A model-based parallel origin and orientation refinement algorithm for cryoTEM and its application to the study of virus structures

Abstract

We present a model-based parallel algorithm for origin and orientation refinement for 3D reconstruction in cryoTEM. The algorithm is based upon the Projection Theorem of the Fourier Transform. Rather than projecting the current 3D model and searching for the best match between an experimental view and the calculated projections, the algorithm computes the Discrete Fourier Transform (DFT) of each projection and searches for the central section ("cut") of the 3D DFT that best matches the DFT of the projection. Factors that affect the efficiency of a parallel program are first reviewed and then the performance and limitations of the proposed algorithm are discussed. The parallel program that implements this algorithm, called PO(2)R, has been used for the refinement of several virus structures, including those of the 500 Angstroms diameter dengue virus (to 9.5 Angstroms resolution), the 850 Angstroms mammalian reovirus (to better than 7A), and the 1800 Angstroms paramecium bursaria chlorella virus (to 15 Angstroms).

Fig. 1

Factors affecting the efficiency of a parallel algorithms and the means to increase the speedup of a parallel program.

Fig. 2

The input and the output of the origin and orientation refinement algorithm.

Fig. 3

The main computation steps for the sequential algorithm (left). The data distribution across nodes for the parallel version of the algorithm (right).

Fig. 4

The “distance” between the 2D DFT of the cross-sections several Δ_θ,ϕ apart from the exact orientation and the 2D DFT of the cross-section with the precise orientation for Δ_θ,ϕ equal to 1°, 0.1°, 0.01°, and 0.001°. In our experiments ω = 0° and 10° ≤ (θ, ϕ) ≤ 80°, the 3D map size is 511 × 511 × 511, and the Fourier space size is 768 × 768 × 768. The “distance” is plotted versus the equi-angular spacing for the two angles θ and ϕ (ω is fixed) in a horizontal plane. The figures show only the range for θ, (−6,…,0,…,+6)× the refinement step size, 1°, 0.1°, 0.01°, and 0.001°. The range for ϕ is identical and it is not shown for lack of space. As we move to increasingly smaller step size we show only the tip of the curve. For example, only the bottom most section of the surface (colored in gray) for the 1° curve is shown at the next smaller step size curve, 0.1°. Notice that the range of the values on the vertical axis is increasingly smaller and viewed at the same scale the surface for 0.001° step size would look perfectly flat. The “distance” converges as we decrease the refinement step Δ_θ,ϕ,ω from 1°, to 0.1°, to 0.01°, and finally to 0.001°.

Fig. 5

DENV Structure. (A) Micrograph of vitrified DENV sample shows particles with a spherical morphology and a smooth outer surface. DENV particles are quite fragile, as evidenced by a significant fraction of disrupted particles (arrows) in the sample. Images of such particles were eliminated from the image reconstruction process. (B) Shaded-surface view of the DENV reconstruction at 14 Å resolution. The outlines of three E protein dimers (arranged in herringbone pattern) are highlighted in color with the dimer at the icosahedral 2-fold axis highlighted in pink and the dimers at a quasi-2-fold axes highlighted in green and blue. The locations of the two carbohydrate moieties in each E protein are depicted with red (residue N153) and yellow (residue N67) circles, respectively. (C) Ribbon diagram of the atomic structure of the E protein dimer (Zhang et al., 2004) with domains I, II and III is colored red, yellow and blue, respectively. The fusion peptide at the tip of domain II is colored green. (D) Atomic modeling of the DENV structure. The a-carbon backbone model of the ecto-domains of the E protein dimer were fitted into the 14 Å DENV reconstruction. The color scheme is the same as that used in (C).

Fig. 6

Comparisons of the DENV₂₄ (A, C, E) and DENV_9.5 (B, D, F) reconstructions without inverse temperature factor. (A) Equatorial section from DENV₂₄ reconstruction. The darkest densities correspond to protein, lipid and nucleocapsid components. Icosahedral 2-, 3-, and 5-fold axes are labeled i2, i3 and i5, respectively. The mark at r1 identifies the center of the membrane bilayer at radius 185 Å. The center of mass in the outer protein shell, comprised of the E protein ecto-domains at radius 221 Å, is labeled r2. (B) Same as (A) for DENV_9.5. The scale bar is the same for all panels. (C) Radial projection of DENV₂₄ at a radius of 221 Å (r2 position in (A)). Three E protein dimers are highlighted in colors (same scheme used in Fig. 5B). (D) Same as (C) for DENV_9.5. The herringbone pattern of E protein dimers is evident at this resolution. (E) Same as (C) at a radius of 185 Å (r1 position in (A)). The transmembrane domains of the E and M proteins are not resolved. (F) Same as (E) for DENV_9.5. The eight dark, punctate features (in four groups of two) associated with each dimer are attributed to eight transmembrane helices, with two copies attributed to each E and M monomer.

Fig. 7

Fourier shell correlation (FSC) plot for MRV T3D virion data. The FSC curve was calculated using T3D maps separately computed from the ‘odd’ and ‘even’ images (e.g., Baker et al., 1999). The effective resolution of the current reconstruction was estimated to be about 7.0 Å (FSC = 0.535 at 7.0 Å). The sharp drop in the FSC curve is a consequence of only using transform data up to approximately 7.0 Å resolution for data refinement, but even and odd maps were computed to higher resolution limits to generate the FSC plot.

Fig. 8

Equatorial section from the cryoTEM T3D_7.0 map. The map was computed to a Fourier cutoff of 1/6.7 Å, with structure factor amplitudes gradually attenuated (Gaussian function) to background level between spatial frequencies of 1/7.0 and 1/6.7 Å. In addition, an inverse temperature factor of 1/400 Ų was imposed to enhance the high spatial frequency features in the map. Icosahedral 2-, 3-, and 5-fold axes are indicated along with the approximate positions of six of the T3D structural proteins. Scale bar = 200 Å.

Fig. 9

Comparison of 3D image reconstructions of PBCV-1 at low (A and B) and moderate (C) resolutions without inverse temperature factor. (A) Shaded-surface view of PBCV₂₆ reconstruction, viewed along a 3-fold axis of symmetry. (B) Magnified view of the area outlined in (A) showing the close-packed arrangement of doughnut-like capsomers. At 26 Å resolution, the trimeric nature of each capsomer is difficult to recognize. (C) Same area as in (B), but for a view of the PBCV₁₅ reconstruction. At 15 Å resolution, the trimeric character of each capsomer becomes obvious as all capsomers exhibit a clearly defined hexagonal base that is topped with three, tower-like protrusions.