Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

Quantitative single-molecule localization microscopy combined with rule-based modeling reveals ligand-induced TNF-R1 reorganization toward higher-order oligomers

We report on the assembly of tumor necrosis factor receptor 1 (TNF-R1) prior to ligand activation and its ligand-induced reorganization at the cell membrane. We apply single-molecule localization microscopy to obtain quantitative information on receptor cluster sizes and copy numbers. Our data suggest a dimeric pre-assembly of TNF-R1, as well as receptor reorganization toward higher oligomeric states with stable populations comprising three to six TNF-R1. Our experimental results directly serve as input parameters for computational modeling of the ligand-receptor interaction. Simulations corroborate the experimental finding of higher-order oligomeric states. This work is a first demonstration how quantitative, super-resolution and advanced microscopy can be used for systems biology approaches at the single-molecule and single-cell level.

A global proteomics approach identifies novel phosphorylated signaling proteins in GPVI-activated platelets: involvement of G6f, a novel platelet Grb2-binding membrane adapter.

Collagen-related peptide (CRP) stimulates powerful activation of platelets through the glycoprotein VI (GPVI)-FcR gamma-chain complex. We have combined proteomics and traditional biochemistry approaches to study the proteome of CRP-activated platelets, focusing in detail on tyrosine phosphorylation. In two separate approaches, phosphotyrosine immunoprecipitations followed by 1-D-PAGE, and 2-DE, were used for protein separation. Proteins were identified by MS. By following these approaches, 96 proteins were found to undergo PTM in response to CRP in human platelets, including 11 novel platelet proteins such as Dok-1, SPIN90, osteoclast stimulating factor 1, and beta-Pix. Interestingly, the type I transmembrane protein G6f was found to be specifically phosphorylated on Tyr-281 in response to platelet activation by CRP, providing a docking site for the adapter Grb2. G6f tyrosine phoshporylation was also found to take place in response to collagen, although not in response to the G protein-coupled receptor agonists, thrombin and ADP. Further, we also demonstrate for the first time that Grb2 and its homolog Gads are tyrosine-phosphorylated in CRP-stimulated platelets. This study provides new insights into the mechanism of platelet activation through the GPVI collagen receptor, helping to build the basis for the development of new drug targets for thrombotic disease.

Coupling liquid MALDI MS to liquid chromatography

Matrix-assisted laser desorption/ionisation (MALDI) coupled with time-of-flight (TOF) mass spectrometry (MS) is a powerful tool for the analysis of biological samples, and nanoflow high-performance liquid chromatography (nanoHPLC) is a useful separation technique for the analysis of complex proteomics samples. The off-line combination of MALDI and nanoHPLC has been extensively investigated and straightforward techniques have been developed, focussing particularly on automated MALDI sample preparation that yields sensitive and reproducible spectra. Normally conventional solid MALDI matrices such as α-cyano-4-hydroxycinnamic acid (CHCA) are used for sample preparation. However, they have limited usefulness in quantitative measurements and automated data acquisition because of the formation of heterogeneous crystals, resulting in highly variable ion yields and desorption/ ionization characteristics. Glycerol-based liquid support matrices (LSM) have been proposed as an alternative to the traditional solid matrices as they provide increased shot-to-shot reproducibility, leading to prolonged and stable ion signals and therefore better results. This chapter focuses on the integration of the liquid LSM MALDI matrices into the LC-MALDI MS/MS approach in identifying complex and large proteomes. The interface between LC and MALDI consists of a robotic spotter, which fractionates the eluent from the LC column into nanoliter volumes, and co-spots simultaneously the liquid matrix with the eluent fractions onto a MALDI target plate via sheath flow. The efficiency of this method is demonstrated through the analysis of trypsin digests of both bovine serum albumin (BSA) and Lactobacillus plantarum WCFS1 proteins.

Quantitative structure of an acetate dye molecule analogue at the TiO2- acetic acid interface

The positions of atoms in and around acetate molecules at the rutile TiO2(110) interface with 0.1 M acetic acid have been determined with a precision of ±0.05 Å. Acetate is used as a surrogate for the carboxylate groups typically employed to anchor monocarboxylate dye molecules to TiO2 in dye-sensitised solar cells (DSSC). Structural analysis reveals small domains of ordered (2 x 1) acetate molecules, with substrate atoms closer to their bulk terminated positions compared to the clean UHV surface. Acetate is found in a bidentate bridge position, binding through both oxygen atoms to two five-fold titanium atoms such that the molecular plane is along the [001] azimuth. Density functional theory calculations provide adsorption geometries in excellent agreement with experiment. The availability of these structural data will improve the accuracy of charge transport models for DSSC.

Transcriptomics and proteomics show that selenium affects inflammation, ctoskeleton, and cancer pathways in human rectal biopsies

Epidemiologic studies highlight the potential role of dietary selenium (Se) in colorectal cancer prevention. Our goal was to elucidate whether expression of factors crucial for colorectal homoeostasis is affected by physiologic differences in Se status. Using transcriptomics and proteomics followed by pathway analysis, we identified pathways affected by Se status in rectal biopsies from 22 healthy adults, including 11 controls with optimal status (mean plasma Se = 1.43 μM) and 11 subjects with suboptimal status (mean plasma Se = 0.86 μM). We observed that 254 genes and 26 proteins implicated in cancer (80%), immune function and inflammatory response (40%), cell growth and proliferation (70%), cellular movement, and cell death (50%) were differentially expressed between the 2 groups. Expression of 69 genes, including selenoproteins W1 and K, which are genes involved in cytoskeleton remodelling and transcription factor NFκB signaling, correlated significantly with Se status. Integrating proteomics and transcriptomics datasets revealed reduced inflammatory and immune responses and cytoskeleton remodelling in the suboptimal Se status group. This is the first study combining omics technologies to describe the impact of differences in Se status on colorectal expression patterns, revealing that suboptimal Se status could alter inflammatory signaling and cytoskeleton in human rectal mucosa and so influence cancer risk.