scholarly journals The Euler characteristic as a basis for teaching topology concepts to crystallographers

2022 ◽  
Vol 55 (1) ◽  
Author(s):  
Bartosz Naskręcki ◽  
Mariusz Jaskolski ◽  
Zbigniew Dauter

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V − E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

2014 ◽  
Vol 70 (a1) ◽  
pp. C336-C336
Author(s):  
Marcin Kowiel ◽  
Mariusz Jaskolski ◽  
Andrzej Gzella ◽  
Zbigniew Dauter

Unique choice of the unit cell and the asymmetric unit are well defined and described in the International Tables for Crystallography vol. A. Unfortunately, the placement of molecules within the unit cell is not standardized. Since structure solution programs often use random numbers in their algorithms, the selected set of atomic coordinates may be different even with successive runs of the same program. Although formally correct, an arbitrary choice of molecular placement within the unit cell is confusing and may lead to interpretation errors [1]. With the use of the anti-Cheshire unit cell introduced by Dauter [2], for all space groups without inversion symmetry, it is possible to transform the molecular model such that its center of gravity falls within the anti-Cheshire asymmetric unit cell. It means that for macromolecular crystal structures it should be possible to standardize the placement of the molecules within the unit cell. In consequence, it should be easier for crystallographers and non-crystallographers to compare similar or related crystal structures. An implementation of the anti-Cheshire concept has been programmed in Python as a web service, aCHESYM. The aCHESYM program takes a PDB file as input and transforms the macromolecular model into the desired anti-Cheshire region. The program can also handle structure factor CIF files if the transformation used requires reindexing of the reflection data. The unit cell, coordinates and displacement parameters of all atoms after transformation are saved in a new PDB file. All the calculated transformations are reversible, so there is no danger of data loss. Moreover, the program helps the user to find the most compact assembly of the molecules (chains) in the structure when there are several chains in the asymmetric unit.


2015 ◽  
Vol 71 (9) ◽  
pp. 1161-1168 ◽  
Author(s):  
Christopher T. Jurgenson ◽  
Thomas D. Pollard

Co-crystals of the bovine Arp2/3 complex with the CA motif from N-WASP in two new space groups were analyzed by X-ray diffraction. The crystals in the orthorhombic space groupP212121contained one complex per asymmetric unit, with unit-cell parametersa= 105.48,b= 156.71,c= 177.84 Å, and diffracted to 3.9 Å resolution. The crystals in the tetragonal space groupP41contained two complexes per asymmetric unit, with unit-cell parametersa=b= 149.93,c = 265.91 Å, and diffracted to 5.0 Å resolution. The electron-density maps of both new crystal forms had densities for small segments of subdomains 1 and 2 of Arp2. Both maps had density at the binding site on Arp3 for the C-terminal EWE tripeptide from N-WASP and a binding site proposed for the C motif of N-WASP in the barbed-end groove of Arp2. The map from the tetragonal crystal form had density near the barbed end of Arp3 that may correspond to the C helix of N-WASP. The noise levels and the low resolution of the maps made the assignment of specific molecular structures for any of these CA peptides impossible.


2021 ◽  
Vol 77 (2) ◽  
pp. 126-129
Author(s):  
Bartosz Naskręcki ◽  
Zbigniew Dauter ◽  
Mariusz Jaskolski

The puzzling observation that the famous Euler's formula for three-dimensional polyhedra V − E + F = 2 or Euler characteristic χ = V − E + F − I = 1 (where V, E, F are the numbers of the bounding vertices, edges and faces, respectively, and I = 1 counts the single solid itself) when applied to space-filling solids, such as crystallographic asymmetric units or Dirichlet domains, are modified in such a way that they sum up to a value one unit smaller (i.e. to 1 or 0, respectively) is herewith given general validity. The proof provided in this paper for the modified Euler characteristic, χm = V m − E m + F m − I m = 0, is divided into two parts. First, it is demonstrated for translational lattices by using a simple argument based on parity groups of integer-indexed elements of the lattice. Next, Whitehead's theorem, about the invariance of the Euler characteristic, is used to extend the argument from the unit cell to its asymmetric unit components.


1998 ◽  
Vol 54 (3) ◽  
pp. 436-436 ◽  
Author(s):  
Hee-Jeong Choi ◽  
Sang Won Kang ◽  
Chul-Hak Yang ◽  
Sue Goo Rhee ◽  
Seong-Eon Ryu

HORF6 is a member of the novel antioxidant enzyme family found in humans. A recombinant form of hORF6 expressed and purified from E. coli has been crystallized by the hanging-drop method using various PEG's as precipitating agents. HORF6 crystallizes in two different monoclinic space groups, P21 and C2. The P21 crystals have unit-cell dimensions of a = 47.85, b = 75.17, c = 63.30 Å and β = 110.21° and contain two monomers per asymmetric unit, while the C2 crystals have unit-cell dimensions of a = 165.27, b = 95.44, c = 166.44 Å and β = 128.97° and contain more than six monomers per asymmetric unit. The P21 crystals with the smaller unit cell diffract X-rays better and behave well for the X-ray analysis. A native data set from a single crystal of the P21 space group gas been collected to 2.0 Å resolution.


2012 ◽  
Vol 68 (8) ◽  
pp. o283-o287 ◽  
Author(s):  
Vasily S. Minkov ◽  
Elena V. Boldyreva

N,N-Dimethylglycine, C4H9NO2, and its hemihydrate, C4H9NO2·0.5H2O, are discussed in order to follow the effect of the methylation of the glycine amino group (and thus its ability to form several hydrogen bonds) on crystal structure, in particular on the possibility of the formation of hydrogen-bonded `head-to-tail' chains, which are typical for the crystal structures of amino acids and essential for considering amino acid crystals as mimics of peptide chains. Both compounds crystallize in centrosymmetric space groups (PbcaandC2/c, respectively) and have twoN,N-dimethylglycine zwitterions in the asymmetric unit. In the anhydrous compound, there are no head-to-tail chains but the zwitterions formR44(20) ring motifs, which are not bonded to each other by any hydrogen bonds. In contrast, in the crystal structure ofN,N-dimethylglycinium hemihydrate, the zwitterions are linked to each other by N—H...O hydrogen bonds into infiniteC22(10) head-to-tail chains, while the water molecules outside the chains provide additional hydrogen bonds to the carboxylate groups.


2010 ◽  
Vol 66 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Mingrun Li ◽  
Junliang Sun ◽  
Peter Oleynikov ◽  
Sven Hovmöller ◽  
Xiaodong Zou ◽  
...  

The structure of a complicated quasicrystal approximant ∊16 was predicted from a known and related quasicrystal approximant ∊6 by the strong-reflections approach. Electron-diffraction studies show that in reciprocal space, the positions of the strongest reflections and their intensity distributions are similar for both approximants. By applying the strong-reflections approach, the structure factors of ∊16 were deduced from those of the known ∊6 structure. Owing to the different space groups of the two structures, a shift of the phase origin had to be applied in order to obtain the phases of ∊16. An electron-density map of ∊16 was calculated by inverse Fourier transformation of the structure factors of the 256 strongest reflections. Similar to that of ∊6, the predicted structure of ∊16 contains eight layers in each unit cell, stacked along the b axis. Along the b axis, ∊16 is built by banana-shaped tiles and pentagonal tiles; this structure is confirmed by high-resolution transmission electron microscopy (HRTEM). The simulated precession electron-diffraction (PED) patterns from the structure model are in good agreement with the experimental ones. ∊16 with 153 unique atoms in the unit cell is the most complicated approximant structure ever solved or predicted.


1999 ◽  
Vol 55 (4) ◽  
pp. 607-616 ◽  
Author(s):  
Martina Walker ◽  
Ehmke Pohl ◽  
Regine Herbst-Irmer ◽  
Martin Gerlitz ◽  
Jürgen Rohr ◽  
...  

The crystal structures of Emycin E (1), di-o-bromobenzoyl-Emycin F (2) and o-bromobenzoyl-Emycin D (3) have been determined by X-ray analysis at low temperature. Emycin E and o-bromobenzoyl-Emycin D both crystallize with two molecules in a triclinic unit cell. These two structures can be solved and refined either in the centrosymmetric space group P\bar 1, with apparent disorder localized at or around the expected chiral centre, or in the non-centrosymmetric space group P1 as mixtures of two diastereomers without disorder. Only the latter interpretation is consistent with the chemical and spectroscopic evidence. Refinements in the centrosymmetric and non-centrosymmetric space groups are compared in this paper and are shown to favour the chemically correct interpretation, more decisively so in the case of the bromo derivative as a result of the anomalous dispersion of bromine. Structures (1) and (3) provide a dramatic warning of the dangers inherent in the conventional wisdom that if a structure can be refined satisfactorarily in both centrosymmetric and non-centrosymmetric space groups, the former should always be chosen. In these two cases, despite apparently acceptable intensity statistics and R factors (5.87 and 3.55%), the choice of the centrosymmetric space group leads to the serious chemical error that the triclinic unit cell contains a racemate rather than two chiral diastereomers! The weakest reflections are shown to be most sensitive to the correct choice of space group, underlining the importance of refining against all data rather than against intensities greater than a specified threshold. The use of similar-distance restraints is shown to be beneficial in both P1 refinements. Di-o-bromobenzoyl-Emycin F crystallizes in the monoclinic space group P21 with one molecule in the asymmetric unit and so does not give rise to these problems of interpretation. The absolute configuration of the two bromo derivatives, and hence the Emycins in general, was determined unambiguously as S at the chiral centre C3.


Author(s):  
Fang Lu ◽  
Bei Zhang ◽  
Yong Liu ◽  
Ying Song ◽  
Gangxing Guo ◽  
...  

Phytases are phosphatases that hydrolyze phytates to less phosphorylatedmyo-inositol derivatives and inorganic phosphate. β-Propeller phytases, which are very diverse phytases with improved thermostability that are active at neutral and alkaline pH and have absolute substrate specificity, are ideal substitutes for other commercial phytases. PhyH-DI, a β-propeller phytase fromBacillussp. HJB17, was found to act synergistically with other single-domain phytases and can increase their efficiency in the hydrolysis of phytate. Crystals of native and selenomethionine-substituted PhyH-DI were obtained using the vapour-diffusion method in a condition consisting of 0.2 Msodium chloride, 0.1 MTris pH 8.5, 25%(w/v) PEG 3350 at 289 K. X-ray diffraction data were collected to 3.00 and 2.70 Å resolution, respectively, at 100 K. Native PhyH-DI crystals belonged to space groupC121, with unit-cell parametersa = 156.84,b = 45.54,c = 97.64 Å, α = 90.00, β = 125.86, γ = 90.00°. The asymmetric unit contained two molecules of PhyH-DI, with a corresponding Matthews coefficient of 2.17 Å3 Da−1and a solvent content of 43.26%. Crystals of selenomethionine-substituted PhyH-DI belonged to space groupC2221, with unit-cell parametersa = 94.71,b= 97.03,c= 69.16 Å, α = β = γ = 90.00°. The asymmetric unit contained one molecule of the protein, with a corresponding Matthews coefficient of 2.44 Å3 Da−1and a solvent content of 49.64%. Initial phases for PhyH-DI were obtained from SeMet SAD data sets. These data will be useful for further studies of the structure–function relationship of PhyH-DI.


1999 ◽  
Vol 55 (2) ◽  
pp. 522-524 ◽  
Author(s):  
Randall L. Oliver ◽  
Jacqueline M. Tremblay ◽  
George M. Helmkamp ◽  
Lynwood R. Yarbrough ◽  
Natalie W. Breakfield ◽  
...  

Phosphatidylinositol-transfer protein (PITP) is a soluble, ubiquitously expressed, highly conserved protein encoded by two genes in humans, rodents and other mammals. A cDNA encoding the alpha isoform of the rat gene was expressed to high levels in Escherichia coli, the protein purified and the homogeneous protein used for crystallization studies. Crystals of rat PITP-α were obtained by vapor-diffusion techniques using the sitting-drop method. Crystals grow within two weeks by vapor-diffusion techniques in the presence of polyethylene glycol 4000. Both crystal forms pack in the monoclinic space group P21. Crystal form I has unit-cell parameters a = 44.75, b = 74.25, c = 48.32 Å and β = 114.14°. Unit-cell parameters for crystal form II are a = 47.86, b = 73.59, c = 80.49 Å and β = 98.54°. Crystal form I has a Vm of 2.295 Å3 Da−1 and an estimated solvent content of 46.4% with one molecule per asymmetric unit, while crystal form II has a Vm of 2.196 Å3 Da−1 and an estimated solvent content of 44.0%, assuming two molecules per asymmetric unit.


2016 ◽  
Vol 19 (2) ◽  
pp. 52
Author(s):  
Milan Maksimović

High-contrast gratings (HCG) are ultra-thin elements operating in sub-wavelength regime with the period of the grating smaller than the wavelength and with the high-index grating material fully surrounded by low-index material. Design of MEMS mirrors made from HCG with specific reflectivity response is of great practical interest in integrated optoelectronics. We theoretically investigate design of the spectral response for HCGs with the complex unit cells. We show that the spectral response can be tailored via the unit cell perturbations and with the asymmetric unit cell perturbations may introduce completely new spectral response. Our results can serve as guidance for the design of the complex HCGs and help with the choice of the efficient initial grating topology prior to global optimization procedure.


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