Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.

Oligomeric pseudorotaxanes adopting infinite-chain lattice superstructures

Above a critical chain length, where oligomers contain five or more recognition units, apparently infinite donor–acceptor polypseudorotaxanes are formed in the solid state. X-ray crystallographic analyses of three different examples have shown that although the oligomeric chains are undoubtedly discrete and monodisperse, they nevertheless appear to be infinite in the crystal.

Liquid AP-UV-MALDI enables stable ion yields of multiply charged peptide and protein ions for sensitive analysis by mass spectrometry

In biological mass spectrometry (MS), two ionization techniques are predominantly employed for the analysis of larger biomolecules, such as polypeptides. These are nano-electrospray ionization [1, 2] (nanoESI) and matrix-assisted laser desorption/ionization [3, 4] (MALDI). Both techniques are considered to be “soft”, allowing the desorption and ionization of intact molecular analyte species and thus their successful mass-spectrometric analysis. One of the main differences between these two ionization techniques lies in their ability to produce multiply charged ions. MALDI typically generates singly charged peptide ions whereas nanoESI easily provides multiply charged ions, even for peptides as low as 1000 Da in mass. The production of highly charged ions is desirable as this allows the use of mass analyzers, such as ion traps (including orbitraps) and hybrid quadrupole instruments, which typically offer only a limited m/z range (< 2000–4000). It also enables more informative fragmentation spectra using techniques such as collisioninduced dissociation (CID) and electron capture/transfer dissociation (ECD/ETD) in combination with tandem MS (MS/MS). [5, 6] Thus, there is a clear advantage of using ESI in research areas where peptide sequencing, or in general, the structural elucidation of biomolecules by MS/MS is required. Nonetheless, MALDI with its higher tolerance to contaminants and additives, ease-of-operation, potential for highspeed and automated sample preparation and analysis as well as its MS imaging capabilities makes it an ionization technique that can cover bioanalytical areas for which ESI is less suitable. [7, 8] If these strengths could be combined with the analytical power of multiply charged ions, new instrumental configurations and large-scale proteomic analyses based on MALDI MS(/MS) would become feasible.

Polymer therapeutics designed for a combination therapy of hormone-dependent cancer

Designer drug: A polymer therapeutic was designed for a combination therapy of breast cancer. N-(2-Hydroxypropyl)methacrylamide was used as the model polymer platform to prepare a unimolecular polymer conjugate (see picture, radius of gyration: 12.8 nm) that combines an endocrine (the aromatase inhibitor aminoglutethimide, blue) and a chemotherapeutic agent (the anthraxcycline antibiotic doxorubicin, red).

Insights into the molecular architecture of a peptide nanotube using FTIR and solid-state NMR spectroscopic measurements on an aligned sample

The self-assembly of proteins and peptides into b-sheet-rich amyloid fibers is a process that has gained notoriety because of its association with human diseases and disorders. Spontaneous self-assembly of peptides into nonfibrillar supramolecular structures can also provide a versatile and convenient mechanism for the bottom-up design of biocompatible materials with functional properties favoring a wide range of practical applications.[1] One subset of these fascinating and potentially useful nanoscale constructions are the peptide nanotubes, elongated cylindrical structures with a hollow center bounded by a thin wall of peptide molecules.[2] A formidable challenge in optimizing and harnessing the properties of nanotube assemblies is to gain atomistic insight into their architecture, and to elucidate precisely how the tubular morphology is constructed from the peptide building blocks. Some of these fine details have been elucidated recently with the use of magic-angle-spinning (MAS) solidstate NMR (SSNMR) spectroscopy.[3] MAS SSNMR measurements of chemical shifts and through-space interatomic distances provide constraints on peptide conformation (e.g., b-strands and turns) and quaternary packing. We describe here a new application of a straightforward SSNMR technique which, when combined with FTIR spectroscopy, reports quantitatively on the orientation of the peptide molecules within the nanotube structure, thereby providing an additional structural constraint not accessible to MAS SSNMR.

AP-UV-MALDI mit flüssigen Matrizes: stabile Ionenausbeuten von mehrfach geladenen Peptid- und Proteinionen für die empfindliche Massenspektrometrie

In der biologischen Massenspektrometrie (MS) werden überwiegend zwei Ionisationstechniken für die Analyse von grçßeren Biomolekfürlen wie Polypeptiden eingesetzt. Dies sind die Nano-Elektrospray-Ionisation[1,2] (nanoESI) und die matrixunterstfürtzte Laserdesorption/-ionisation[3, 4] (MALDI). Beide Techniken werden als „sanft“ bezeichnet, weil sie die Desorption und Ionisation von intakten Analytmolekfürlen und damit ihre erfolgreiche massenspektrometrische Analyse erlauben. Einer der wichtigsten Unterschiede zwischen diesen beiden Ionisationstechniken liegt in ihrer F�higkeit, mehrfach geladene Ionen zu erzeugen. MALDI erzeugt typischerweise einfach geladene Peptidionen, w�hrend nano- ESI leicht mehrfach geladene Ionen produziert, sogar für Peptide mit einer Masse von weniger als 1000 Da. Die Erzeugung von hoch geladenen Ionen ist wünschenswert, da dies die Verwendung von Massenanalysatoren wie Ionenfallen (inkl. Orbitraps) und Hybrid-Quadrupolinstrumenten ermçglicht, die typischerweise nur einen begrenzten m/z- Bereich (<2000–4000) bieten. Hohe Ladungszust�nde ermçglichen auch die Aufnahme von informativeren Fragmentionenspektren, wenn Methoden wie die kollisionsinduzierte Dissoziation (CID), die Elektroneneinfang-Dissoziation (ECD) und die Elektronentransfer-Dissoziation (ETD) in Kombination mit der Tandem-MS (MS/MS) verwendet werden.

Switchable amplification of vibrational circular dichroism as a probe of local chiral structure

A new method to detect the vibrational circular dichroism (VCD) of a localized part of a chiral molecular system is reported. A local VCD amplifier was implemented, and the distance dependence of the amplification was investigated in a series of peptides. The results indicate a characteristic distance of 2.0±0.3 bonds, which suggests that the amplification is a localized phenomenon. The amplifier can be covalently coupled to a specific part of a molecule, and can be switched ON and OFF electrochemically. By subtracting the VCD spectra obtained when the amplifier is in the ON and OFF states, the VCD of the local environment of the amplifier can be separated from the total VCD spectrum. Switchable local VCD amplification thus makes it possible to “zoom in” on a specific part of a chiral molecule.

Reversal of a single base pair step controls guanine photo-oxidation by an intercalating Ru(II) dipyridophenazine complex

Small changes in DNA sequence can often have major biological effects. Here the rates and yields of guanine photo-oxidation by Λ [Ru(TAP)2(dppz)]2+ have been compared in 5′-{CCGGATCCGG}2 and 5′-{CCGGTACCGG}2 using ps/ns transient visible and time-resolved IR (TRIR) spectroscopy. The inefficiency of electron transfer in the TA sequence is consistent with the 5′-TA-3′ vs. 5′-AT-3′ binding preference predicted by X-ray crystallography. The TRIR spectra also reveal the differences in binding sites in the two oligonucleotides.