A functional equation and its application to the characterization of gamma distributions

The functional equation $$ f\left(x\right)g\left(y\right)=p\left(x+y\right)q\left(\frac{x}{y} \right) $$ is investigated for almost all ${\left(x,\,y\right)\in\mathbb{R}^{2}_{+}}$ and for the measurable functions ${f,\,g,\,p,\,q:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}}$ . This equation is related to the Lukács characterization of gamma distribution.     

Activity-Directed Synthesis of Inhibitors of the p53/hDM2 Protein–Protein Interaction

Protein–protein interactions (PPIs) provide a rich source of potential targets for drug discovery and biomedical science research. However, the identification of structural-diverse starting points for discovery of PPI inhibitors remains a significant challenge. Activity-directed synthesis (ADS), a function-driven discovery approach, was harnessed in the discovery of the p53/hDM2 PPI. Over two rounds of ADS, 346 microscale reactions were performed, with prioritisation on the basis of the activity of the resulting product mixtures. Four distinct and novel series of PPI inhibitors were discovered that, through biophysical characterisation, were shown to have promising ligand efficiencies. It was thus shown that ADS can facilitate ligand discovery for a target that does not have a defined small-molecule binding site, and can provide distinctive starting points for the discovery of PPI inhibitors.     

Multiplicative type functional equations arising from characterization problems

We give the general and the so-called density function solutions of equation

Glucocorticoids Directly Affect Hyaluronan Production of Orbital Fibroblasts; A Potential Pleiotropic Effect in Graves’ Orbitopathy

Orbital connective tissue expansion is a hallmark of Graves’ orbitopathy (GO). In moderate-to-severe active GO, glucocorticoids (GC) are the first line of treatment. Here we show that hydrocortisone (HC), prednisolone (P), methylprednisolone (MP), and dexamethasone (DEX) inhibit the hyaluronan (HA) production of orbital (OF) and dermal (DF) fibroblasts. HA production of GO OFs (n = 4), NON-GO OFs (n = 4) and DFs (n = 4) was measured by ELISA. mRNA expression of enzymes of HA metabolism and fibroblast proliferation was examined by RT-PCR and BrdU incorporation, respectively. After 24 h of GC treatment (1µM) HA production decreased by an average of 67.9 ± 3.11% (p < 0.0001) in all cell cultures. HAS2, HAS3 and HYAL1 expression in OFs also decreased (p = 0.009, p = 0.0005 and p = 0.015, respectively). Ten ng/mL PDGF-BB increased HA production and fibroblast proliferation in all cell lines (p < 0.0001); GC treatment remained effective and reduced HA production under PDGF-BB-stimulated conditions (p < 0.0001). MP and DEX reduced (p < 0.001, p = 0.002, respectively) PDGF-BB-induced HAS2 expression in OFs. MP and DEX treatment decreased PDGF-BB stimulated HAS3 expression (p = 0.035 and p = 0.029, respectively). None of the GCs tested reduced the PDGF-BB stimulated proliferation rate. Our results confirm that GCs directly reduce the HA production of OFs, which may contribute to the beneficial effect of GCs in GO.     

Surface characteristics of layered silicates: Influence on the properties of clay/polymer nanocomposites

A sodium montmorillonite and six organophilic montmorillonites coated with different surfactants were characterized in various ways in the study. Particle and surface characteristics were determined by nitrogen adsorption and inverse gas chromatography, respectively. The gallery structure of organophilic montmorillonite, the orientation of surfactants in the galleries, and surface coverage were estimated by X-ray diffraction measurements and model calculations. The effect of organophilization on the properties of polypropylene/clay composites was determined by the measurement of tensile properties. The results showed that the surface energy of uncoated layered silicates is large; thus, the forces keeping the layers together are very strong. The long chain surfactants used for the coating of montmorillonite orientate more or less parallel to the surface and usually cover the platelets in a single layer in commercial silicates. Surplus surfactant is not located in the galleries, but among the particles, and might influence the properties of composites negatively. Organophilization leads to the drastic decrease of surface free energy. Surface tension of all coated clays is practically the same, irrespective of the type of the surfactant used for treatment. Low surface energy leads to weaker forces between the layers, which might facilitate exfoliation. This effect can be further enhanced by the use of surfactants with two long aliphatic chains, one of which orientates vertically to the surface, leading to larger gallery distance. Polymer/silicate interaction also decreases as an effect of decreasing surface tension proved by the decrease of tensile yield stress of polypropylene/montmorillonite composites. Besides surface tension, the exfoliation of layered silicates is influenced by several other factors as well, like gallery distance, mutual solubility of the components, competitive adsorption, or possible chemical reactions.     

Synthesis of Four Orthogonally Protected Rare l-Hexose Thioglycosides from d-Mannose by C-5 and C-4 Epimerization

l-Hexoses are important components of biologically relevant compounds and precursors of some therapeuticals. However, they typically cannot be obtained from natural sources and due to the complexity of their synthesis, their commercially available derivatives are also very expensive. Starting from one of the cheapest d-hexoses, d-mannose, using inexpensive and readily available chemicals, we developed a reaction pathway to obtain two orthogonally protected l-hexose thioglycoside derivatives, l-gulose and l-galactose, through the corresponding 5,6-unsaturated thioglycosides by C-5 epimerization. From these derivatives, the orthogonally protected thioglycosides of further two l-hexoses (l-allose and l-glucose) were synthesized by C-4 epimerization. The preparation of the key intermediates, the 5,6-unsaturated derivatives, was systematically studied using various protecting groups. By the method developed, we are able to produce highly functionalized l-gulose derivatives in 9 steps (total yields: 21–23%) and l-galactose derivatives in 12 steps (total yields: 6–8%) starting from d-mannose.     

Identification of β-strand mediated protein–protein interaction inhibitors using ligand-directed fragment ligation

β-Strand mediated protein–protein interactions (PPIs) represent underexploited targets for chemical probe development despite representing a significant proportion of known and therapeutically relevant PPI targets. β-Strand mimicry is challenging given that both amino acid side-chains and backbone hydrogen-bonds are typically required for molecular recognition, yet these are oriented along perpendicular vectors. This paper describes an alternative approach, using GKAP/SHANK1 PDZ as a model and dynamic ligation screening to identify small-molecule replacements for tranches of peptide sequence. A peptide truncation of GKAP functionalized at the N- and C-termini with acylhydrazone groups was used as an anchor. Reversible acylhydrazone bond exchange with a library of aldehyde fragments in the presence of the protein as template and in situ screening using a fluorescence anisotropy (FA) assay identified peptide hybrid hits with comparable affinity to the GKAP peptide binding sequence. Identified hits were validated using FA, ITC, NMR and X-ray crystallography to confirm selective inhibition of the target PDZ-mediated PPI and mode of binding. These analyses together with molecular dynamics simulations demonstrated the ligands make transient interactions with an unoccupied basic patch through electrostatic interactions, establishing proof-of-concept that this unbiased approach to ligand discovery represents a powerful addition to the armory of tools that can be used to identify PPI modulators.

Dynamic ligation screening is used to identify acylhydrazone-linked peptide-fragment hybrids which bind to the SHANK1 PDZ domain with comparable affinity to the native GKAP peptide as shown by biophysical and structural analyses.     

Optical Resolution of Two Pharmaceutical Bases with Various Uses of Tartaric Acid Derivatives and Their Sodium Salts: Racemic Ephedrine and Chloramphenicol Base

The optically active dibenzoyltartaric acid, tartaric acid, and its sodium salts were successfully applied to the optical resolution of (1R,2S)(1S,2R)-2-(methylamino)-1-phenylpropan-1-ol (EPH) and (1R,2R)(1S,2S)-2-amino-1-(4-nitrophenyl)propane-1,3-diol (AD) as resolving agents. It was observed that both compounds’ resolution using a mixture of salts of quasi-racemic resolving agents showed a change in chiral recognition under the same conditions compared to the result of the use of the single enantiomeric resolving agent. The changes are followed by detailed analytical (XRD, FTIR, and DSC) studies. Meanwhile, the DASH indexing software package was also tested on powder XRD patterns of pure initial materials and intermediate salt samples of high diastereomeric excess.