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  • Other Sources  (19)
  • MDPI  (16)
  • American Chemical Society (ACS)
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  • 2015-2019  (19)
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  • 2018  (19)
  • 1
    Publication Date: 2021-02-08
    Description: The structural repertoire of bioactive naphthacene quinones is expanded by engineering Streptomyces albus to express the lysolipin minimal polyketide synthase II (PKS II) genes from Streptomyces tendae Tü 4042 (llpD-F) with the corresponding cyclase genes llpCI-CIII. Fermentation of the recombinant strain revealed the two new polyaromatic tridecaketides lysoquinone-TH1 (7, identified) and TH2 (8, postulated structure) as engineered congeners of the dodecaketide lysolipin (1). The chemical structure of 7, a benzo[a]naphthacene-8,13-dione, was elucidated by NMR and HR-MS and confirmed by feeding experiments with [1,2-13C2]-labeled acetate. Lysoquinone-TH1 (7) is a pentangular polyphenol and one example of such rare extended polyaromatic systems of the benz[a]napthacene quinone type produced by the expression of a minimal PKS II in combination with cyclases in an artificial system. While the natural product lysolipin (1) has antimicrobial activity in nM-range, lysoquinone-TH1 (7) showed only minor potency as inhibitor of Gram-positive microorganisms. The bioactivity profiling of lysoquinone-TH1 (7) revealed inhibitory activity towards phosphodiesterase 4 (PDE4), an important target for the treatment in human health like asthma or chronic obstructive pulmonary disease (COPD). These results underline the availability of pentangular polyphenolic structural skeletons from biosynthetic engineering in the search of new chemical entities in drug discovery
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  • 2
    Publication Date: 2021-03-19
    Description: The combination of LC-MS/MS based metabolomics approach and anti-MRSA activity-guided fractionation scheme was applied on the Gram-negative bacterium Aequorivita sp. isolated from shallow Antarctic sea sediment using a miniaturized culture chip technique. This methodology afforded the isolation of three new (1–3) and four known (4–7) N-terminal glycine- or serine-bearing iso-fatty acid amides esterified with another iso-fatty acid through their C-3 hydroxy groups. The chemical structures of the new compounds were elucidated using a set of spectroscopic (NMR, [α]D and FT-IR) and spectrometric (HRMS, HRMS/MS) methods. The aminolipids possessing an N-terminal glycine unit (1, 2, 4, 5) showed moderate in vitro antimicrobial activity against MRSA (IC50 values 22–145 µg/mL). This is the first in-depth chemistry and biological activity study performed on the microbial genus Aequorivita.
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  • 3
    Publication Date: 2021-02-08
    Description: Habitat-forming species sustain biodiversity and ecosystem functioning in harsh environments through the amelioration of physical stress. Nonetheless, their role in shaping patterns of species distribution under future climate scenarios is generally overlooked. Focusing on coastal systems, we assess how habitat-forming species can influence the ability of stress-sensitive species to exhibit plastic responses, adapt to novel environmental conditions, or track suitable climates. Here, we argue that habitat-former populations could be managed as a nature-based solution against climate-driven loss of biodiversity. Drawing from different ecological and biological disciplines, we identify a series of actions to sustain the resilience of marine habitat-forming species to climate change, as well as their effectiveness and reliability in rescuing stress-sensitive species from increasingly adverse environmental conditions.
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  • 4
    Publication Date: 2021-02-08
    Description: The ultramafic-hosted Kairei vent field is located at 25°19′ S, 70°02′ E, towards the Northern end of segment 1 of the Central Indian Ridge (CIR-S1) at a water depth of ~2450 m. This study aims to investigate the distribution of trace elements among sulfide minerals of differing textures and to examine the possible factors controlling the trace element distribution in those minerals using LA-ICP-MS spot and line scan analyses. Our results show that there are distinct systematic differences in trace element distributions throughout the different minerals, as follows: (1) pyrite is divided into three types at Kairei, including early-stage euhedral pyrite (py-I), sub-euhedral pyrite (py-II), and colloform pyrite (py-III). Pyrite is generally enriched with Mo, Au, As, Tl, Mn, and U. Pyrite-I has high contents of Se, Te, Bi, and Ni when compared to the other types; py-II is enriched in Au relative to py-I and py-III, but poor in Ni; py-III is enriched in Mo, Pb, and U but is poor in Se, Te, Bi, and Au relative to py-I and py-II. Variations in the concentrations of Se, Te, and Bi in pyrite are most likely governed by the strong temperature gradient. There is generally a lower concentration of nickel than Co in pyrite, indicating that our samples precipitated at high temperatures, whereas the extreme Co enrichment is likely from a magmatic heat source combined with an influence of serpentinization reactions. (2) Chalcopyrite is characterized by high concentrations of Co, Se, and Te. The abundance of Se and Te in chalcopyrite over the other minerals is interpreted to have been caused by the high solubilities of Se and Te in the chalcopyrite lattice at high temperatures. The concentrations of Sb, As, and Au are relatively low in chalcopyrite from the Kairei vent field. (3) Sphalerite from Zn-rich chimneys is characterized by high concentrations of Sn, Co, Ga, Ge, Ag, Pb, Sb, As, and Cd, but is depleted in Se, Te, Bi, Mo, Au, Ni, Tl, Mn, Ba, V, and U in comparison with the other minerals. The high concentrations of Cd and Co are likely caused by the substitution of Cd2+ and Co2+ for Zn2+ in sphalerite. A high concentration of Pb accompanied by a high Ag concentration in sphalerite indicates that Ag occurs as Pb–Ag sulfosalts. Gold is generally low in sphalerite and strongly correlates with Pb, suggesting its presence in microinclusions of galena. The strong correlation of As with Ge in sphalerite from Kairei suggests that they might precipitate at medium temperatures and under moderately reduced conditions. (4) Bornite–digenite has very low concentrations of most trace elements, except for Co, Se, and Bi. Serpentinization in ultramafic-hosted hydrothermal systems might play an important role in Au enrichment in pyrite with low As contents. Compared to felsic-hosted seafloor massive sulfide deposits, sulfide minerals from ultramafic-hosted deposits show higher concentrations of Se and Te, but lower As, Sb, and Au concentrations, the latter often attributed to the contribution of magmatic volatiles. As with typical ultramafic-hosted seafloor massive sulfide deposits, Se enrichment in chalcopyrite from Kairei indicates that the primary factor that controls the Se enrichment is temperature-controlled mobility in vent fluids.
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  • 5
    Publication Date: 2021-02-08
    Description: An UPLC-qTOF-MS-based dereplication study led to the targeted isolation of seven bromoindole alkaloids from the sub-Arctic sponge Geodia barretti. This includes three new metabolites, namely geobarrettin A–C (1–3) and four known compounds, barettin (4), 8,9-dihydrobarettin (5), 6-bromoconicamin (6), and l-6-bromohypaphorine (7). The chemical structures of compounds 1–7 were elucidated by extensive analysis of the NMR and HRESIMS data. The absolute stereochemistry of geobarrettin A (1) was assigned by ECD analysis and Marfey’s method employing the new reagent l-Nα-(1-fluoro-2,4-dinitrophenyl)tryptophanamide (l-FDTA). The isolated compounds were screened for anti-inflammatory activity using human dendritic cells (DCs). Both 2 and 3 reduced DC secretion of IL-12p40, but 3 concomitantly increased IL-10 production. Maturing DCs treated with 2 or 3 before co-culturing with allogeneic CD4+ T cells decreased T cell secretion of IFN-γ, indicating a reduction in Th1 differentiation. Although barettin (4) reduced DC secretion of IL-12p40 and IL-10 (IC50 values 11.8 and 21.0 μM for IL-10 and IL-12p40, respectively), maturing DCs in the presence of 4 did not affect the ability of T cells to secrete IFN-γ or IL-17, but reduced their secretion of IL-10. These results indicate that 2 and 3 may be useful for the treatment of inflammation, mainly of the Th1 type.
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  • 6
    Publication Date: 2021-02-08
    Description: This study presents a novel approach, based on high-dimensionality hydro-acoustic data, for improving the performance of angular response analysis (ARA) on multibeam backscatter data in terms of acoustic class separation and spatial resolution. This approach is based on the hyper-angular cube (HAC) data structure which offers the possibility to extract one angular response from each cell of the cube. The HAC consists of a finite number of backscatter layers, each representing backscatter values corresponding to single-incidence angle ensonifications. The construction of the HAC layers can be achieved either by interpolating dense soundings from highly overlapping multibeam echo-sounder (MBES) surveys (interpolated HAC, iHAC) or by producing several backscatter mosaics, each being normalized at a different incidence angle (synthetic HAC, sHAC). The latter approach can be applied to multibeam data with standard overlap, thus minimizing the cost for data acquisition. The sHAC is as efficient as the iHAC produced by actual soundings, providing distinct angular responses for each seafloor type. The HAC data structure increases acoustic class separability between different acoustic features. Moreover, the results of angular response analysis are applied on a fine spatial scale (cell dimensions) offering more detailed acoustic maps of the seafloor. Considering that angular information is expressed through high-dimensional backscatter layers, we further applied three machine learning algorithms (random forest, support vector machine, and artificial neural network) and one pattern recognition method (sum of absolute differences) for supervised classification of the HAC, using a limited amount of ground truth data (one sample per seafloor type). Results from supervised classification were compared with results from an unsupervised method for inter-comparison of the supervised algorithms. It was found that all algorithms (regarding both the iHAC and the sHAC) produced very similar results with good agreement (〉0.5 kappa) with the unsupervised classification. Only the artificial neural network required the total amount of ground truth data for producing comparable results with the remaining algorithms.
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  • 7
    Publication Date: 2021-03-19
    Description: The metabolism of seaweeds depends on environmental parameters, the availability of nutrients, and biotic/abiotic stresses; therefore, their chemical composition fluctuates throughout the year. This study investigated seasonal variations in the metabolome of the Baltic Sea brown alga Fucus vesiculosus and its potential relation to the bioactivity profile. By using a definitive screening design (DSD) combined with pressurised liquid extraction (PLE), an optimised protocol was developed to extract algal biomass monthly for a full calendar year. An untargeted metabolomics approach using ultra-high performance liquid chromatography-tandem mass spectrometry (UPLC-MSn)-based molecular networking and manual dereplication was employed. The extracts were simultaneously screened for their in vitro antimicrobial, anticancer/apoptotic, and free radical scavenging activities. 44 compounds were putatively dereplicated in the metabolome. Many compounds were found to vary with the sampling month; phlorotannin total ion count (TIC) was highest in summer, whilst chlorophylls, lipids, and carotenoids peaked in winter and spring. The greatest radical scavenging and apoptotic activities against pancreas cancer cells observed in the summer months were attributed to high phlorotannin TIC. Methicillin-resistant Staphylococcus aureus (MRSA) inhibitory activity was produced year-round without a clear seasonal trend. This is the first study applying DSD-based optimised PLE extraction combined with a metabolome analysis of F. vesiculosus for the identification of seasonal variations in both metabolome and bioactivity.
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  • 8
    Publication Date: 2020-01-02
    Description: Microplastics (MPs, 〈5 mm) have been reported as emerging environmental contaminants, but reliable data are still lacking. We compared the two most promising techniques for MP analysis, namely, Raman and Fourier transform infrared (FTIR) spectroscopy, by analyzing MPs extracted from North Sea surface waters. Microplastics 〉500 μm were visually sorted and manually analyzed by μ-Raman and attenuated total reflection (ATR)-FTIR spectroscopy. Microplastics ≤500 μm were concentrated on gold-coated filters and analyzed by automated single-particle exploration coupled to μ-Raman (ASPEx-μ-Raman) and FTIR imaging (reflection mode). The number of identified MPs 〉500 μm was slightly higher for μ-Raman (+23%) than ATR-FTIR analysis. Concerning MPs ≤500 μm, ASPEx-μ-Raman quantified two-times higher MP numbers but required a four-times higher analysis time compared to FTIR imaging. Because ASPEx-μ-Raman revealed far higher MP concentrations (38–2621 particles m–3) compared to the results of previous water studies (0–559 particles m–3), the environmental concentration of MPs ≤500 μm may have been underestimated until now. This may be attributed to the exceptional increase in concentration with decreasing MP size found in this work. Our results demonstrate the need for further research to enable time-efficient routine application of ASPEx-μ-Raman for reliable MP counting down to 1 μm.
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  • 9
    Publication Date: 2018-12-17
    Description: Many scientific institutions are faced with the question of how they should inform their scientists and scientific coordinators about the option of publishing open access. This task is one that libraries have taken upon themselves: libraries are familiar with the market participants and have years of experience in teaching information and publication literacy. This case report looks at two approaches taken by the Central Library of Forschungszentrum Jülich in 2017. It highlights the motivation,strategy,resourcesandimplementation,aswellasthefirstevaluationofbothapproaches. ThefirstapproachwasaredesignofthetrainingcoursesofferedbytheCentralLibrarywithafocuson the target groups and new contents. The second approach was implemented as part of International Open Access Week and involved offering an information event tailored to each scientific institute. Theeventwascustomizedtomeettheneedsofthetargetgroupdefinedbyeachinstitute,theinstitute itself, and was organized individually. As a result of these efforts, the open access rate increased over the last few months and at 48% open access in 2017, Forschungszentrum Jülich is well on its way to achieving the open access goals set by the Helmholtz Association.
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  • 10
    Publication Date: 2021-01-08
    Description: The cage occupancy plays a crucial role in the thermodynamic stability of clathrate hydrates and is an important quantity for understanding the CO2–CH4 replacement phenomenon. In this work, the occupancy isotherms of pure CH4, pure CO2, and their mixture in sI and sII hydrates are studied by GCMC + MD simulations. The adsorption of CH4 and CO2 + CH4 in the sI and sII hydrates can be categorized as the one-site Langmuir type. The calculated occupancy ratio θL/θS and the abundance ratio of CO2 to CH4 vary with the temperature and pressure, which provide the prerequisite information for the prediction of CH4 recovery yield at different conditions in the CO2–CH4 gas exchange process. The phase equilibria of clathrate hydrates of pure gases and mixtures are explored and the corresponding heat of dissociation and hydration numbers are determined. The current investigation provides new perspectives to understand the mechanism behind the gas adsorption behavior of clathrate hydrates.
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  • 11
    Publication Date: 2021-03-18
    Description: This article presents gas hydrate experimental measurements for mixtures containing methane (CH4), carbon dioxide (CO2) and nitrogen (N2) with the aim to better understand the impact of water (H2O) on the phase equilibrium. Some of these phase equilibrium experiments were carried out with a very high water-to-gas ratio that shifts the gas hydrate dissociation points to higher pressures. This is due to the significantly different solubilities of the different guest molecules in liquid H2O. A second experiment focused on CH4-CO2 exchange between the hydrate and the vapor phases at moderate pressures. The results show a high retention of CO2 in the gas hydrate phase with small pressure variations within the first hours. However, for our system containing 10.2 g of H2O full conversion of the CH4 hydrate grains to CO2 hydrate is estimated to require 40 days. This delay is attributed to the shrinking core effect, where initially an outer layer of CO2-rich hydrate is formed that effectively slows down the further gas exchange between the vapor phase and the inner core of the CH4-rich hydrate grain.
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  • 12
    Publication Date: 2021-03-19
    Description: The production of secondary metabolites by a new isolate of the purple sulfur bacterium Allochromatium vinosum, which had shown antibiotic activities during a preliminary study, revealed the production of several metabolites. Growth conditions suitable for the production of one of the compounds shown in the metabolite profile were established and compound 1 was purified. The molecular formula of compound 1 (C20H28O2) was determined by high resolution mass spectra, and its chemical structure by means of spectroscopic methods. The evaluation of these data revealed that the structure of the compound was identical to dehydroabietic acid, a compound known to be characteristically produced by conifer trees, but so far not known from bacteria, except cyanobacteria. The purified substance showed weak antibiotic activities against Bacillus subtilis and Staphylococcus lentus with IC50 values of 70.5 µM (±2.9) and 57.0 µM (±3.3), respectively.
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  • 13
    Publication Date: 2021-02-08
    Description: The decarbonisation of our energy supply is reliant on new technologies that are raw material intensive and will require a significant increase in the production of metals to sustain them. Ferromanganese (FeMn) crusts are seafloor precipitates, enriched in metals such as cobalt and tellurium, both of which have a predicted future demand above current production rates. In this study, we investigate the texture and composition of FeMn crusts on Tropic Seamount, a typical Atlantic guyot off the coast of western Africa, as a basis for assessing the future mineral resource potential of Atlantic Seamounts. The majority of the summit is flat and covered by FeMn crusts with average thicknesses of 3–4 cm. The crusts are characterized by two dominant textures consisting of either massive pillared growth or more chaotic, cuspate sections of FeMn oxides, with an increased proportion of detrital and organic material. The Fe, Mn, and Co contents in the FeMn oxide layers are not affected by texture. However, detrital material and bioclasts can form about 50% of cuspate areas, and the dilution effect of this entrained material considerably reduces the Fe, Mn, and Co concentrations if the bulk samples are analyzed. Whilst Tropic Seamount meets many of the prerequisites for a crust mining area, the thickness of the crusts and their average metal composition means extraction is unlikely to be viable in the near future. The ability to exploit more difficult terrains or multiple, closely spaced edifices would make economic feasibility more likely.
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  • 14
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    In:  , ed. by Domingues, R., Domingues, P. and Calado, R. Marine Drugs, Special Issue . MDPI, Basel, Switzerland, 157 pp. ISBN 978-3-03842-800-8
    Publication Date: 2021-02-08
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  • 15
    Publication Date: 2021-03-19
    Description: Modern volcano monitoring commonly involves Interferometric Synthetic Aperture Radar (InSAR) measurements to identify ground motions caused by volcanic activity. However, InSAR is largely affected by changes in atmospheric refractivity, in particular by changes which can be attributed to the distribution of water (H2O) vapor in the atmospheric column. Gas emissions from continuously degassing volcanoes contain abundant water vapor and thus produce variations in the atmospheric water vapor content above and downwind of the volcano, which are notably well captured by short-wavelength X-band SAR systems. These variations may in turn cause differential phase errors in volcano deformation estimates due to excess radar path delay effects within the volcanic gas plume. Inversely, if these radar path delay effects are better understood, they may be even used for monitoring degassing activity, by means of the precipitable water vapor (PWV) content in the plume at the time of SAR acquisitions, which may provide essential information on gas plume dispersion and the state of volcanic and hydrothermal activity. In this work we investigate the radar path delays that were generated by water vapor contained in the volcanic gas plume of the persistently degassing Láscar volcano, which is located in the dry Atacama Desert of Northern Chile. We estimate water vapor contents based on sulfur dioxide (SO2) emission measurements from a scanning UV spectrometer (Mini-DOAS) station installed at Láscar volcano, which were scaled by H2O/SO2 molar mixing ratios obtained during a multi-component Gas Analyzer System (Multi-GAS) survey on the crater rim of the volcano. To calculate the water vapor content in the downwind portion of the plume, where an increase of water vapor is expected, we further applied a correction involving estimation of potential evaporation rates of water droplets governed by turbulent mixing of the condensed volcanic plume with the dry atmosphere. Based on these estimates we obtain daily average PWV contents inside the volcanic gas plume of 0.2–2.5 mm equivalent water column, which translates to a slant wet delay (SWD) in DInSAR data of 1.6–20 mm. We used these estimates in combination with our high resolution TerraSAR-X DInSAR observations at Láscar volcano, in order to demonstrate the occurrence of repeated atmospheric delay patterns that were generated by volcanic gas emissions. We show that gas plume related refractivity changes are significant and detectable in DInSAR measurements. Implications are two-fold: X-band satellite radar observations also contain information on the degassing state of a volcano, while deformation signals need to be interpreted with care, which has relevance for volcano observations at Láscar and for other sites worldwide.
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  • 16
    Publication Date: 2021-04-23
    Description: Increasing atmospheric CO2 concentrations are resulting in a reduction in seawater pH, with potential detrimental consequences for marine organisms. Improved efforts are required to monitor the anthropogenically driven pH decrease in the context of natural pH variations. We present here a high resolution surface water pH data set obtained in summer 2011 in North West European Shelf Seas. The aim of our paper is to demonstrate the successful deployment of the pH sensor, and discuss the carbonate chemistry dynamics of surface waters of Northwest European Shelf Seas using pH and ancillary data. The pH measurements were undertaken using spectrophotometry with a Lab-on-Chip pH sensor connected to the underway seawater supply of the ship. The main processes controlling the pH distribution along the ship's transect, and their relative importance, were determined using a statistical approach. The pH sensor allowed 10 measurements h(-1) with a precision of 0.001 pH units and a good agreement with pH calculated from a pair of discretely sampled carbonate variables dissolved inorganic carbon (DIC), total alkalinity (TA) and partial pressure of CO2 (pCO(2)) (e.g., pH(DICpCO2)). For this summer cruise, the biological activity formed the main control on the pH distribution along the cruise transect. This study highlights the importance of high quality and high resolution pH measurements for the assessment of carbonate chemistry dynamics in marine waters.
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  • 17
    Publication Date: 2021-03-19
    Description: Optical communication promises to be a high-rate supplement for acoustic communication in short-range underwater applications. In the photic zone of oceanic and coastal waters, underwater optical communication systems are exposed by remaining sunlight. This ambient light generates additional noise in photodetectors, thus degrading system performance. This effect can be diminished by the use of optical filters. This paper investigates light field characteristics of different water types and potential interactions with optical underwater communication. A colored glass and different thin film bandpass filters are examined as filter/detector combinations under varying light and water conditions, and their physical constraints are depicted. This is underlined by various spectral measurements as well as optical signal-to-noise ratio calculations. The importance of matching the characteristics of the light emitting diode (LED) light source, the photodetector, and the filter on the ambient conditions using wider angle of incidents is emphasized.
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  • 18
    Publication Date: 2023-05-04
    Description: The Antarctic deep-sea sponge Latrunculia (Latrunculia) biformis Kirkpatrick, 1908 (Class Demospongiae Sollas, Order Poecilosclerida Topsent, Latrunculiidae Topsent) was selected for chemical analyses due to its potent anticancer activity. Metabolomic analysis of its crude extract by HRMS/MS-based molecular networking showed the presence of several clusters of pyrroloiminoquinone alkaloids, i.e., discorhabdin and epinardin-type brominated pyridopyrroloquinolines and tsitsikammamines, the non-brominated bis-pyrroloiminoquinones. Molecular networking approach combined with a bioactivity-guided isolation led to the targeted isolation of the known pyrroloiminoquinone tsitsikammamine A (1) and its new analog 16,17-dehydrotsitsikammamine A (2). The chemical structures of the compounds 1 and 2 were elucidated by spectroscopic analysis (one-dimensional (1D) and two-dimensional (2D) NMR, HR-ESIMS). Due to minute amounts, molecular modeling and docking was used to assess potential affinities to potential targets of the isolated compounds, including DNA intercalation, topoisomerase I-II, and indoleamine 2,3-dioxygenase enzymes. Tsitsikammamines represent a small class of pyrroloiminoquinone alkaloids that have only previously been reported from the South African sponge genus Tsitsikamma Samaai & Kelly and an Australian species of the sponge genus Zyzzya de Laubenfels. This is the first report of tsitsikammamines from the genus Latrunculia du Bocage and the successful application of molecular networking in the identification of comprehensive chemical inventory of L.biformis followed by targeted isolation of new molecules. This study highlights the high productivity of secondary metabolites of Latrunculia sponges and may shed new light on their biosynthetic origin and chemotaxonomy
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  • 19
    Publication Date: 2023-11-03
    Description: Mastermind is a two players zero sum game of imperfect information. Starting with Erdős and Rényi (1963), its combinatorics have been studied to date by several authors, e.g., Knuth (1977), Chvátal (1983), Goodrich (2009). The first player, called “codemaker”, chooses a secret code and the second player, called “codebreaker”, tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. For variants that allow color repetition, Doerr et al. (2016) showed optimal results. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and 𝑘≥𝑛 colors where no repetition of colors is allowed. We present upper and lower bounds on the number of guesses necessary to break the secret code. For the case 𝑘=𝑛, the secret code can be algorithmically identified within less than (𝑛−3)⌈log2𝑛⌉+52𝑛−1 queries. This result improves the result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2. For the case 𝑘〉𝑛, we prove an upper bound of (𝑛−2)⌈log2𝑛⌉+𝑘+1. Furthermore, we prove a new lower bound of n for the case 𝑘=𝑛, which improves the recent 𝑛−loglog(𝑛) bound of Berger et al. (2016). We then generalize this lower bound to k queries for the case 𝑘≥𝑛.
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