V. Yu. Lunin

R-Free Likelihood-Based Estimates of Errors for Phases Calculated from Atomic Models

Reasonable assumptions about the statistical properties of errors in an atomic model lead to the probability distributions for the values of structure-factor phases. These distributions contain some generally unknown parameters reflecting how large the model errors are. These parameters must be determined properly to give realistic estimates of phase errors. Maximum-likelihood-based estimates suggested by Lunin & Urzhumtsev [Acta Cryst. (1984), A40, 269–277] are good for models not subjected to refinement but underestimate the errors when being used for refined models. The R-free methodology of Brunger [Nature (London), (1992), 355, 472–474] applied to the likelihood-function calculation allows realistic phase-error estimates to be obtained for both unrefined and refined models. These estimates may be used as an additional indicator in the refinement process.

Direct low-resolution phasing from electron-density histograms in protein crystallography

An approach to direct phasing of low-resolution reflections is proposed. It is based on the generation of a large number of phase sets and selection of those variants whose electron-density-synthesis histograms are close to a prescribed standard. Classifying them into clusters and averaging them inside every cluster restricts their number to one to three usually, in which a phase set close to the standard is contained. The best variant can be recognized by the properties of its cluster. Test phasing of 29 low-resolution reflections has resulted in a correlation coefficient of 0.94 and a mean phase difference of 40° compared with the true phases.

Use of the information on electron density distribution in macromolecules

A new type of information on the distribution of electron density in crystals of biological macromolecules is proposed. This is a quasihistogram of the image of the function of electron density distribution at a finite resolution. It is shown how this information should be used to restore the values of low-angle structure factors whose amplitudes have not been measured during X-ray experiments.

Three-dimensional structure of Serratia marcescens nuclease at 1.7 A resolution and mechanism of its action.

The three‐dimensional crystal structure of Serratia marcescens (Sm) nuclease has been refined at 1.7 Å resolution to the R‐factor of 17.3% and R‐free of 22.2%. The final model consists of 3678 non‐hydrogen atoms and 443 water molecules. The analysis of the secondary and the tertiary structures of the Sm nuclease suggests a topology which reveals essential inner symmetry in all the three layers forming the monomer. We propose the plausible mechanism of its action based on a concerted participation of the catalytically important amino acid residues of the enzyme active site.

Phase Improvement in Protein Crystallography Using a Mixed Electron Density Model

The proposed technique for phase improvement is based on the refinement of a so-called mixed electron density model. This model consists of two parts. The first is a partial stereochemically correct atomic model of the protein molecule related to the interpreted part of electron density of the unit cell. The second is an artificial atomic model describing the uninterpreted part of the residual electron density of the unit cell. The conventional free-atom crystallographic refinement of such a mixed model results in phase improvement. No attention is paid to the structural sense of the refined atomic positions of the mixed electron density model. This method of phase improvement has been applied for eye lens protein y-crystallin Illb at 2-7 A resolution. A starting partial model of the molecule contained about 56% of the total number of residues. Phase refinement is done in two stages. Each stage leads to a significant increase in the quality of the electron density map, so that the partial atomic model of the protein molecule can be improved and expanded.

Structure of γ-crystallin IIIb from calf lens at 5 Å resolution

Crystallins are water-soluble proteins from eye lens of vertebrates. By electrophoretic mobility they are usually subdivided into three main groups 01, p and y. y-Crystallins have the lowest relative molecular mass of -20 000 and are separated into several fractions [ 1,2]. Proteins of these fractions are very similar in amino acid composition and sequence [3,4]. y-Crystallins are actively synthesized at early stages of the organism’s development being subjected to only small modifications during the whole life span. One of the remarkable differences of ycrystallins as compared with (Yand /3-crystallins is the large amount of SH groups which are converted into S-S bonds on ageing and in eye cataract. Recent X-ray studies of ycrystallin fraction II have allowed its structure to be obtained at 5.5 a [5] and 5 .O A [6] resolution. The structure of the homologous ycrystallin fraction IIIb also appears to be of interest in the sense of a possible change of the protein space structure. We had reported preliminary crystallographic data on ycrystallin IIIb [7] which were confirmed later in [5].

New possibilities of X-ray nanocrystallography of biological macromolecules based on X-ray free-electron lasers

X-ray serial nanocrystallography is a new technique for determining the three-dimensional structure of biological macromolecules from data on the diffraction of ultrashort pulses generated by X-ray free-electron lasers. The maximum achievable resolution for a set of experimental data as a function of the sample sizes and parameters of the equipment is estimated based on simulations of the diffraction process with allowance for changes in the electronic structure of the atoms of the sample under the influence of X-rays. Estimates show that nanocrystallography greatly enhances the possibilities of X-ray analysis, reducing the requirements for the minimum permitted size of the crystals and enabling to explore poorly crystallizable molecular objects, such as many membrane proteins and complexes of macromolecules.

Program Construction for Macromolecule Atomic Model Refinement Based on the Fast Fourier Transform and Fast Differentiation Algorithms

Structure refinement may be considered as a minimization of a function R(X) of a large number ofrefineable parameters. A new type of function incorporating phase probability distribution is proposed. The minimization of the function utilizing gradient methods requires the computation of gradient V R, as well as the product of the gradient and the matrix of second derivatives with some direction. The algorithm of Kim, Nesterov & Cherkassky [Dokl. Akad. Nauk SSSR (1984), 275, 1306-1309] adapted to macromolecular structure refinement takes about four times longer for the computation of these values compared to the computation of the value of the minimized function. The matrix of second derivatives is used without any approximation.

Reliability of maximum likelihood-based figures of merit

Various schemes for determining the maximum likelihood-based figures of merit for phases of structure factors have been considered. It is shown that the use of the likelihood function of all the available structure factors provides the adequate estimates of the accuracy of phases calculated for the atomic models with independent errors in the coordinates, but, at the same time, systematically overestimates the figures of merit for models preliminarily refined in the reciprocal space. It is shown that the use of the marginal likelihood function calculated from the control set of reflections allows the elimination of the systematic bias estimates. A method for reducing the statistical dispersion of the estimates based on a small number of control reflections is suggested.

Frequency-restrained structure-factor refinement. II, Comparison of methods

The frequency distribution of electron-density-function values encountered in a protein crystal has a characteristic shape and may be predicted for a protein with unknown spatial structure. It is shown that various methods of refinement of structure-factor phases (frequency-restrained refinement, histogram matching, density modification) may be regarded as various approaches to the same problem of obtaining the electron-density distribution which agrees with the X-ray experimental data and has a prescribed histogram. Test computations illustrate the relative efficiency of the methods analyzed.