Stationary Solutions and Connecting Orbits for <Emphasis Type="Italic">p</Emphasis>-Laplace Equation

We deal with one dimensional p-Laplace equation of the form

$$\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \ x\in (0,l), \ t>0, \end{aligned}$$

under Dirichlet boundary condition, where \(p>2\) and \(f:[0,l]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a continuous function with \(f(x,0)=0\). We will prove that if there is at least one eigenvalue of the p-Laplace operator between \(\lim _{u\rightarrow 0} f(x,u)/|u|^{p-2}u\) and \(\lim _{|u|\rightarrow +\infty } f(x,u)/|u|^{p-2}u\), then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

     

Influence of Open Chain and Cyclic Structure of Peptidomimetics on Antibacterial Activity in E. coli Strains

An efficient method for the synthesis of functionalized peptidomimetics via multicomponent Ugi reaction has been developed. The application of trifluoroethanol (TFE) as a reaction medium provided desired products with good yields. Further, using the developed cyclisation reaction, the obtained peptidomimetics were transformed into the cyclic analogues (diketopiperazines, DKPs). The goal of the performed studies was to revised and compare whether the structure of the obtained structurally flexible acyclic peptidomimetics and their rigid cycling analogue DKPs affect antimicrobial activity. We studied the potential of synthesized peptidomimetics, both cyclic and acyclic, as antimicrobial drugs on model E. coli bacteria strains (k12, R2–R4). The biological assays reveal that DKPs hold more potential as antimicrobial drugs compared to open chain Ugi peptidomimetics. We believe that it can be due to the rigid cyclic structure of DKPs which promotes the membrane penetration in the cell of studied pathogens. The obtained data clearly indicate the high antibiotic potential of synthesized diketopiperazine derivatives over tested antibiotics.     

Isoquinoline Alkaloid Contents in Macleaya cordata Extracts and Their Acetylcholinesterase and Butyrylcholinesterase Inhibition

An important strategy for treating neurodegenerative disorders is to maintain the levels of acetylcholine in the synaptic cleft by blocking the cholinesterases. Searching for new effective compounds with inhibited acetylcholinesterase and butyrylcholinesterase activity is one of the most significant challenges of the modern scientific research. The aim of this study was the optimization of the condition for cholinesterase activity determination by high-performance liquid chromatography coupled with diode array detector (HPLC-DAD) in terms of concentrations of enzymatic reaction mixture components, temperature of incubation, and incubation time. In vitro investigation of acetylcholinesterase and butyrylcholinesterase activity inhibition by some isoquinoline alkaloids and extracts obtained from the aerial part and roots of Macleaya cordata collected in May, July, and September. Acetylcholinesterase and butyrylcholinesterase activity inhibition of the extracts obtained from the plant had not been tested previously. The application of the HPLC method allowed eliminating absorption of interfering components, for example, alkaloids such as sanguinarine and berberine. The HPLC method was successfully applied for the evaluation of the acetylcholinesterase inhibitory activity in samples such as plant extracts, especially those containing colored components adsorbing at the same wavelength as the adsorption wavelength of 5-thio-2-nitro-benzoic acid, which is the product of the reaction between thiocholine (product of the hydrolysis of acetyl/butyrylthiocholine reaction) with Ellman’s reagent. Moreover, liquid chromatography coupled with a triple quadrupole mass spectrometer (LC–QqQ–ESI–MS/MS) analysis allowed evaluating the identification of relevant bioactive compounds in the obtained plant extracts. The investigated alkaloids, especially sanguinarine and chelerythrine, and all the Macleaya cordata extracts, especially the extract obtained from the aerial part collected in May, exhibited very high cholinesterase activity inhibition. HPLC-DAD was also applied for the kinetics study of the most active alkaloids sanguinarine and chelerythrine. Our investigations demonstrated that these plant extracts can be recommended for further in vivo experiments to confirm their cholinesterase inhibition activity.     

Decoding the Fucose Migration Product during Mass-Spectrometric analysis of Blood Group Epitopes

Fucose is a signaling carbohydrate that is attached at the end of glycan processing. It is involved in a range of processes, such as the selectin-dependent leukocyte adhesion or pathogen-receptor interactions. Mass-spectrometric techniques, which are commonly used to determine the structure of glycans, frequently show fucose-containing chimeric fragments that obfuscate the analysis. The rearrangement leading to these fragments—often referred to as fucose migration—has been known for more than 25 years, but the chemical identity of the rearrangement product remains unclear. In this work, we combine ion-mobility spectrometry, radical-directed dissociation mass spectrometry, cryogenic IR spectroscopy of ions, and density-functional theory calculations to deduce the product of the rearrangement in the model trisaccharides Lewis x and blood group H2. The structural search yields the fucose moiety attached to the galactose with an α(1→6) glycosidic bond as the most likely product.