Electromagnetism,Local Covariance,the Aharonov–Bohm Effect and Gauss’ Law

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov–Bohm effect. By Peierls’ method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss’ law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov–Bohm effect and Gauss’ law – a relationship which seems to have gone unnoticed so far. Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p <  dim(M)), modulo exact fields, for the Lagrangian density ${\mathcal{L} = \frac{1}{2} dA\wedge*dA+ A\wedge*j}$ . In conclusion we note that the theory is not locally covariant, in the sense of Brunetti–Fredenhagen–Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.     

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.     

Scope and mechanism for lewis acid-catalyzed cycloadditions of aldehydes and donor-acceptor cyclopropanes: evidence for a stereospecific intimate ion pair pathway

In this work, the one-step diastereoselective synthesis of cis-2,5-disubstituted tetrahydrofurans via Lewis acid catalyzed [3 + 2] cycloadditions of donor-acceptor (D-A) cyclopropanes and aldehydes is described. The scope and limitations with respect to both reaction partners are provided. A detailed examination of the mechanism has been performed, including stereochemical analysis and electronic profiling of both reactants. Experimental evidence supports an unusual stereospecific intimate ion pair mechanism wherein the aldehyde functions as a nucleophile and malonate acts as the nucleofuge. The reaction proceeds with inversion at the cyclopropane donor site and allows absolute stereochemical information to be transferred to the products with high fidelity. The mechanism facilitates the stereospecific synthesis of a range of optically active tetrahydrofuran derivatives from enantioenriched D-A cyclopropanes.     

Coordination polymers based on aluminum(III) porphyrins

Aluminum(III) porphyrin carboxylate complexes have shown an affinity for a sixth nitrogenous ligand. The use of isonicotinic or nicotinic acid, which offers both a carboxylate and a nitrogen donor in the same molecule, resulted in the formation of one-dimensional (1-D) coordination polymers. The complexes and their linear oligomers have been characterized by (1)H NMR spectroscopy and nanoelectrospray ionization spectrometry. X-ray analyses confirmed the formation of the 1-D polymers in the solid state.     

Molecular Dynamics Simulations Guide Chimeragenesis and Engineered Control of Chemoselectivity in Diketopiperazine Dimerases

In the biosynthesis of the tryptophan-linked dimeric diketopiperazines (DKPs), cytochromes P450 selectively couple DKP monomers to generate a variety of intricate and isomeric frameworks. To determine the molecular basis for selectivity of these biocatalysts we obtained a high-resolution crystal structure of selective Csp²−N bond forming dimerase, AspB. Overlay of the AspB structure onto C−C and C−N bond forming homolog NzeB revealed no significant structural variance to explain their divergent chemoselectivities. Molecular dynamics (MD) simulations identified a region of NzeB with increased conformational flexibility relative to AspB, and interchange of this region along with a single active site mutation led to a variant that catalyzes exclusive C−N bond formation. MD simulations also suggest that intermolecular C−C or C−N bond formation results from a change in mechanism, supported experimentally through use of a substrate mimic.     

Application of pulsed field gradient NMR with high gradient strength for studies of self-diffusion in lipid membranes on the nanoscale

This work demonstrates the feasibility of noninvasive studies of lipid self-diffusion in model lipid membranes on the nanoscale using proton pulsed field gradient (PFG) NMR spectroscopy with high (up to 35 T/m) gradient amplitudes. Application of high gradients affords for the use of sufficiently small diffusion times under the conditions when the width of the gradient pulses is much smaller than the diffusion time. As a result, PFG NMR studies of partially restricted or anomalous diffusion in lipid bilayers become possible over length scales as small as 100 nm. This length scale is important because it is comparable to the size of membrane domains, or lipid rafts, which are believed to exist in biomembranes. In this work, high-gradient PFG NMR has been applied to study lipid self-diffusion in three-component planar-supported multibilayers (1,2-dioleoyl- sn-glycerol-3-phosphocholine/sphingomyelin/cholesterol). The degree of lipid orientation in the bilayers was determined with (31)P NMR. A special insert was designed to mechanically align the multibilayer stack at the magic angle with respect to the direction of the constant magnetic field to address the detrimental effects of proton dipole-dipole interactions on the NMR signal. This insert is an alternative to the conventional method of magic angle orientation of lipid membranes, the goniometer probe, which is not compatible with commercial high-gradient coils because of the lack of space in the magnet bore. Macroscopic orientation of the multibilayer stacks using the insert was confirmed with (1)H NMR spectroscopic studies and the comparison of results obtained from identical experiments using a goniometer probe for orientation. Diffusion studies were carried out at three different constant magnetic field strengths ( B 0) over a range of temperatures and diffusion times. The measured diffusivities were found to be in agreement with the data obtained previously by techniques that are limited to much larger length scales of diffusion observation than high-gradient PFG NMR.     

Monitoring the thermodynamically-controlled formation of diimide-based resin-attached rotaxanes by gel-phase HR MAS 1H NMR spectroscopy

The thermodynamically controlled self-assembly of rotaxane and pseudorotaxane systems consisting of (i) a naphthodiimide thread unit terminated at one end with a pyridine ligand, and covalently linked at the other to a gel-phase polystyrene resin support, (ii) a dinaphtho-crown ether shuttle unit, and (iii) a ruthenium carbonyl metalloporphyrin stopper unit, is investigated by high resolution magic angle spinning proton (HR MAS 1H) NMR spectroscopy. The effects of variable concentration of the solution-phase components, the temperature, and added Li+ and Na+ ions are described, and the limitations of the technique are addressed. The dynamic behaviour is compared directly to the solution-phase analogues, where a bulky stopper group is substituted for the polystyrene resin bead.     

Structural investigations of magnesium silicate glasses by Si 2D Magic-Angle Flipping NMR

The two-dimensional Magic Angle Flipping Nuclear Magnetic Resonance (2D MAF NMR) experiment on ²⁹Si nuclei is used to determine the distribution of Q⁽ⁿ⁾ sites in two ²⁹Si-enriched magnesium silicate glasses with compositions 2MgO·SiO₂ and MgO·SiO₂. A significant degree of polymerization is observed in the 2MgO·SiO₂ glass, supporting previous studies using Raman and ²⁹Si NMR spectroscopy. Relative abundances of 0.629 ± 0.001 for Q⁽⁰⁾ and 0.371 ± 0.001 for Q⁽¹⁾ were obtained from spectral fits of the 2D MAF spectrum of the 2MgO·SiO₂ glass. Mole fractions for the free oxygen anion and each Q⁽ⁿ⁾-species were calculated and used in a thermodynamic model of Q⁽ⁿ⁾ disproportionation to calculate an equilibrium constant of k₀ = 0.04 ± 0.02 in 2MgO·SiO₂. In the MgO·SiO₂ glass relative abundance of 0.014 ± 0.001 for Q⁽⁰⁾, 0.191 ± 0.003 for Q⁽¹⁾, 0.530 ± 0.004 for Q⁽²⁾, 0.252 ± 0.003 for Q⁽³⁾, and 0.014 ± 0.001 for Q⁽⁴⁾ were measured. The mole fractions for the free oxygen anion and each Q⁽ⁿ⁾-species in MgO·SiO₂ were used to calculate corresponding disproportionation equilibrium constants of k₁ = 0.19 ± 0.02, k₂ = 0.174 ± 0.009, and k₃ = 0.11 ± 0.01. A comparison of k₃ values from previous MAF studies of various alkali and alkaline earth silicate glasses indicate an exponential increase in k₃ with the increasing modifying cation potential. Using the van't Hoff relation, we show that differences in both thermal history and modifier cation potential contribute to this spread in k₃ values. Nuclear shielding tensor anisotropy, ζ, and asymmetry, η, values of ζ = 0.0 ppm and η = 0.0 for Q⁽⁰⁾ and ζ = 33.0 ± 0.1 ppm, and η = 0.4 ± 0.1 for Q⁽¹⁾ in 2MgO·SiO₂ glass were determined from its 2D MAF spectrum. These values were used in obtaining the remaining values of ζ = − 36.0 ± 0.5 ppm and η = 0.99 ± 0.01 for Q⁽²⁾, and ζ = − 27.5 ± 0.5 ppm and η = 0.45 ± 0.11 for Q⁽³⁾, ζ = 0.0 ppm and η = 0.0 for Q⁽⁴⁾ in the MgO·SiO₂ glass from its 2D MAF spectrum. The magnitude of ζ values observed are lower than those reported in previous MAF studies of alkali and alkaline earth silicate glasses containing different modifier cations, consistent with previously reported trends in ζ versus modifying cation potential.     

Direct observation of surface-mediated thioacetyl deprotection: covalent tethering of a thiol-terminated porphyrin to the Ag(100) surface

Covalent tethering of macromolecules such as porphyrins to metal surfaces underpins bottom-up fabrication of systems intended for a variety of applications. Thiol linkages are especially useful but often need protection during macromolecule synthesis. By means of scanning tunneling microscopy, we directly observe the spontaneous deprotection of an acetyl-protected thioporphyrin upon contact with a silver surface, without the intervention of any solution-mediated chemistry.     

The Dirac delta function in two settings of Reverse Mathematics

The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak K?nig’s Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL₀ as one of the ‘Big’ systems of Reverse Mathematics. In the context of ERNA’s Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA’s Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA’s Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.