Enantiomorphic Pair Research Articles

Programmed Self-Assembly of Single Colloidal Gyroids for Chiral Photonic Crystals.

Gyroid structures are of extensive interest because they provide a rich platform for chiroptics as well as topological photonics. While the double-gyroid morphology as a bicontinuous structure is not uncommon in self-assembled soft materials, direct self-assembly of single-network gyroids has proven elusive. Here, an enantiomorphic pair of single-gyroid crystals comprising colloidal spheres is presented, and two distinct routes are demonstrated for programmed self-assembly of each single colloidal gyroid enantiomorph from rationally designed patchy spheres. The designer colloidal patchy spheres, which closely hew to their synthetic feasibility, are chiral, having either two staggered rectangular patches at opposite poles or four circular patches arranged in a well-defined geometry. The single colloidal gyroid, as well as its inverse structure, is shown to support a wide complete photonic bandgap in addition to exhibiting rich chiroptical properties, making them attractive chiral photonic crystals. The versatility of this single colloidal gyroid, the bottom-up routes devised here in silico, and the robustness of the design space for the chiral colloidal patchy spheres together make a strong case for single colloidal gyroids to supersede colloidal diamond, as a target for programmed self-assembly, in the quest for a photonic crystals operating at opticalfrequencies.

Molecular crystals with a sole bearing contact: structural classes and statistical data

Abstract All the structural classes (SCs) of molecular crystals, which are possible to be formed by a sole intermolecular bearing contact, are derived from 1,1- and 2,1-transitive nets. There are 160 (167 including enantiomorphic pairs) monosystemic SCs (in which the molecules occupy one crystallographic orbit) and 244 (248) bisystemic SCs (in which the molecules occupy two crystallographic orbits), i.e. 404 (411) SCs in all. These are the only SCs which can correspond to a homodesmic crystal structure, either organic, or inorganic, in which the structural units are bonded in a single way. Statistical investigation of the Cambridge Structural Database (CSD) shows that ∼1% of monosystemic molecular crystals can be formed by a sole intermolecular bearing contact.

Structural chirality of β -Mn

A theoretical study of Bragg diffraction by an enantiomorphic pair of structures should apply to patterns gathered on beta-Mn using resonance enhanced x-ray diffraction. The chiral polymorph of manganese, and structurally related compounds, will be more convenient than trigonal crystals of low-quartz, tellurium and berlinite used in previous structural studies of this type, because cubic symmetry removes some complications of correction for absorption and macroscopic birefringence. Intensity of a Bragg spot engaged by circular polarization in the primary beam of photons Y(h, k, l) is proposed as a chiral signature of the illuminated material. The partial intensity requires a knowledge of scattering amplitudes in all four channels of polarization, which are reported as functions of an azimuthal angle (rotation of the crystal about the axis of the reflection vector). Unlike trigonal chiral crystals, Y(0, 0, l) = 0 for beta-Mn and it is (h, k, 0), and symmetry related Bragg spots, that epitomize structural chirality. Specifically, reflection vectors (h, k, 0) with odd Miller indices and the chiral axis of beta-Mn mesh in terms of helicity, with the corresponding Y(h, k, 0) equal in magnitude and opposite in sign for partners in the enantiomorphic pair. Dependence of Y(h, k, 0) on the azimuthal angle does not mirror the dyad or tetrad axes of rotation symmetry in the cubic crystal structure.

Hydrodynamic interactions between two forced objects of arbitrary shape. II. Relative translation.

We study the relative translation of two arbitrarily shaped objects, caused by their hydrodynamic interaction as they are forced through a viscous fluid in the limit of zero Reynolds number. It is well known that in the case of two rigid spheres in an unbounded fluid, the hydrodynamic interaction does not produce relative translation. More generally, such an effective pair-interaction vanishes in configurations with spatial inversion symmetry; for example, an enantiomorphic pair in mirror image positions has no relative translation. We show that the breaking of inversion symmetry by boundaries of the system accounts for the interactions between two spheres in confined geometries, as observed in experiments. The same general principle also provides new predictions for interactions in other object configurations near obstacles. We examine the time-dependent relative translation of two self-aligning objects, extending the numerical analysis of our preceding publication [Goldfriend, Diamant, and Witten, Phys. Fluids 27, 123303 (2015)]PHFLE61070-663110.1063/1.4936894. The interplay between the orientational interaction and the translational one, in most cases, leads over time to repulsion between the two objects. The repulsion is qualitatively different for self-aligning objects compared to the more symmetric case of uniform prolate spheroids. The separation between the two objects increases with time t as t^{1/3} in the former case, and more strongly, as t, in the latter.

Comparison of structures applying the tools available at the Bilbao Crystallographic Server

A quantitative comparison of similar crystal structures is often convenient to cross-check different experimental and/or theoretical structural models of the same phase coming from different sources. It is also important for the identification of different phases with the same symmetry, and it is fundamental for the still open problem of the classification of structures into structure types. In most cases, even if the setting of its space group is fixed, there is more than one equivalent description for a given structure. The existence of various equivalent structure descriptions makes the comparison of different structural models a non-trivial task in general. To deal with it, the programCOMPSTRUhas been developed, available at the Bilbao Crystallographic Server (http://www.cryst.ehu.es). The program measures the similarity between two structures having the same space-group symmetry (or space groups that form an enantiomorphic pair) with the same or different compositions, and under the condition that the sequence of the occupied Wyckoff positions is the same in both structures (isopointal structures). The efficiency and utility of the program are demonstrated by a number of illustrative examples. It is also shown how the program can be used to outline different structure types within a set of isopointal structures.

The ash heap of crystallography: restoring forgotten basic knowledge

A critical analysis of some basic notions often overlooked in crystallographic education is presented to correct some common oversights occurring both in the literature and in textbooks. The crystal forms (face forms), defined in terms of their geometric eigensymmetry, are 47 in number, not 48 as often found in the literature. The split of the dihedron into dome and sphenoid calls for the consideration of the physical properties of the faces building a form; in that case, however, the same criterion should be used for all forms. By taking the handedness of the faces as representative of the physical properties of the faces, the occurrence of 130 crystallographic face forms (97 affine face forms and 33 enantiomorphic pairs) is demonstrated. Next, the correct use of non-coprime Miller indices when a centred unit cell is adopted is shown, and the inconsistent multiplication of Miller indices in the Bravais–Friedel–Donnay–Harker law is pointed out. A geometric derivation of the reflection conditions is reviewed. Finally, the inconsistent presentation of metric restrictions imposed by the structural symmetry is pointed out and corrected.

Using electron vortex beams to determine chirality of crystals in transmission electron microscopy

We investigate electron vortex beams elastically scattered on chiral crystals. After deriving a general expression for the scattering amplitude of a vortex electron, we study its diffraction on point scatterers arranged on a helix. We derive a relation between the handedness of the helix and the topological charge of the electron vortex on one hand, and the symmetry of the Higher Order Laue Zones in the diffraction pattern on the other for kinematically and dynamically scattered electrons. We then extend this to atoms arranged on a helix as found in crystals which belong to chiral space groups and propose a new method to determine the handedness of such crystals by looking at the symmetry of the diffraction pattern. Contrary to alternative methods, our technique does not require multiple scattering which makes it possible to also investigate extremely thin samples in which multiple scattering is suppressed. In order to verify the model, elastic scattering simulations are performed and an experimental demonstration on Mn$_2$Sb$_2$O$_7$ is given where we find the sample to belong to the right handed variant of its enantiomorphic pair. This demonstrates the usefulness of electron vortex beams to reveal the chirality of crystals in a transmission electron microscope and provides the required theoretical basis for further developments in this field.

Coexistence of a pair of enantiomorphic forms of chiral quartz nets with an interpenetrating mode in a centrosymmetric coordination polymer

2-Fold interpenetrating chiral quartz nets with opposite handedness have been observed in a centrosymmetric coordination polymer {[Zn3(bmzbc)6]·11.5DMF}n (Hbmzbc = 4-(benzimidazole-1-yl)benzoic acid), which is a unique example of an interpenetrating enantiomorphic pair of quartz nets that have never been realized before.

Crystallization and preliminary X-ray analysis of the receiver domain of a putative response regulator, BPSL0128, from Burkholderia pseudomallei.

bpsl0128, a gene encoding a putative response regulator from Burkholderia pseudomallei strain D286, has been cloned into a pETBLUE-1 vector system, overexpressed in Escherichia coli and purified. The full-length protein is degraded during purification to leave a fragment corresponding to the putative receiver domain, and crystals of this protein that diffracted to beyond 1.75 Å resolution have been grown by the hanging-drop vapour-diffusion technique using PEG 6000 as the precipitant. The crystals belonged to one of the enantiomorphic pair of space groups P3(1)21 and P3(2)21, with unit-cell parameters a = b = 65.69, c = 105.01 Å and either one or two molecules in the asymmetric unit.

Dynamic NMR spectroscopy of three 2,2′-dialkyl/aryl-1-picrylhydrazines in liquid solution

We have studied and compared the proton nuclear magnetic resonance spectra of a series of three 2,2'-di(alkyl/aryl)-1-(2,4,6-trinitrophenyl)hydrazines at 500 MHz in liquid toluene-d(8), with a view to learning details of the internal conformation changes taking place, as a function of temperature. Herein "alkyl" denotes 'methyl', and "aryl" denotes 'phenyl'. The synthesis and characterization of 2-phenyl,2'-methyl-1-picrylhydrazine is reported. One goal of the present study was to afford a comparison with the oxidation products, i.e., the three corresponding hydrazyl free radicals - which show striking differences. All the hydrazine molecules in liquid solution occur as interconverting enantiomorphic pairs (atropisomers). Advanced techniques for obtaining the relevant activation energies and pseudo-thermodynamic parameters (Arrhenius and Eyring) have been utilized, and these parameters are listed and discussed. The three hydrazines turn out be very similar with respect to their internal dynamics.

2-tert-Butylamino-4-chloro-6-ethylamino-1,3,5-triazine: a structure withZ′ = 4 containing two different molecular conformations and two independent chains of hydrogen-bondedR22(8) rings

The title compound (trivial name terbutylazine), C(9)H(16)ClN(5), (I), crystallizes with Z' = 4 in the space group Pca2(1), and equal numbers of molecules adopt two different conformations for the ethylamine groups. The four independent molecules form two approximately enantiomorphic pairs. Eight independent N-H...N hydrogen bonds link the molecules into two independent chains of R(2)(2)(8) rings, in which the arrangement of the alkylamine substituents in the independent molecules precludes any further crystallographic symmetry. The significance of this study lies in its finding of two distinct molecular conformations within the structure and two distinct ways in which the molecules are organized into hydrogen-bonded chains, and in its comparison of the hydrogen-bonded structure of (I) with those of analogous 1,3,5-triazines and pyrimidines.

Comparative architecture of octahedral protein cages. I. Indexed enclosing forms

The architectural elements of four protein cages (bacterio ferritin, human mitochondrial ferritin, sulfur oxygenase reductase and small heat-shock protein) are compared top-to-bottom. The starting points are polyhedra with octahedral symmetry 432 enclosing the cage and delimiting the central cavity, respectively, which have vertices at points of a species-dependent cubic form lattice. The approach is extended from the whole cage to axial-symmetric clusters down to polyhedral forms of single monomers viewed along the fourfold, the threefold and the twofold axes, respectively. The corresponding projected monomeric forms can be approximated by two-dimensional tiles, 13 enantiomorphic pairs in total. The determination of the cubic indices of the monomeric vertices opens the possibility of the way back analysis, bottom-to-top, as discussed in paper II [Janner (2008). Acta Cryst. A64, 503-512].

Enantiomorph identification of crystals belonging to the point groups 321 and 312 by convergent-beam electron diffraction

The recently proposed CBED (convergent-beam electron diffraction) method for enantiomorph identification has been successfully applied to crystals belonging to the point groups 321 and 312. The intensity asymmetry of zeroth-order Laue zone and/or first-order Laue zone reflections of Bijvoet pairs is easily recognized in CBED patterns with the incidence along appropriate zone-axis orientations for each member of the enantiomorphic pair. The intensity asymmetry with respect to the symmetry line is reversed upon changing the space group (handedness) from one to the other. Thus, enantiomorph identification can be easily performed in principle for all crystals belonging to the point groups 321 and 312.

Epitaxy of Organic Crystal Films: Phenanthrene on Potassium Acid Phthalate

The growth of epitaxial layers of phenanthrene from the vapor on cleaved (010) potassium acid phthalate (KAP) single-crystal substrates is reported. Using a growth cell designed for precise control of supersaturation and substrate temperature, the deposition process was followed in situ using optical microscopy. After epitaxial three-dimensional (3D) nucleation at high supersaturation and subsequent growing together of the nuclei at low supersaturation, closed crystalline phenanthrene layers were obtained. The produced crystal film was composed of four orientational domains, being two pairs of enantiomorphs 180° rotated with respect to each other. The epitaxy is explained by a structural match at the KAP (010)−phenanthrene (001) contact faces, which are both terminated by a pseudo-hexagonal pattern of more or less perpendicular hexagonal aromatic rings.

Dynamic NMR spectroscopy of 2,2′-dimethyl-1-picrylhydrazine in various solvents

We have studied the 1H NMR spectra of 2,2′-dimethyl-1-(2,4,6-trinitrophenyl)hydrazine at 300 and 500 MHz in seven liquid solvents, with a view to learning details of the internal conformational changes taking place as a function of temperature and of the solvent. These molecules in liquid solution occur as interconverting enantiomorphic pairs (atropisomers). Advanced techniques for obtaining the correct activation energies and pseudo-thermodynamic parameters have been utilized, and these parameters are listed and discussed. Our results point to a transformation between the pair of atropisomers that is not quite as complicated as one might have encountered in that the solvent does not affect ΔG‡ in any major fashion. Molecular orbital calculations clarified some of the chemical shifts observed for both 1H and 13C. One goal of this study was to enable a detailed comparison with similar results available for 2,2′-diphenyl-1-(2,4,6-trinitrophenyl)hydrazine.Key words: dynamic NMR, dimethylpicrylhydrazine, hindered rotation, atropisomers, activation parameters.

Cloning, purification and preliminary crystallographic analysis of a putative pyridoxal kinase from Bacillus subtilis.

Pyridoxal kinases (PdxK) are able to catalyse the phosphorylation of three vitamin B(6) precursors, pyridoxal, pyridoxine and pyridoxamine, to their 5'-phosphates and play an important role in the vitamin B(6) salvage pathway. Recently, the thiD gene of Bacillus subtilis was found to encode an enzyme which has the activity expected of a pyridoxal kinase despite its previous assignment as an HMPP kinase owing to higher sequence similarity. As such, this enzyme would appear to represent a new class of ;HMPP kinase-like' pyridoxal kinases. B. subtilis thiD has been cloned and the protein has been overexpressed in Escherichia coli, purified and subsequently crystallized in a binary complex with ADP and Mg(2+). X-ray diffraction data have been collected from crystals to 2.8 A resolution at 100 K. The crystals belong to a primitive tetragonal system, point group 422, and analysis of the systematic absences suggest that they belong to one of the enantiomorphic pair of space groups P4(1)2(1)2 or P4(3)2(1)2. Consideration of the space-group symmetry and unit-cell parameters (a = b = 102.9, c = 252.6 A, alpha = beta = gamma = 90 degrees ) suggest that the crystals contain between three and six molecules in the asymmetric unit. A full structure determination is under way to provide insights into aspects of the enzyme mechanism and substrate specificity.

Formation probability for enantiomorphic crystals (with the space groups of P6 222 and P6 422) in transition-metal disilicides with the C40 structure as determined by convergent-beam electron diffraction

The new CBED (convergent-beam electron diffraction) method we proposed recently for enantiomorph identification has successfully been applied to polycrystals of TaSi 2 and NbSi 2 with the C40 structure in order to determine the probability for the formation of enantiomorphic crystals (those belonging to the space groups of P6 222 and P6 422). To the best of the authors' knowledge, this is the first assessment for the formation probability of enantiomorphic crystals for optically opaque inorganic crystals. Although the formation probability varies from sample to sample, it tends to be statistically equal for the enantiomorphic pairs of crystals as the number of samples investigated increases, in agreement with other results so far obtained for many organic crystals.

Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6.

This paper gives classification results for crystallographic groups in dimensions up to 6 which refine earlier enumeration results. Based on the classification data, the asymptotic growth of the number of space-group types is discussed. The classification scheme for crystallographic groups is revisited and a new classification level in between that of geometric and arithmetic crystal classes is introduced and denoted as harmonic crystal classes. Enantiomorphic pairs are determined on all classification levels from space-group types to crystal families and the enantiomorphic pairs of fixed-point-free space groups are given. A general algorithm to compute enantiomorphic pairs is described.

Corrections toCrystallographic Groups of Four-Dimensional Spaceby Brownet al.(1978) [New York: Wiley and Sons

There is an error in the tabulation of enantiomorphic space-group types given in the book Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978). It has recently been found by one of us (BS) while extending the derivation of the enantiomorphism of crystallographic groups up to dimension 6. We refer to the terminology of the book. In the book, it is wrongly claimed that the (af®ne) four-dimensional space-group type 08/01/01/002 splits into an enantiomorphic pair of (proper) space-group types. This is indicated in the book by an * preceding the space-group number on p. 118. This asterisk should be removed. As a consequence of this error, the following changes have also to be made: 1. Page 11, last line of the section `Enantiomorphism': replace 112 by 111. 2. Page 52, in List 3.1A, Overall statistics: replace in the line `Number of space-group types' 4783 (112) by 4783 (111). 3. Page 53, in the line starting with VII: replace 24 (1) by 24 and 2 (1) by 2. 4. Page 407, line 2: delete the line `Family VII 08/01/01/002'. In addition to the calculations by BS that led to the discovery of this error, J. N. and V. Felsch have also recalculated the classi®cation of the four-dimensional crystallographic groups (including their enantiomorphism) using computer programs different from those used by BS. No further errors have been found. We cannot reconstruct how the error in the original listings occurred. While reporting this error, we also want to point out another fact. In the theoretical introduction of the book, the de®nition of the term `crystal system', while working for dimensions up to four, was not dimension independent. A correction has been given by NeubuE ser et al. (1981).

Crystallization of NAD+-dependent phenylalanine dehydrogenase from Nocardia sp239.

The NAD+-dependent phenylalanine dehydrogenase from Nocardia sp239 has been crystallized by the hanging-drop method of vapour diffusion using ammonium sulfate as the precipitant. Two crystal forms were obtained in the presence and absence of the enzyme substrates phenylpyruvic acid or phenylalanine and its coenzyme NADH. Crystals of the native protein belong to the hexagonal system, with the space group being one of the enantiomorphic pair P6122 or P6522. The cell dimensions are a = b = 111.0, c = 174.5 A, alpha = beta = 90 and gamma = 120 degrees. Crystals grown from the protein co-crystallized with its substrates all belong to the trigonal system, space group P3121 or P3221, with unit-cell dimensions of a = b = 88.1, c = 112.6 A, alpha = beta = 90 and gamma = 120 degrees. Preliminary protein-sequencing experiments have established that this enzyme is related to the octameric PheDH's which are members of the wider superfamily of amino-acid dehydrogenases. However, gel-filtration studies suggest that this enzyme is active as a monomer. The full determination of the three-dimensional structure of this phenylalanine dehydrogenase will add to the understanding of the molecular basis of the differential substrate specificity within this enzyme superfamily. In turn this will contribute to the rational design of an amino-acid dehydrogenase which could be used for the diagnosis of phenylketonuria and for the chiral synthesis of high-value pharmaceuticals.