Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

A first low‐resolution difference Fourier map of phosphorus in a membrane protein from near‐edge anomalous diffraction

Crystal diffraction of three membrane proteins (cytochrome bc₁ complex, sarcoplasmic reticulum Ca²⁺ ATPase, ADP‐ATP carrier) and of one nucleoprotein complex (leucyl tRNA synthetase bound to tRNAleu, leuRS:tRNAleu) was tested at wavelengths near the X‐ray K‐absorption edge of phosphorus using a new set‐up for soft X‐ray diffraction at the beamline ID01 of the ESRF. The best result was obtained from crystals of Ca²⁺ ATPase [adenosin‐5′‐(β,γ‐methylene) triphosphate complex] which diffracted out to 7 Å resolution. Data were recorded at a wavelength at which the real resonant scattering factor of phosphorus reaches the extreme value of ?20 electron units. The positions of the four triphosphates of the monoclinic unit cell of the ATPase have been obtained from a difference Fourier synthesis based on a limited set of anomalous diffraction data.     

Studies on the Stability of the Human Cone Visual Pigments†

The retina of vertebrates contains two kinds of photoreceptor cells, rods and cones, which contain their specific visual pigments that are responsible for scotopic and photopic vision, respectively. In cone photoreceptor cells, there are three types of color pigments: blue, green and red, each with a distinctive absorption maximum. The goal of this investigation was to identify optimal conditions under which these pigments could be obtained and isolated in a stable form, thereby facilitating structural studies using high‐resolution approaches. For this purpose, all three human cone opsins were initially expressed in mammalian cells, reconstituted with 11‐cis retinal, detergent solubilized, purified and their stability compared with rod rhodopsin. As all three pigments showed dramatically reduced stability relative to rhodopsin, site‐directed mutagenesis was used in an attempt to engineer stability into the green cone pigment. The mutations introduced some structural motifs and sites of posttranslational modification present in rhodopsin, as well as amino acid substitutions that have been found to stabilize the rod opsin apo‐protein. We also modified the hydrophobic environment of the green cone pigment by varying the detergent and detergent/lipid composition used during solubilization and purification, and compared them with the retinal reconstituted pigment in membranes. Our results show that these changes do not significantly improve the inherent instability of the human cone pigments, and in some cases, lead to a decrease in stability and protein aggregation. We conclude that further efforts are required to stabilize the human cone pigments in a form suitable for high‐resolution structural studies.     

Preparation of (1R,1′R)-1,1′-(anthracene-9,10-diyl)bis(2,2,2-trifluoroethanamine): a chiral diamine with low basicity

A new chiral diamine with low basicity was synthesized in enantiopure form. (1R,1′R)-1,1′-(Anthracene-9,10-diyl)bis(2,2,2-trifluoroethanamine) was obtained by means of several stereochemically controlled reactions. The structures of the title compound and several intermediates were studied.     

Surface modification of thermally expandable microspheres by grafting poly(glycidyl methacrylate) using ARGET ATRP

This study demonstrates the surface modification of thermally expandable core/shell microspheres by grafting glycidyl methacrylate (GMA) using activators regenerated by electron transfer (ARGET) ATRP. To retain the expansion properties it was essential to minimize the shear forces, use solvents compatible with the microspheres and keep the reaction times short (three hours or less). Using microspheres with hydroxyl groups on the surface, it was found that after converting these to α-bromo esters, GMA could be grafted by ARGET ATRP using only 50 ppm of copper catalyst in toluene at 30 °C. Decent control of the polymerization was achieved with PMDETA as ligand reaching PDIs of 1.4 for the solution polymerization of GMA. When microspheres were present, the polymerization was less controlled with higher PDIs. The epoxide groups of the grafted microspheres were hydrolyzed by HCl in THF providing a hydrophilic surface of the microsphere. The expansion property of the microspheres was studied after each reaction step by thermal mechanical analysis, and it was found that the expansion capacity was well preserved with only limited negative effect on the microspheres.     

Chemometric deconvolution of gas chromatographic unresolved conjugated linoleic acid isomers triplet in milk samples

A generally known problem of GC separation of trans-7;cis-9; cis-9,trans-11; and trans-8,cis-10 CLA (conjugated linoleic acid) isomers was studied by GC–MS on 100 m capillary column coated with cyanopropyl silicone phase at isothermal column temperatures in a range of 140–170 °C. The resolution of these CLA isomers obtained at given conditions was not high enough for direct quantitative analysis, but it was, however, sufficient for the determination of their peak areas by commercial deconvolution software. Resolution factors of overlapped CLA isomers determined by the separation of a model CLA mixture prepared by mixing of a commercial CLA mixture and CLA isomer fraction obtained by the HPLC semi-preparative separation of milk fatty acids methyl esters were used to validate the deconvolution procedure. Developed deconvolution procedure allowed the determination of the content of studied CLA isomers in ewes’ and cows’ milk samples, where dominant isomer cis-9,trans-11 is eluted between two small isomers trans-7,cis-9 and trans-8,cis-10 (in the ratio up to 1:100).     

Zinc azaphthalocyanines with pyridin-3-yloxy peripheral substituents

Phthalocyanines (Pc), which are peripherally substituted with pyridin-3-yloxy groups, have shown promise as sensitizers for photodynamic cancer therapy (PDT). Some aza-analogues (AzaPc) are reported here. Four monomers were synthesized, i.e. 5,6-di(pyridin-3-yloxy)pyrazine-2,3-dicarbonitrile, and three pyrazine-2,3-dicarbonitriles, substituted with pyridin-3-yloxy- in combination with H, Me and Ph groups. Cyclotetramerizations of these monomers with the reagent Zn(quinoline)₂Cl₂ yielded the targeted ZnAzaPcs in 20–40% yields.The cyclotetramerizations were accompanied, and apparently initiated, by complexation between zinc(II) and the pyridin-3-yloxy groups attached to the pyrazine-dicarbonitriles. Two such zinc(II) complexes were isolated and characterized. Identifications of all new substances were primarily based on NMR spectra, where the pulse techniques COSY, NOESY, HSQC and HMBC were applied. Molecular ions of the ZnAzaPcs were determined by mass spectrometry (MALDI-TOF). The UV–Vis spectra of these macrocycles were as expected, with Q-band absorptions at 630–650 nm and molar extinction coefficients, ε, 70 000–100 000. Eight peripheral pyridin-3-yloxy groups induced a small blue shift of the Q-band, from 636 nm for unsubstituted ZnAzaPc, to 630 nm, whereas a red shifted Q-band at 650 nm resulted from the combination of phenyl and pyridin-3-yloxy substituents. Improved solubilities were observed for the unsymmetrical ZnAzaPcs compared to octa(pyridin-3-yloxy)ZnAzaPc.     

Structural and Magnetic Insights into the Trinuclear Ferrocenophane and Unexpected Hydrido Inverse Crown Products of Alkali‐Metal‐Mediated Manganation(II) of Ferrocene

With the aim of introducing the diisopropylamide [NiPr₂] ^? ligand to alkali‐metal‐mediated manganation (AMMMn) chemistry, the temperature‐dependent reactions of a 1:1:3 mixture of butylsodium, bis(trimethylsilylmethyl)manganese(II), and diisopropylamine with ferrocene in hexane/toluene have been investigated. Performed at reflux temperature, the reaction affords the surprising, ferrocene‐free, hydrido product [Na₂Mn₂ (μ‐H)₂{N(iPr)₂}₄]?2 toluene ( 1 ), the first Mn hydrido inverse crown complex. Repeating the reaction rationally, excluding ferrocene, produces 1 in an isolated crystalline yield of 62 %. At lower temperatures, the same bimetallic amide mixture leads to the manganation of ferrocene to generate the first trimanganese, trinuclear ferrocenophane, [{Fe(C₅H₄)₂}₃{Mn₃Na₂(NiPr₂)₂ (HNiPr₂)₂}] ( 2 ) in an isolated crystalline yield of 81 %. Both 1 and 2 have been characterised by X‐ray crystallographic studies. The magnetic properties of paramagnetic 1 and 2 have also been examined by variable‐temperature magnetisation measurements on powdered samples. For 1 , the room‐temperature value for χT is 3.45 cm³ K mol^?1, and on lowering the temperature a strong antiferromagnetic coupling between the two Mn ions is observed. For 2 , the room‐temperature value for χT is 4.06 cm³ K mol^?1, which is significantly lower than the expected value for three isolated paramagnetic Mn^II ions.     

Structurally Defined Potassium‐Mediated Zincation of Pyridine and 4‐R‐Substituted Pyridines (R=Et,iPr, tBu,Ph, and Me2N) by Using Dialkyl–TMP–Zincate Bases

Two potassium–dialkyl–TMP–zincate bases [(pmdeta)K(μ‐Et)(μ‐tmp)Zn(Et)] ( 1 ) (PMDETA=N,N,N′,N′′,N′′‐pentamethyldiethylenetriamine, TMP=2,2,6,6‐tetramethylpiperidide), and [(pmdeta)K(μ‐nBu)(μ‐tmp)Zn(nBu)] ( 2 ), have been synthesized by a simple co‐complexation procedure. Treatment of 1 with a series of substituted 4‐R‐pyridines (R=Me₂N, H, Et, iPr, tBu, and Ph) gave 2‐zincated products of the general formula [{2‐Zn(Et)₂‐μ‐4‐R‐C₅H₃N}₂ ? 2{K(pmdeta)}] ( 3 – 8 , respectively) in isolated crystalline yields of 53, 16, 7, 23, 67, and 51 %, respectively; the treatment of 2 with 4‐tBu‐pyridine gave [{2‐Zn(nBu)₂‐μ‐4‐tBu‐C₅H₃N}₂ ? 2{K(pmdeta)}] ( 9 ) in an isolated crystalline yield of 58 %. Single‐crystal X‐ray crystallographic and NMR spectroscopic characterization of 3 – 9 revealed a novel structural motif consisting of a dianionic dihydroanthracene‐like tricyclic ring system with a central diazadicarbadizinca (ZnCN)₂ ring, face‐capped on either side by PMDETA‐wrapped K⁺ cations. All the new metalated pyridine complexes share this dimeric arrangement. As determined by NMR spectroscopic investigations of the reaction filtrates, those solutions producing 3 , 7 , 8 , and 9 appear to be essentially clean reactions, in contrast to those producing 4 , 5 , and 6 , which also contain laterally zincated coproducts. In all of these metalation reactions, the potassium–zincate base acts as an amido transfer agent with a subsequent ligand‐exchange mechanism (amido replacing alkyl) inhibited by the coordinative saturation, and thus, low Lewis acidity of the 4‐coordinate Zn centers in these dimeric molecules. Studies on analogous trialkyl–zincate reagents in the absence and presence of stoichiometric or substoichiometric amounts of TMP(H) established the importance of Zn? N bonds for efficient zincation.