Toward Real Real-Space Refinement of Atomic Models.

High-quality atomic models providing structural information are the results of their refinement versus diffraction data (reciprocal-space refinement), or versus experimental or experimentally based maps (real-space refinement). A proper real-space refinement can be achieved by comparing such a map with a map calculated from the atomic model. Similar to density distributions, the maps of a limited and even inhomogeneous resolution can also be calculated as sums of terms, known as atomic images, which are three-dimensional peaky functions surrounded by Fourier ripples. These atomic images and, consequently, the maps for the respective models, can be expressed analytically as functions of coordinates, atomic displacement parameters, and the local resolution. This work discusses the practical feasibility of such calculation for the real-space refinement of macromolecular atomic models.

Program VUE: analysing distributions of cryo-EM projections using uniform spherical grids.

Three-dimensional cryo electron microscopy reconstructions are obtained by extracting information from a large number of projections of the object. These projections correspond to different 'views' or 'orientations', i.e. directions in which these projections show the reconstructed object. Uneven distribution of these views and the presence of dominating preferred orientations may distort the reconstructed spatial images. This work describes the program VUE (views on uniform grids for cryo electron microscopy), designed to study such distributions. Its algorithms, based on uniform virtual grids on a sphere, allow an easy calculation and accurate quantitative analysis of the frequency distribution of the views. The key computational element is the Lambert azimuthal equal-area projection of a spherical uniform grid onto a disc. This projection keeps the surface area constant and represents the frequency distribution with no visual bias. Since it has multiple tunable parameters, the program is easily adaptable to individual needs, and to the features of a particular project or of the figure to be produced. It can help identify problems related to an uneven distribution of views. Optionally, it can modify the list of projections, distributing the views more uniformly. The program can also be used as a teaching tool.

Accounting for nonuniformity of bulk-solvent: A mosaic model.

A flat mask-based model is almost universally used in macromolecular crystallography to account for disordered (bulk) solvent. This model assumes any voxel of the crystal unit cell that is not occupied by the atomic model is occupied by the solvent. The properties of this solvent are assumed to be exactly the same across the whole volume of the unit cell. While this is a reasonable approximation in practice, there are a number of scenarios where this model becomes suboptimal. In this work, we enumerate several of these scenarios and describe a new generalized approach to modeling the bulk-solvent which we refer to as mosaic bulk-solvent model. The mosaic bulk-solvent model allows nonuniform features of the solvent in the crystal to be accounted for in a computationally efficient way. It is implemented in the computational crystallography toolbox and the Phenix software.

Local heterogeneity analysis of crystallographic and cryo-EM maps using shell-approximation.

In X-ray crystallography and cryo-EM, experimental maps can be heterogeneous, showing different level of details in different regions. In this work we interpret heterogeneity in terms of two parameters, assigned individually for each atom, combining the conventional atomic displacement parameter with the resolution of the atomic image in the map. We propose a local real-space procedure to estimate the values of these heterogeneity parameters, assuming that a fragment of the density map and atomic positions are given. The procedure is based on an analytic representation of the atomic image, as a function of the inhomogeneity parameters and atomic coordinates. In this article, we report the results of the tests both with maps simulated and those derived from experimental data. For simulated maps containing regions with different resolutions, the method determines the local map resolution around the atomic centers and the values of the displacement parameter with reasonable accuracy. For experimental maps, obtained as a Fourier synthesis of a given global resolution, estimated values of the local resolution are close to the global one, and the values of the estimated displacement parameters are close to the respective values of the closest atoms in the refined model. Shown successful applications of the proposed method to experimental crystallographic and cryo-EM maps can be seen as a practical proof of method.

Efficient structure-factor modeling for crystals with multiple components.

Diffraction intensities from a crystallographic experiment include contributions from the entire unit cell of the crystal: the macromolecule, the solvent around it and eventually other compounds. These contributions cannot typically be well described by an atomic model alone, i.e. using point scatterers. Indeed, entities such as disordered (bulk) solvent, semi-ordered solvent (e.g. lipid belts in membrane proteins, ligands, ion channels) and disordered polymer loops require other types of modeling than a collection of individual atoms. This results in the model structure factors containing multiple contributions. Most macromolecular applications assume two-component structure factors: one component arising from the atomic model and the second one describing the bulk solvent. A more accurate and detailed modeling of the disordered regions of the crystal will naturally require more than two components in the structure factors, which presents algorithmic and computational challenges. Here an efficient solution of this problem is proposed. All algorithms described in this work have been implemented in the computational crystallography toolbox (CCTBX) and are also available within Phenix software. These algorithms are rather general and do not use any assumptions about molecule type or size nor about those of its components.

Direct calculation of cryo-EM and crystallographic model maps for real-space refinement.

This work addresses the problem of the calculation of limited-resolution maps from an atomic model in cryo-electron microscopy and in X-ray and neutron crystallography, including cases where the resolution varies from one molecular region to another. Such maps are necessary in real-space refinement for comparison with the experimental maps. For an appropriate numeric comparison, the calculated maps should reproduce not only the structural features contained in the experimental maps but also the principal map distortions. These model maps can be obtained with no use of Fourier transforms but, similar to density distributions, as a sum of individual atomic contributions. Such contributions, referred to as atomic density images, are atomic densities morphed to reflect distortions of the experimental map, in particular the loss of resolution. They are described by functions composed of a central peak surrounded by Fourier ripples. For practical calculations, atomic images should be cut at some distance. It is shown that to reach a reasonable accuracy such a distance should be significantly larger than the distance customarily applied when calculating density distributions. This is a consequence of the slow rate with which the amplitude of the Fourier ripples decreases. Such a large distance means that at least a few ripples should be included in calculations in order to obtain a map that is sufficiently accurate. Oscillating functions describing these atomic contributions depend, for a given atomic type, on the resolution and on the atomic displacement parameter values. To express both the central peak and the Fourier ripples of the atomic images, these functions are represented by the sums of especially designed terms, each concentrated in a spherical shell and depending analytically on the atomic parameters. In this work, the strength of the dependence of the accuracy of resulting map on the accuracy of the atomic displacement parameters and on the truncation distance, i.e. the number of ripples included in atomic density images, is analyzed. This analysis is completed by practical aspects of the calculation of maps of inhomogeneous resolution. Tests show that the calculation of limited-resolution maps from an atomic model as a sum of atomic contributions requires a large truncation radius extending beyond the central peak of an atomic image and the first Fourier ripples. The article discusses the practical details of such calculations expressing atomic contributions as analytic functions of the atomic coordinates, the atomic displacement parameters and the local resolution.

Implementation of the riding hydrogen model in CCTBX to support the next generation of X-ray and neutron joint refinement in Phenix.

A fundamental prerequisite for implementing new procedures of atomic model refinement against neutron diffraction data is the efficient handling of hydrogen atoms. The riding hydrogen model, which constrains hydrogen atom parameters to those of the non-hydrogen atoms, is a plausible parameterization for refinements. This work describes the implementation of the riding hydrogen model in the Computational Crystallography Toolbox and in Phenix. Riding hydrogen atoms can be found in several different configurations that are characterized by specific geometries. For each configuration, the hydrogen atom parameterization and the expressions for the gradients of refinement target function with respect to non-hydrogen parameters are described.

Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria.

A new type of mask-selection criterion is suggested for mask-based phasing. In this phasing approach, a large number of connected molecular masks are randomly generated. Structure-factor phases corresponding to a trial mask are accepted as an admissible solution of the phase problem if the mask satisfies some specified selection rules that are key to success. The admissible phase sets are aligned and averaged to give a preliminary solution of the phase problem. The new selection rule is based on the likelihood of the generated mask. It is defined as the probability of reproducing the observed structure-factor magnitudes by placing atoms randomly into the mask. While the result of the direct comparison of mask structure-factor magnitudes with observed ones using a correlation coefficient is highly dominated by a few very strong low-resolution reflections, a new method gives higher weight to relatively weak high-resolution reflections that allows them to be phased accurately. This mask-based phasing procedure with likelihood-based selection has been applied to simulated single-particle diffraction data of the photosystem II monomer. The phase set obtained resulted in a 16 Šresolution Fourier synthesis (more than 4000 reflections) with 98% correlation with the exact phase set and 69% correlation for about 2000 reflections in the highest resolution shell (20-16 Å). This work also addresses another essential problem of phasing methods, namely adequate estimation of the resolution achieved. A model-trapping analysis of the phase sets obtained by the mask-based phasing procedure suggests that the widely used `50% shell correlation' criterion may be too optimistic in some cases.

Macromolecular structure determination using X-rays, neutrons and electrons: recent developments in Phenix.

Diffraction (X-ray, neutron and electron) and electron cryo-microscopy are powerful methods to determine three-dimensional macromolecular structures, which are required to understand biological processes and to develop new therapeutics against diseases. The overall structure-solution workflow is similar for these techniques, but nuances exist because the properties of the reduced experimental data are different. Software tools for structure determination should therefore be tailored for each method. Phenix is a comprehensive software package for macromolecular structure determination that handles data from any of these techniques. Tasks performed with Phenix include data-quality assessment, map improvement, model building, the validation/rebuilding/refinement cycle and deposition. Each tool caters to the type of experimental data. The design of Phenix emphasizes the automation of procedures, where possible, to minimize repetitive and time-consuming manual tasks, while default parameters are chosen to encourage best practice. A graphical user interface provides access to many command-line features of Phenix and streamlines the transition between programs, project tracking and re-running of previous tasks.

Mask-based approach to phasing of single-particle diffraction data.

A Monte Carlo-type approach for low- and medium-resolution phasing of single-particle diffraction data is suggested. Firstly, the single-particle phase problem is substituted with the phase problem for an imaginary crystal. A unit cell of this crystal contains a single isolated particle surrounded by a large volume of bulk solvent. The developed phasing procedure then generates a large number of connected and finite molecular masks, calculates their Fourier coefficients, selects the sets with magnitudes that are highly correlated with the experimental values and finally aligns the selected phase sets and calculates the averaged phase values. A test with the known structure of monomeric photosystem II resulted in phases that have 97% correlation with the exact phases in the full 25 Å resolution shell (1054 structure factors) and correlations of 99, 94, 81 and 79% for the resolution shells ∞-60, 60-40, 40-30 and 30-25 Å, respectively. The same procedure may be used for crystallographic ab initio phasing.