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In this paper we present a method for computing the dihedral angle sums (and their two-sided estimates) of cartesian and skew product polytopes provided the sums of dihedral angles (or their estimates) are known for the factors. In addition, a formula for computing the number of facets of such product polytopes is derived. The method proposed is very universal and illustrated by several examples. The estimates are in a full agreement with known results for prisms and hexahedra.

We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval (3π, 5π). Moreover, for any number in (3π, 5π) there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound 4π is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation and analysis.

Intrinsically disordered proteins (IDPs) are challenging established structural biology perception and urge a reassessment of the conventional understanding of the subtle interplay between protein structure and dynamics. Due to their importance in eukaryotic life and central role in protein interaction networks, IDP research is a fascinating and highly relevant research area in which NMR spectroscopy is destined to be a key player. The flexible nature of IDPs, as a result of the sampling of a vast conformational space, however, poses a tremendous scientific challenge, both technically and theoretically. Pronounced signal averaging results in narrow signal dispersion and requires higher dimensionality NMR techniques. Moreover, a fundamental problem in the structural characterization of IDPs is the definition of the conformational ensemble sampled by the polypeptide chain in solution, where often the interpretation relies on the concept of ‘residual structure’ or ‘conformational preference’. An important source of structural information is information-rich NMR experiments that probe protein backbone dihedral angles in a unique manner. Cross-correlated relaxation experiments have proven to fulfil this task as they provide unique information about protein backbones, particularly in IDPs. Here we present a novel cross-correlation experiment that utilizes non-uniform sampling detection schemes to resolve protein backbone dihedral ambiguities in IDPs. The sensitivity of this novel technique is illustrated with an application to the prototypical IDP α-Synculein for which unexpected deviations from random-coil-like behaviour could be observed.

Many mainstream dielectric energy storage technologies in the emergent applications, such as renewable energy, electrified transportations and advanced propulsion systems, are usually required to operate under harsh-temperature conditions. However, excellent capacitive performance and thermal stability tend to be mutually exclusive in the current polymer dielectric materials and applications. Here, we report a strategy to tailor structural units for the design of high-temperature polymer dielectrics. A library of polyimide-derived polymers from diverse combinations of structural units are predicted, and 12 representative polymers are synthesized for direct experimental investigation. This study provides important insights into decisive structural factors necessary to achieve robust and stable dielectrics with high energy storage capabilities at elevated temperature. We also find that the high-temperature insulation performance would experience diminishing marginal utility as the bandgap increases beyond a critical point, which is strongly correlated to the dihedral angle between neighboring planes of conjugation in these polymers. By experimentally testing the optimized and predicted structures, an increased energy storage at temperatures up to 250 °C is observed. We discuss the possibility for this strategy to be generally applied to other polymer dielectrics to achieve further performance enhancement. Polymer dielectrics face huge challenges in the harsh environments of emergent applications. Now, increased energy storage of polymer dielectrics at temperatures up to 250 °C by designing tailored combinations of structural units is reported.

[...]the authors present the solutions offered for this problem by the computer programs when 3D model is used. [...]they agreed that it necessary to analyse which tools provide software such Aut°CAD or Inventor to solve this problem when a part is designed, from the start, using 3D model. First one is based on the method of substituting planes of projection, and the second is based on the folding method of the planes on a level plane [1], [2], [3]. The new projection axis (O2X2) is choose in such way in that, in the new projection system O2X2[Vi][Hi], the intersection line is a vertical line. [...]the projection line O2X2 is perpendicular on the (d'i) projection, and the intersection line (D) has, in the projection system (O2X2)[Vi][Hi], the projections (d'i) and (di).

Intrinsically disordered proteins (IDPs) do not autonomously adopt a stable unique 3D structure and exist as an ensemble of rapidly interconverting structures. They are characterized by significant conformational plasticity and are associated with several biological functions and dysfunctions. The rapid conformational fluctuation is governed by the backbone segmental dynamics arising due to the dihedral angle fluctuation on the Ramachandran ϕ–ψ conformational space. We discovered that the intrinsic backbone torsional mobility can be monitored by a sensitive fluorescence readout, namely fluorescence depolarization kinetics, of tryptophan in an archetypal IDP such as α-synuclein. This methodology allows us to map the site-specific torsional mobility in the dihedral space within picosecond-nanosecond time range at a low protein concentration under the native condition. The characteristic timescale of ~ 1.4 ns, independent of residue position, represents collective torsional dynamics of dihedral angles (ϕ and ψ) of several residues from tryptophan and is independent of overall global tumbling of the protein. We believe that fluorescence depolarization kinetics methodology will find broad application to study both short-range and long-range correlated motions, internal friction, binding-induced folding, disorder-to-order transition, misfolding and aggregation of IDPs.

Doping has been widely used to control the charge carrier concentration in organic semiconductors. However, in conjugated polymers, n-doping is often limited by the tradeoff between doping efficiency and charge carrier mobilities, since dopants often randomly distribute within polymers, leading to significant structural and energetic disorder. Here, we screen a large number of polymer building block combinations and explore the possibility of designing n-type conjugated polymers with good tolerance to dopant-induced disorder. We show that a carefully designed conjugated polymer with a single dominant planar backbone conformation, high torsional barrier at each dihedral angle, and zigzag backbone curvature is highly dopable and can tolerate dopant-induced disorder. With these features, the designed diketopyrrolopyrrole (DPP)-based polymer can be efficiently n-doped and exhibit high n-type electrical conductivities over 120 S cm −1 , much higher than the reference polymers with similar chemical structures. This work provides a polymer design concept for highly dopable and highly conductive polymeric semiconductors. In conjugated polymers, n-doping is often limited by the tradeoff between doping efficiency and charge carrier mobilities, since dopants often randomly distribute within polymers, leading to significant structural and energetic disorder. Here, the authors screen a large number of polymer building block combinations and explore the possibility of designing n-type conjugated polymers with good tolerance to dopant-induced disorder.

The process of coalescence of various metal nanoparticles has been studied by the Monte Carlo method. The interaction of nanoparticles is described by a multiparticle Gupta type potential. An algorithm of recognizing and estimating a dihedral angle at the neck is developed. The dihedral angle between metal nanoparticles during their sintering is estimated.

Perylene bisimide (PBI) dyes are known as red, maroon and black pigments, whose colors depend on the close π−π stacking arrangement. However, contrary to the luminescent monomers, deep-red and black PBI pigments are commonly non- or only weakly fluorescent due to (multiple) quenching pathways. Here, we introduce N -alkoxybenzyl substituted PBIs that contain close π stacking arrangement (exhibiting d π−π ≈ 3.5 Å, and longitudinal and transversal displacements of 3.1 Å and 1.3 Å); however, they afford deep-red emitters with solid-state fluorescence quantum yields ( Φ F ) of up to 60%. Systematic photophysical and computational studies in solution and in the solid state reveal a sensitive interconversion of the PBI-centred locally excited state and a charge transfer state, which depends on the dihedral angle ( θ ) between the benzyl and alkoxy groups. This effectively controls the emission process, and enables high Φ F by circumventing the common quenching pathways commonly observed for perylene black analogues. Perylene bisimide dyes are high-performance pigments that have red, maroon or black colors, but with typically only weak fluorescence in the solid state. Here, the authors report a series of π-stacked PBI derivatives that show highly efficient deep-red fluorescence in the solid state.

Diatomic-site catalysts (DACs) garner tremendous attention for selective CO 2 photoreduction, especially in the thermodynamical and kinetical mechanism of CO 2 to C 2+ products. Herein, we first engineer a novel Zn-porphyrin/RuCu-pincer complex DAC (ZnPor-RuCuDAC). The heteronuclear ZnPor-RuCuDAC exhibits the best acetate selectivity (95.1%), while the homoatomic counterparts (ZnPor-Ru 2 DAC and ZnPor-Cu 2 DAC) present the best CO selectivity. In-situ spectroscopic measurements reveal that the heteronuclear Ru–Cu sites easily appear C 1 intermediate coupling. The in-depth analyses confirm that due to the strong gradient orbital coupling of Ru4 d –Cu3 d resonance, two formed * CO intermediates of Ru–Cu heteroatom show a significantly weaker electrostatic repulsion for an asymmetric charge distribution, which result from a side-to-side absorption and narrow dihedral angle distortion. Moreover, the strongly overlapped Ru/Cu- d and CO molecular orbitals split into bonding and antibonding orbitals easily, resulting in decreasing energy splitting levels of C 1 intermediates. These results collectively augment the collision probability of the two * CO intermediates on heteronuclear DACs. This work first provides a crucial perspective on the symmetry-forbidden coupling mechanism of C 1 intermediates on diatomic sites. Molecular insights into the selectivity mechanism of dual-atom sites are required to engineer efficient solar-fuel catalysts. Here, the authors reveal symmetry-forbidden coupling mechanism of C1 intermediates on diatomic sites by manipulating metal gradient orbital interaction over diatomic COFs.