Fourier transform, null variety, and Laplacian's eigenvalues

We consider a quantity κ(Ω)—the distance to the origin from the null variety of the Fourier transform of the characteristic function of Ω. We conjecture, firstly, that κ(Ω) is maximised, among all convex balanced domains of a fixed volume, by a ball, and also that κ(Ω) is bounded above by the square root of the second Dirichlet eigenvalue of Ω. We prove some weaker versions of these conjectures in dimension two, as well as their validity for domains asymptotically close to a disk, and also discuss further links between κ(Ω) and the eigenvalues of the Laplacians.

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.

Phylogeny, function, and evolution of the cupins, a structurally conserved, functionally diverse superfamily of proteins

The cupin superfamily is a group of functionally diverse proteins that are found in all three kingdoms of life, Archaea, Eubacteria, and Eukaryota. These proteins have a characteristic signature domain comprising two histidine- containing motifs separated by an intermotif region of variable length. This domain consists of six beta strands within a conserved beta barrel structure. Most cupins, such as microbial phosphomannose isomerases (PMIs), AraC- type transcriptional regulators, and cereal oxalate oxidases (OXOs), contain only a single domain, whereas others, such as seed storage proteins and oxalate decarboxylases (OXDCs), are bi-cupins with two pairs of motifs. Although some cupins have known functions and have been characterized at the biochemical level, the majority are known only from gene cloning or sequencing projects. In this study, phylogenetic analyses were conducted on the conserved domain to investigate the evolution and structure/function relationships of cupins, with an emphasis on single- domain plant germin-like proteins (GLPs). An unrooted phylogeny of cupins from a wide spectrum of evolutionary lineages identified three main clusters, microbial PMIs, OXDCs, and plant GLPs. The sister group to the plant GLPs in the global analysis was then used to root a phylogeny of all available plant GLPs. The resulting phylogeny contained three main clades, classifying the GLPs into distinct subfamilies. It is suggested that these subfamilies correlate with functional categories, one of which contains the bifunctional barley germin that has both OXO and superoxide dismutase (SOD) activity. It is proposed that GLPs function primarily as SODs, enzymes that protect plants from the effects of oxidative stress. Closer inspection of the DNA sequence encoding the intermotif region in plant GLPs showed global conservation of thymine in the second codon position, a character associated with hydrophobic residues. Since many of these proteins are multimeric and enzymatically inactive in their monomeric state, this conservation of hydrophobicity is thought to be associated with the need to maintain the various monomer- monomer interactions. The type of structure-based predictive analysis presented in this paper is an important approach for understanding gene function and evolution in an era when genomes from a wide range of organisms are being sequenced at a rapid rate.

Microbial relatives of the seed storage proteins of higher plants: conservation of structure and diversification of function during evolution of the cupin superfamily.

This review summarizes the recent discovery of the cupin superfamily (from the Latin term "cupa," a small barrel) of functionally diverse proteins that initially were limited to several higher plant proteins such as seed storage proteins, germin (an oxalate oxidase), germin-like proteins, and auxin-binding protein. Knowledge of the three-dimensional structure of two vicilins, seed proteins with a characteristic beta-barrel core, led to the identification of a small number of conserved residues and thence to the discovery of several microbial proteins which share these key amino acids. In particular, there is a highly conserved pattern of two histidine-containing motifs with a varied intermotif spacing. This cupin signature is found as a central component of many microbial proteins including certain types of phosphomannose isomerase, polyketide synthase, epimerase, and dioxygenase. In addition, the signature has been identified within the N-terminal effector domain in a subgroup of bacterial AraC transcription factors. As well as these single-domain cupins, this survey has identified other classes of two-domain bicupins including bacterial gentisate 1, 2-dioxygenases and 1-hydroxy-2-naphthoate dioxygenases, fungal oxalate decarboxylases, and legume sucrose-binding proteins. Cupin evolution is discussed from the perspective of the structure-function relationships, using data from the genomes of several prokaryotes, especially Bacillus subtilis. Many of these functions involve aspects of sugar metabolism and cell wall synthesis and are concerned with responses to abiotic stress such as heat, desiccation, or starvation. Particular emphasis is also given to the oxalate-degrading enzymes from microbes, their biological significance, and their value in a range of medical and other applications.

Secoiridoids delivered as olive leaf extract induce acute improvements in human vascular function and reduction of an inflammatory cytokine: a randomised, double-blind, placebo-controlled, cross-over trial

The leaves of the olive plant (Olea europaea) are rich in polyphenols, of which oleuropein and hydroxytyrosol (HT) are most characteristic. Such polyphenols have been demonstrated to favourably modify a variety of cardiovascular risk factors. The aim of the present intervention was to investigate the influence of olive leaf extract (OLE) on vascular function and inflammation in a postprandial setting and to link physiological outcomes with absorbed phenolics. A randomised, double-blind, placebo-controlled, cross-over, acute intervention trial was conducted with eighteen healthy volunteers (nine male, nine female), who consumed either OLE (51 mg oleuropein; 10mg HT), or a matched control (separated by a 4-week wash out) on a single occasion. Vascular function was measured by digital volume pulse (DVP), while blood collected at baseline, 1, 3 and 6 h was cultured for 24 h in the presence of lipopolysaccharide in order to investigate effects on cytokine production. Urine was analysed for phenolic metabolites by HPLC. DVP-stiffness index and ex vivo IL-8 production were significantly reduced (P < 0.05) after consumption of OLE compared to the control. These effects were accompanied by the excretion of several phenolic metabolites, namely HT and oleuropein derivatives, which peaked in urine after 8-24 h. The present study provides the first evidence that OLE positively modulates vascular function and IL-8 production in vivo, adding to growing evidence that olive phenolics could be beneficial for health.