A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels

On the solvability of a class of second kind integral equations on unbounded domains

We consider integral equations of the form ψ(x) = φ(x) + ∫Ωk(x, y)z(y)ψ(y) dy(in operator form ψ = φ + Kzψ), where Ω is some subset ofRn(n ≥ 1). The functionsk,z, and φ are assumed known, withz ∈ L∞(Ω) and φ ∈ Y, the space of bounded continuous functions on Ω. The function ψ ∈ Yis to be determined. The class of domains Ω and kernelskconsidered includes the case Ω = Rnandk(x, y) = κ(x − y) with κ ∈ L1(Rn), in which case, ifzis the characteristic function of some setG, the integral equation is one of Wiener–Hopf type. The main theorems, proved using arguments derived from collectively compact operator theory, are conditions on a setW ⊂ L∞(Ω) which ensure that ifI − Kzis injective for allz ∈ WthenI − Kzis also surjective and, moreover, the inverse operators (I − Kz)−1onYare bounded uniformly inz. These general theorems are used to recover classical results on Wiener–Hopf integral operators of21and19, and generalisations of these results, and are applied to analyse the Lippmann–Schwinger integral equation.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Impact of probiotics, prebiotics and synbiotics on lipid metabolism in humans

Public health strategies for reducing the risk of coronary heart disease have focused on lowering plasma lipids, particularly cholesterol levels, with recent studies also highlighting triacylglycerol (TAG) as an important modifiable risk factor. One approach is to supplement the diet with probiotics, prebiotics or synbiotics. Probiotics are live microorganisms which when administered in adequate amounts confer a health benefit on the host. Putative health benefits include improved resistance to gastrointestinal infections, reduction in lipid levels and stimulation of the immune system. Prebiotics are selectively fermented dietary components that are aimed at improving host health through selective fermentation by the gut microbiota, such as bifidobacteria and lactobacilli. Animal studies have shown prebiotics to markedly reduce circulating TAG and to a lesser extent cholesterol concentrations, with favourable but inconsistent findings with respect to changes in lipid levels in human studies. Here we provide an overview of the effects, and possible mechanisms, of probiotics, prebiotics and synbiotics (combination of a probiotic and prebiotic) on circulating lipeamia in humans.

Selected wheat seed defense proteins exhibit competitive binding to model microbial lipid interfaces

Puroindolines (Pins) and purothionins (Pths) are basic, amphiphilic, cysteine-rich wheat proteins that play a role in plant defense against microbial pathogens. We have examined the co-adsorption and sequential addition of Pins (Pin-a, Pin-b and a mutant form of Pin-b with Trp-44 to Arg-44 substitution) and β-purothionin (β-Pth) model anionic lipid layers, using a combination of surface pressure measurements, external reflection FTIR spectroscopy and neutron reflectometry. Results highlighted differences in the protein binding mechanisms, and in the competitive binding and penetration of lipid layers between respective Pins and β-Pth. Pin-a formed a blanket-like layer of protein below the lipid surface that resulted in the reduction or inhibition of β-Pth penetration of the lipid layer. Wild-type Pin-b participated in co-operative binding with β-Pth, whereas the mutant Pin-b did not bind to the lipid layer in the presence of β-Pth. The results provide further insight into the role of hydrophobic and cationic amino acid residues in antimicrobial activity.

Interaction between a cationic surfactant-like peptide and lipid vesicles and its relationship to antimicrobial activity

We investigate the properties of an antimicrobial surfactant-like peptide (Ala)6(Arg), A6R, containing a cationic headgroup. The interaction of this peptide with zwitterionic (DPPC) lipid vesicles is investigated using a range of microscopic, X-ray scattering, spectroscopic, and calorimetric methods. The β-sheet structure adopted by A6R is disrupted in the presence of DPPC. A strong effect on the small-angle X-ray scattering profile is observed: the Bragg peaks from the DPPC bilayers in the vesicle walls are eliminated in the presence of A6R and only bilayer form factor peaks are observed. All of these observations point to the interaction of A6R with DPPC bilayers. These studies provide insight into interactions between a model cationic peptide and vesicles, relevant to understanding the action of antimicrobial peptides on lipid membranes. Notably, peptide A6R exhibits antimicrobial activity without membrane lysis.

The integration of lipid-sensing and anti-inflammatory effects: how the PPARs play a role in metabolic balance

The peroxisomal proliferating-activated receptors (PPARs) are lipid-sensing transcription factors that have a role in embryonic development, but are primarily known for modulating energy metabolism, lipid storage, and transport, as well as inflammation and wound healing. Currently, there is no consensus as to the overall combined function of PPARs and why they evolved. We hypothesize that the PPARs had to evolve to integrate lipid storage and burning with the ability to reduce oxidative stress, as energy storage is essential for survival and resistance to injury/infection, but the latter increases oxidative stress and may reduce median survival (functional longevity). In a sense, PPARs may be an evolutionary solution to something we call the 'hypoxia-lipid' conundrum, where the ability to store and burn fat is essential for survival, but is a 'double-edged sword', as fats are potentially highly toxic. Ways in which PPARs may reduce oxidative stress involve modulation of mitochondrial uncoupling protein (UCP) expression (thus reducing reactive oxygen species, ROS), optimising forkhead box class O factor (FOXO) activity (by improving whole body insulin sensitivity) and suppressing NFkB (at the transcriptional level). In light of this, we therefore postulate that inflammation-induced PPAR downregulation engenders many of the signs and symptoms of the metabolic syndrome, which shares many features with the acute phase response (APR) and is the opposite of the phenotype associated with calorie restriction and high FOXO activity. In genetically susceptible individuals (displaying the naturally mildly insulin resistant 'thrifty genotype'), suboptimal PPAR activity may follow an exaggerated but natural adipose tissue-related inflammatory signal induced by excessive calories and reduced physical activity, which normally couples energy storage with the ability to mount an immune response. This is further worsened when pancreatic decompensation occurs, resulting in gluco-oxidative stress and lipotoxicity, increased inflammatory insulin resistance and oxidative stress. Reactivating PPARs may restore a metabolic balance and help to adapt the phenotype to a modern lifestyle.

Control and analysis of oriented thin films of lipid inverse bicontinuous cubic phases using grazing incidence small-angle X‑ray scattering

Lipid cubic phases are complex nanostructures that form naturally in a variety of biological systems, with applications including drug delivery and nanotemplating. Most X-ray scattering studies on lipid cubic phases have used unoriented polydomain samples as either bulk gels or suspensions of micrometer-sized cubosomes. We present a method of investigating cubic phases in a new form, as supported thin films that can be analyzed using grazing incidence small-angle X-ray scattering (GISAXS). We present GISAXS data on three lipid systems: phytantriol and two grades of monoolein (research and industrial). The use of thin films brings a number of advantages. First, the samples exhibit a high degree of uniaxial orientation about the substrate normal. Second, the new morphology allows precise control of the substrate mesophase geometry and lattice parameter using a controlled temperature and humidity environment, and we demonstrate the controllable formation of oriented diamond and gyroid inverse bicontinuous cubic along with lamellar phases. Finally, the thin film morphology allows the induction of reversible phase transitions between these mesophase structures by changes in humidity on subminute time scales, and we present timeresolved GISAXS data monitoring these transformations.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

Variation in genes related to hepatic lipid metabolism and changes in waist circumference and body weight.

We analysed single nucleotide polymorphisms (SNPs) tagging the genetic variability of six candidate genes (ATF6, FABP1, LPIN2, LPIN3, MLXIPL and MTTP) involved in the regulation of hepatic lipid metabolism, an important regulatory site of energy balance for associations with body mass index (BMI) and changes in weight and waist circumference. We also investigated effect modification by sex and dietary intake. Data of 6,287 individuals participating in the European prospective investigation into cancer and nutrition were included in the analyses. Data on weight and waist circumference were followed up for 6.9 ± 2.5 years. Association of 69 tagSNPs with baseline BMI and annual changes in weight as well as waist circumference were investigated using linear regression analysis. Interactions with sex, GI and intake of carbohydrates, fat as well as saturated, monounsaturated and polyunsaturated fatty acids were examined by including multiplicative SNP-covariate terms into the regression model. Neither baseline BMI nor annual weight or waist circumference changes were significantly associated with variation in the selected genes in the entire study population after correction for multiple testing. One SNP (rs1164) in LPIN2 appeared to be significantly interacting with sex (p = 0.0003) and was associated with greater annual weight gain in men (56.8 ± 23.7 g/year per allele, p = 0.02) than in women (-25.5 ± 19.8 g/year per allele, p = 0.2). With respect to gene-nutrient interaction, we could not detect any significant interactions when accounting for multiple testing. Therefore, out of our six candidate genes, LPIN2 may be considered as a candidate for further studies.

Genetic, structural, and molecular insights into the function of ras of complex proteins domains

Ras of complex proteins (ROC) domains were identified in 2003 as GTP binding modules in large multidomain proteins from Dictyostelium discoideum. Research into the function of these domains exploded with their identification in a number of proteins linked to human disease, including leucine-rich repeat kinase 2 (LRRK2) and death-associated protein kinase 1 (DAPK1) in Parkinson’s disease and cancer, respectively. This surge in research has resulted in a growing body of data revealing the role that ROC domains play in regulating protein function and signaling pathways. In this review, recent advances in the structural informa- tion available for proteins containing ROC domains, along with insights into enzymatic function and the integration of ROC domains as molecular switches in a cellular and organismal context, are explored.

Pathogenic Parkinson’s disease mutations across the functional domains of LRRK2 alter the autophagic/lysosomal response to starvation

LRRK2 is one of the most important genetic contributors to Parkinson’s disease (PD). Point mutations in this gene cause an autosomal dominant form of PD, but to date no cellular phenotype has been consis- tently linked with mutations in each of the functional domains (ROC, COR and Kinase) of the protein product of this gene. In this study, primary fibroblasts from individuals carrying pathogenic mutations in the three central domains of LRRK2 were assessed for alterations in the autophagy/lysosomal pathway using a combination of biochemical and cellular approaches. Mutations in all three domains resulted in alterations in markers for autophagy/lysosomal function compared to wild type cells. These data high- light the autophagy and lysosomal pathways as read outs for pathogenic LRRK2 function and as a marker for disease, and provide insight into the mechanisms linking LRRK2 function and mutations.

O6-carboxymethylguanine DNA adduct formation and lipid peroxidation upon in vitro gastrointestinal digestion of haem-rich meat

Scope Epidemiological and clinical studies have demonstrated that the consumption of red haem-rich meat may contribute to the risk of colorectal cancer. Two hypotheses have been put forward to explain this causal relationship, i.e. N-nitroso compound (NOC) formation and lipid peroxidation (LPO). Methods and Results In this study, the NOC-derived DNA adduct O6-carboxymethylguanine (O6-CMG) and the LPO product malondialdehyde (MDA) were measured in individual in vitro gastrointestinal digestions of meat types varying in haem content (beef, pork, chicken). While MDA formation peaked during the in vitro small intestinal digestion, alkylation and concomitant DNA adduct formation was observed in seven (out of 15) individual colonic digestions using separate faecal inocula. From those, two haem-rich meat digestions demonstrated a significantly higher O6-CMG formation (p < 0.05). MDA concentrations proved to be positively correlated (p < 0.0004) with haem content of digested meat. The addition of myoglobin, a haem-containing protein, to the digestive simulation showed a dose–response association with O6-CMG (p = 0.004) and MDA (p = 0.008) formation. Conclusion The results suggest the haem-iron involvement for both the LPO and NOC pathway during meat digestion. Moreover, results unambiguously demonstrate that DNA adduct formation is very prone to inter-individual variation, suggesting a person-dependent susceptibility to colorectal cancer development following haem-rich meat consumption.

Interactions between lipid-free apolipoprotein-AI and a lipopeptide incorporating the RGDS cell adhesion motif

The interaction of a designed bioactive lipopeptide C16-GGGRGDS, comprising a hexadecyl lipid chain attached to a functional heptapeptide, with the lipid-free apoliprotein, Apo-AI, is examined. This apolipoprotein is a major component of high density lipoprotein and it is involved in lipid metabolism and may serve as a biomarker for cardiovascular disease and Alzheimers’ disease. We find via isothermal titration calorimetry that binding between the lipopeptide and Apo-AI occurs up to a saturation condition, just above equimolar for a 10.7 μM concentration of Apo-AI. A similar value is obtained from circular dichroism spectroscopy, which probes the reduction in α-helical secondary structure of Apo-AI upon addition of C16-GGGRGDS. Electron microscopy images show a persistence of fibrillar structures due to self-assembly of C16-GGGRGDS in mixtures with Apo-AI above the saturation binding condition. A small fraction of spheroidal or possibly “nanodisc” structures was observed. Small-angle X-ray scattering (SAXS) data for Apo-AI can be fitted using a published crystal structure of the Apo-AI dimer. The SAXS data for the lipopeptide/ Apo-AI mixtures above the saturation binding conditions can be fitted to the contribution from fibrillar structures coexisting with flat discs corresponding to Apo-AI/lipopeptide aggregates.