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.

The ability to theoretically predict accurate NMR chemical shifts in solids is increasingly important due to the role such shifts play in selecting among proposed model structures. Herein, two theoretical methods are evaluated for their ability to assign 15 N shifts from guanosine dihydrate to one of the two independent molecules present in the lattice. The NMR data consist of 15 N shift tensors from 10 resonances. Analysis using periodic boundary or fragment methods consider a benchmark dataset to estimate errors and predict uncertainties of 5.6 and 6.2 ppm, respectively. Despite this high accuracy, only one of the five sites were confidently assigned to a specific molecule of the asymmetric unit. This limitation is not due to negligible differences in experimental data, as most sites exhibit differences of >6.0 ppm between pairs of resonances representing a given position. Instead, the theoretical methods are insufficiently accurate to make assignments at most positions.

The famous Euler\'s rule for three-dimensional polyhedra, F - E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn - En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.

Sulfonucleotide reductases catalyse the first reductive step of sulfate assimilation. Their substrate specificities generally correlate with the requirement for a [Fe4S4] cluster, where adenosine 5\'-phosphosulfate (APS) reductases possess a cluster and 3\'-phosphoadenosine 5\'-phosphosulfate reductases do not. The exception is the APR-B isoform of APS reductase from the moss Physcomitrella patens, which lacks a cluster. The crystal structure of APR-B, the first for a plant sulfonucleotide reductase, is consistent with a preference for APS. Structural conservation with bacterial APS reductase rules out a structural role for the cluster, but supports the contention that it enhances the activity of conventional APS reductases.

Tn916-like conjugative transposons carrying antibiotic resistance genes are found in a diverse range of bacteria. Orf14 within the conjugation module encodes a bifunctional cell wall hydrolase CwlT that consists of an N-terminal bacterial lysozyme domain (N-acetylmuramidase, bLysG) and a C-terminal NlpC/P60 domain (γ-d-glutamyl-l-diamino acid endopeptidase) and is expected to play an important role in the spread of the transposons. We determined the crystal structures of CwlT from two pathogens, Staphylococcus aureus Mu50 (SaCwlT) and Clostridium difficile 630 (CdCwlT). These structures reveal that NlpC/P60 and LysG domains are compact and conserved modules, connected by a short flexible linker. The LysG domain represents a novel family of widely distributed bacterial lysozymes. The overall structure and the active site of bLysG bear significant similarity to other members of the glycoside hydrolase family 23 (GH23), such as the g-type lysozyme (LysG) and Escherichia coli lytic transglycosylase MltE. The active site of bLysG contains a unique structural and sequence signature (DxxQSSES+S) that is important for coordinating a catalytic water. Molecular modeling suggests that the bLysG domain may recognize glycan in a similar manner to MltE. The C-terminal NlpC/P60 domain contains a conserved active site (Cys-His-His-Tyr) that appears to be specific to murein tetrapeptide. Access to the active site is likely regulated by isomerism of a side chain atop the catalytic cysteine, allowing substrate entry or product release (open state), or catalysis (closed state).