The Automatic Solution of Macromolecular Crystal Structures via Molecular Replacement Techniques: REMO22 and Its Pipeline

A description of REMO22, a new molecular replacement program for proteins and nucleic acids, is provided. This program, as with REMO09, can use various types of prior information through appropriate conditional distribution functions. Its efficacy in model searching has been validated through several test cases involving proteins and nucleic acids. Although REMO22 can be configured with different protocols according to user directives, it has been developed primarily as an automated tool for determining the crystal structures of macromolecules. To evaluate REMO22's utility in the current crystallographic environment, its experimental results must be compared favorably with those of the most widely used Molecular Replacement (MR) programs. To accomplish this, we chose two leading tools in the field, PHASER and MOLREP. REMO22, along with MOLREP and PHASER, were included in pipelines that contain two additional steps: phase refinement (SYNERGY) and automated model building (CAB). To evaluate the effectiveness of REMO22, SYNERGY and CAB, we conducted experimental tests on numerous macromolecular structures. The results indicate that REMO22, along with its pipeline REMO22 + SYNERGY + CAB, presents a viable alternative to currently used phasing tools.

Towards the automatic crystal structure solution of nucleic acids: automated model building using the new CAB program

CAB, a recently described automated model-building (AMB) program, has been modified to work effectively with nucleic acids. To this end, several new algorithms have been introduced and the libraries have been updated. To reduce the input average phase error, ligand heavy atoms are now located before starting the CAB interpretation of the electron-density maps. Furthermore, alternative approaches are used depending on whether the ligands belong to the target or to the model chain used in the molecular-replacement step. Robust criteria are then applied to decide whether the AMB model is acceptable or whether it must be modified to fit prior information on the target structure. In the latter case, the model chains are rearranged to fit prior information on the target chains. Here, the performance of the new AMB program CAB applied to various nucleic acid structures is discussed. Other well documented programs such as Nautilus, ARP/wARP and phenix.autobuild were also applied and the experimental results are described.

How far are we from automatic crystal structure solution via molecular-replacement techniques?

Although the success of molecular-replacement techniques requires the solution of a six-dimensional problem, this is often subdivided into two three-dimensional problems. REMO09 is one of the programs which have adopted this approach. It has been revisited in the light of a new probabilistic approach which is able to directly derive conditional distribution functions without passing through a previous calculation of the joint probability distributions. The conditional distributions take into account various types of prior information: in the rotation step the prior information may concern a non-oriented model molecule alone or together with one or more located model molecules. The formulae thus obtained are used to derive figures of merit for recognizing the correct orientation in the rotation step and the correct location in the translation step. The phases obtained by this new version of REMO09 are used as a starting point for a pipeline which in its first step extends and refines the molecular-replacement phases, and in its second step creates the final electron-density map which is automatically interpreted by CAB, an automatic model-building program for proteins and DNA/RNA structures.

CAB: a cyclic automatic model-building procedure

The program Buccaneer, a well known fast and efficient automatic model-building program, is also a tool for phase refinement: indeed, input phases are used to calculate electron-density maps that are interpreted in terms of a molecular model, from which new phase estimates may be obtained. This specific property is shared by all other automatic model-building programs and allows their cyclic use, as is usually performed in other phase-refinement methods (for example electron-density modification techniques). Buccaneer has been included in a cyclic procedure, called CAB, aimed at increasing the rate of success of Buccaneer and the quality of the molecular models provided. CAB has been tested on 81 protein structures that were solved via molecular-replacement, anomalous dispersion and ab initio methods. The corresponding phases were submitted to a phase-refinement process that synergically combines current phase-refinement techniques and out-of-mainstream refinement methods [Burla et al. (2017), Acta Cryst. D73, 877-888]. The phases thus obtained were used as input for CAB. The experimental results were compared with those obtained by the sole use of Buccaneer: it is shown that CAB improves the Buccaneer results, both in completeness and in accuracy.

Phasingviapure crystallographic least squares: an unexpected feature

Crystallographic least-squares techniques, the main tool for crystal structure refinement of small and medium-size molecules, are for the first time used forab initiophasing. It is shown that the chief obstacle to such use, the least-squares severe convergence limits, may be overcome by a multi-solution procedure able to progressively recognize and discard model atoms in false positions and to include in the current model new atoms sufficiently close to correct positions. The applications show that the least-squares procedure is able to solve many small structures without the use of important ancillary tools:e.g.no electron-density map is calculated as a support for the least-squares procedure.

About difference electron densities and their properties

Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with theVLD(Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and forab initiocrystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted asF−Fp,mF−FpandmF−DFpsyntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which theVLDapproach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

Synergy among phase-refinement techniques in macromolecular crystallography

Ab initio and non-ab initio phasing methods are often unable to provide phases of sufficient quality to allow the molecular interpretation of the resulting electron-density maps. Phase extension and refinement is therefore a necessary step: its success or failure can make the difference between solution and nonsolution of the crystal structure. Today phase refinement is trusted to electron-density modification (EDM) techniques, and in practice to dual-space methods which try, via suitable constraints in direct and in reciprocal space, to generate higher quality electron-density maps. The most popular EDM approaches, denoted here as mainstream methods, are usually part of packages which assist crystallographers in all of the structure-solution steps from initial phasing to the point where the molecular model perfectly fits the known features of protein chemistry. Other phase-refinement approaches that are based on different sources of information, denoted here as out-of-mainstream methods, are not frequently employed. This paper aims to show that mainstream and out-of-mainstream methods may be combined and may lead to dramatic advances in the present state of the art. The statement is confirmed by experimental tests using molecular-replacement, SAD-MAD and ab initio techniques.

Solving proteins at non-atomic resolution by direct methods: update

Direct methods can be used to solve proteins of great structural complexity even when diffraction data are at non-atomic resolution. However, one of the main obstacles to the wider application of direct methods is that they reliably phase only a small fraction of the observed reflections, those with a sufficiently large value of the normalized structure factor amplitude. The subsequent phase expansion and refinement required for full structure solution are difficult. Here a new phase refinement procedure is described, which combines (1–2) difference Fourier synthesis with electron density modification techniques and thevive la differenceand Free Lunch algorithms. This procedure is able to solve data resistant to other direct space refinement procedures.

The phantom derivative method when a structure model is available: about its theoretical basis

This study clarifies why, in the phantom derivative (PhD) approach, randomly created structures can help in refining phases obtained by other methods. For this purpose the joint probability distribution of target, model, ancil and phantom derivative structure factors and its conditional distributions have been studied. Since PhD may usenphantom derivatives, withn≥ 1, a more general distribution taking into account all the ancil and derivative structure factors has been considered, from which the conditional distribution of the target phase has been derived. The corresponding conclusive formula contains two components. The first is the classical Srinivasan & Ramachandran term, relating the phases of the target structure with the model phases. The second arises from the combination of two correlations: that between model and derivative (the first is a component of the second) and that between derivative and target. The second component mathematically codifies the information on the target phase arising from model and derivative electron-density maps. The result is new, and explains why a random structure, uncorrelated with the target structure, adds useful information on the target phases, provided a model structure is known. Some experimental tests aimed at checking if the second component really provides information on φ (the target phase) were performed; the favourable results confirm the correctness of the theoretical calculations and of the corresponding analysis.

MPF, a multipurpose figure of merit for phasing procedures

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 ifab initioor nonab initio.

Phase improvementviathePhantom Derivativetechnique: ancils that are related to the target structure

Density modification is a general standard technique which may be used to improve electron density derived from experimental phasing and also to refine densities obtained byab initioapproaches. Here, a novel method to expand density modification is presented, termed thePhantom derivativetechnique, which is based on non-existent structure factors and is of particular interest in molecular replacement. ThePhantom derivativeapproach uses randomly generated ancil structures with the same unit cell as the target structure to create non-existent derivatives of the target structure, called phantom derivatives, which may be used forab initiophasing or for refining the available target structure model. In this paper, it is supposed that a model electron density is available: it is shown that ancil structures related to the target obtained by shifting the target by origin-permissible translations may be employed to refine model phases. The method enlarges the concept of the ancil, is as efficient as the canonical approach using random ancils and significantly reduces the CPU refinement time. The results from many real test cases show that the proposed methods can substantially improve the quality of electron-density maps from molecular-replacement-based phases.

Solving proteins at non-atomic resolution by direct methods

A new probabilistic formula estimating triplet invariants and capable of exploiting a model electron-density map gradually created during theab initiophasing process has been tested on a set of protein structures with data at non-atomic resolution. All the structures contain heavy atoms larger than Ca, and show a structural complexity which may attain the level of about 8000 non-hydrogen atoms in the asymmetric unit. It is shown that the majority of such structures may be solved by our direct methods procedure, which, in combination with a number of ancillary phasing tools, weakens the traditional expectation that atomic resolution is a necessary ingredient for the success of a directab initiophasing.

Advances in molecular-replacement procedures: theREVANpipeline

TheREVANpipeline aiming at the solution of protein structuresviamolecular replacement (MR) has been assembled. It is the successor toREVA, a pipeline that is particularly efficient when the sequence identity (SI) between the target and the model is greater than 0.30. TheREVANandREVAprocedures coincide when the SI is >0.30, but differ substantially in worse conditions. To treat these cases,REVANcombines a variety of programs and algorithms (REMO09,REFMAC,DM,DSR,VLD,free lunch,Coot,Buccaneerandphenix.autobuild). The MR model, suitably rotated and positioned, is first refined by a standardREFMACrefinement procedure, and the corresponding electron density is then submitted to cycles ofDM–VLD–REFMAC. The nextREFMACapplications exploit the better electron densities obtained at the end of theVLD–EDM sections (a procedure called vector refinement). In order to make the model more similar to the target, the model is submitted to mutations, in whichCootplays a basic role, and it is then cyclically resubmitted toREFMAC–EDM–VLDcycles. The phases thus obtained are submitted tofree lunchand allow most of the test structures studied by DiMaioet al.[(2011),Nature (London),473, 540–543] to be solved without using energy-guided programs.

Refining a model electron-density mapviathePhantom Derivativemethod

ThePhantom Derivative(PhD) method [Giacovazzo (2015),Acta Cryst.A71, 483–512] has recently been described forab initioand non-ab initiophasing. It is based on the random generation of structures with the same unit cell and the same space group as the target structure (called ancil structures), which are used to create derivatives devoid of experimental diffraction amplitudes. In this paper, the non-ab initiovariant of the method was checked using phase sets obtained by molecular-replacement techniques as a starting point for phase extension and refinement. It has been shown that application ofPhDis able to extend and refine phases in a way that is competitive with other electron-density modification techniques.

Crystal structure determination and refinementviaSIR2014

SIR2014is the latest program of theSIRsuite for crystal structure solution of small, medium and large structures. A variety of phasing algorithms have been implemented, bothab initio(standard or modern direct methods, Patterson techniques,Vive la Différence) and non-ab initio(simulated annealing, molecular replacement). The program contains tools for crystal structure refinement and for the study of three-dimensional electron-density mapsviasuitable viewers.

Protein phasing at non-atomic resolution by combining Patterson and VLD techniques

Phasing proteins at non-atomic resolution is still a challenge for any ab initio method. A variety of algorithms [Patterson deconvolution, superposition techniques, a cross-correlation function (C map), the VLD (vive la difference) approach, the FF function, a nonlinear iterative peak-clipping algorithm (SNIP) for defining the background of a map and the free lunch extrapolation method] have been combined to overcome the lack of experimental information at non-atomic resolution. The method has been applied to a large number of protein diffraction data sets with resolutions varying from atomic to 2.1 Å, with the condition that S or heavier atoms are present in the protein structure. The applications include the use of ARP/wARP to check the quality of the final electron-density maps in an objective way. The results show that resolution is still the maximum obstacle to protein phasing, but also suggest that the solution of protein structures at 2.1 Å resolution is a feasible, even if still an exceptional, task for the combined set of algorithms implemented in the phasing program. The approach described here is more efficient than the previously described procedures: e.g. the combined use of the algorithms mentioned above is frequently able to provide phases of sufficiently high quality to allow automatic model building. The method is implemented in the current version of SIR2014.

VLD algorithm and hybrid Fourier syntheses

The VLD (vive la difference) phasing algorithm combines the model electron density with the difference electron densityviareciprocal space relationships to obtain new phase values and drive them to the correct values. The process is iterative and has been applied to small and medium-size structures and to proteins. Hybrid Fourier syntheses show properties that are intermediate between those of the observed synthesis (whose peaks should correspond to the most probable atomic positions) and those of the difference synthesis (whose positive and negative peaks should correspond to missed atomic positions and to false atoms of the model, respectively). Thanks to these properties some hybrid syntheses can be used in the phase extension and refinement step, to reduce the model bias and more rapidly move to the target structure. They have been recently revisitedviathe method of joint probability distribution functions [Burla, Carrozzini, Cascarano, Giacovazzo & Polidori (2011).Acta. Cryst. A67, 447–455]. The results suggested that VLD could be usefully combined, forab initiophasing, with the hybrid rather than with the difference Fourier synthesis. This paper explores the feasibility of such a combination and shows that the original VLD algorithm is only one of several variants, all with relevant phasing capacity. The study explores the role of several parameters in order to design a standard procedure with optimized phasing power.

Automated determination of the extinction symbolviaelectron diffraction data

The identification of the extinction symbol is routine for single-crystal X-ray data. Because of peak overlap and possible preferred orientation the task is more difficult for powder diffraction data, but recent computer programs based on probabilistic approaches have made it more automatic. This paper describes a new algorithm for the automatic identification of the Laue group and of the extinction symbol from electron diffraction intensities. The algorithm has a statistical basis and tries to deal with the severe problems arising from the often nonkinematical nature of the diffraction intensities and from the limited accuracy of the lattice parameters determinedviaelectron diffraction. The approach has been checked using a wide set of test structures.

SIR2011: a new package for crystal structure determination and refinement

SIR2011, the successor ofSIR2004, is the latest program of theSIRsuite. It can solveab initiocrystal structures of small- and medium-size molecules, as well as protein structures, using X-ray or electron diffraction data. With respect to the predecessor the program has several new abilities:e.g.a new phasing method (VLD) has been implemented, it is able to exploit prior knowledge of the molecular geometryviasimulated annealing techniques, it can use molecular replacement methods for solving proteins, it includes new tools like free lunch and new approaches for electron diffraction data, and it visualizes three-dimensional electron density maps. The graphical interface has been further improved and allows the straightforward use of the program even in difficult cases.

Automatic α-helix identification in Patterson maps

α-Helices are peculiar atomic arrangements characterizing protein structures. Their occurrence can be used within crystallographic methods as minimal a priori information to drive the phasing process towards solution. Recently, brute-force methods have been developed which search for all possible positions of α-helices in the crystal cell by molecular replacement and explore all of them systematically. Knowing the α-helix orientations in advance would be a great advantage for this kind of approach. For this purpose, a fully automatic procedure to find α-helix orientations within the Patterson map has been developed. The method is based on Fourier techniques specifically addressed to the identification of helical shapes and operating on Patterson maps described in spherical coordinates. It supplies a list of candidate orientations, which are then refined by using a figure of merit based on a rotation function calculated for a template polyalanine helix oriented along the current direction. The orientation search algorithm has been optimized to work at 3 Å resolution, while the candidates are refined against all measured reflections. The procedure has been applied to a large number of protein test structures, showing an overall efficiency of 77% in finding α-helix orientations, which decreases to 48% on limiting the number of candidate solutions (to 13 on average). The information obtained may be used in many aspects in the framework of molecular-replacement phasing, as well as to constrain the generation of models in computational modelling programs. The procedure will be accessible through the next release of IL MILIONE and could be decisive in the solution of new unknown structures.

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.