Real-space Averaging Research Articles

CalAtom: A software for quantitatively analysing atomic columns in a transmission electron microscope image

We developed a software tool named CalAtom in MATLAB™ for quantitative analyses of atomic columns in (scanning) transmission electron microscopy, (S)TEM, images. This software provides three algorithms to determine the position of an atomic column with high precision: the moment method, the model-based method and the multiple-ellipse fitting method. Based on the measured positions, the software offers several options for further analyses, such as in-plane scale local environments of the atomic columns, local elementary composition and real-space averaging of image motifs. As an example, we use the CalAtom software to determine atomic positions and local elemental composition of the MoS2(1−x)Se2x alloy monolayers.

Preliminary X-ray data analysis of crystalline hibiscus chlorotic ringspot virus

Hibiscus chlorotic ringspot virus (HCRSV) is a positive-sense monopartite single-stranded RNA virus that belongs to the Carmovirus genus of the Tombusviridae family, which includes carnation mottle virus (CarMV). The HCRSV virion has a 30 nm diameter icosahedral capsid with T = 3 quasi-symmetry containing 180 copies of a 38 kDa coat protein (CP) and encapsidates a full-length 3.9 kb genomic RNA. Authentic virus was harvested from infected host kenaf leaves and was purified by saturated ammonium sulfate precipitation, sucrose density-gradient centrifugation and anion-exchange chromatography. Virus crystals were grown in multiple conditions; one of the crystals diffracted to 3.2 A resolution and allowed the collection of a partial data set. The crystal belonged to space group R32, with unit-cell parameters a = b = 336.4, c = 798.5 A. Packing considerations and rotation-function analysis determined that there were three particles per unit cell, all of which have the same orientation and fixed positions, and resulted in tenfold noncrystallography symmetry for real-space averaging. The crystals used for the structure determination of southern bean mosaic virus (SBMV) have nearly identical characteristics. Together, these findings will greatly aid the high-resolution structure determination of HCRSV.

Three-dimensional structure of the KdpFABC complex of Escherichia coli by electron tomography of two-dimensional crystals

The KdpFABC complex (Kdp) functions as a K+ pump in Escherichia coli and is a member of the family of P-type ATPases. Unlike other family members, Kdp has a unique oligomeric composition and is notable for segregating K+ transport and ATP hydrolysis onto separate subunits (KdpA and KdpB, respectively). We have produced two-dimensional crystals of the KdpFABC complex within reconstituted lipid bilayers and determined its three-dimensional structure from negatively stained samples using a combination of electron tomography and real-space averaging. The resulting map is at a resolution of 2.4 nm and reveals a dimer of Kdp molecules as the asymmetric unit; however, only the cytoplasmic domains are visible due to the lack of stain penetration within the lipid bilayer. The sizes of these cytoplasmic domains are consistent with Kdp and, using a pseudo-atomic model, we have described the subunit interactions that stabilize the Kdp dimer within the larger crystallographic array. These results illustrate the utility of electron tomography in structure determination of ordered assemblies, especially when disorder is severe enough to hamper conventional crystallographic analysis.

Structure determination of tubular crystals of membrane proteins. IV. Distortion correction and its combined application with real-space averaging and solvent flattening

A method for correction of three-dimensional distortions has been developed for helical assemblies and applied to tubular crystals of Ca 2 + -ATPase. This method approximates distorted helical particles with short straight segments of different orientation parameters, which are determined by fitting them to the reference data in reciprocal space. Thus, the method follows Beroukhim and Unwin [Ultramicroscopy 70 (1997) 57], but is more extended to achieve better alignment and to cope with images of poor S/N ratio. Substantial improvements were achieved by dividing the reference image into the segments of optimal lengths in exactly the same way as the test, and treating the distortions in the near and far sides of a helical particle separately. The improvement was further enhanced when combined with real-space averaging [Yonekura, Toyoshima, Ultramicroscopy 84 (2000) 15] and solvent flattening [Yonekura, Toyoshima, Ultramicroscopy 84 (2000) 29], and most pronounced when all these three were applied iteratively.

Averaging data derived from images of helical structures with different symmetries

There are many examples of macromolecules that form helical tubes or crystals, which are useful for structure determination by electron microscopy and image processing. Helical crystals can be thought of as two-dimensional crystals that have been rolled into a cylinder such that two lattice points are superimposed. In many real cases, helical crystals of a particular macromolecule derive from an identical two-dimensional lattice but have different lattice points superimposed, thus producing different helical symmetries which cannot be simply averaged in Fourier-space. When confronted with this situation, one can select images corresponding to one of the observed symmetries at the expense of reducing the number of images that can be used for data collection and averaging, or one can calculate separate density maps from each symmetry, then align and average them together in real-space. Here, we present a third alternative, which is based on averaging of the Fourier-Bessel coefficients,gn,l(r), and which allows the inclusion of data from all symmetry groups derived from a common two-dimensional lattice. The method is straightforward and simple in practice and is shown, through a specific example with real data, to give results comparable to real-space averaging.

Structure determination of Nudaurelia capensis omega virus.

The structure of Nudaurelia capensis omega virus (NomegaV), a single-stranded RNA virus, was determined to 2.8 A resolution. Triclinic crystals (a = 413.6, b = 410.2, c = 419.7 A, alpha = 59.13, beta = 58.9, gamma = 64.0 degrees ) diffracted X-rays beyond 2.7 A resolution. The unit cell contained one icosahedral virus particle, providing 60-fold non-crystallographic symmetry (n.c.s.) and structural redundancy. The particle orientation in the unit cell was determined by self-rotation function analyses. Initial phases to 18 A resolution were derived from a hollow spherical model of 192 A outer radius and 139 A inner radius, filled with uniform electron density. Radii of the model were determined by maximizing the correlation of the model-based calculated data with the low-resolution X-ray diffraction and solution-scattering data. Phases were refined by 60-fold non-crystallographic electron-density averaging and extended in small steps to a resolution of 5 A. The phases obtained represented a mixture of four different phase sets, each consistent with the icosahedral symmetry constraints. The resulting electron density was not interpretable. A difference Fourier map computed with the native and an isomorphous heavy-atom derivative data sets and phases refined by real-space averaging was interpretable only if data within the 10 A resolution shell were used. Maps calculated with data significantly higher than 10 A resolution failed to display a constellation of heavy-atom sites consistent with the T = 4 icosahedral symmetry. Attempts to extend the phases beyond 10 A resolution, starting with either phases based on a model or single isomorphous replacement, were unsuccessful. Successful phase extension was achieved by computing the phases for the higher resolution reflections from a partial atomic model (poly gly) built into the averaged 10 A electron-density map. Phases from this model served as the starting point for n.c.s. phase refinement and extension to slightly higher resolution. The atomic model was improved at each extension interval and these phases were used for the subsequent phase calculation and extension. The entire polypeptide backbone corresponding to the NomegaV structure was built into the map at 4 A. The same procedure for phase refinement was used to extend the phases to 2.8 A in small increments of resolution. The overall molecular averaging R factor and correlation coefficient at 2.8 A resolution were 18.4% and 0.87, respectively.

Human class II MHC molecule HLA-DR1: X-ray structure determined from three crystal forms.

The three-dimensional structure of the extracellular region of a 60 kDa class II major histocompatibility glycoprotein, HLA-DR1, was determined to 3.3 A by X-ray crystallography using three crystal forms, each containing two molecules per asymmetric unit. Phases were initially determined to 4.2 A using two crystal forms both containing DR1 from human lymphocytes complexed with a mixture of endogenous peptides. One of these crystal forms also contained a 28 kDa superantigen, Staphylococcus aureus enterotoxin B (SEB), bound to each DR1 molecule. Single-isomorphous replacement phasing followed by iterative two- and fourfold non-crystallographic real-space averaging between the two crystal forms resulted in 4.2 A resolution electron-density maps from which the paths of the polypeptides could be traced. Cryocrystallography and synchrotron radiation were then used to extend the resolution to 3.3 A for the two lymphocyte-derived crystal forms and for a third crystal form grown from DR1 produced in insect cells and complexed in vitro with a specific antigenic peptide. Iterative sixfold non-crystallographic real-space averaging resulted in an electron- density map into which 340 of 371 residues could be fit unambiguously. Crystal contacts and the existence of a parallel dimer of the DR1 alphabeta heterodimer in the three crystal forms are discussed.

Reciprocal-space molecular-replacement averaging.

The molecular-replacement equations, in which electron-density averaging and skew averaging have been unified, were used in reciprocal space to refine and extend the resolution of phased reflections. A procedure has been developed for the treatment of molecular envelopes of general shape. The equations were successfully applied to the reflection data of bacteriophage phiX174 (60-fold redundancy). Truncation of the G diffraction function beyond the first few nodes did not have a significant effect on the quality of the molecular-replacement equations. Reciprocal-space molecular-replacement averaging should prove to be a useful alternative to real-space averaging. Strategies are discussed that are possible only in reciprocal space.

Quantitative imaging and diffraction of zeolites using a slow-scan CCD camera

We demonstrate ways in which structural information on zeolites can be obtained from high-resolution images and electron diffraction patterns. We employ a commercial slow-scan CCD camera to record the data and demonstrate the advantage of on-line data processing and analysis. Quantitative agreement is obtained between experimental electron diffraction intensities and theoretical calculations. Real-space averaging is a versatile technique which can be employed to reduce the effects of both noise and radiation damage. Under favorable circumstances the weak-phase object approximation can be used to simplify image interpretation and quantification.

Quantitative imaging and diffraction of zeolites using a slow-scan CCD camera

Abstract We demonstrate ways in which structural information on zeolites can be obtained from high-resolution images and electron diffraction patterns. We employ a commercial slow-scan CCD camera to record the data and demonstrate the advantage of on-line data processing and analysis. Quantitative agreement is obtained between experimental electron diffraction intensities and theoretical calculations. Real-space averaging is a versatile technique which can be employed to reduce the effects of both noise and radiation damage. Under favorable circumstances the weak-phase object approximation can be used to simplify image interpretation and quantification.

Molecular replacement real-space averaging

Structure determination of macromolecules often depends on phase improvement and phase extension by use of real-space averaging of electron density related by noncrystallographic symmetry. Although techniques for such procedures have been described previously [Bricogne (1976). Acta Cryst. A32, 832–847; Johnson (1978). Acta Cryst. B34, 576–577], modern computer architecture and experience with these methods have suggested changes and improvements. Two unit cells are considered: (1) the p-cell corresponding to the actual crystal structure(s) being determined (there would be more than one of these if the molecule crystallizes in more than one crystal form) and (2) the h-cell corresponding to the molecule in a standard orientation with respect to which the molecular symmetry axes are defined. Averaging can proceed entirely within the p-cell, referring to the h-cell only in as far as knowledge of the molecular symmetry is required. It is also possible to place the averaged molecule back into the h-cell, where it can be used to redefine the molecular envelope or for displaying a suitably chosen asymmetric unit of the molecule. Techniques are discussed for automatically selecting a molecular envelope which is consistent with packing considerations within the p-cell and which retains the symmetry of the molecular point group. The electron density map to be averaged is divided into bricks for storage in virtual memory. Roughly as many bricks as there are noncrystallographic asymmetric units per crystallographic asymmetric unit need to be retained in memory at one time. This procedure minimizes paging problems and avoids double sorting. Use of eight-point interpolation permits storing the map at grid points separated by no more than 1/2.5 of the resolution limit to obtain rapid convergence.

Structure determination of monoclinic canine parvovirus.

The three-dimensional structure of the single-stranded DNA canine parvovirus has been determined to 3.25 A resolution. Monoclinic crystals belonging to space group P2(1) (a = 263.1, b = 348.9, c = 267.2 A, beta = 90.82 degrees) were selected for data collection using primarily the Cornell High Energy Synchrotron Source and oscillation photography. There was one icosahedral particle per crystallographic asymmetric unit, giving 60-fold redundancy. The particle orientations in the unit cell were determined with a rotation function. The rough positions of the particles in the unit cell were estimated by considering the packing of spheres into the P2(1) crystal cell. More accurate particle centers were determined from Harker peaks in a Patterson function. Hollow-shell models were used to compute phases to 20 A resolution. The radii of the models were based on packing considerations, the fit of spherical shells to the low-resolution X-ray data and low-angle solution scattering data. The phases were extended to 9 A resolution using molecular replacement real-space averaging. These were then used to determine the heavy-atom position of a K2PtBr6 derivative, for which only 5% of the theoretically observable reflections had been recorded. The center of gravity of the 60 independent heavy-atom sites gave an improved particle center position. Single isomorphous replacement phases to 8 A resolution were then calculated with the platinum derivative. These were used to initiate phase improvement and extension to 3.25 A resolution using density averaging and Fourier back-transformation in steps of one reciprocal lattice point at a time. The resulting electron density map was readily interpretable and an atomic model was built into the electron density map on a PS390 graphics system using the FRODO program. The R factor prior to structure refinement for data between 5.0 and 3.25 A was 36%.

Refined structure of the human histocompatibility antigen HLA-A2 at 2.6 Å resolution

The three-dimensional structure of the human histocompatibility antigen HLA-A2 was determined at 3.5 Å resolution by a combination of isomorphous replacement and iterative real-space averaging of two crystal forms. The monoclinic crystal form has now been refined by least-squares methods to an R-factor of 0.169 for data from 6 to 2.6 Å resolution. A superposition of the structurally similar domains found in the heterodimer, α 1onto α 2 and α 3onto β 2m, as well as the latter pair onto the ancestrally related immunoglobulin constant domain, reveals that differences are mainly in the turn regions. Structural features of the α 1 and α 2 domains, such as conserved salt-bridges that contribute to stability, specific loops that form contacts with other domains, and the antigen-binding groove formed from two adjacent helical regions on top of an eight-stranded β-sheet, are analyzed. The interfaces between the domains, especially those between β 2m and the HLA heavy chain presumably involved in β 2m exchange and heterodimer assembly, are described in detail. A detailed examination of the binding groove confirms that the solvent-accessible amino acid side-chains that are most polymorphic in mouse and human alleles fill up the central and widest portion of the binding groove, while conserved side-chains are clustered at the narrower ends of the groove. Six pockets or sub-sites in the antigen-binding groove, of diverse shape and composition, appear suited for binding side-chains from antigenic peptides. Three pockets contain predominantly non-polar atoms; but others, especially those at the extreme ends of the groove, have clusters of polar atoms in close proximity to the “extra” electron density in the binding site. A possible role for β 2m in stabilizing permissible peptide complexes during folding and assembly is presented.

Two structural states of the Z-band in cardiac and skeletal muscle

We have recently established the existence of two structural states of the Z band lattice in cross section in cardiac as well as in skeletal muscle. The two structural states are related to the contractile state of the muscle. In skeletal muscle at rest, the Z band is in the small square (ss) lattice form, but tetanized muscle exhibits the basket weave (bw) form. In contrast, unstimu- lated cardiac muscle exhibits the bw form, but cardiac muscles exposed to EGTA show the ss form.We have used two-dimensional computer enhancement techniques on digitized electron micrographs to compare each lattice form as it appears in both cardiac and skeletal muscle. Both real space averaging and fourier filtering methods were used. Enhanced images were displayed as grey-scale projections, as contour maps, and in false color.There is only a slight difference between the lattices produced by the two different enhancement techniques. Thus the information presented in these images is not likely to be an artifact of the enhancement algorithm.

Reconstruction of imperfectly ordered zinc-induced tubulin sheets using cross-correlation and real space averaging

Large negatively-stained two-dimensional tubulin arrays induced by incubation with zinc were examined using electron microscopy. When Fourier methods are applied for the reconstruction, the resolution depends critically on the order of these arrays. The real space cross-correlation method as described in this paper, however, can be satisfactorily applied to disordered arrays. The method involves selection of a small area which is used as a reference to locate correlation peaks in other regions. In this way the locations of various unit cells are determined, often at points which deviate from the exact lattice. The unit cells at their determined locations are averaged, and the average is used as an improved reference in another search for correlation peaks. The process can be iterated to refine the reference and improve the correlation mapping. If necessary, poorly correlating unit cells can be ignored in the averaging, and rotational as well as translational corrections can be considered; but neither of these steps were found to be necessary in the studies on the zinc-induced tubulin sheets reported here. When both correlation and Fourier methods were compared for the same areas of selected sheets, the two methods gave comparable results for relatively well ordered regions, but the correlation images were superior for the more poorly ordered regions. In addition, even for the relatively well ordered regions the correlation images revealed some details that were lacking in the simple Fourier reconstructions but did appear in more extensive studies involving three-dimensional reconstruction. A direct comparison of the efficacy of real space and Fourier space reconstruction methods is presented using disordered model data.

Methods and programs for direct-space exploitation of geometric redundancies

Phases can be determined for geometrically redundant amplitudes by iteration of the following procedure: compute an electron density map from the currently available phases; average the electron densities of all the crystallographically independent molecules; rebuild the crystal(s) from this averaged subunit, setting the density outside the molecular boundaries to its average value; obtain phase information from the resulting structure, and combine it with that given by isomorphous replacement to produce the phases to be used in the next iteration. This algorithm converges very rapidly, and has proved to be a powerful tool in the solution of two large unknown protein structures. This paper describes the computational techniques developed to implement it, which include a swift and general method for real-space averaging of electron density maps.