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.

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.

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.

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.

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.

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.