The Fermat-Torricelli problem in normed planes and spaces

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

On the multiple Borsuk numbers of sets

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

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

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.

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

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

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

Monitoring one-electron photo-oxidation of guanine in DNA crystals using ultrafast infrared spectroscopy

To understand the molecular origins of diseases caused by ultraviolet and visible light, and also to develop photodynamic therapy, it is important to resolve the mechanism of photoinduced DNA damage. Damage to DNA bound to a photosensitizer molecule frequently proceeds by one-electron photo-oxidation of guanine, but the precise dynamics of this process are sensitive to the location and the orientation of the photosensitizer, which are very difficult to define in solution. To overcome this, ultrafast time-resolved infrared (TRIR) spectroscopy was performed on photoexcited ruthenium polypyridyl–DNA crystals, the atomic structure of which was determined by X-ray crystallography. By combining the X-ray and TRIR data we are able to define both the geometry of the reaction site and the rates of individual steps in a reversible photoinduced electron-transfer process. This allows us to propose an individual guanine as the reaction site and, intriguingly, reveals that the dynamics in the crystal state are quite similar to those observed in the solvent medium

SS18 Together with Animal-Specific Factors Defines Human BAF-Type SWI/SNF Complexes

Contains fulltext : 94049.pdf (publisher's version ) (Open Access

Native structure of photosystem II at 1.95 Å resolution viewed by femtosecond X-ray pulses

Photosynthesis converts light energy into biologically useful chemical energy vital to life on Earth. The initial reaction of photosynthesis takes place in photosystem II (PSII), a 700-kilodalton homodimeric membrane protein complex which catalyses photo-oxidation of water into dioxygen through an S-state cycle of the oxygen evolving complex (OEC). The structure of PSII has been solved by X-ray diffraction (XRD) at 1.9-ångström (Å) resolution, which revealed that the OEC is a Mn4CaO5-cluster coordinated by a well-defined protein environment1. However, extended X-ray absorption fine structure (EXAFS) studies showed that the manganese cations in the OEC are easily reduced by X-ray irradiation2, and slight differences were found in the Mn–Mn distances between the results of XRD1, EXAFS3–7 and theoretical studies8–14. Here we report a ‘radiation-damage-free’ structure of PSII from Thermosynechococcus vulcanus in the S1 state at a resolution of 1.95 Å using femtosecond X-ray pulses of the SPring-8 ångström compact free-electron laser (SACLA) and a huge number of large, highly isomorphous PSII crystals. Compared with the structure from XRD, the OEC in the X-ray free electron laser structure has Mn–Mn distances that are shorter by 0.1–0.2 Å. The valences of each manganese atom were tentatively assigned as Mn1D(III), Mn2C(IV), Mn3B(IV) and Mn4A(III), based on the average Mn–ligand distances and analysis of the Jahn–Teller axis on Mn(III). One of the oxo-bridged oxygens, O5, has significantly longer Mn–O distances in contrast to the other oxo-oxygen atoms, suggesting that it is a hydroxide ion instead of a normal oxygen dianion and therefore may serve as one of the substrate oxygen atoms. These findings provide a structural basis for the mechanism of oxygen evolution, and we expect that this structure will provide a blueprint for design of artificial catalysts for water oxidation

Electrically stimulated droplet injector for reduced sample consumption in serial crystallography

15 pags., 6 figs., 1 tab.With advances in X-ray free-electron lasers (XFELs), serial femtosecond crystallography (SFX) has enabled the static and dynamic structure determination for challenging proteins such as membrane protein complexes. In SFX with XFELs, the crystals are typically destroyed after interacting with a single XFEL pulse. Therefore, thousands of new crystals must be sequentially introduced into the X-ray beam to collect full data sets. Because of the serial nature of any SFX experiment, up to 99% of the sample delivered to the X-ray beam during its "off-time" between X-ray pulses is wasted due to the intrinsic pulsed nature of all current XFELs. To solve this major problem of large and often limiting sample consumption, we report on improvements of a revolutionary sample-saving method that is compatible with all current XFELs. We previously reported 3D-printed injection devices coupled with gas dynamic virtual nozzles (GDVNs) capable of generating samples containing droplets segmented by an immiscible oil phase for jetting crystal-laden droplets into the path of an XFEL. Here, we have further improved the device design by including metal electrodes inducing electrowetting effects for improved control over droplet generation frequency to stimulate the droplet release to matching the XFEL repetition rate by employing an electrical feedback mechanism. We report the improvements in this electrically triggered segmented flow approach for sample conservation in comparison with a continuous GDVN injection using the microcrystals of lysozyme and 3-deoxy-D-manno-octulosonate 8-phosphate synthase and report the segmented flow approach for sample injection applied at the Macromolecular Femtosecond Crystallography instrument at the Linear Coherent Light Source for the first time.Financial support from the STC Program of the National Science Foundation through BioXFEL under agreement no. 1231306, NSF ABI Innovations award no. 1565180, and the National Institutes of Health award no. R01GM095583 is gratefully acknowledged. The use of the Linac Coherent Light Source (LCLS), SLAC National Accelerator Laboratory, is generously supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-76SF00515. The HERA system for in-helium experiments at MFX was developed by Bruce Doak and funded by the Max Planck Institute for Medical Research. This work was also supported by The Center for Structural Dynamics in Biology, NIH grant P41GM139687.Peer reviewe