Advances in the VLD algorithm

The VLD algorithm relies on the properties of the difference Fourier synthesis and is designed for solving crystal structures in the correct space group, starting from random models. The standard approach has been modified by integrating it with the RELAX procedure, for translating to the correct position misplaced but correctly oriented models. A better control of the parameters and additional phase refinement cycles were able to improve the quality of the solutions and to make superfluous, for macromolecules and medium-sized molecules, the least-squares refinement cycles that, in the standard VLD approach, follow the phasing step. As a result, the efficiency of the new VLD algorithm is strongly increased; it has been checked using a wide variety of practical cases and compared with the effectiveness of direct methods.

About the hybrid Fourier syntheses: a probabilistic approach

The difference electron density has recently been revisited via the method of joint probability distribution functions [Burla et al. (2010). Acta Cryst. A 66, 347-361]. New Fourier coefficients were devised which were the basis of a new ab initio method for the solution of the phase problem (i.e. VLD, vive la difference). In this paper we study the joint probability distribution functions P(F, F(p), F(Q)), where F(Q) is the structure factor corresponding to the ideal hybrid Fourier synthesis ρ(Q) = τρ - ωρ(p) and τ and ω are any pair of real numbers. New Fourier coefficients for the calculations of any hybrid synthesis are obtained, and the properties of the corresponding electron-density maps are discussed. The first applications show the correctness of our theoretical approach and suggest possible applications in phasing procedures.

The cross-correlation function: main properties and first applications

When a model structure, and more generally a model electron density ρM(r), is available, its cross-correlation functionC(u) with the unknown true structure ρ(r) cannot be exactly calculated. A useful approximation ofC(u) is obtained by replacing exp[i(φh − φMh)] by its expected value. In this caseC′(u), a potentially useful approximation of the functionC(u), is obtained. In this paper the main crystallographic properties of the functionsC(u) andC′(u) are established. It is also shown that such functions may be useful for the success of the phasing process.

Crystal structure solution of small-to-medium-sized molecules at non-atomic resolution

Data resolution limits the information carried by diffraction data and is therefore the most critical limit for the success ofab initiocrystal structure solution. To overcome this limit, two methods have recently been proposed, namely the correction of resolution bias in electron-density maps and extrapolation of the structure factors beyond the data resolution limit. The first method has successfully been applied to powder data and the second to protein data. Neither of them has been applied to single-crystal data from small or medium-sized molecules. A third technique, the active use of the PSI-0 triplets in a tangent procedure, was applied to small molecules in the early days of crystallography, but it soon became obsolete because of the great success of methods combining reciprocal and direct space techniques. This paper explores the role of data resolution for small-to-medium-sized molecules and studies the usefulness of three auxiliary techniques,i.e.active use of the PSI-0 triplets, resolution bias correction and extrapolation of the structure factors.

The (Fo-Fc) Fourier synthesis: a probabilistic study

(F(o)-F(c)) and (2F(o)-F(c)) Fourier syntheses are considered the most powerful tools for recovering the remainder of a structure and for correcting crystal structure models. A probabilistic approach has been applied to derive the formula for the variance for the expected value of the coefficient (F(o)-F(c)). This has allowed a better understanding of the features of the difference Fourier synthesis; in particular, a subset of well phased reflections has been separated from the subset of reflections best phased by the standard F(o) Fourier synthesis. An iterative procedure, based on the electron-density modification of the difference Fourier map, has been devised which aims to improve phase and modulus estimates of the reflections with higher variance value, by using as lever arm the set of reflections with lower variance value. The new procedure (DEDM) has been implemented and verified on a wide set of test structures, the partial models of which were obtained by molecular replacement or by automatic model-building routines applied to experimental electron-density maps. Phase and modulus estimates of the difference Fourier syntheses improve in all the test cases; as a consequence, the quality of the difference Fourier maps also improves in the region where the target structure deviates from the partial model. A new procedure is suggested, combining DEDM with standard electron-density modification techniques, which leads to significant reduction of the phase errors. The procedure may be considered a starting point for further developments.

Ab initiophasing of proteins with heavy atoms at non-atomic resolution: pushing the size limit of solvable structures up to 7890 non-H atoms in the asymmetric unit

The success of theab initiophasing 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 thatab initiotechniques 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.

Advances inab initioprotein phasing by Patterson deconvolution techniques

New algorithms have been devised and implemented in the programSIR2007for the deconvolution of Patterson mapsviathe use of implication transformations and of the minimum superposition function. The new algorithms concern several practical aspects, such as the use of weighted Patterson syntheses to simplify the recognition of useful pivot peaks, the definition of a ranking criterion for them, the introduction of an early figure of merit to discard useless pivots and the fast Fourier transform transposition of theRELAXalgorithm, aiming to find the correct location when the current molecular model is translated with respect to the true position. The advantages of the new procedure have been assessed using a large set of test structures. The results have been analysed with respect to four parameters: the CPU time necessary for obtaining and recognizing the correct solution, the presence of heavy atoms, the complexity limit of the structures solvableab initio, and the data resolution limit.

The revenge of the Patterson methods. III.Ab initiophasing from powder diffraction data

In the present paper, the third and last of a series (the first two papers were dedicated to the crystal structure solution of proteins), the Patterson superposition method, based on the use of the symmetry minimum function, has been applied to powder diffraction patterns. The method has been modified to take into account the special challenges of this kind of data and to optimize the performance of the approach. The new algorithms have been implemented in a computer program and applied also to single-crystal data of small and medium-size crystal structures. The experimental results have been compared with those obtainedviadirect methods, so enabling the role and the perspectives of these two approaches in the global phasing problem to be established, no matter what the experimental technique (powder or single-crystal diffraction) or the size of the structures (small, medium or macro-molecules).

Advances in thefree lunchmethod

The most critical limit of macromolecular crystallography, the experimental data resolution, is partially `tricked' by the `free lunchmethod' (non-measured reflection extrapolation). The best electron density map available when only observed data are used may be employed to extrapolate moduli and phases of unobserved reflections behind and beyond the experimental resolution limit. The method is able to reduce the mean phase error of the observed reflections and to produce a more interpretable (in terms of a molecular model) electron density map. The main features of thefree lunchmethod have been studied and its performance has been enhanced; it is beneficial even if data resolution is about 2 Å. Furthermore, the technique has been parameterized so that it may be routinely used by other phasing programs.

The revenge of the Patterson methods. II. Substructure applications

The Patterson techniques, recently developed by the same authors for theab initiocrystal 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 forab initiophasing, and disproving the common belief that they could only be applied to determine large equal-atom substructures with difficulty.

Use of Patterson-based methods automatically to determine the structures of heavy-atom-containing proteins with up to 6000 non-hydrogen atoms in the asymmetric unit

The Patterson superposition methods described by Burlaet 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 programSIR2004and 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).

The revenge of the Patterson methods. I. Proteinab initiophasing

Direct methods combined with direct-space refinement procedures are the standard tools forab initiocrystal structure solution of macromoleculesviadiffraction data collected up to atomic or quasi-atomic resolution. An entirely direct-space approach is described here: it includes an automated Patterson deconvolution method, based on the minimum superposition function, followed by an effective direct-space refinement, consisting of cycles of electron density modification. The new approach has been implemented in a new version of theSIR2004program and tested on a large set of test structures selected from the Protein Data Bank, with data resolution better than 1.6 Å and number of non-hydrogen atoms in the asymmetric unit up to 2000. The new procedure proved to be extremely efficient and very fast in solving crystal structures with atomic resolution data and heavy atoms: their solution and refinement requires a computing time roughly comparable with that necessary for solving small-molecule crystal structuresviaa modern computer program. It markedly overcomes direct methods, even for crystal structures with atomic data resolution and heaviest atomic species up to calcium, as well as for crystal structures with quasi-atomic data resolution (i.e.1.2–1.6 Å). The Patterson approach proved to be loosely dependent on the structure complexity.

Molecular replacement: the approach of the programREMO

A new program for molecular replacement,REMO, has been written. In the rotation step, the orientation of the model molecule is found by rotating the weighted reciprocal lattice of the protein with respect to the calculated transform of the model structure: the fitting is searched in the reciprocal space. The space group of the model structure is assumed to be the symmorphic variant of the protein space group. The algebra necessary to optimize the correlation factor between protein and model structure-factor moduli is described. The oriented model molecule is located by using the correlation function coupled with a translation function calculated by fast Fourier transforms.REMOhas been successfully applied to a variety of test problems and extensively compared with other currently available molecular replacement programs.

The partial structure with errors: a probabilistic treatment

The method of the joint probability distribution functions has been applied to the case in which observed (with errors) and calculated structure factors are available, the latter referred to a part of the structure with finite errors in the coordinates, the thermal parameters and the scattering factors. Results obtained by other authors are confirmed and generalized. A new relationship is found to estimate the parameter sigmaA, affecting the reliability of the estimates of cos(varphi-varphip). Some practical applications are described.

MAD phasing: choosing the most informative wavelength combination

Two algorithms are described for limiting data resolution and for predicting the most informative wavelength combinations in MAD techniques. Both have been successfully tested using experimental data from a large set of test structures.

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.