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Central to crystallographic structure solution is obtaining accurate phases in order to build a molecular model, ultimately followed by refinement of that model to optimize its fit to the experimental diffraction data and prior chemical knowledge. Recent advances in phasing and model refinement and validation algorithms make it possible to arrive at better electron density maps and more accurate models.
X-ray crystallography is one of the most content-rich methods available for providing high resolution information about biological macromolecules. The goal of the crystallographic experiment is to obtain a three-dimensional map of the electron density in the macromolecular crystal. Given sufficient resolution this map can be interpreted to build an atomic model of the macromolecule (Figure 1 summarizes the process of crystallographic structure solution). To calculate the distribution of electron density in the crystal requires a Fourier synthesis using complex numbers derived from the diffraction experiment. Each complex number is composed of a diffraction amplitude and an associated phase (i.e. the magnitude and direction of a vector in a complex plane). However, experimentally it is only possible to measure the amplitude; the phase information is lost. Therefore, one of the central problems in the crystallographic experiment is the indirect derivation of phase information. Multiple methods have been developed to obtain phase information. In the article we review recent advances in the computational aspects of phase determination.
After a map has been obtained and an atomic model built it is necessary to optimize the model with respect to the experimental diffraction data and prior chemical knowledge, achieved by multiple cycles of refinement and model rebuilding. Efficient and accurate optimization of the atomic model is desirable in order to rapidly generate the best models for biological interpretation. We review the recent advances in the computational methods used to optimize atomic models with respect to X-ray diffraction data.
Experimental, and associated computational, methods have been developed to obtain phase information for structures with no prior structural information. These isomorphous replacement and anomalous scattering methods rely on information that can be derived from small differences between diffraction datasets. The first step in both methods is the location of the heavy atoms or anomalous scatterers, generally termed the substructure, in the crystallographic asymmetric unit.
The location of heavy atoms in isomorphous replacement or the location of anomalous scatterers was traditionally performed by manual inspection of Patterson maps. However, in recent years labeling techniques such as seleno-methionyl incorporation have become widely used. This leads to an increase in the number of atoms to be located, rendering manual interpretation of Patterson maps extremely difficult. As a result automated heavy atom location methods have proliferated, including automated Patterson methods . The most successful substructure location methods today are dual-space recycling algorithms [2,3,4,5]. These algorithms are characterized by the use of Fourier transforms to switch between crystal (direct) space and diffraction (reciprocal) space, with a variety of modification procedures working in both spaces. The dual-space recycling procedures are designed to converge to a self-consistent state. The likelihood that this is a correct solution depends on the quality of the experimental data and the size of the problem. Some algorithms combine Patterson and dual-space recycling methods [2,3]. In the recent work of Dumas and van der Lee , the charge flipping method of Oszlányi & Süto [6**] is used instead of the more traditional squaring method  underlying the direct methods steps in the other dual-space recycling programs. The new charge-flipping approach is promising and appears to be competitive.
After successful substructure location complex structure factors for the substructure can be calculated. This information can be used to bootstrap the calculation of phase information for the whole contents of the asymmetric unit in a process termed phasing. In macromolecular crystallography a thorough statistical treatment of errors is crucial. The magnitudes of structure factors are measured relatively accurately but the phases are not measured directly at all. This leads to combinations of experimental and model errors that are not simple Gaussian distributions. In the phasing step, maximum-likelihood based methods (MLPHARE , CNS , SHARP ), have for some time been the most effective techniques for modeling the crystallographic experiment.
The development of tractable likelihood targets for experimental phasing generally requires some assumptions to be made about the independence of sources of error . However, the case of single-wavelength anomalous diffraction (SAD) can be handled without such assumptions . SAD phasing has recently undergone a renaissance, due to a combination of more sensitive detectors measuring data more precisely, the absence of non-isomorphism, and SAD’s reduced problems with radiation damage compared to MAD phasing . However, in cases of weak anomalous signal or a single scattering site in polar space groups it may still be advantageous to perform a MAD experiment, to maximize the amount of information obtained and resolve phase ambiguities. Clearly, the likelihood of success decreases as crystal sensitivity to radiation damage increases, which at an extreme can require the merging of data from multiple, possibly non-isomorphous, crystals.
Initial substructures supplied to phasing programs are generally incomplete, so effective substructure completion is an essential element of an optimal phasing strategy. Log-likelihood-gradient maps are highly sensitive in detecting new sites or signs of anisotropy, whether for general experimental phasing methods  or specifically for the SAD target in Phaser .
In addition to the intrinsic complication of modelling errors in crystallography due to not measuring phases directly, there are significant complications due to the physics of the diffraction process and to damage of the sample during the experiment. Recently statistical methods have been developed that account for these effects and in some cases even make virtues of them. The correlations of errors in the modelling of the anomalously-scattering atoms at various wavelengths in a MAD experiment and of non-isomorphism errors in various heavy-atom derivatives are included in experimental phasing with SOLVE . The radiation-induced damage at heavy-atom sites in MAD and SAD experiments is used as a source of phase information in the RIP (radiation-damage-induced phasing) method . The interactions between the polarization of synchrotron X-ray beams and the anisotropy of anomalous diffraction have long been recognized but only recently has a systematic and practical treatment been developed that allows a full use of this source of phase information [16*].
The method of molecular replacement is commonly used to solve structures for which a homologous structure is already known. As the database of known structures increases, the number of new folds drops and the proportion of structures that can be solved by molecular replacement increases. About two-thirds of structures deposited in the PDB are currently solved by molecular replacement, and the proportion could probably be higher [17*].
The introduction of likelihood targets [18,19,20] has increased the sensitivity of molecular replacement searches, thus allowing structures to be solved with more distant homologues. These targets, implemented in Phaser , allow the information from partial solutions to be exploited, which improves success in solving structures of complexes or crystals containing multiple copies.
Automation is also playing a major role. Even within a single program such as Phaser, the ability to test multiple choices of model for multiple choices of possible space group, following a tree of potential solutions, allows problems to be solved when they would exhaust the patience of a user. Molecular replacement pipelines can extend this power even further, by testing alternative strategies for model preparation (CaspR , MrBUMP [22*]) or building up models using automated databases of domains (BALBES [17*]). In some cases, the target structure differs from the molecular replacement model by deformations that can be modeled by a normal modes calculation (elNemo ).
As the level of sequence identity drops below about 30%, the success rate of molecular replacement drops precipitously. It might be expected that homology modelling could improve distant templates for molecular replacement, but until recently this was not the case. The best strategy was to use sensitive profile-profile alignment techniques to determine which parts of the template would not be preserved, and then to trim off loops and side chains . However, modelling techniques have now matured to the point where value can be added to the template, and it is possible to improve homology models or NMR structures for use in molecular replacement [25**]. At least in favorable circumstances, similar modelling techniques can generate ab initio models that are sufficiently accurate to succeed in molecular replacement calculations [25**,26].
Once experimental or molecular replacement phase information is available, electron density maps can be calculated. Often these initial maps are very noisy, but powerful density modification techniques can be used to greatly improve their quality [27,28,29]. The essence of density modification procedures is that electron density maps have recognizable features such as a flat solvent region, and that enhancing these features generally improves the accuracy of the crystallographic phases. Key to this process is the fact that improving one region in a map (flattening the solvent) improves the phases, which then improves other parts of the map as well (the image of the macromolecule). Recently density modification procedures have been greatly improved through statistical treatments that reduce bias towards the starting map by solvent flipping  or maximum-likelihood techniques [31,32,33].
With an electron density map of sufficient resolution and quality it is possible to build an atomic model. Traditionally this has been a time consuming subjective manual process prone to human error . More recently, automated methods have been developed that work at higher and lower resolution limits [35*,36*]. These methods iterate automated model building, model refinement, and, in the case of the Phenix  Autobuild procedure, density modification [36*]. This method has been extended to calculate minimally biased electron density maps by iterative build omit maps [38**]. In general an atomic model obtained by automatic or manual methods contains some errors and must be optimized to best fit the experimental data and prior chemical information. In addition, the initial model is often incomplete and refinement is carried out to generate improved phases that can then be used to compute a more accurate electron density map.
Over the last ten years there have been great improvements in the targets for refinement of incomplete, error-containing models by making use of the more general maximum likelihood formulation [39,40]. The resulting maximum likelihood refinement targets have been successfully combined with the powerful optimization method of simulated annealing to provide a very robust and efficient refinement scheme . For many structures, some initial experimental phase information is available from either isomorphous heavy atom replacement or anomalous diffraction methods. These phases represent additional observations that can be incorporated in the refinement target. Tests have shown that the addition of experimental phase information greatly improves the results of refinement . Recently there have been efforts to use phasing targets in structure refinement, thus more correctly accounting for the anomalous scattering and the relationship between structure factors . Results indicate that improved models and electron density maps are produced.
Of central importance in model refinement is the parameterization of the atomic model. The two main features of the model that are optimized are the coordinates (position of the atoms) and atomic displacement parameters (movements of the atoms from their average position). In both cases restraints may be used to provide additional information, or constraints applied to reduce the number of parameters being refined. The application of constraints to anisotropic displacement parameters, using the Translation-Libration-Screw (TLS) formalism , is now widely used to refine the displacements of large rigid bodies. This method typically improves the fit between the model and experimental data, as judged by the free R-factor, although significant improvements in the electron density maps are typically less common. Very recently an alternative approach to modeling concerted atomic displacements has been introduced: refinement of normal modes against the experimental diffraction data [45**]. Although only applied to a few structures to date the method has demonstrated improvements in models and in some challenging cases sufficient changes in the electron density maps to permit correction of model errors [46*].
The refinement of molecular models is particularly challenging when the upper limit of diffraction is low, 4Å or worse. At this resolution atomic features are typically not visible, although secondary structure elements are usually recognizable. However, by application of the collection of tools currently available, researchers have been able to push the limits of meaningful refinement to 4.7Å resolution [47,48**]. To achieve this, it was essential to include experimental phase information in the refinement target, apply B-factor sharpening to the data, apply non-crystallographic symmetry restraints, and optimize the bulk solvent scattering model . As more and more crystallographic experiments are focused on large molecular complexes, increasing the likelihood of poor diffraction, tools for low resolution refinement and validation are becoming more important.
Macromolecular crystals can also be studied using neutron diffraction. The neutrons interact with the nucleus of the atoms rather than the electrons. This allows neutron diffraction experiments to provide information about the position of hydrogen atoms, something usually not possible with X-ray diffraction methods unless ultra high resolution diffraction is obtained. Recent work has demonstrated that the joint (simultaneous) refinement of a single model against X-ray and neutron data leads to an improved model, as judged by both X-ray and neutron R-factors [50**]. In addition the electron density maps are improved for both types of data. The method has been successfully applied to the high resolution structure of aldose reductase leading to a quantum model of catalysis that depends on accurate modeling of the hydrogen atoms in the structure [51*].
Since the inception of Rfree for model-to-data fit  and ProCheck for model quality assessment  in the early 1990’s, structure validation has been considered a necessary final step before deposition, occasionally prompting correction of an individual problem but chiefly serving a gatekeeping function to ensure professional standards for publication of crystal structures. For true cross-validation, the criteria should be independent of the refinement target function, as engineered into the definition of Rfree and nearly always true for the Ramachandran measure important in ProCheck. However, local measures are typically more important to end users than global ones, since no level of global quality can protect against a large local error at the specific region of interest. Local measures can also enable the crystallographer to make specific local corrections to the model.
In the intervening years, further measures of model-to-data agreement have been developed, for example in WhatCheck , SFCheck , and the Electron-Density Server . Stereochemical validation such as rotamer  and Ramachandran  criteria have benefitted from great increases in high-resolution data, B-factor filtering at the residue level, and evaluation as multi-dimensional distributions. All-atom contacts  contributed a new major source of independent information by adding, optimizing, and analyzing the other half of the atoms – the hydrogens – and analyzing their steric clashes as well as their H-bonding.
Recently, validation criteria have been developed for carbohydrates  and for RNA backbone [61**], as well as a promising new evaluation of under-packing in proteins [62*]. The MolProbity validation web site [63**] combines all-atom contacts (especially the “clashscore”) with geometric and dihedral-angle criteria for proteins, nucleic acids, and ligands, to produce numerical and graphical local evaluations as well as global scores. The local results can guide manual [64,65] or automated [66**] rebuilding to correct systematic errors such as backward-fit sidechains trapped in the wrong local minimum, thereby improving refinement behavior, electron density quality, and chemical reasonableness, and also lowering R and Rfree by small amounts. Such procedures have become standard in many structural genomics and industrial labs that do high-throughput crystallography, and are being built into software such as ARP/wARP [35*], Coot , Buster , and PHENIX [36*,37]. In general, there are now many fewer “false alarms”, and outliers flagged by validation are nearly always worth examining.
The most significant overall development currently happening in the area of validation is the application of these local criteria much earlier in the structure solution process, their integration into a cycle of repeated refinement, correction, and re-refinement, and the gradual automation of more aspects of that cycle. A useful level of independence for cross-validation can still be preserved, for stereochemistry by the wealth of interdependent criteria that must be satisfied simultaneously with data match, and for all-atom steric clashes by avoiding refinement of explicit hydrogen contacts. The benefits are a significant increase in the accuracy of structures treated in this new manner.
The advances in data collection hardware, development of anomalous diffraction-based phasing methods, and new structure solution, refinement and validation algorithms have made it easier to arrive at high quality macromolecular structures. Routine structures can often be solved rapidly using a variety of automated tools. Many challenging structures can now be solved that otherwise would be intractable using the tools available 15 years ago. While the availability of better validation tools makes the existence of local conformational, steric, and geometric errors less likely, the global fit of models to the experimental data has not improved substantially, as judged by a criterion such as the R-free value. A brief inspection of the Protein Data Bank reveals that structures solved at a typical resolution of 2.2Å still routinely have final R and R-free values in the range of 20% to 30%, which is much larger than the data measurement errors. Inasmuch as a cross-validated measure such as R-free reflects the degree to which the atomic model adequately models the true contents of the crystal, we can suppose that our current atomic models are still in some way deficient. The most likely culprit is motion and multiple conformations, at all size scales. This highlights the need for yet more sophisticated model parameterizations. There have been recent advances in the modeling of domain motion using TLS  and normal mode methods [45**], both of which have been observed to decrease R and R-free values. However, obvious areas for further improvement are in the modeling of alternate conformations, domain motion/disorder, local macromolecular disorder, ordered solvent disorder, and the current relatively crude bulk solvent models in use. Unfortunately, as the resolution of the experimental data worsens these features become less and less easy to detect and model but are still present in the crystal, probably to an even greater degree. In addition, our current harmonic parameterizations are in many cases approximate; progress will require the use of anharmonic models to better capture the underlying molecular reality. One of the main challenges of developing better models will be arriving at efficient parameterizations that provide improved physical meaning while only requiring a small number of refinable parameters.
The last ten years has seen a dramatic improvement in the computational tools available for the determination of macromolecular crystal structures. The routine inclusion of likelihood-based algorithms in experimental phasing, molecular replacement, and structure refinement serve to generate better electron density maps and atomic models. These in turn serve to improve the efficiency and success rate of automated model building methods. Structure validation methods are starting to be applied throughout the crystallographic process, further improving the quality of models. Many challenges still remain. In particular, the generation of accurate structures when only low resolution data are available, and improved parameterizations of macromolecular models and their motions to best fit the experimental data at all resolutions.
The authors would like to thank the NIH for their support (1P01 GM063210). This work was partially supported by the US Department of Energy under Contract No. DE-AC02–05CH11231. RJR is supported by a Principal Research Fellowship from the Wellcome Trust (UK).
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