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Although the ambiguity of the crystal structures determined directly from diffraction intensities has been historically recognized, it is not well understood in quantitative terms. Bernstein's theorem has recently been used to obtain the number of one-dimensional crystal structures of equal point atoms, given a minimum set of diffraction intensities. By a similar approach, the number of two- and three-dimensional crystal structures that can be determined from a minimum intensity data set is estimated herein. The ambiguity of structure determination from the algebraic minimum of data increases at least exponentially fast with the increasing structure size. Substituting lower-resolution intensities by higher-resolution ones in the minimum data set has little or no effect on this ambiguity if the number of such substitutions is relatively small.

Direct methods have practically solved the phase problem for small–medium‐size molecules but have substantially failed in macromolecular crystallography. They have two main limitations: a strong dependence on structural complexity and the need to work with atomic‐resolution data. Many attempts have been made to broaden their field of applicability, for example the use of some a priori information to make the estimate of the triplet invariant phases more effective. Unfortunately none of these new approaches allowed the successful application of direct methods to proteins and nucleic acids. Direct methods are still a niche tool in macromolecular crystallography. In a recent publication [Giacovazzo (2019). Acta Cryst. A75, 142–157] the method of joint probability distributions has been modified to take into account new sources of prior information, one of which is relevant to this article: the Patterson map. In practice, it has been shown that with prior knowledge of the interatomic vectors one is able to modify the classic Cochran reliability parameter for estimating the triplet invariant phases. The article was essentially theoretical in nature, and no attempt was described to test the practical usefulness of the new probabilistic formulas. This work is therefore the first application of the new method. It is shown that the use of the Patterson map as prior information substantially improves the Cochran estimate of triplet phases; the phase error distribution for the new estimates, even if it is related to macromolecular structures, becomes similar to that obtained for medium‐size structures. In some ways, it is as if the use of the Patterson information reduces the structural complexity, thus allowing a more general use of direct methods in macromolecular crystallography. Atomic resolution no longer seems to be a necessary ingredient for the applicability of direct methods; tests show that the apparent reduction in structural complexity also occurs in macromolecular structures with experimental data having a resolution of 2.3 Å. A number of test structures have been used to show the potential of the new technique. [ABSTRACT FROM AUTHOR]

The marvel of X-ray crystallography is the beauty and precision of the atomic structures deduced from diffraction patterns. Since these patterns record only amplitudes, phases for the diffracted waves must also be evaluated for systematic structure determination. Thus, we have the phase problem as a central complication, both intellectually for the field and practically so for many analyses. Here, I discuss how we – myself, my laboratory and the diffraction community – have faced the phase problem, considering the evolution of methods for phase evaluation as structural biology developed to the present day. During the explosive growth of macromolecular crystallography, practice in diffraction analysis evolved from a universal reliance on isomorphous replacement to the eventual domination of anomalous diffraction for de novo structure determination. As the Protein Data Bank (PDB) grew and familial relationships among proteins became clear, molecular replacement overtook all other phasing methods; however, experimental phasing remained essential for molecules without obvious precedents, with multi- and single-wavelength anomalous diffraction (MAD and SAD) predominating. While the mathematics-based direct methods had proved to be inadequate for typical macromolecules, they returned to crack substantial selenium substructures in SAD analyses of selenomethionyl proteins. Native SAD, exploiting the intrinsic S and P atoms of biomolecules, has become routine. Selenomethionyl SAD and MAD were the mainstays of structural genomics efforts to populate the PDB with novel proteins. A recent dividend has been paid in the success of PDB-trained artificial intelligence approaches for protein structure prediction. Currently, molecular replacement with AlphaFold models often obviates the need for experimental phase evaluation. For multiple reasons, we are now unfazed by the phase problem. Cryo-EM analysis is an attractive alternative to crystallography for many applications faced by today's structural biologists. It simply finesses the phase problem; however, the principles and procedures of diffraction analysis remain pertinent and are adopted in single-particle cryo-EM studies of biomolecules.

Obituary for Professor Dr Hendrik (Henk) Schenk. [ABSTRACT FROM AUTHOR]

In the title molecule, C24H25N3O2, the piperazine ring adopts a chair conformation. The methoxyphenyl-substituted ring makes a dihedral angle of 6.79 (5)° with the quinoline ring system. In the crystal, molecules are consolidated in the crystal packing by a combination of weak C—H...N and C—H...O interactions. π–π stacking interactions also occur.

The aim of this article is a general description of the so-called Patterson-function direct methods (PFDM), from their origin to their present state. It covers a 20-year period of methodological contributions to crystal structure solution, most of them published in Acta Crystallographica Section A. The common feature of these variants of direct methods is the introduction of the experimental intensities in the form of the Fourier coefficients of origin-free Patterson-type functions, which allows the active use of both strong and weak reflections. The different optimization algorithms are discussed and their performances compared. This review focuses not only on those PFDM applications related to powder diffraction data but also on some recent results obtained with electron diffraction tomography data.

The efficient multipurpose figure of merit MPF has been defined and characterized. It may be very helpful in phasing procedures. Indeed, it might be used for establishing the centric or acentric nature of an unknown structure, for identifying the presence of some pseudotranslational symmetry, for recognizing the correct solution in multisolution approaches and for estimating the quality of structure models as they become available during the phasing process. Thus, phase improvement or deterioration may be monitored and useless models may be discarded to save computing time. It is also shown that MPF may be applied in different phasing approaches, no matter if ab initio or non ab initio. [ABSTRACT FROM AUTHOR]

The present communication deals with the eco-friendly synthesis, spectral properties and X-ray crystal structure of an indole derivative - Ethyl 2'-amino-3'-cyano-6'-methyl-5-nitro-2-oxospiro [indoline-3,4'-pyran]-5'-carboxylate. The title compound was synthesized in 87% yield. The crystal structure of the molecule is stabilized by intermolecular N H … N, N H … O and C H … π interactions. The molecule is organized in the crystal lattice forming sheet like structure. To interpret the experimental data, ab initio computations of the vibrational frequencies were carried out using the Gaussian 09 program followed by the full optimizations done using Density Functional Theory (DFT) at B3LYP/6-31 + G(d,p) level. The combined use of experiments and computations allowed a firm assignment of the majority of observed bands for the compound. The calculated highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) with frontier orbital gap were presented. The electronic and charge transfer properties have been explained on the basis of highest occupied molecular orbitals (HOMOs), lowest unoccupied molecular orbitals (LUMOs) and density of states (DOS). From the optimized geometry of the molecule, molecular electrostatic potential (MEP) distribution, frontier molecular orbitals (FMOs) of the title compound have been calculated in the ground state theoretically. The theoretical results showed good agreement with the experimental values. First hyperpolarizability values have been calculated to describe the nonlinear optical (NLO) property of the synthesized compound. [ABSTRACT FROM AUTHOR]

A fast and fully automatic procedure for collecting electron diffraction tomography data is presented. In the case of a very stable goniometer it is demonstrated how, by variation of the tilting speed and the CCD detector parameters, it is possible to obtain fully automatic precession-assisted electron diffraction tomography data collections, rotation electron diffraction tomography data collections or new integrated electron diffraction tomography data collections, in which the missing wedge of the reciprocal space between the patterns is recorded by longer exposures during the crystal tilt. It is shown how automatic data collection of limited tilt range can be used to determine the unit-cell parameters, while data of larger tilt range are suitable to solve the crystal structure ab initio with direct methods. The crystal structure of monoclinic MgMoO4 has been solved in this way as a test structure. In the case where the goniometer is not stable enough to guarantee a steady position of the crystal over large tilt ranges, an automatic method for tracking the crystal during continuous rotation of the sample is proposed. [ABSTRACT FROM AUTHOR]

No deterministic approach to obtaining a crystal structure from a set of diffraction intensities exists, despite significant progress in traditional probabilistic direct methods. One of the biggest hurdles in determining a crystal structure algebraically is solving a system of many polynomial equations of high power on intensities in terms of atomic coordinates. In this study, homotopy continuation is used for exhaustive investigation of such systems and an optimized homotopy continuation method is developed with random restarts to determine small ( N < 5) crystal structures from a minimum set of error-free intensities. [ABSTRACT FROM AUTHOR]

Apart from solving the heavy-atom substructure in proteins and ab initio phasing of protein diffraction data at atomic resolution, direct methods have also been successfully combined with other protein crystallographic methods in dealing with diffraction data far below atomic resolution, leading to significantly improved results. In this respect, direct methods provide phase constraints in reciprocal space within a dual-space iterative framework rather than solve the phase problem independently. Applications of this type of direct methods to difficult SAD phasing, model completion and low-resolution phase extension will be described in detail. [ABSTRACT FROM AUTHOR]

Recent experiments at free-electron laser X-ray sources have been able to resolve the intensity distributions about Bragg peaks in nanocrystals of large biomolecules. Information derived from small shifts in the peak positions augment the Bragg samples of the particle intensity with samples of its gradients. Working on the assumption that the nanocrystal is entirely generated by lattice translations of a particle, an algorithm is developed that reconstructs the particle from intensities and intensity gradients. Unlike traditional direct phasing methods that require very high resolution data in order to exploit sparsity of the electron density, this method imposes no constraints on the contrast other than positivity and works well at low resolution. Successful reconstructions are demonstrated with simulated P1 lysozyme nanocrystal data down to a signal-to-noise ratio of 2 in the intensity gradients. [ABSTRACT FROM AUTHOR]

The triplet structure invariant is estimated via the method of joint probability distribution functions when a model structure is available. The six-variate probability distribution function P( E h, E k, E− h− k, E p h, E p k, E p,− h− k) is studied under the condition that imperfect isomorphism between the target and model structures exist. The results are compared with those available in the literature, which were obtained under the condition of perfect isomorphism. It is shown that the new formalism is more suitable for real cases, where perfect isomorphism is very rare. [ABSTRACT FROM AUTHOR]

A central problem in crystallography is crystal structure determination directly from diffraction intensities. For structures of small molecules, this problem has been addressed by probabilistic direct methods that allow one to obtain the structure coordinates with a high degree of certainty given a sufficiently large set of intensities. In contrast, deterministic algebraic methods that could guarantee a solution and may be applicable to macromolecules have not yet emerged. In this study a basic algebraic question is posed: how many crystal structures can be obtained from a given set of intensities? Recently, by using a new origin definition and the method of elementary symmetrical polynomials, all small ( N≤ 4 atoms) one-dimensional crystal structures that could be obtained from the minimum set of N− 1 lowest-resolution intensities were enumerated. Here, by using methods of modern algebraic geometry the maximum number of one-dimensional crystal structures that can be determined from the minimum set of intensities for N > 4 is obtained. It is demonstrated that this ambiguity increases exponentially with the increasing number of atoms in the structure N (∼4 N/ N3/2 for N >> 1) and includes non-homometric structures. Therefore, a minimum set of intensities, even in principle, is insufficient for structure determination for all but very small structures. [ABSTRACT FROM AUTHOR]

Crystal structure analysis of the novel 2-amino-4,6,6-trimethyl-cyclohex-2-en-1,1,3,4( S)-tetracarbonitrile, obtained in 70% yield, by the Bi(OTf) catalyzed reaction of acetone and malononitrile, at room temperature, is described. For the first time Bi(OTf)-EtN has been used in this inverse electron demand Diels-Alder cycloaddition reaction involving the participation of carbonitrile π-bond. The structure of the molecule was established by spectral analysis and X-ray diffraction studies. The compound crystallizes in the monoclinic space group P2/n with unit cell parameters: a = 8.0580 (17), b = 13.038 (3), c = 12.641 (3) Å, β = 101.883 (4)°, Z = 4. The crystal structure was solved by direct methods and refined to R = 0.0506 for 2,456 observed reflections. The cyclohexene ring of the molecule adopts a distorted sofa conformation. The molecules in the unit cell are arranged in layers. The crystal structure in stabilized by C-H⋯N and N-H⋯N interactions. Graphical Abstract: Crystal structure analysis of the novel 2-amino-4,6,6-trimethyl-cyclohex-2-en-1,1,3,4( S)-tetracarbonitrile.[Figure not available: see fulltext.] [ABSTRACT FROM AUTHOR]

Abstract: The statistical features of the amplitudes obtained via precession electron diffraction have been studied, with particular concern with their effects on direct phasing procedures. A new algorithm, denoted by BEA, is described: according to it, the average amplitude of the symmetry equivalent reflections is used in the Direct Methods step. Once an even imperfect structural model is available, the best amplitude among the equivalent reflections is used to improve the model. It is shown that BEA is able to provide more complete structural models, to make the phasing process more straightforward and to end with crystallographic residual much better than those usually obtained by electron diffraction. [ABSTRACT FROM AUTHOR]

The `charge-flipping' method proposed by Oszlányi & Sütő [ Acta Cryst. (2004), A 60, 134–141] has been extended to include the direct-methods tangent formula within the iterative process. The tangent formula acts as a corrective influence allowing for solutions at resolutions poorer than 1 Å. The resulting algorithm solves difficult structures in minutes rather than days or not at all. Modifications include (i) flipping a percentage of charge rather than charge below a threshold value and critically (ii) dampening the magnitude of charge above the threshold; this impedes tangent-formula solutions comprising one or two very intense peaks in the electron density which is commonly known as the `uranium atom solution'. For data at poor resolution, an alternate charge-flipping regime avoids uranium atom solutions by truncating electron-density pixels that are greater than half the maximum value. [ABSTRACT FROM AUTHOR]

Abstract: The prospect for improving the success of ab initio zeolite structure investigations with electron diffraction data is evaluated. First of all, the quality of intensities obtained by precession electron diffraction at small hollow cone illumination angles is evaluated for seven representative materials: ITQ-1, ITQ-7, ITQ-29, ZSM-5, ZSM-10, mordenite, and MCM-68. It is clear that, for most examples, an appreciable fraction of a secondary scattering perturbation is removed by precession at small angles. In one case, ZSM-10, it can also be argued that precession diffraction produces a dramatically improved ‘kinematical’ data set. There seems to no real support for application of a Lorentz correction to these data and there is no reason to expect for any of these samples that a two-beam dynamical scattering relationship between structure factor amplitude and observed intensity should be valid. Removal of secondary scattering by the precession mode appears to facilitate ab initio structure analysis. Most zeolite structures investigated could be solved by maximum entropy and likelihood phasing via error-correcting codes when precession data were used. Examples include the projected structure of mordenite that could not be determined from selected area data alone. One anomaly is the case of ZSM-5, where the best structure determination in projection is made from selected area diffraction data. In a control study, the zonal structure of SSZ-48 could be determined from selected area diffraction data by either maximum entropy and likelihood or traditional direct methods. While the maximum entropy and likelihood approach enjoys some advantages over traditional direct methods (non-dependence on predicted phase invariant sums), some effort must be made to improve the figures of merit used to identify potential structure solutions. [Copyright &y& Elsevier]

Abstract: The application of electron crystallography to the study of organic materials is reviewed, mainly in context of the author''s own experience. Direct methods for crystallographic phase determination have been shown to be very effective for ab initio structure analyses with electron diffraction intensities, permitting the elucidation of previously uncharacterized crystal structures. Fruitful applications areas have included chain-folded linear polymers, pigments, polydisperse linear chain arrays and, surprisingly, the subgroup assembly of certain proteins. [Copyright &y& Elsevier]

The electron density map produced after the application of direct methods to powder diffraction data of organic compounds is usually very approximated: some atoms are missed, other atoms are in false positions, some atoms are imperfectly located and the connectivity is quite low. A new procedure able to recover the complete structure model is described. In this procedure, a better interpretation of the map is combined with geometrical techniques for generating new atomic positions. The application of the new procedure may lead to the recovery of the complete crystal structure. [ABSTRACT FROM AUTHOR]

Precession electron diffraction (PED) is a method that considerably reduces dynamical effects in electron diffraction data, potentially enabling more straightforward solution of structures using the transmission electron microscope. This study focuses upon the characterization of PED data in an effort to improve the understanding of how experimental parameters affect it in order to predict favorable conditions. A method for generating simulated PED data by the multislice method is presented and tested. Data simulated for a wide range of experimental parameters are analyzed and compared to experimental data for the (Ga,In)2SnO4 (GITO) and ZSM-5 zeolite (MFI) systems. Intensity deviations between normalized simulated and kinematical data sets, which are bipolar for dynamical diffraction data, become unipolar for PED data. Three-dimensional difference plots between PED and kinematical data sets show that PED data are most kinematical for small thicknesses, and as thickness increases deviations are minimized by increasing the precession cone semi-angle φ. Lorentz geometry and multibeam dynamical effects explain why the largest deviations cluster about the transmitted beam, and one-dimensional diffraction is pointed out as a strong mechanism for deviation along systematic rows. R factors for the experimental data sets are calculated, demonstrating that PED data are less sensitive to thickness variation. This error metric was also used to determine the experimental specimen thickness. R1 (unrefined) was found to be about 12 and 15% for GITO and MFI, respectively. [ABSTRACT FROM AUTHOR]

The maximum-likelihood method is applied to direct methods to derive a more general probability density function of the triple-phase sums which is capable of predicting negative values. This study also proves that maximization of the origin-free modulus sum function S yields, within the limitations imposed by the assumed approximations, the maximum-likelihood estimates of the phases. It thus represents the formal theoretical justification of the S function that was initially derived from Patterson-function arguments [Rius (1993). ]. [ABSTRACT FROM AUTHOR]

Abstract: The electron precession diffraction technique is employed to provide quasi-kinematical data for determination of atom positions in the (Ga,In)2SnO5 m-phase. Precession data are compared with conventional diffraction data captured under identical conditions and show a distinct superiority because they exhibit kinematical characteristics in the structure-defining reflections. Precessed data are not usable within a kinematical interpretation in all cases, and a simple basis is presented for omission of errant reflections to improve adherence to kinematical behavior. A second approach is demonstrated where intensities are used with direct methods instead of amplitudes, enhancing the contrast between strong and weak beams. The unrefined atom positions recovered a priori via direct methods are consistent between the two approaches and fall on average within 4 picometers of positions in the previously refined structure. [Copyright &y& Elsevier]

EAFP2 is a novel antifungal protein isolated from the bark of the tree Eucommia ulmoides Oliver. It consists of 41 residues and is characterized with a five-disulfide motif and the inhibitory effects on the growth of both cell wall chitin-containing and chitin-free fungi. The crystal structure of EAFP2 at an atomic resolution of 0.84 Å has been determined by using Shake-and-Bake direct methods with the program SnB. The phases obtained were of sufficient quality to permit the initial model built automatically and the structural refinement carried out using anisotropic displacement parameters resulted in a final crystallographic R factor of 6.8%. In the resulting structural model, all non-hydrogen protein atoms including an unusual pyroglutamyl acid residue at the N-terminal can fit to the articulated electron densities with one centre and more than 65% of the hydrogen atoms in the protein can be observed as individual peaks in the difference map. The general fold of EAFP2 is composed of a 310 helix (Cys3–Arg6), an α-helix (Ala27–Cys31) and a three-stranded antiparallel β-sheet (Cys16–Ser18, Cys23–Ser25, and Cys35–Cys37) and cross-linked by five disulfide bridges. The tertiary structure of EAFP2 can be divided into two structural sectors, A and B. Sector A composed of residues 11–30 adopts a conformation similar to the chitin-binding domain in the hevein-like proteins and features a hydrophobic surface embraced a chitin-binding site (Tyr20, 22, 29, and Ser18). The distinct disulfide bridge Cys7–Cys37 connects the N-terminal ten residues with the C-terminal segment 35–41 to form the sector B, which features a cationic surface distributing all four positively charged residues, Arg6, 9, 36, and 40. Based on these structural features, the possible structural basis of the functional properties of EAFP2 is discussed. [Copyright &y& Elsevier]

The study of atomic structure of surfaces is fundamental to the understanding of electronic, chemical and mechanical properties of surfaces and numerous techniques have been developed to this end. Transmission Electron Microscopy techniques, namely transmission electron imaging (TEM) and diffraction (TED), due to their ability to provide structural information at very high resolutions, have emerged as powerful tools for the study of surface structure. In this article we review the experimental method alongside the various post-processing routines that are necessary to extract vital structural information from experimental data. [Copyright &y& Elsevier]

Electron crystallography uses the electron microscope to generate diffraction patterns or high resolution images which can be used to solve crystal structures from nano-sized crystallites. The diffraction data collected are subject to systematic errors in intensity arising from n-beam dynamical scattering, secondary scattering, sample bending and radiation damage, but nonetheless the technique yields structural information when all other diffraction methods fail. Solving such structures can be routine especially from inorganic materials and intermetallic compounds, but, in general, the process can be difficult as can the validation of the proposed model. Direct methods, maximum entropy, Patterson techniques, model building and the use of image data can all be employed, and structures of high complexity can be solved and refined in favourable cases. Even low resolution protein data can be analysed in this way, and it is an important tool in surface crystallography. [ABSTRACT FROM AUTHOR]

We have recently drawn the attention of crystallographers to the maximum entropy method (MEM) of image reconstruction which has found application in radio astronomy and geophysics. In this article, we review the theory of the mem as applied to the crystallographic phase problem and discuss points of similarity, and differences, with conventional techniques. We then present a number of illustrative calculations in two and three dimensions on simulated and real structures. These examples show that the MEM can be used for phase refinement from partially known phases, say in macromolecular crystallography, and also for solving crystal structures ab initio using only measured Fourier amplitudes. With improved algorithms the method promises to become competitive with current techniques in crystal structure analysis. [ABSTRACT FROM AUTHOR]