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

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.

Dr. Alexander Wlodawer-celebrating five decades of service to the structural biology community.

This 75th birthday tribute to our Editorial Board member Alexander Wlodawer recounts his decades-long service to the community of structural biology researchers. His former and current colleagues tell the story of his upbringing and education, followed by an account of his dedication to quality and rigor in crystallography and structural science. The FEBS Journal Editor-in-Chief Seamus Martin further highlights Alex's outstanding contributions to the journal's success over many years.

A topological proof of the modified Euler characteristic based on the orbifold concept.

The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.

Rapid response to emerging biomedical challenges and threats.

As part of the global mobilization to combat the present pandemic, almost 100 000 COVID-19-related papers have been published and nearly a thousand models of macromolecules encoded by SARS-CoV-2 have been deposited in the Protein Data Bank within less than a year. The avalanche of new structural data has given rise to multiple resources dedicated to assessing the correctness and quality of structural data and models. Here, an approach to evaluate the massive amounts of such data using the resource https://covid19.bioreproducibility.org is described, which offers a template that could be used in large-scale initiatives undertaken in response to future biomedical crises. Broader use of the described methodology could considerably curtail information noise and significantly improve the reproducibility of biomedical research.

Arithmetic proof of the multiplicity-weighted Euler characteristic for symmetrically arranged space-filling polyhedra.

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 = Vm - Em + Fm - Im = 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.

Crystallographic models of SARS-CoV-2 3CLpro: in-depth assessment of structure quality and validation.

The appearance at the end of 2019 of the new SARS-CoV-2 coronavirus led to an unprecedented response by the structural biology community, resulting in the rapid determination of many hundreds of structures of proteins encoded by the virus. As part of an effort to analyze and, if necessary, remediate these structures as deposited in the Protein Data Bank (PDB), this work presents a detailed analysis of 81 crystal structures of the main protease 3CLpro, an important target for the design of drugs against COVID-19. The structures of the unliganded enzyme and its complexes with a number of inhibitors were determined by multiple research groups using different experimental approaches and conditions; the resulting structures span 13 different polymorphs representing seven space groups. The structures of the enzyme itself, all determined by molecular replacement, are highly similar, with the exception of one polymorph with a different inter-domain orientation. However, a number of complexes with bound inhibitors were found to pose significant problems. Some of these could be traced to faulty definitions of geometrical restraints for ligands and to the general problem of a lack of such information in the PDB depositions. Several problems with ligand definition in the PDB itself were also noted. In several cases extensive corrections to the models were necessary to adhere to the evidence of the electron-density maps. Taken together, this analysis of a large number of structures of a single, medically important protein, all determined within less than a year using modern experimental tools, should be useful in future studies of other systems of high interest to the biomedical community.

Celebrating the 75th birthday of Professor Wladek Minor, one of the most accomplished Polish-American structural biologists.

This paper describes the scientific career and accomplishments of Professor Wladek Minor, who holds an endowed chair at the University of Virginia in Charlottesville, USA. Prof. Minor is a coauthor of data processing software used by macromolecular crystallographers world-wide, as well as of structural biology servers and of a repository for raw diffraction data. He made major contributions to the validation of biostructural data, with special focus on drug design targets and reproducibility in biomedical research. He is among the most highly cited molecular biologists ever.

Covid-19.bioreproducibility.org: A web resource for SARS-CoV-2-related structural models.

The COVID-19 pandemic has triggered numerous scientific activities aimed at understanding the SARS-CoV-2 virus and ultimately developing treatments. Structural biologists have already determined hundreds of experimental X-ray, cryo-EM, and NMR structures of proteins and nucleic acids related to this coronavirus, and this number is still growing. To help biomedical researchers, who may not necessarily be experts in structural biology, navigate through the flood of structural models, we have created an online resource, covid19.bioreproducibility.org, that aggregates expert-verified information about SARS-CoV-2-related macromolecular models. In this article, we describe this web resource along with the suite of tools and methodologies used for assessing the structures presented therein.

Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra.

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.

A new modulated crystal structure of the ANS complex of the St John's wort Hyp-1 protein with 36 protein molecules in the asymmetric unit of the supercell.

Superstructure modulation, with violation of the strict short-range periodic order of consecutive crystal unit cells, is well known in small-molecule crystallography but is rarely reported for macromolecular crystals. To date, one modulated macromolecular crystal structure has been successfully determined and refined for a pathogenesis-related class 10 protein from Hypericum perforatum (Hyp-1) crystallized as a complex with 8-anilinonaphthalene-1-sulfonate (ANS) [Sliwiak et al. (2015), Acta Cryst. D71, 829-843]. The commensurate modulation in that case was interpreted in a supercell with sevenfold expansion along c. When crystallized in the additional presence of melatonin, the Hyp-1-ANS complex formed crystals with a different pattern of structure modulation, in which the supercell shows a ninefold expansion of c, manifested in the diffraction pattern by a wave of reflection-intensity modulation with crests at l = 9n and l = 9n ± 4. Despite complicated tetartohedral twinning, the structure has been successfully determined and refined to 2.3 Šresolution using a description in a ninefold-expanded supercell, with 36 independent Hyp-1 chains and 156 ANS ligands populating the three internal (95 ligands) and five interstitial (61 ligands) binding sites. The commensurate superstructures and ligand-binding sites of the two crystal structures are compared, with a discussion of the effect of melatonin on the co-crystallization process.

Ligand-centered assessment of SARS-CoV-2 drug target models in the Protein Data Bank.

A bright spot in the SARS-CoV-2 (CoV-2) coronavirus pandemic has been the immediate mobilization of the biomedical community, working to develop treatments and vaccines for COVID-19. Rational drug design against emerging threats depends on well-established methodology, mainly utilizing X-ray crystallography, to provide accurate structure models of the macromolecular drug targets and of their complexes with candidates for drug development. In the current crisis, the structural biological community has responded by presenting structure models of CoV-2 proteins and depositing them in the Protein Data Bank (PDB), usually without time embargo and before publication. Since the structures from the first-line research are produced in an accelerated mode, there is an elevated chance of mistakes and errors, with the ultimate risk of hindering, rather than speeding up, drug development. In the present work, we have used model-validation metrics and examined the electron density maps for the deposited models of CoV-2 proteins and a sample of related proteins available in the PDB as of April 1, 2020. We present these results with the aim of helping the biomedical community establish a better-validated pool of data. The proteins are divided into groups according to their structure and function. In most cases, no major corrections were necessary. However, in several cases significant revisions in the functionally sensitive area of protein-inhibitor complexes or for bound ions justified correction, re-refinement, and eventually reversioning in the PDB. The re-refined coordinate files and a tool for facilitating model comparisons are available at https://covid-19.bioreproducibility.org. DATABASE: Validated models of CoV-2 proteins are available in a dedicated, publicly accessible web service https://covid-19.bioreproducibility.org.

On the evolution of the quality of macromolecular models in the PDB.

Crystallographic models of biological macromolecules have been ranked using the quality criteria associated with them in the Protein Data Bank (PDB). The outcomes of this quality analysis have been correlated with time and with the journals that published papers based on those models. The results show that the overall quality of PDB structures has substantially improved over the last ten years, but this period of progress was preceded by several years of stagnation or even depression. Moreover, the study shows that the historically observed negative correlation between journal impact and the quality of structural models presented therein seems to disappear as time progresses.

S-adenosylmethionine synthases in plants: Structural characterization of type I and II isoenzymes from Arabidopsis thaliana and Medicago truncatula.

S-adenosylmethionine synthases (MATs) are responsible for production of S-adenosylmethionine, the cofactor essential for various methylation reactions, production of polyamines and phytohormone ethylene, etc. Plants have two distinct MAT types (I and II). This work presents the structural analysis of MATs from Arabidopsis thaliana (AtMAT1 and AtMAT2, both type I) and Medicago truncatula (MtMAT3a, type II), which, unlike most MATs from other domains of life, are dimers where three-domain subunits are sandwiched flat with one another. Although MAT types are very similar, their subunits are differently oriented within the dimer. Structural snapshots along the enzymatic reaction reveal the exact conformation of precatalytic methionine in the active site and show a binding niche, characteristic only for plant MATs, that may serve as a lock of the gate loop. Nevertheless, plants, in contrary to mammals, lack the MAT regulatory subunit, and the regulation of plant MAT activity is still puzzling. Our structures open a possibility of an allosteric activity regulation of type I plant MATs by linear compounds, like polyamines, which would tighten the relationship between S-adenosylmethionine and polyamine biosynthesis.

Molecular structure of a U•A-U-rich RNA triple helix with 11 consecutive base triples.

Three-dimensional structures have been solved for several naturally occurring RNA triple helices, although all are limited to six or fewer consecutive base triples, hindering accurate estimation of global and local structural parameters. We present an X-ray crystal structure of a right-handed, U•A-U-rich RNA triple helix with 11 continuous base triples. Due to helical unwinding, the RNA triple helix spans an average of 12 base triples per turn. The double helix portion of the RNA triple helix is more similar to both the helical and base step structural parameters of A'-RNA rather than A-RNA. Its most striking features are its wide and deep major groove, a smaller inclination angle and all three strands favoring a C3'-endo sugar pucker. Despite the presence of a third strand, the diameter of an RNA triple helix remains nearly identical to those of DNA and RNA double helices. Contrary to our previous modeling predictions, this structure demonstrates that an RNA triple helix is not limited in length to six consecutive base triples and that longer RNA triple helices may exist in nature. Our structure provides a starting point to establish structural parameters of the so-called 'ideal' RNA triple helix, analogous to A-RNA and B-DNA double helices.

Structural basis of methotrexate and pemetrexed action on serine hydroxymethyltransferases revealed using plant models.

Serine hydroxymethyltransferases (SHMTs) reversibly transform serine into glycine in a reaction accompanied with conversion of tetrahydrofolate (THF) into 5,10-methylene-THF (5,10-meTHF). In vivo, 5,10-meTHF is the main carrier of one-carbon (1C) units, which are utilized for nucleotide biosynthesis and other processes crucial for every living cell, but hyperactivated in overproliferating cells (e.g. cancer tissues). SHMTs are emerging as a promising target for development of new drugs because it appears possible to inhibit growth of cancer cells by cutting off the supply of 5,10-meTHF. Methotrexate (MTX) and pemetrexed (PTX) are two examples of antifolates that have cured many patients over the years but target different enzymes from the folate cycle (mainly dihydrofolate reductase and thymidylate synthase, respectively). Here we show crystal structures of MTX and PTX bound to plant SHMT isozymes from cytosol and mitochondria-human isozymes exist in the same subcellular compartments. We verify inhibition of the studied isozymes by a thorough kinetic analysis. We propose to further exploit antifolate scaffold in development of SHMT inhibitors because it seems likely that especially polyglutamylated PTX inhibits SHMTs in vivo. Structure-based optimization is expected to yield novel antifolates that could potentially be used as chemotherapeutics.

Sulfur-SAD phasing from microcrystals utilizing low-energy X-rays.

A practical approach for obtaining S-SAD data from native protein microcrystals with low-wavelength synchrotron radiation [Guo et al. (2019), IUCrJ, 6, 532-542] is presented in this issue of IUCrJ.

Spermidine Synthase (SPDS) Undergoes Concerted Structural Rearrangements Upon Ligand Binding - A Case Study of the Two SPDS Isoforms From Arabidopsis thaliana.

Spermidine synthases (SPDSs) catalyze the production of the linear triamine, spermidine, from putrescine. They utilize decarboxylated S-adenosylmethionine (dc-SAM), a universal cofactor of aminopropyltransferases, as a donor of the aminopropyl moiety. In this work, we describe crystal structures of two SPDS isoforms from Arabidopsis thaliana (AtSPDS1 and AtSPDS2). AtSPDS1 and AtSPDS2 are dimeric enzymes that share the fold of the polyamine biosynthesis proteins. Subunits of both isoforms present the characteristic two-domain structure. Smaller, N-terminal domain is built of the two ß-sheets, while the C-terminal domain has a Rossmann fold-like topology. The catalytic cleft composed of two main compartments, the dc-SAM binding site and the polyamine groove, is created independently in each AtSPDS subunits at the domain interface. We also provide the structural details about the dc-SAM binding mode and the inhibition of SPDS by a potent competitive inhibitor, cyclohexylamine (CHA). CHA occupies the polyamine binding site of AtSPDS where it is bound at the bottom of the active site with the amine group placed analogously to the substrate. The crystallographic snapshots show in detail the structural rearrangements of AtSPDS1 and AtSPDS2 that are required to stabilize ligands within the active site. The concerted movements are observed in both compartments of the catalytic cleft, where three major parts significantly change their conformation. These are (i) the neighborhood of the glycine-rich region where aminopropyl moiety of dc-SAM is bound, (ii) the very flexible gate region with helix η6, which interacts with both, the adenine moiety of dc-SAM and the bound polyamine or inhibitor, and (iii) the N-terminal ß-hairpin, that limits the putrescine binding grove at the bottom of the catalytic site.

Structural Study of Agmatine Iminohydrolase From Medicago truncatula, the Second Enzyme of the Agmatine Route of Putrescine Biosynthesis in Plants.

Plants are unique eukaryotes that can produce putrescine (PUT), a basic diamine, from arginine via a three-step pathway. This process starts with arginine decarboxylase that converts arginine to agmatine. Then, the consecutive action of two hydrolytic enzymes, agmatine iminohydrolase (AIH) and N-carbamoylputrescine amidohydrolase, ultimately produces PUT. An alternative route of PUT biosynthesis requires ornithine decarboxylase that catalyzes direct putrescine biosynthesis. However, some plant species lack this enzyme and rely only on agmatine pathway. The scope of this manuscript concerns the structural characterization of AIH from the model legume plant, Medicago truncatula. MtAIH is a homodimer built of two subunits with a characteristic propeller fold, where five αßßαß repeated units are arranged around the fivefold pseudosymmetry axis. Dimeric assembly of this plant AIH, formed by interactions of conserved structural elements from one repeat, is drastically different from that observed in dimeric bacterial AIHs. Additionally, the structural snapshot of MtAIH in complex with 6-aminohexanamide, the reaction product analog, presents the conformation of the enzyme during catalysis. Our structural results show that MtAIH undergoes significant structural rearrangements of the long loop, which closes a tunnel-shaped active site over the course of the catalytic event. This conformational change is also observed in AIH from Arabidopsis thaliana, indicating the importance of the closed conformation of the gate-keeping loop for the catalysis of plant AIHs.