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The origin‐free modulus sum function SM refines the set Φ of phases of the structure factors by maximizing the coincidence between the experimental origin‐free modulus function and the calculated one in terms of the ρ(Φ)2 density function [Rius, J. (1993). Acta Cryst. A49, 406–409]. Maximization is normally achieved through the recursive application of a Fourier‐based algorithm. The purpose of the present study is: (i) to show that ρ(Φ)2 can be replaced by |ρ(Φ)| in SM; (ii) to illustrate the viability of the corresponding phasing algorithm with experimental data. [ABSTRACT FROM AUTHOR]

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two‐, three‐ and four‐phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures. Direct methods techniques are revisited and new mathematical approaches are described. [ABSTRACT FROM AUTHOR]

Coherent X-ray diffraction was used to measure the type, quantity and the relative distances between stacking faults along the growth direction of two individual wurtzite GaAs nanowires grown by metalorganic vapour epitaxy. The presented approach is based on the general property of the Patterson function, which is the autocorrelation of the electron density as well as the Fourier transformation of the diffracted intensity distribution of an object. Partial Patterson functions were extracted from the diffracted intensity measured along the direction in the vicinity of the wurtzite Bragg peak. The maxima of the Patterson function encode both the distances between the fault planes and the type of the fault planes with the sensitivity of a single atomic bilayer. The positions of the fault planes are deduced from the positions and shapes of the maxima of the Patterson function and they are in excellent agreement with the positions found with transmission electron microscopy of the same nanowire. [ABSTRACT FROM AUTHOR]

This paper reviews the number-theoretic concept of diaphony, a measure of uniform distribution for number sequences and point sets based on a Fourier theory approach, and its relation to crystallographic concepts like the largest interplanar spacing of a lattice, the structure-factor equation and the Patterson function. [ABSTRACT FROM AUTHOR]

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and a complex number is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant in P1. Calculation of the joint probability distribution of the triplet is not performed here but will be done in the future. [ABSTRACT FROM AUTHOR]

X-ray crystallography and NMR are complementary tools in structural biology. However, it is often difficult to use NMR structures as search models in molecular replacement (MR) to phase crystallographic data. In this study, a new approach is reported utilizing a molecular envelope of NMR structures for MR phasing with the program FSEARCH at low resolution (about 6 Å). Several targets with both crystallographic and NMR structures available have been tested. FSEARCH was able to find the correct translation and orientation of the search model in the crystallographic unit cell, while conventional MR procedures were unsuccessful. [ABSTRACT FROM AUTHOR]

The Src homology 3 (SH3) domain is a small, noncatalytic domain with a conserved sequence of about 60 amino-acid residues that interacts with proline-rich peptides to form a protein complex. In this study, the C-terminal SH3 domain of human Tks4 (residues 853-911) was expressed, purified and crystallized. X-ray diffraction data were collected to 2.3 Å resolution. The crystal belonged to the trigonal space group P3121 (or P3221), with unit-cell parameters a = b = 83.87, c = 108.44 Å, α = β = 90, γ = 120°. Calculating the self-rotation and the native Patterson function did not lead to the detection of any noncrystallographic translational symmetry. Six, seven or eight protein molecules are likely to be present in the asymmetric unit, resulting in a Matthews coefficient and approximate solvent content of 2.71 Å3 Da−1 and 55%, 2.32 Å3 Da−1 and 47%, and 2.03 Å3 Da−1 and 39%, respectively. To solve the crystal structure of the C-terminal SH3 domain of human Tks4, the isomorphous replacement method is presently being utilized. [ABSTRACT FROM AUTHOR]

Monalysin was recently described as a novel pore-forming toxin (PFT) secreted by the Drosophila pathogen Pseudomonas entomophila. Recombinant monalysin is multimeric in solution, whereas PFTs are supposed to be monomeric until target membrane association. Monalysin crystals were obtained by the hanging-drop vapour-diffusion method using PEG 8000 as precipitant. Preliminary X-ray diffraction analysis revealed that monalysin crystals belonged to the monoclinic space group C2, with unit-cell parameters a = 162.4, b = 146.2, c = 144.4 Å, β = 122.8°, and diffracted to 2.85 Å resolution using synchrotron radiation. Patterson self-rotation analysis and Matthews coefficient calculation indicate that the asymmetric unit contains nine copies of monalysin. Heavy-atom derivative data were collected and a Ta6Br14 cluster derivative data set confirmed the presence of ninefold noncrystallographic symmetry. [ABSTRACT FROM AUTHOR]

The cross-correlation function between the target and a model electron density, denoted as the C map, has been crystallographically characterized. In particular, a study of its interatomic vectors and of their relation with the Patterson vectors has been undertaken. Since the C map is not available during the phasing process, the C′ map, its centric modification, is considered. It may be computed at any stage of the phasing process and shows properties that are very useful for the crystal structure determination process. It has been combined with the implication transformation method and with vector-superposition techniques for performing the Patterson deconvolution and obtaining an initial model for dual-space recycling. While Patterson methods are traditionally considered to be more efficient for structures containing heavy atoms, the C map extends their potential to light-atom structures ( i.e. containing atoms not heavier than O). [ABSTRACT FROM AUTHOR]

This paper presents a shuffled frog leaping algorithm(SFLA) for the single-mode resource-constrained project scheduling problem where activities can be divided into equant units and interrupted during processing. Each activity consumes 0-3 types of resources which are renewable and temporarily not available due to resource vacations in each period. The presence of scarce resources and precedence relations between activities makes project scheduling a difficult and important task in project management. A recent popularmetaheuristic shuffled frog leaping algorithm, which is enlightened by the predatory habit of frog group in a small pond, is adopted to investigate the project makespan improvement on Patterson benchmark sets which is composed of different small and medium size projects. Computational results demonstrate the effectiveness and efficiency of SFLA in reducing project makespan and minimizing activity splitting number within an average CPU runtime, 0.521 second. This paper exposes all the scheduling sequences for each project and shows that of the 23 best known solutions have been improved. [ABSTRACT FROM AUTHOR]

The direct methods origin-free modulus sum function [Rius (1993). Acta Cryst. A 49, 406-409] includes in its definition the structure factor G(Φ) of the squared crystal structure expressed in terms of Φ, the set of ϕ phases of the normalized structure factors E's of the crystal structure of unit-cell volume V. Here the simpler sum function variant = ∑ H E− H∫ VδP,Δ(Φ)exp( i2π Hr)d V extended over all H reflections is introduced which involves no G's and in which the δP,Δ function corresponds to δP = FT−1{()exp[ iϕ H(Φ)]} (where FT = Fourier transform) with all values smaller than Δ = 2.5σP equated to zero ( is the variance of δP calculable from the experimental intensities). The new phase estimates are obtained by Fourier transforming δP,Δ. This iterative phasing method (δ recycling) only requires calculation of Fourier transforms at two stages. Since δM≃δP/2, similar arguments are valid for δM = FT−1[( E H−〈 E〉)exp( iϕ H)] from which the corresponding phasing function follows. [ABSTRACT FROM AUTHOR]

A fast Fourier transform algorithm is introduced into the method recently defined for calculating powder diffraction patterns by means of the Debye scattering equation (DSE) [Thomas (2010). Acta Cryst. A 66, 64-77]. For this purpose, conventionally used histograms of interatomic distances are replaced by compound transmittance functions. These may be Fourier transformed to partial diffraction patterns, which sum to give the complete diffraction pattern. They also lead to an alternative analytical expression for the DSE sum, which reveals its convergence behaviour. A means of embedding the DSE approach within the reciprocal-lattice-structure-factor method is indicated, with interpolation methods for deriving the peak profiles of nanocrystalline materials outlined. Efficient calculation of transmittance functions for larger crystallites requires the Patterson group symmetry of the crystals to be taken into account, as shown for α- and β-quartz. The capability of the transmittance functions to accommodate stacking disorder is demonstrated by reference to kaolinite, with a fully analytical treatment of disorder described. Areas of future work brought about by these developments are discussed, specifically the handling of anisotropic atomic displacement parameters, inverse Fourier transformation and the incorporation of instrumental (diffractometer) parameters. [ABSTRACT FROM AUTHOR]

The article discusses the study on the use of Patterson-function direct methods (DM) in determining the structure of organic compounds from powder diffraction data. It says that the method is based on the structure factors (Φ) maximization of the sum function of Sp. It cites the optimization of Sp sum with Patterson-function tangent formula (TF) by the Sliding-Fast Fourier Transform (S-FFT) algorithm. It shows that Patterson-function DM can deal with powder data.

The article presents a study on the use of Patterson vectors molecular-replacement (PVMR) method in tracking helix fragments. It states that PVMR is calculated by equating a search model in a real space with Patterson cross and self vectors. It mentions that the equal matches of Patterson vectors are added to get the signal-to-noise ratio. Moreover, findings of the study show that PVMR is suitable in setting small helix fragments as structural root for molecular substitution.

The article offers information on a study which aims to investigate the intensity difference in Friedel opposites on the analysis of small-molecule single-crystal structure using the Patterson functions. It says that Patterson functions give clear view with regards to resonant scattering of the crystal structures and on the difference on the model and observed intensities. The study used least square refinement on the space group determination in the crystal structure.

The symmetry inherent to many biological macromolecular assemblies has been implicated in a range of crystal pathologies, including lattice-translocation defects (LTDs). Crystals suffering from classic LTDs contain two lattices that are shifted with respect to each other but nonetheless remain within the length of coherent interference. LTD introduces an undesirable intensity modulation into diffraction data, resulting in scrambled or partially scrambled electron densities. In this report, LTD theory is extended and a new general method for determining defect fractions is developed based on the heights of the non-origin peaks observed in native Patterson maps. The application of this method to crystals of lentiviral integrase in complex with its cofactor, where the observed translocation vector does not equal a small integral fraction of a unit-cell edge, is reported and its general application to all classic LTD cases is predicted. [ABSTRACT FROM AUTHOR]

A new joint probability distribution function method is described to find the anomalous scatterer substructure from powder data. The method requires two wavelengths; the conclusive formulas provide estimates of the substructure structure factor moduli, from which the anomalous scatterer positions can be found by Patterson or direct methods. The theory has been preliminarily applied to two compounds, the first having Pt and the second having Fe as anomalous scatterer. Both substructures were correctly identified. [ABSTRACT FROM AUTHOR]

A cross-platform program, VESTA, has been developed to visualize both structural and volumetric data in multiple windows with tabs. VESTA represents crystal structures by ball-and-stick, space-filling, polyhedral, wireframe, stick, dot-surface and thermal-ellipsoid models. A variety of crystal-chemical information is extractable from fractional coordinates, occupancies and oxidation states of sites. Volumetric data such as electron and nuclear densities, Patterson functions, and wavefunctions are displayed as isosurfaces, bird's-eye views and two-dimensional maps. Isosurfaces can be colored according to other physical quantities. Translucent isosurfaces and/or slices can be overlapped with a structural model. Collaboration with external programs enables the user to locate bonds and bond angles in the 'graphics area', simulate powder diffraction patterns, and calculate site potentials and Madelung energies. Electron densities determined experimentally are convertible into their Laplacians and electronic energy densities. [ABSTRACT FROM AUTHOR]

The success of the ab initio phasing process mainly depends on two parameters: data resolution and structural complexity. In agreement with the Sheldrick rule, the presence of heavy atoms can also play a nonnegligible role in the success of direct methods. The increased efficiency of the Patterson methods and the advent of new phasing techniques based on extrapolated reflections have changed the state of the art. In particular, it is not clear how much the resolution limit and the structural complexity may be pushed in the presence of heavy atoms. In this paper, it is shown that the limits fixed by the Sheldrick rule may be relaxed if the structure contains heavy atoms and that ab initio techniques can succeed even when the data resolution is about 2 Å, a limit unthinkable a few years ago. The method is successful in solving a structure with 7890 non-H atoms in the asymmetric unit at a resolution of 1.65 Å, a considerable advance on the previous record of 6319 atoms at atomic resolution. [ABSTRACT FROM AUTHOR]

The Patterson techniques, recently developed by the same authors for the ab initio crystal structure solution of proteins, have been applied to single and multiple anomalous diffraction (SAD and MAD) data to find the substructure of the anomalous scatterers. An automatic procedure has been applied to a large set of test structures, some of which were originally solved with remarkable difficulty. In all cases, the procedure automatically leads to interpretable electron density maps. Patterson techniques have been compared with direct methods; the former seem to be more efficient than the latter, so confirming the results obtained for ab initio phasing, and disproving the common belief that they could only be applied to determine large equal-atom substructures with difficulty. [ABSTRACT FROM AUTHOR]

The Patterson superposition methods described by Burla et al. [ J. Appl. Cryst. (2006), 39, 527–535], based on the use of the `multiple implication functions', have been enriched by supplementary filtering techniques based on some general (resolution-dependent) features of both the Patterson and the electron density maps. The method has been implemented in a modified version of the program SIR2004 and tested using a set of 20 crystal structures selected from the Protein Data Bank, having a number of non-hydrogen atoms in the asymmetric unit larger than 2000, atomic resolution data and some heavy atoms (equal to or heavier than Ca). The new phasing procedure is able to solve most of the test structures, among which there are two proteins with more than 6000 non-hydrogen atoms in the asymmetric unit, so extending by far the complexity today commonly considered as the limit for Patterson-based methods ( i.e. about 2000 non-hydrogen atoms). [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]

The success of ab initio crystal structure solution by powder diffraction data is strictly related to the quality of the integrated intensity estimates. A new method that is able to improve the pattern decomposition step has been developed. It combines the inversion of a suitably modified Patterson map with the use of the Hamming codes [13,10] and [40,36] in order to explore more decomposition trials. The new approach has been introduced in EXPO2005, an updated version of EXPO2004, and successfully applied to a set of known organic and inorganic structures. [ABSTRACT FROM AUTHOR]

Complete photosystem I from a higher plant (pea; Pisum sativum) was isolated, purified and crystallized. The crystals diffract to 4 Å resolution using synchrotron radiation and belong to the monoclinic crystal system, space group P2[SUB1], with unit-cell parameters a = 181.90, b = 190.24, c = 219.66 Å, Β = 90.484°. Data sets from three crystals have been collected and were merged into a 4.4 Å resolution data set, taing advantage of the high isomorphicity observed for these crystals. Analysis of the self-rotation Patterson function suggests the possible presence of two PSI complexes in the asymmetric unit. [ABSTRACT FROM AUTHOR]

A method for improving crystallographic phases is presented that is based on the preferential occurrence of certain local patterns of electron density in macromolecular electron density maps. The method focuses on the relationship between the value of electron density at a point in the map and the pattern of density that can be superimposed by rotation about the central point are considered equivalent. Standard templates are created from experimental or model electron-density maps by clustering and averaging local patterns of electron density. The clustering is based on correlation coefficients after rotation to maximize the correlation. Experimental or model maps are also used to create histograms relating the value of electron density at the central point to the correlation coefficient of the density surrounding this point with each member of the set of standard patterns. These histograms are then used to estimate the electron density at each point in a new experimental electron-density map using the pattern of electron density at points surrounding this point with each member of the set of standard patterns. These histograms are then used to estimate the electron density at each point in a new experimental electron-density map using the pattern of electron density at points surrounding that point and the correlation coefficient of this density to each of the set of standard templates, again after rotation to maximize the correlation. The method is strengthened by excluding any information from the point in question from both the templates and the local pattern of density in the calculation. A function based on the origin of the Patterson function is used to remove information about the electron density at the point in question from nearby electron density. This allows an estimation of the electron density at each point in map, using only information from other points in the process. The resulting estimates of electron density are shown to have errors that are nearly independent of the errors in the original map using model data and templates calculated at a resolution of 2.6Å. Owing to this independence of errors, information from the new map can be combined in a simple fashion with information from the original map to create an improved map An iterative Phase-improvement process using this approach and other applications of the image-reconstruction method are described and applied to experimental data at resolutions ranging from 2.4 to 2.8 &Aring. [ABSTRACT FROM AUTHOR]

Iterative dual-space direct methods based on phase refinement in reciprocal space and peak picking in real space are able to locate relatively large numbers of anomalous scatterers efficiently from MAD or SAD data. Truncation of the data at a particular resolution, typically in the range 3.0-3.5 Å, can be critical to success. The efficiency can be improved by roughly an order of magnitude by Patterson-based seeding instead of starting from random phases or sites; Patterson superposition methods also provide useful validation. The program SHELXD implementing this approach is available as part of the SHELX package. [ABSTRACT FROM AUTHOR]

An automatic structure determination method based on an interpretation of the Patterson function has been improved by utilizing information on superposition pseudosymmetry. This pseudosymmetry can be detected using MIF (multiple implication function) groups. A new method of structure determination based on an enantioselective cross-vector function and on a sum atomic minimum superposition is introduced. This new method seems to be suitable for equalatom structures. The reliability and speed of the structure determination has been increased. [ABSTRACT FROM AUTHOR]

This paper presents a review of the principles of molecular replacement with the audience of the CCP4 Study Weekend in mind. A complementary presentation with animated Patterson maps is available online (http://cci.lbl.gov/∼rwgk/ccp4sw2001/). The implementation of molecular-replacement methods in the Crystallography and NMR System (CNS) is presented and discussed in some detail. The three principal components are the direct rotation function, Patterson correlation refinement and the fast translation function. CNS is available online and is free of charge for academic users. [ABSTRACT FROM AUTHOR]

A method is described which attempts to simultaneously and independently determine the positional and orientational parameters of all molecules present in the asymmetric unit of a target crystal structure. This is achieved through a reverse Monte Carlo optimization of a suitable statistic (such as the R factor or the linear correlation coefficient between the observed and calculated amplitudes of the structure factors) in the 6n-dimensional space defined by the rotational and translational parameters of the n search models. Results from the application of this stochastic method -- obtained with a space-group-general computer program which has been developed for this purpose -- indicate that with present-day computing capabilities the method may be applied successfully to molecular-replacement problems for which the target crystal structure contains up to three molecules per asymmetric unit. It is also shown that the method may be useful in cases where the assumption of topological segregation of the self- and cross-vectors in the Patterson function is violated (as may happen, for example, in closely packed crystal structures). [ABSTRACT FROM AUTHOR]

Examines the derivation of a tangent formula from Patterson-function arguments. Solution of inorganic structures from powder data with accidental overlap; Application of the direct-methods sum function; Estimation of phases and overlapped amplitudes.

A new computer program, ShakePSD (filtering using the Patterson function search procedures and direct methods), has been developed to solve the structures of organic compounds automatically on personal computers (PCs) operating under Windows. ShakePSD is a single program that can discriminate whether a heavy-atom is included or not and can employ heavy-atom methods (Patterson and search tot all remaining atoms) or direct methods (tangent formula and/or minimal function). A new subroutine, MINIMAL, adopts the Karle recycling method based on the theory. The input data are cell constants, space group, number of atoms with atomic types and observed structure factors F[sub D](h). Compared to the TANGENT produced (employing the tangent formula). MINIMAL is able to obtain the same or a larger number of atoms in a run set and the number of run sets is distinctly larger in a Single computer execution. MINIMAL is a powerful procedure, but it takes a considerable amount of processing time on a Windows PC. The output is in the form of two-dimensional projection diagrams of the molecules in a unit cell. ShakePSD is an automatic general phase solver, being able to handle both heavy-atom methods and direct methods within a single program, and determines, with certainty, molecular fragments. [ABSTRACT FROM AUTHOR]

A program for inspection and interpretation of the Patterson function is described. The program is mainly intended for finding heavy-atom positions from difference Patterson maps, but may also be used to locate molecules with non-crystallographic symmetry when the local axis is nearly parallel to a crystallographic symmetry axis. Options are available for vector-based methods to locate heavy-atom sites, for finding sets from a list of possible heavy-atom positions and for checking of potential solutions. Both crystallographic and non-crystallographic symmetry may be used, either independently or in conjunction. [ABSTRACT FROM AUTHOR]

A pancreatic polypeptide with some hormonal properties has been purified from chicken and turkey pancreas using acid-ethanol extraction, gel filtration and anion-exchange chromatography. The material has been crystallised. The crystals are monoclinic with space group C2. Preliminary isomorphous replacement experiments have so far provided a single-site derivative with Hg(NO3)2. A low-resolution electron density map phased with this derivative using anomalous scattering considered together with Patterson function calculations suggest that the molecules are partly helical and are arranged as a compact dimer about the crystallographic two-fold axis. The structure and association of these molecules are compared with those of insulin and glucagon, pancreatic protein and polypeptide hormones respectively, which have been studied in great detail. [ABSTRACT FROM AUTHOR]

Two phasing equations based on the Fourier syntheses δP = T−1[( E2−〈 E2〉)exp( iϕ)] and δM = T−1[( E−〈 E〉)exp( iϕ)] were recently described [Rius (2012). Acta Cryst. A 68, 77-81] ( E is the quasi-normalized structure factor and 〈 E〉 is the average over all reflections). These equations were found by comparison with the direct methods origin-free modulus sum function and constitute the core of the `δ recycling' phasing procedure. The derivation of these phasing equations from the minimization of a residual ( RP) between two differently calculated density functions (one of them including the positivity constraint) is shown. [ABSTRACT FROM AUTHOR]