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.

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.

Analytic modeling of inhomogeneous-resolution maps in cryo-electron microscopy and crystallography.

Refinement of macromolecular atomic models versus experimental maps in crystallography and cryo-electron microscopy is a critical step in structure solution. For an appropriate comparison, model maps should mimic the imperfections in the experimental maps, mainly atomic disorder and limited resolution, which are often inhomogeneous over the molecular region. In the suggested method, these model maps are calculated as the sum of atomic contributions expressed through a specifically designed function describing a solitary spherical wave. Thanks to this function, atomic contributions are analytically expressed through their atomic displacement parameter and local resolution, a value now associated with each atom. Such a full analytic dependence of inhomogeneous-resolution map values on model parameters permits the refinement of all of these parameters together.

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.

On the suitability of using the diagonal Gaussian approximation in the calculation of the magnitude-based likelihood function.

Statistical likelihood maximization is currently one of the main tools in computational procedures in biological crystallography. In these procedures, the likelihood function is calculated, as a rule, within the framework of a diagonal Gaussian approximation (DGA) of the joint probability distribution of the real and imaginary parts of a set of structure factors. This approximation assumes pairwise uncorrelated values of various structure-factor components. In this paper, exact formulas are derived for pairwise correlations of structure factors, and conditions under which these correlations can be considered to be negligible are discussed. It is shown that in the case where the probability distribution of the atomic coordinates is related to the region of the molecule or its domains, the correlation of the structure factors of reflections s and w is determined mostly by the magnitudes of the Fourier transform of the probability distribution calculated at the points 2s, 2w, s - w and s + w. However, in the case where the probability distribution describes small corrections to the coordinates of the existing preliminary atomic model, the correlation is determined by the values of the structure factors of the preliminary model that correspond to the 2s, 2w, s - w and s + w reflections rather than by the Fourier transform of the probability distribution. Test cases demonstrate that the practice of using the DGA for calculation of the likelihood when based on sets containing neighbouring reflections may be unjustified in some crystallographic applications, especially in single-particle studies.

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.

Signaling ammonium across membranes through an ammonium sensor histidine kinase.

Sensing and uptake of external ammonium is essential for anaerobic ammonium-oxidizing (anammox) bacteria, and is typically the domain of the ubiquitous Amt/Rh ammonium transporters. Here, we report on the structure and function of an ammonium sensor/transducer from the anammox bacterium "Candidatus Kuenenia stuttgartiensis" that combines a membrane-integral ammonium transporter domain with a fused histidine kinase. It contains a high-affinity ammonium binding site not present in assimilatory Amt proteins. The levels of phosphorylated histidine in the kinase are coupled to the presence of ammonium, as conformational changes during signal recognition by the Amt module are transduced internally to modulate the kinase activity. The structural analysis of this ammonium sensor by X-ray crystallography and small-angle X-ray-scattering reveals a flexible, bipartite system that recruits a large uptake transporter as a sensory module and modulates its functionality to achieve a mechanistic coupling to a kinase domain in order to trigger downstream signaling events.

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.

Efficient calculation of diffracted intensities in the case of nonstationary scattering by biological macromolecules under XFEL pulses.

The calculation of diffracted intensities from an atomic model is a routine step in the course of structure solution, and its efficiency may be crucial for the feasibility of the study. An intense X-ray free-electron laser (XFEL) pulse can change the electron configurations of atoms during its action. This results in time-dependence of the diffracted intensities and complicates their calculation. An algorithm is suggested that enables this calculation with a computational cost comparable to that for the time-independent case. The intensity is calculated as a sum of the `effective' intensity and a finite series of `correcting' intensities. These intensities are calculated in the conventional way but with modified atomic scattering factors that are specially derived for a particular XFEL experiment. The total number of members of the series does not exceed the number of chemically different elements present in the object under study. This number is small for biological molecules; in addition, the correcting terms are negligible within the parameter range and accuracy acceptable in biological crystallography. The time-dependent atomic scattering factors were estimated for different pulse fluence levels by solving the system of rate equations. The simulation showed that the changes in a diffraction pattern caused by the time-dependence of scattering factors are negligible if the pulse fluence does not exceed the limit that is currently achieved in experiments with biological macromolecular crystals (10(4) photons Å(-2) per pulse) but become significant with an increase in the fluence to 10(6) or 10(8) photons Å(-2) per pulse.

Metrics for comparison of crystallographic maps.

Numerical comparison of crystallographic contour maps is used extensively in structure solution and model refinement, analysis and validation. However, traditional metrics such as the map correlation coefficient (map CC, real-space CC or RSCC) sometimes contradict the results of visual assessment of the corresponding maps. This article explains such apparent contradictions and suggests new metrics and tools to compare crystallographic contour maps. The key to the new methods is rank scaling of the Fourier syntheses. The new metrics are complementary to the usual map CC and can be more helpful in map comparison, in particular when only some of their aspects, such as regions of high density, are of interest.

Detection of alternative conformations by unrestrained refinement.

Unrestrained refinement is stable for the vast majority of atoms when working at atomic resolution. Nevertheless, geometrical restraints should be retained in refinement for residues that are present in several (alternative) conformations in the crystal used for the X-ray experiment; otherwise, such residues deteriorate significantly. The authors believe that a large distortion of a residue in unrestrained refinement may hint at the presence of alternative conformations of this residue. To obtain these hints in a routine way, two methods of analyzing the shifts of atomic centres resulting from several cycles of unrestrained refinement are described. A simple diagram plotting the values of the atomic shifts against the residue number may give an idea of the crystallographic order of different parts of the structure at a qualitative level. To put the analysis on a more quantitative basis, several decision-making procedures were developed and tested which compose a list of residues that are likely to be present in alternative conformations or to be disordered and so should be checked thoroughly using Fourier syntheses and included in the model with alternative conformations when necessary. The parameters and performance of the suggested procedures were estimated by the use of 203 PDB structures refined at resolutions better than 1.2 Å. Decision-making procedures based on analysis of atomic shifts were found to be more reliable than similar procedures based on atomic displacement parameters or density values calculated at atomic centres.

Low-resolution structure determination of Na(+)-translocating NADH:ubiquinone oxidoreductase from Vibrio cholerae by ab initio phasing and electron microscopy.

A low-resolution structure of the Na(+)-translocating NADH:ubiquinone oxidoreductase from the human pathogen Vibrio cholerae was determined by ab initio phasing and independently confirmed by electron microscopy. This multi-subunit membrane-protein complex (molecular weight 210 kDa) generates an Na(+) gradient that is essential for substrate uptake, motility, pathogenicity and efflux of antibiotics. The obtained 16 Å resolution electron density-map revealed an asymmetric particle with a central region of low electron density and a putative detergent region, and allowed the identification of the transmembrane regions of the complex.

A noncrystallographic screw axis parallel to a twin axis can corrupt intensity statistics.

A number of methods to detect twinning are based upon the assumption that the statistical properties of diffracted intensities are different for twinned and untwinned specimens. This may not be true for a large portion of the reflections in a twinned specimen if a noncrystallographic screw axis parallel to the twinning axis is present. In this case, up to half of all reflections can obey Wilson statistics, which are typical of untwinned crystals. The distribution corresponding to a whole set of observed intensities is biased towards the Wilson distribution in this case.

X-ray-induced deterioration of disulfide bridges at atomic resolution.

Overall and site-specific X-ray-induced damage to porcine pancreatic elastase was studied at atomic resolution at temperatures of 100 and 15 K. The experiments confirmed that irradiation causes small movements of protein domains and bound water molecules in protein crystals. These structural changes occur not only at 100 K but also at temperatures as low as 15 K. An investigation of the deterioration of disulfide bridges demonstrated the following. (i) A decrease in the occupancy of S(γ) atoms and the appearance of new cysteine rotamers occur simultaneously. (ii) The occupancy decrease is observed for all S(γ) atoms, while new rotamers arise for some of the cysteine residues; the appearance of new conformations correlates with the accessibility to solvent. (iii) The sum of the occupancies of the initial and new conformations of a cysteine residue is approximately equal to the occupancy of the second cysteine residue in the bridge. (iv) The most pronounced changes occur at doses below 1.4 × 10(7) Gy, with only small changes occurring at higher doses. Comparison of the radiation-induced changes in an elastase crystal at 100 and 15 K suggested that the dose needed to induce a similar level of deterioration of the disulfide bonds and atomic displacements at 15 K to those seen at 100 K is more than two times higher.

Cluster analysis for phasing with molecular replacement: a feasibility study.

Molecular replacement can fail to find a solution, namely a unique orientation and position of a search model, even when many search models are tested under various conditions. Simultaneous use of the results of these searches may help in the solution of such difficult structures. A closeness between the peaks of several calculated rotation functions may identify the model orientation. The largest and most compact cluster of such peaks usually corresponds to models which are oriented similarly to the molecule under study. A search for the optimal translation may be more problematic and both individual translation functions and straightforward cluster analysis in the space of geometric parameters such as rotation angles and translation vectors may give no result. An improvement may be obtained by performing cluster analysis of the peaks of several translation functions in phase-set space. In this case, the Fourier maps computed using the observed structure-factor magnitudes and the phases calculated from differently positioned models are compared. Again, as a rule, the largest and the most compact cluster corresponds to the correct solution. The result of the updated procedure is no longer a single search model but an averaged Fourier map.

X-ray-radiation-induced cooperative atomic movements in protein.

X-rays interact with biological matter and cause damage. Proteins and other macromolecules are damaged primarily by ionizing X-ray photons and secondarily by reactive radiolytic chemical species. In particular, protein molecules are damaged during X-ray diffraction experiments with protein crystals, which is, in many cases, a serious hindrance to structure solution. The local X-ray-induced structural changes of the protein molecule have been studied using a number of model systems. However, it is still not well understood whether these local chemical changes lead to global structural changes in protein and what the mechanism is. We present experimental evidence at atomic resolution indicating the movement of large parts of the protein globule together with bound water molecules in the early stages of radiation damage to the protein crystal. The data were obtained from a crystal cryocooled to approximately 100 K and diffracting to 1 A. The movement of the protein structural elements occurs simultaneously with the decarboxylation of several glutamate and aspartate residues that mediate contacts between moving protein structural elements and with the rearrangement of the water network. The analysis of the anisotropy of atomic displacement parameters reveals that the observed atomic movements occur at different rates in different unit cells of the crystal. Thus, the examination of the cooperative atomic movement enables us to better understand how radiation-induced local chemical and structural changes of the protein molecule eventually lead to disorder in protein crystals.

Estimates of the twinning fraction for macromolecular crystals using statistical models accounting for experimental errors.

An advanced statistical model is suggested that is designed to estimate the twinning fraction in merohedrally (or pseudo-merohedrally) twinned crystals. The model takes experimental errors of the measured intensities into account and is adapted to the accuracy of a particular X-ray experiment through the standard deviations of the reflection intensities. The theoretical probability distributions for the improved model are calculated using a Monte Carlo-type simulation procedure. The use of different statistical criteria (including likelihood) to estimate the optimal twinning-fraction value is discussed. The improved model enables better agreement of theoretical and observed cumulative distribution functions to be obtained and produces twinning-fraction estimates that are closer to the refined values in comparison to the conventional model, which disregards experimental errors. The results of the two approaches converge when applied to selected subsets of measured intensities of high accuracy.

On macromolecular refinement at subatomic resolution with interatomic scatterers.

A study of the accurate electron-density distribution in molecular crystals at subatomic resolution (better than approximately 1.0 A) requires more detailed models than those based on independent spherical atoms. A tool that is conventionally used in small-molecule crystallography is the multipolar model. Even at upper resolution limits of 0.8-1.0 A, the number of experimental data is insufficient for full multipolar model refinement. As an alternative, a simpler model composed of conventional independent spherical atoms augmented by additional scatterers to model bonding effects has been proposed. Refinement of these mixed models for several benchmark data sets gave results that were comparable in quality with the results of multipolar refinement and superior to those for conventional models. Applications to several data sets of both small molecules and macromolecules are shown. These refinements were performed using the general-purpose macromolecular refinement module phenix.refine of the PHENIX package.

Ultrahigh-resolution study of protein atomic displacement parameters at cryotemperatures obtained with a helium cryostat.

Two X-ray data sets for a complex of human aldose reductase (h-AR) with the inhibitor IDD 594 and the cofactor NADP(+) were collected from two different parts of the same crystal to a resolution of 0.81 A at 15 and 60 K using cold helium gas as cryogen. The contribution of temperature to the atomic B values was estimated by comparison of the independently refined models. It was found that although being slightly different for different kinds of atoms, the differences (deltaB) in the isotropic equivalents B of atomic displacement parameters (ADPs) were approximately constant (about 1.7 A(2)) for well ordered atoms as the temperature was increased from 15 to 60 K. The mean value of this difference varied according to the number of non-H atoms covalently bound to the parent atom. Atoms having a B value of higher than 8 A(2) at 15 K showed much larger deviations of deltaB from the average value, which might reflect partial occupancy of atomic sites. An analysis of the anisotropy of ADPs for individual atoms revealed an increase in the isotropy of ADPs with the increase of the temperature from 15 to 60 K. In a separate experiment, a 0.93 A resolution data set was collected from a different crystal of the same complex at 100 K using cold nitrogen as a cryogen. The effects of various errors on the atomic B values were estimated by comparison of the refined models and the temperature-dependent component was inferred. It was found that both decreasing the data redundancy and increasing the resolution cutoff led to an approximately constant increase in atomic B values for well ordered atoms.

Low-resolution ab initio phasing of Sarcocystis muris lectin SML-2.

Structural analysis of the lectin SML-2 faced difficulties when applying standard crystallographic phasing methods. The connectivity-based ab initio phasing method allowed the computation of a 16 A resolution Fourier synthesis and the derivation of primary structural information. It was found that SML-2 crystals have three dimers in the asymmetric part of the unit cell linked by a noncrystallographic symmetry close to translation by (0, 0, 1/3). A clear identification of the noncrystallographic twofold axis explains the space-group transformation from the primitive P2(1)2(1)2(1) to the C-centred C222(1) observed during annealing procedures within an N(2) cryostream for cocrystals of SML-2 and galactose. Related packing considerations predict a possible arrangement of SML-2 molecules in a tetragonal unit cell. Multiple noncrystallographic symmetries and crystal forms provide a basis for further image improvements.