Figure 1. Electron microscopic images and electron diffraction of 2D crystals. (a) Overview image of a vitrified 2D crystal. Scale bar, 2 m. (b) Power spectrum of a high-resolution image of a 2D crystal. The periodic arrangement of the protein in the 2D crystal leads to a periodic signal in the recorded image. Therefore, the |modulus|² of the Fourier transform, i.e. the power spectrum, shows discrete spots, in which the structural information of the protein is concentrated. Furthermore, noise (all information outside the spots) and the CTF (see text) is visible, (c) Electron diffraction of a highly ordered 2D crystal which is not affected by the CTF but does not carry phase information. The red circle indicates order 26,0 at 3.7 Å resolution.

Figure 2. Fourier peak-filtering and unbending of 2D crystals. The raw image (a) is Fourier transformed (1) and the crystal lattice is indexed in the power spectrum (b) of the raw image. Note that in this case, two crystalline layers of the flattened crystalline vesicle can be separated as they are rotated with respect to each other. For the Fourier peak-filtering, the spots containing all the crystal information are cut out, and the amplitudes outside the mask-area (containing the other crystal layer and noise) is set to 0 (2) as illustrated in (c). The inverse Fourier transform (step (3), as shown in (d)) reveals the packing of the crystal (inset). To unbend the 2D crystal, a reference (e) is generated (4) and a cross-correlation (5) with the raw image is calculated. The cross-correlation (f) reveals the positions of the unit cells. These can be compared to the index of the crystal (ideal crystal) (6) to generate distortion vectors (g). This information is then used to interpolate the raw image to unbend the crystal, but this step is only executed after the correction of the CTF (7). As a result, the spots of the power spectrum are better focused (h). In inset (h1), peak 5,3 (indicated with a red circle) is depicted before unbending and in (h2), after unbending. Amplitudes and phases of the spots are read out and combined with the data of other crystals (8) to yield a final projection map. Such a result can be seen in the 3.7 Å map of GlpF (i) revealing the typical tetrameric structures of an aquaglyceroporin. This map can also be used to generate a synthetic reference for the unbending procedure (9).

Figure 3. 3D reconstruction of membrane proteins by 2D crystallography. (a) To obtain a 3D reconstruction from 2D crystals, projections are recorded at different tilt angles (1). The images are Fourier-filtered and processed (2), and the Fourier transforms are combined in the 3D Fourier space according to the central section theorem (3). The inverse Fourier transform reveals the unit cell structure (4). The discrete orders in the Fourier transform from the crystal are aligned in continuous lattice lines since the sample is not periodic in the z direction. The lattice lines are regularly interpolated to sample the 3D Fourier space on a cubic raster. Back-transformation of the combined data finally leads to the representation of the 3D unit cell (4). (b) Azimuthal projection of the sampling in z* direction. The different tilt angles can be distinguished. In this case, a maximal nominal tilt angle of 60° was applied indicated with a black line revealing the missing cone. The lattice lines are visible; an example is given in (c). Amplitude and phase of lattice line 1,12 revealing a z resolution of (1/7 Å)^–1. The plotted curve indicates the interpolation of the lattice line. (d) Power spectra of an untilted (0°) and 60° tilted 2D crystal. The inset shows the Fourier-filtered projection map from an unbent image. Perpendicular to the tilt axis (line in the 60° panel), the resolution is reduced as a result of support nonflatness and charging.