Structure Solution Method Research Articles

Structure Determination Feasibility of Three-Dimensional Electron Diffraction in Case of Limited Data

During the last two decades, three-dimensional electron diffraction (3D ED) has undergone a renaissance, starting with the introduction of precession (Precession Electron Diffraction Tomography, PEDT) that led to variations on the idea of collecting as much of the diffraction space as possible in order to solve crystal structures from sub-micron sized crystals. The most popular of these acquisition methods is based on the continuous tilting/rotation of the crystal (so-called Microcrystal Electron Diffraction, MicroED) akin to the oscillating crystal method in X-ray crystallography, which was enabled by the increase of sensitivity and acquisition speed in electron detectors. While 3D ED data is more complex than the equivalent X-ray data due to the higher proportion of dynamical scattering, the same basic principles of what is required in terms of data quality and quantity in order to solve a crystal structure apply; high completeness, high data resolution and good signal-to-noise statistics on measured reflection intensities. However, it may not always be possible to collect data in these optimum conditions, the most common limitations being the tilt range of the goniometer stage, often due to a small pole piece gap or the use of a non-tomography holder, or the position of the sample on the TEM grid, which may be too close to a grid bar and then the specimen of interest becomes occluded during tilting. Other factors that can limit the quality of the acquired data include the limited dynamic range of the detector, which can result on truncated intensities, or the sensitivity of the crystal to the electron beam, whereby the crystallinity of the particle is changing under the illumination of the beam. This limits the quality and quantity of the measured intensities and makes structure analysis of such data challenging. Under these circumstances, traditional approaches may fail to elucidate crystal structures, and global optimization methods may be used here as an alternative powerful tool. In this context, this work presents a systematic study on the application of a global optimization method to crystal structure determination from 3D ED data. The results are compared with known structure models and crystal phases obtained from traditional ab initio structure solution methods demonstrating how this strategy can be reliably applied to the analysis of partially complete 3D ED data.

Solving inorganic crystal structures from X-ray powder diffraction using a generative first-principles framework

X-ray powder diffraction (XRD) is a powerful structure characterization technique, but solving unknown inorganic crystal structures from powder diffraction patterns can often be labor-intensive or speculative. We introduce an Automated XRD to Structure (AXS) solution method based on the crystal symmetry obtained from XRD patterns, an efficient search of candidate structures spanning the available degrees of freedom, and density functional theory (DFT). This methodology is completely agnostic to structural prototypes and robust in solving inorganic structures of various chemistries, crystal systems, and unit cell sizes; 92% of all crystal structures were accurately determined from the simulated XRD patterns in our benchmark set. In addition, we demonstrate the efficacy of this methodology on experimental XRD patterns by solving the crystal structures of Li8Hf6O8, Li3CrO4, and LiFeO2.

On-line three-dimensional packing problems: A review of off-line and on-line solution approaches

Three-Dimensional Packing Problems (3D-PPs) can be applied to effectively reduce logistics costs in various areas, such as airline cargo management and warehouse management. In general, 3D-PP studies can be divided into two different streams: those tackling the off-line problem, where full knowledge about items is available beforehand; and those tackling the on-line (real-time) problem, where items arrive one by one and should be packed immediately without having full prior knowledge about them. During the past decades, off-line and on-line 3D-PPs have been studied in the literature with various constraints and solution approaches. However, and despite the numerous practical applications of on-line problems in real-world situations, most of the literature to date has focused on off-line problems and is quite sparse when it comes to on-line solution methods. In this regard, and despite the different nature of on-line and off-line problems, some approaches can be applied in both environments. Hence, we conducted an in-depth and updated literature review to identify and structure various constraints and solution methods employed by researchers in off-line and on-line 3D-PPs. Building on this, by bringing together the two separate streams of the literature, we identified several off-line approaches that can be adopted in on-line environments. Additionally, we addressed relevant research gaps and ways to bridge them in the future, which can help to develop this research field.

New Horizons in Structural Biology of Membrane Proteins: Experimental Evaluation of the Role of Conformational Dynamics and Intrinsic Flexibility.

A plethora of membrane proteins are found along the cell surface and on the convoluted labyrinth of membranes surrounding organelles. Since the advent of various structural biology techniques, a sub-population of these proteins has become accessible to investigation at near-atomic resolutions. The predominant bona fide methods for structure solution, X-ray crystallography and cryo-EM, provide high resolution in three-dimensional space at the cost of neglecting protein motions through time. Though structures provide various rigid snapshots, only an amorphous mechanistic understanding can be inferred from interpolations between these different static states. In this review, we discuss various techniques that have been utilized in observing dynamic conformational intermediaries that remain elusive from rigid structures. More specifically we discuss the application of structural techniques such as NMR, cryo-EM and X-ray crystallography in studying protein dynamics along with complementation by conformational trapping by specific binders such as antibodies. We finally showcase the strength of various biophysical techniques including FRET, EPR and computational approaches using a multitude of succinct examples from GPCRs, transporters and ion channels.

Notes of a protein crystallographer: the advantages of combining new integrated methods of structure solution with traditional data visuals.

The suggestion is made that combining analysis using the most advanced crystallographic software with the integrated visual tools of the field will result in more knowledgeable and better trained future generations of structural biologists. The use of integrated visuals could also expedite the structure solution of some recalcitrant and complex macromolecular crystal structures that resist automatic workflows.

High-throughput crystal structure solution using prototypes

Numerous experimentally observed inorganic compounds are missing from materials databases due to challenges in solving their crystal structures from diffraction data. The authors present a rapid DFT-based structure solution method and use it to conduct high-throughput structure solutions for hundreds of previously unsolved compounds. This effort has yielded 520 experimentally observed, newly solved crystal structures, a remarkable expansion that could prove valuable across a wide range of applications.

Leveraging glycomics data in glycoprotein 3D structure validation with Privateer.

The heterogeneity, mobility and complexity of glycans in glycoproteins have been, and currently remain, significant challenges in structural biology. These aspects present unique problems to the two most prolific techniques: X-ray crystallography and cryo-electron microscopy. At the same time, advances in mass spectrometry have made it possible to get deeper insights on precisely the information that is most difficult to recover by structure solution methods: the full-length glycan composition, including linkage details for the glycosidic bonds. The developments have given rise to glycomics. Thankfully, several large scale glycomics initiatives have stored results in publicly available databases, some of which can be accessed through API interfaces. In the present work, we will describe how the Privateer carbohydrate structure validation software has been extended to harness results from glycomics projects, and its use to greatly improve the validation of 3D glycoprotein structures.

Insitu crystallography as an emerging method for structure solution of membrane proteins: the case of CCR2A.

The inmeso insitu serial X-ray crystallization method (Huang et al., (2015) Acta Crystallogr D Biol Crystallogr 71, 1238) combines lipid cubic phase crystallization, direct freezing of the crystallization droplet without handling of the crystals, and data collection insitu. Recently, this method was used to overcome the mechanical fragility of crystals which enabled the X-ray structure determination of chemokine receptor 2A (Apel et al., (2019) Structure 27, 427) at 2.7Å resolution. The CCR2 structure provides the structural basis for ligand selectivity of CCR2 against chemokine receptor 5 and provides insights into the residence time of MK-0812 analogs based on molecular dynamics simulations. These findings offer new opportunities for drug discovery targeting chemokine receptors.

Surface Induced Phenytoin Polymorph. 2. Structure Validation by Comparing Experimental and Density Functional Theory Raman Spectra

A method for structure solution in thin films that combines grazing incidence X-ray diffraction data analysis and crystal structure prediction was presented in a recent work (Braun et al. Cryst. Growth Des.2019, DOI: 10.1021/acs.cgd.9b00857). Applied to phenytoin form II, which is only detected in films, the approach gave a very reasonable, but not fully confirmed, candidate structure with Z = 4 and Z′ = 2. In the present work, we demonstrate how, by calculating and measuring the crystal Raman spectrum in the low wavenumber energy region with the aim of validating the candidate structure, this can be further refined. In fact, we find it to correspond to a saddle point of the energy landscape of the system, from which a minimum of lower symmetry may be reached. With the new structure, with Z = 4 and Z′ = 2, we finally obtain an excellent agreement between experimental and calculated Raman spectra.

Visualising newsvendor profits: the single-period problem with location-scale demand distributions

The single-period, or newsvendor, problem is typically formulated to maximise expected profits or minimise expected costs. Recent research has suggested that decision-makers tend not to act in accordance with this objective, and extensions to the model have been introduced that take into account other objectives, such as risk tolerance or achieving target profits. The purpose of this research is to provide decision support via visualisation to assist in the determination of an appropriate order quantity. A visual characterisation of the single-period profit distribution is provided for problems with location-scale demand distributions. From this, bounds on the profit as well as stochastically dominated solutions are identified, and the profit distribution is formulated as a mixture distribution of two extremes. This visualisation provides novel insight into the structure and alternative solution methods for single-period problems. Alternative decision criteria are then presented as a direct consequence of the visualisation, including quantile optimisation.

Single-Crystal Automated Refinement (SCAR): A Data-Driven Method for Determining Inorganic Structures.

Single-crystal diffraction is one of the most common experimental techniques in chemistry for determining a crystal structure. However, the process of crystal structure determination and refinement is not always straightforward. Methods for simplifying and rationalizing the path to the most optimal crystal structure model have been incorporated into various data processing and crystal structure solution software, with the focus generally on aiding macromolecular or protein structure determination. In this work, we propose a new method that uses single-crystal data to determine the crystal structures of inorganic, extended solids called "single-crystal automated refinement" (SCAR). The approach was developed using data mining and machine learning methods and considers several structural features common in inorganic solids, like atom assignment based on physically reasonable distances, atomic statistical mixing, and crystallographic site deficiency. The output is a tree of possible solutions for the data set with a corresponding fit score indicating the most reasonable crystal structure. Here, the foundation for SCAR is presented followed by the implementation of SCAR to determine two newly synthesized and previously unreported phases, ZrAu0.5Os0.5 and Nd4Mn2AuGe4. The structure solutions are found to be comparable with those produced by manually solving the data set, including the same refined mixed occupancies and atomic deficiency, supporting the validity of this automatic structure solution method. The proposed SCAR program is thus verified as being a fast and reliable assistant in determining even complex single-crystal diffraction data for extended inorganic solids.

In-house high-energy-remote SAD phasing using the magic triangle: how to tackle the P1 low symmetry using multiple orientations of the same crystal of human IBA57 to increase the multiplicity.

This article describes the approach used to solve the structure of human IBA57 in-house by 5-amino-2,4,6-triiodoisophthalic acid (I3C) high-energy-remote single-wavelength anomalous dispersion (SAD) phasing. Multiple orientations of the same triclinic crystal were exploited to acquire sufficient real data multiplicity for phasing. How the collection of an in-house native data set and its joint use with the I3C derivative through a SIRAS approach decreases the data multiplicity needed by almost 50% is described. Furthermore, it is illustrated that there is a clear data-multiplicity threshold value for success and failure in phasing, and how adding further data does not significantly affect substructure solution and model building. To our knowledge, this is the only structure present in the PDB that has been solved in-house by remote SAD phasing in space group P1 using only one crystal. All of the raw data used, derived from the different orientations, have been uploaded to Zenodo in order to enable software developers to improve methods for data processing and structure solution, and for educational purposes.

Electron crystallography with the EIGER detector.

Electron crystallography is a discipline that currently attracts much attention as method for inorganic, organic and macromolecular structure solution. EIGER, a direct-detection hybrid pixel detector developed at the Paul Scherrer Institut, Switzerland, has been tested for electron diffraction in a transmission electron microscope. EIGER features a pixel pitch of 75 × 75µm2, frame rates up to 23kHz and a dead time between frames as low as 3µs. Cluster size and modulation transfer functions of the detector at 100, 200 and 300keV electron energies are reported and the data quality is demonstrated by structure determination of a SAPO-34 zeotype from electron diffraction data.

Automated crystal structure solution from powder diffraction data: Validation of the first-principles-assisted structure solution method

The authors demonstrate an automated method to solve crystal structures from powder diffraction data. Originally developed by Meredig and Wolverton in 2013, their First-Principles-Assisted Structure Solution (FPASS) method uses $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ energetics to guide the search for reasonable structures. Here, the authors find that FPASS accurately recovers the known solution for nearly 100 crystals and employ it to solve 10 structures that were previously incomplete in crystallographic databases. The authors identified several potential semiconducting materials from these new solutions, and also have released their automation software to allow others to use FPASS.

Validation and extraction of molecular-geometry information from small-molecule databases.

A freely available small-molecule structure database, the Crystallography Open Database (COD), is used for the extraction of molecular-geometry information on small-molecule compounds. The results are used for the generation of new ligand descriptions, which are subsequently used by macromolecular model-building and structure-refinement software. To increase the reliability of the derived data, and therefore the new ligand descriptions, the entries from this database were subjected to very strict validation. The selection criteria made sure that the crystal structures used to derive atom types, bond and angle classes are of sufficiently high quality. Any suspicious entries at a crystal or molecular level were removed from further consideration. The selection criteria included (i) the resolution of the data used for refinement (entries solved at 0.84 Å resolution or higher) and (ii) the structure-solution method (structures must be from a single-crystal experiment and all atoms of generated molecules must have full occupancies), as well as basic sanity checks such as (iii) consistency between the valences and the number of connections between atoms, (iv) acceptable bond-length deviations from the expected values and (v) detection of atomic collisions. The derived atom types and bond classes were then validated using high-order moment-based statistical techniques. The results of the statistical analyses were fed back to fine-tune the atom typing. The developed procedure was repeated four times, resulting in fine-grained atom typing, bond and angle classes. The procedure will be repeated in the future as and when new entries are deposited in the COD. The whole procedure can also be applied to any source of small-molecule structures, including the Cambridge Structural Database and the ZINC database.

Towards solution and refinement of organic crystal structures by fitting to the atomic pair distribution function.

A method towards the solution and refinement of organic crystal structures by fitting to the atomic pair distribution function (PDF) is developed. Approximate lattice parameters and molecular geometry must be given as input. The molecule is generally treated as a rigid body. The positions and orientations of the molecules inside the unit cell are optimized starting from random values. The PDF is obtained from carefully measured X-ray powder diffraction data. The method resembles `real-space' methods for structure solution from powder data, but works with PDF data instead of the diffraction pattern itself. As such it may be used in situations where the organic compounds are not long-range-ordered, are poorly crystalline, or nanocrystalline. The procedure was applied to solve and refine the crystal structures of quinacridone (β phase), naphthalene and allopurinol. In the case of allopurinol it was even possible to successfully solve and refine the structure in P1 with four independent molecules. As an example of a flexible molecule, the crystal structure of paracetamol was refined using restraints for bond lengths, bond angles and selected torsion angles. In all cases, the resulting structures are in excellent agreement with structures from single-crystal data.

Three new crystal structures in the Na-Pb system: solving structures without additional experimental input.

The structures of three Na-Pb compounds, γ, δ and δ', have remained incompletely solved for nearly 60 years. The space group, lattice parameters and positions of the Pb atoms of these three structures have been determined, but the positions of the Na atoms are still unknown. In this work, the First-Principles Assisted Structure Solution (FPASS) method [Meredig & Wolverton (2013). Nat. Mater. 12, 123-127] has been used to complete the description of these three structures using only experimental information available from the literature as input. The paper also discusses the relative advantages of constrained crystal structure prediction tools, like FPASS, in comparison to conventional crystal structure prediction methods in reference to their abilities to complete the solution of other unsolved structures.

Mathematical programming approaches to pricing problems

There are many real cases where a company needs to determine the price of its products so as to maximise its revenue or profit.To do so, the company must consider customers' reactions to these prices, as they may refuse to buy a given product or service if its price is too high. This is commonly known in literature as a pricing problem.This class of problems, which is typically bilevel, was first studied in the 1990s and is NP-hard, although polynomial algorithms do exist for some particular cases. Many questions are still open on this subject.The aim of this thesis is to investigate mathematical properties of pricing problems, in order to find structural properties, formulations and solution methods that are as efficient as possible. In particular, we focus our attention on pricing problems over a network. In this framework, an authority owns a subset of arcs and imposes tolls on them, in an attempt to maximise his/her revenue, while users travel on the network, seeking for their minimum cost path.First, we provide a detailed review of the state of the art on bilevel pricing problems. Then, we consider a particular case where the authority is using an unit toll scheme on his/her subset of arcs, imposing either the same toll on all of them, or a toll proportional to a given parameter particular to each arc (for instance a per kilometre toll). We show that if tolls are all equal then the complexity of the problem is polynomial, whereas in case of proportional tolls it is pseudo-polynomial.We then address a robust approach taking into account uncertainty on parameters. We solve some polynomial cases of the pricing problem where uncertainty is considered using an interval representation.Finally, we focus on another particular case where toll arcs are connected such that they constitute a path, as occurs on highways. We develop a Dantzig-Wolfe reformulation and present a Branch-and-Cut-and-Price algorithm to solve it. Several improvements are proposed, both for the column generation algorithm used to solve the linear relaxation and for the branching part used to find integer solutions. Numerical results are also presented to highlight the efficiency of the proposed strategies. This problem is proved to be APX-hard and a theoretical comparison between our model and another one from the literature is carried out.

Novel Coordination Polymers with (Pyrazolato)-Based Tectons: Catalytic Activity in the Peroxidative Oxidation of Alcohols and Cyclohexane

Coupling five rigid or flexible bis(pyrazolato)-based tectons with late transition metal ions allowed us to isolate 18 coordination polymers (CPs). As assessed by thermal analysis, all of them possess a remarkable thermal stability, their decomposition temperatures lying in the range of 340–500 °C. As demonstrated by N2 adsorption measurements at 77 K, their Langmuir specific surface areas span the rather vast range of 135–1758 m2/g, in agreement with the porous or dense polymeric architectures retrieved by powder X-ray diffraction structure solution methods. Two representative families of CPs, built up with either rigid or flexible spacers, were tested as catalysts in (i) the microwave-assisted solvent-free peroxidative oxidation of alcohols by t-BuOOH, and (ii) the peroxidative oxidation of cyclohexane to cyclohexanol and cyclohexanone by H2O2 in acetonitrile. Those CPs bearing the rigid spacer, concurrently possessing higher specific surface areas, are more active than the corresponding ones with the flexible spacer. Moreover, the two copper(I)-containing CPs investigated exhibit the highest efficiency in both reactions, leading selectively to a maximum product yield of 92% (and TON up to 1.5 × 103) in the oxidation of 1-phenylethanol and of 11% in the oxidation of cyclohexane, the latter value being higher than that granted by the current industrial process.

Synthesis, structural features and luminescence properties of a 1-D poly(azolato)-based coordination polymer

The novel coordination polymer [Mn(Andolast)(H2O)] was isolated in the form of air- and moisture-stable polycrystalline powders by reacting Mn(CH3COO)2·4H2O and the sodium salt of N-4-(1H-tetrazol-5-yl)phenyl-4-(1H-tetrazol-5-yl) benzamide (Na2Andolast). The crystal structure of [Mn(Andolast)(H2O)], determined by powder X-ray diffraction (PXRD) structure solution methods, is composed by 1-D double-chain motifs. The thermal behavior of [Mn(Andolast)(H2O)] was investigated by coupling thermal analyses to variable-temperature PXRD: water evolution takes place in the range 80–250°C with no change of the unit cell parameters up to 180°C; at 200°C, a low-crystallinity phase is formed, surviving up to 300°C. The electronic transition spectroscopic properties of both Na2Andolast and the coordination polymer were assessed, with particular care to luminescence: [Mn(Andolast)(H2O)] is luminescent already at room temperature, even if its emission is reduced with respect to that of Na2Andolast, mainly as a consequence of the enhanced rate of non-radiative decays.