A new interpretation of the σA parameter.

A new study of the σA parameter has been undertaken to understand its behaviour when the diffraction amplitude distributions are far from the standard Wilson distributions. The study has led to the formulation of a new statistical interpretation of σA, expressed in terms of a correlation factor. The new formulas allow a more accurate use of σA in electron-density modification procedures.

Phasing diffuse scattering. Application of the SIR2002 algorithm to the non-crystallographic phase problem.

A new phasing algorithm has been used to determine the phases of diffuse elastic X-ray scattering from a non-periodic array of gold balls of 50 nm diameter. Two-dimensional real-space images, showing the charge-density distribution of the balls, have been reconstructed at 50 nm resolution from transmission diffraction patterns recorded at 550 eV energy. The reconstructed image fits well with a scanning-electron-microscope (SEM) image of the same sample. The algorithm, which uses only the density modification portion of the SIR2002 program, is compared with the results obtained via the Gerchberg-Saxton-Fienup HiO algorithm. The new algorithm requires no knowledge of the object's boundary and proceeds from low to high resolution. In this way, the relationship between density modification in crystallography and the HiO algorithm used in signal and image processing is elucidated.

Solving non-periodic structures using direct methods: phasing diffuse scattering.

The problem of reconstructing the charge density of a non-periodic sample from its diffuse X-ray scattering is considered. For a sample known to be isolated, an artificial superlattice may be assumed and the numerical direct methods of crystallography applied to the continuous distribution of diffuse scattering in order to solve the phase problem. This method is applied to simulated soft-X-ray transmission speckle patterns from a two-dimensional array of gold balls of 50 nm diameter. The results are relevant to efforts to phase the scattering from many individual macromolecules that cannot be crystallized, and to the scattering from individual inorganic nanoparticles.

The method of joint probability distribution functions applied to MAD techniques. The two-wavelength case for acentric crystals.

MAD (multiple-wavelength anomalous dispersion) techniques are often considered as a special MIR (multiple isomorphous replacement) case. The rigorous method of the joint probability distribution functions is applied to solve the phase problem for acentric crystals, on the assumption that the anomalous scatterer's substructure is a priori known. The two-wavelength case is considered: errors in measurements and in the model substructure are handled. The probabilistic approach provides a very simple and efficient formula for estimating structure-factor phases.

On integrating direct methods and isomorphous-replacement techniques: triplet estimation and treatment of errors.

The method of joint probability distribution functions has been generalized in order to include and treat different sources of error. The probability distributions of the isomorphous pairs (E(p), E(d)) and of the two triples (E(ph), E(pk), E(ph+k), E(dh), E(dk), E(dh+k)) are obtained, on the assumption that the lack of isomorphism and the errors in measurements cumulate on the E(d) variables. The conditional distributions of the two-phase and the three-phase structure invariants are derived, showing how the reliability of the probabilistic estimates depends on the errors.

The method of joint probability distribution functions applied to MAD techniques. The centric case.

Traditional probabilistic approaches consider MAD (multiple-wavelength anomalous-dispersion) data as a special MIR (multiple isomorphous replacement) case. The rigorous method of the joint probability distribution functions has been applied to solve the phase problem, with the assumption that the anomalous scatterers' substructure is a priori known. The probabilistic approach is able to handle measurement errors: it has been applied to symmetry-restricted phases and provides simple and efficient formulas.

The method of joint probability distribution functions applied to the one-wavelength anomalous-scattering (OAS) case.

The method of the joint probability distribution function is applied to the case in which the positions of the anomalous scatterers are fully or partially known. The mathematical technique is able to handle errors both in the model structure of the located anomalous scatterers and in measurements. A criterion for ranking the more accurate phase estimates is given.

SIR2000, a program for the automatic ab initio crystal structure solution of proteins.

A new phasing procedure is described working both in direct and in reciprocal space. The procedure has been implemented into the program SIR2000, the heir to SIR99, and it is able routinely to solve ab initio crystal structures of proteins without any use of prior information and any user intervention. The moduli and the flow diagram of SIR2000 are also described and its efficiency tested on several protein diffraction data sets. Success has been attained for crystal structures with up to almost 2000 non-hydrogen atoms in the asymmetric unit and resolution higher than 1.2 A. The phasing process is analysed to provide a better insight into the role of the various steps of the procedure.

Structures of nanometre-size crystals determined from selected-area electron diffraction data.

The structure of a new modification of Ti2Se, the beta-phase, and several related inorganic crystal structures containing elements with atomic numbers between 16 and 40 have been solved by quasi-automatic direct methods from single-crystal electron diffraction patterns of nanometre-size crystals, using the kinematical approximation. The crystals were several thousand times smaller than the minimum size required for single-crystal X-ray diffraction. Atomic coordinates were found with an average accuracy of 0.2 A or better. Experimental data were obtained by standardized techniques for recording and quantifying electron diffraction patterns. The SIR97 program for solving crystal structures from three-dimensional X-ray diffraction data by direct methods was modified to work also with two-dimensional electron diffraction data.

The joint probability distribution function of structure factors with rational indices. IV. The P1 case.

The method of the joint probability distribution functions of structure factors has been extended to reflections with rational indices. The most general case, space group P1, has been considered. The positional parameters are the primitive random variables of our probabilistic approach, while the reflection indices are kept fixed. Quite general joint probability distributions have been considered from which conditional distributions have been derived: these proved applicable to the accurate estimation of the real and imaginary parts of a structure factor, given prior information on other structure factors. The method is also discussed in relation to the Hilbert-transform techniques.

The joint probability distribution function of structure factors with rational indices. V. The estimates.

The probabilistic formulae [Giacovazzo, Siliqi & Fernández-Castaño (1999). Acta Cryst. A55, 512-524] relating standard and half-integral index reflections are modified for practical applications. The experimental tests prove the reliability of the probabilistic relationships. The approach is further developed to explore whether the moduli of the half-integral index reflections can be evaluated in the absence of phase information; i.e. by exploiting the moduli of the standard reflections only. The final formulae indicate that estimates can be obtained, even though the reliability factor is a constant.

The probability distribution of structure factors with non-integral indices. II. The P1; case.

The probability distribution of the structure factors with non-integral indices is derived in P1;. For integral values of at least one of the indices, the intensity distribution coincides with that provided by Wilson's statistics, but may strongly differ when the indices are (or are close to) half-integers. The deviations are stronger when the integral part of the indices is small, and increase with the size of the structure. In favourable circumstances, moduli and phases of the reflections may be accurately estimated.

The probability distribution function of structure factors with non-integral indices. III. The joint probability distribution in the P1; case.

The joint probability distribution function method has been developed in P1; for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F(p1), F(p2)) are found to be useful for phasing purposes, where p1 and p2 are any pair of vectorial indices. A variety of conditional distributions like P(|F(p1)| | |F(p2)|), P(|F(p1)| |F(p2)), P(varphi(p1)| |F(p1)|, F(p2)) are derived, which are able to estimate the modulus and phase of F(p1) given the modulus and/or phase of F(p2). The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F(1), F(2),ellipsis, F(n+1)), P(|F(1)| |F(2),ellipsis, F(n+1)) and P(varphi(1)| |F|(1), F(2),ellipsis, F(n+1)) have been obtained. Some practical tests prove the efficiency of the method.

The joint probability distribution function of structure factors with rational indices. VI. Cases with reduced dimensionality.

The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

SIR99, a program for the automatic solution of small and large crystal structures.

The moduli and flow diagram of the program SIR99 are described. New phasing algorithms are proposed working both in direct and in reciprocal space. Their cooperative work is able to solve the structures of both small and large molecules. In particular, small proteins can be solved ab initio without any use of prior information and any user intervention. The efficiency of the various algorithms employed by SIR99 has been tested, and the role of the tangent formula clarified. The user is also provided with some practical information concerning the computer power needed.

The ab initio crystal structure solution of proteins by direct methods. VI. Complete phasing up to derivative resolution.

The procedure described in the papers I-V of this series [Giacovazzo, Siliqi & Ralph (1994). Acta Cryst. A50, 503-505; Giacovazzo, Siliqi & Spagna (1994). Acta Cryst. A50, 609-621; Giacovazzo, Siliqi & Zanotti (1995). Acta Cryst. A51, 177-188; Giacovazzo & Gonzalez Platas (1995). Acta Cryst. A51, 398-404; Giacovazzo, Siliqi & Gonzalez Platas (1995). Acta Cryst. A51, 811-820], aiming at estimating protein phases via a single heavy-atom derivative, has been improved so as to extend phase determination to all the reflections up to derivative resolution. The quality of the resulting electron-density maps is checked for a number of test strutures. Some of the maps are immediately interpretable, and some can be interpreted after some cycles of solvent flattening and/or histogram matching. The correlation with classical SIR techniques is also discussed.

The ab initio crystal structure solution of proteins by direct methods. II. The procedure and its first applications.

A direct phasing method is described that is potentially able to solve ab initio protein structures. The method uses the information contained in diffraction data of the native structure and of one isomorphous derivative. The various steps of the procedure are analysed in order to estimate their robustness against experimental errors in measurements and lack of isomorphism. Experimental tests involve four typical protein structures and show that crystal structure solution is attained in a rather straightforward way.

Solid-state conformations of aminosuccinyl peptides: structure of tert-butyloxycarbonyl-L-prolyl-L-aminosuccinyl-glycyl-L-alanine methyl ester (Boc-L-Pro-L-Asu-Gly-L-Ala-OMe). A case of pseudo-translational symmetry.

C20H30N4O8, M(r) = 454.48, monoclinic, P2(1), a = 13.411 (2), b = 12.592 (2), c = 14.710 (1) A, beta = 104.30 (1) degrees, V = 2407 (6) A3, Z = 4, Dx = 1.254 Mg m-3, lambda (Cu K alpha) = 1.5418 A, mu = 0.783 mm-1, F(000) = 968, room temperature, final R = 0.086, wR = 0.080 for 4055 observed reflections. The title compound is a model for the intermediate formed in the deamidation reaction of porcine adrenocorticotropin hormone. The structure presents a pseudo-translational symmetry and was solved by using a modified version of the SIR88 package. In the refinement, few stereochemical restraints were needed to handle the static disorder shown by the C-terminal fragment of one molecule in the asymmetric unit. The conformation of the two independent molecules is almost identical and is a II' beta-bend, stabilized by an intramolecular hydrogen bond. In the crystal, screw-related molecules are linked by hydrogen bonds. The two molecules in the independent unit are related by the translation vector u = 0.4962 (2)a + 0.7310 (2)b + 0.5075 (2)c.