Volume of Tubes,Non Polynomial Behavior

The behavior of the volume of the tube around a given compact subset in finite dimension, depending on the radius r, is an old and important question. It is related to many fields, like differential geometry, geometric measure theory, integral geometry, and also probability and statistics. Federer (Trans. Amer. Math. Soc. 93, 418–491, 1959), introduces the class of sets with positive reach, for which the volume is given by a polynomial in the radius r. For applications, in numerical analysis and statistics for example, an “almost” polynomial behavior is of equal interest. We exhibit an example showing how far to a polynomial can be the volume of the tube, when the radius r tends to 0, for the simplest extension of the class of sets with positive reach, namely the class of (locally finite) union of sets with positive reach -satisfying a tangency condition- as introduced by Zähle (I. Math. Nachr. 119, 327–339, 1984).     

Effects of salts and ethanol on the population and morphology of triblock copolymer micelles in solution

The morphological changes of micelles composed of triblock copolymer of ethylene oxide and propylene oxide (EO20PO70EO20) in the presence of different inorganic salts and ethanol have been investigated using dynamic light scattering (DLS), rheometry, and cryogenic transmission electron microscopy (cryo-EM). The following salts were studied: KF, KCl, KI, LiCl, and CsCl. In the presence of KF, KCl, and CsCl, spherical and wormlike micelles coexist. LiCl and KI have little influence on the morphology of the micelles, whereas KF has the most pronounced effect. In agreement with the well-known Hoffmeister anion salt series, F- has the strongest effect of the three anions studied (F-, Cl-, I-). In contrast, the effectiveness of the cation type does not follow the original Hoffmeister cation series. The addition of ethanol to the KCl micellar solutions leads to the formation of more or longer wormlike micelles, which start to interact at certain copolymer concentrations depending on the volume fraction of ethanol added. Both the dilute and the semidilute regimes of the wormlike micelles were studied. The length of the micelles reaches a maximum value at around 8-10 vol % ethanol, after which it decreases again. At higher ethanol concentrations (18 vol %), spherical micelles are formed. Conclusions from this study enhance our understanding of the role played by ethanol and salts in the formation of micelle-templated mesoporous materials, such as SBA-15.     

Molecular simulations of solute transport in xylose isomerase crystals

Cross-linked enzyme crystals (CLECs) enclose an extensive regular matrix of chiral solvent-filled nanopores, via which ions and solutes travel in and out. Several cross-linked enzyme crystals have recently been used for chiral separation and as biocatalysts. We studied the dynamics of solute transport in orthorhombic d-xylose isomerase (XI) crystals by means of Brownian dynamics (BD) and molecular dynamics (MD) simulations, which show how the protein residues influence the dynamics of solute molecules in confined regions inside the lattice. In the BD simulations, coarse-grained beads represent solutes of different sizes. The diffusion of S-phenylglycine molecules inside XI crystals is investigated by long-time MD simulations. The computed diffusion coefficients within a crystal are found to be orders of magnitude lower than in bulk water. The simulation results are compared to the recent experimental studies of diffusion and reaction inside XI crystals. The insights obtained from simulations allow us to understand the nature of solute-protein interactions and transport phenomena in CLECs, which is useful for the design of novel nanoporous biocatalysts and bioseparations based on CLECs.     

Helical content of a β(3)-octapeptide in methanol: molecular dynamics simulations explain a seeming discrepancy between conclusions derived from CD and NMR data

Connecting experimental observables with the underlying conformational ensemble is a long-standing problem in the structure determination of biomolecules. The simulations described in this article attempt to resolve a seeming discrepancy between the conformational features derived from measured NOE intensities, (3)J-coupling constants, and circular dichroism (CD) spectra for two β-peptides differing in a linker between two side-chains. Although both peptides are very similar in terms of the r(-6) averaged distances between atom pairs involved in the observed NOEs, the molecular dynamics trajectories suggest why the CD spectra show a greater 3(14)-helical propensity for the linked, cyclic peptide than for the linear one, whereas slightly more NMR NOE peaks are observed and assigned for the latter. The nine 100 ns unrestrained simulations show better agreement with the observed experimental data than the single conformations derived from the published NMR structures by additional energy minimization with the GROMOS force field. They show why the seemingly contradictory quantities obtained by NMR and CD spectroscopy can arise from a single conformational ensemble.     

Redox-Switchable Resorcin[4]arene Cavitands: Molecular Grippers

Diquinone-based resorcin[4]arene cavitands that open to a kite and close to a vase form upon changing their redox state, thereby releasing and binding guests, have been prepared and studied. The switching mechanism is based on intramolecular H-bonding interactions that stabilize the vase form and are only present in the reduced hydroquinone state. The intramolecular H-bonds were characterized using X-ray, IR, and NMR spectroscopies. Guests were bound in the closed, reduced state and fully released in the open, oxidized state.     

Approximation and regularization of Lipschitz functions: Convergence of the gradients

We examine the possible extensions to the Lipschitzian setting of the classical result on -convergence: first (approximation), if a sequence of functions of class from to converges uniformly to a function of class , then the gradient of is a limit of gradients of in the sense that ; second (regularization), the functions can be chosen to be of class and -converging to in the sense that . In other words, the space of functions is dense in the space of functions endowed with the pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if is a locally Lipschitz function, we build a sequence of smooth functions such that

In other words, the space of functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality , which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

     

The New Photocrystallography

Shine your light : Recent work in photocrystallography demonstrates a large photoinduced increase in the magnetic susceptibility of a solid, which is accompanied by significant changes in the bond lengths of the Nd? NC? Fe moiety (see picture; black are ground‐state bond lengths in Å, red are photoinduced bond lengths).

     

From Biomimicking to Bioinspired Design of Electrocatalysts for CO2 Reduction to C1 Products

The electrochemical reduction of CO₂ (CO₂RR) is a promising approach to maintain a carbon cycle balance and produce value-added chemicals. However, CO₂RR technology is far from mature, since the conventional CO₂RR electrocatalysts suffer from low activity (leading to currents <10 mA cm⁻² in an H-cell), stability (<120 h), and selectivity. Hence, they cannot meet the requirements for commercial applications (>200 mA cm⁻², >8000 h, >90 % selectivity). Significant improvements are possible by taking inspiration from nature, considering biological organisms that efficiently catalyze the CO₂ to various products. In this minireview, we present recent examples of enzyme-inspired and enzyme-mimicking CO₂RR electrocatalysts enabling the production of C₁ products with high faradaic efficiency (FE). At present, these designs do not typically follow a methodical approach, but rather focus on isolated features of biological systems. To achieve disruptive change, we advocate a systematic design methodology that leverages fundamental mechanisms associated with desired properties in nature and adapts them to the context of engineering applications.