and spaces

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    We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

    Quantum Imaging with Incoherently Scattered Light from a Free-Electron Laser

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    The advent of accelerator-driven free-electron lasers (FEL) has opened new avenues for high-resolution structure determination via diffraction methods that go far beyond conventional x-ray crystallography methods. These techniques rely on coherent scattering processes that require the maintenance of first-order coherence of the radiation field throughout the imaging procedure. Here we show that higher-order degrees of coherence, displayed in the intensity correlations of incoherently scattered x-rays from an FEL, can be used to image two-dimensional objects with a spatial resolution close to or even below the Abbe limit. This constitutes a new approach towards structure determination based on incoherent processes, including Compton scattering, fluorescence emission or wavefront distortions, generally considered detrimental for imaging applications. Our method is an extension of the landmark intensity correlation measurements of Hanbury Brown and Twiss to higher than second-order paving the way towards determination of structure and dynamics of matter in regimes where coherent imaging methods have intrinsic limitations

    Crystal structure of rhodopsin bound to arrestin by femtosecond X-ray laser.

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    G-protein-coupled receptors (GPCRs) signal primarily through G proteins or arrestins. Arrestin binding to GPCRs blocks G protein interaction and redirects signalling to numerous G-protein-independent pathways. Here we report the crystal structure of a constitutively active form of human rhodopsin bound to a pre-activated form of the mouse visual arrestin, determined by serial femtosecond X-ray laser crystallography. Together with extensive biochemical and mutagenesis data, the structure reveals an overall architecture of the rhodopsin-arrestin assembly in which rhodopsin uses distinct structural elements, including transmembrane helix 7 and helix 8, to recruit arrestin. Correspondingly, arrestin adopts the pre-activated conformation, with a ∼20° rotation between the amino and carboxy domains, which opens up a cleft in arrestin to accommodate a short helix formed by the second intracellular loop of rhodopsin. This structure provides a basis for understanding GPCR-mediated arrestin-biased signalling and demonstrates the power of X-ray lasers for advancing the frontiers of structural biology

    Ternary structure reveals mechanism of a membrane diacylglycerol kinase

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    Diacylglycerol kinase catalyses the ATP-dependent conversion of diacylglycerol to phosphatidic acid in the plasma membrane of Escherichia coli. The small size of this integral membrane trimer, which has 121 residues per subunit, means that available protein must be used economically to craft three catalytic and substrate-binding sites centred about the membrane/cytosol interface. How nature has accomplished this extraordinary feat is revealed here in a crystal structure of the kinase captured as a ternary complex with bound lipid substrate and an ATP analogue. Residues, identified as essential for activity by mutagenesis, decorate the active site and are rationalized by the ternary structure. The g-phosphate of the ATP analogue is positioned for direct transfer to the primary hydroxyl of the lipid whose acyl chain is in the membrane. A catalytic mechanism for this unique enzyme is proposed. The active site architecture shows clear evidence of having arisen by convergen

    Atomic structure of granulin determined from native nanocrystalline granulovirus using an X-ray free-electron laser

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    To understand how molecules function in biological systems, new methods are required to obtain atomic resolution structures from biological material under physiological conditions. Intense femtosecond-duration pulses from X-ray free-electron lasers (XFELs) can outrun most damage processes, vastly increasing the tolerable dose before the specimen is destroyed. This in turn allows structure determination from crystals much smaller and more radiation sensitive than previously considered possible, allowing data collection from room temperature structures and avoiding structural changes due to cooling. Regardless, high-resolution structures obtained from XFEL data mostly use crystals far larger than 1 μm3 in volume, whereas the X-ray beam is often attenuated to protect the detector from damage caused by intense Bragg spots. Here, we describe the 2 Å resolution structure of native nanocrystalline granulovirus occlusion bodies (OBs) that are less than 0.016 μm3 in volume using the full power of the Linac Coherent Light Source (LCLS) and a dose up to 1.3 GGy per crystal. The crystalline shell of granulovirus OBs consists, on average, of about 9,000 unit cells, representing the smallest protein crystals to yield a high-resolution structure by X-ray crystallography to date. The XFEL structure shows little to no evidence of radiation damage and is more complete than a model determined using synchrotron data from recombinantly produced, much larger, cryocooled granulovirus granulin microcrystals. Our measurements suggest that it should be possible, under ideal experimental conditions, to obtain data from protein crystals with only 100 unit cells in volume using currently available XFELs and suggest that single-molecule imaging of individual biomolecules could almost be within reach

    Changes in an Enzyme Ensemble During Catalysis Observed by High Resolution XFEL Crystallography

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    Enzymes populate ensembles of structures with intrinsically different catalytic proficiencies that are difficult to experimentally characterize. We use time-resolved mix-and-inject serial crystallography (MISC) at an X-ray free electron laser (XFEL) to observe catalysis in a designed mutant (G150T) isocyanide hydratase (ICH) enzyme that enhances sampling of important minor conformations. The active site exists in a mixture of conformations and formation of the thioimidate catalytic intermediate selects for catalytically competent substates. A prior proposal for active site cysteine charge-coupled conformational changes in ICH is validated by determining structures of the enzyme over a range of pH values. A combination of large molecular dynamics simulations of the enzyme in crystallo and timeresolved electron density maps shows that ionization of the general acid Asp17 during catalysis causes additional conformational changes that propagate across the dimer interface, connecting the two active sites. These ionization-linked changes in the ICH conformational ensemble permit water to enter the active site in a location that is poised for intermediate hydrolysis. ICH exhibits a tight coupling between ionization of active site residues and catalysis-activated protein motions, exemplifying a mechanism of electrostatic control of enzyme dynamics
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