## Data Processing

4.1. Selecting Micrographs — Optical Diffraction

Although data collection by low dose electron microscopy is a critical element in determining the structure of a membrane protein, data processing is equally important. The first step is to identify which of the many micrographs recorded are suitable for further processing; generally, only a minority of micrographs contain high reso

Figure 6. A comparison of the structure of cytochrome c reductase (cytochrome bci complex) determined by electron crystallography and by X-ray crystallography. (a) A drawing interpreting the positions of 5 subunits in the dimeric complex with respect to the lipid bilayer in the center. Core subunits I and II face the matrix space. (b) Balsa wood models of the low resolution structures determined by electron microscopy—crystallography of: (i) a subcomplex lacking core subunits I and II on the left, and (ii) the intact complex. (iii) A ribbon diagram based upon the atomic coordinants of all subunits determined by X-ray crystallography.

Figure 6. A comparison of the structure of cytochrome c reductase (cytochrome bci complex) determined by electron crystallography and by X-ray crystallography. (a) A drawing interpreting the positions of 5 subunits in the dimeric complex with respect to the lipid bilayer in the center. Core subunits I and II face the matrix space. (b) Balsa wood models of the low resolution structures determined by electron microscopy—crystallography of: (i) a subcomplex lacking core subunits I and II on the left, and (ii) the intact complex. (iii) A ribbon diagram based upon the atomic coordinants of all subunits determined by X-ray crystallography.

lution information. The simplest method to screen micrographs for quality is optical diffraction, since the diffraction pattern is the Fourier transform of the object and is a diagnostic of several important image characteristics. Formation of an optical diffraction pattern is readily accomplished with an optical diffractometer consisting of a laser (generally a 1-5 mW He/Ne laser), beam expansion-spatial filter, and a single diffraction lens to collect the diverging beam and focus it to a point at some distance (24,65). When an electron micrograph is placed in the optical path just after the diffraction lens, the focused diffraction pattern of the illuminated portion of the micrograph can be observed at the focus of the lens. The undiffracted beam appears as a bright spot in the center of the diffraction pattern and is surrounded by the diffraction pattern of the object image. In the case of crystalline objects, the diffraction pattern consists of peaks of light at the points of a 2D lattice whose spacings are the reciprocal of the crystal lattice; thus, the lattice in the diffraction image is called the "reciprocal lattice". The structural information common to all unit cells within the illuminated area of the micrograph is contained at the points of the diffraction pattern's reciprocal lattice, while nonperi-odic noise is distributed throughout the diffraction pattern. Although one can obtain equivalent information by digitizing the electron micrograph and calculating its diffraction pattern, an optical diffractome-ter accomplishes this instantaneously, allowing one to move the micrograph around in the beam to select the best area quickly. One generally looks for two criteria revealed by the optical diffraction pattern:

1. Does the image contain high resolution information about the crystal structure? The diffraction pattern is the Fourier transform of the object illuminated, representing its structure in frequency space, with points furthest from the center representing higher frequency components contributing higher resolution information. One therefore looks for micrographs whose diffraction patterns display diffraction spots on the reciprocal lattice that extend relatively far from the origin of the diffraction pattern. Once the diffraction constant of a particular optical diffractometer is calibrated, usually with a diffraction object or grating of known spacing, the resolution of the information from each micrograph can be calculated based upon the distance from the origin of the furthest diffraction spot.

2. Is the micrograph properly focused with astigmatism corrected? Characteristics of the transfer of information from the object to the image are described by the CTF as described below. A plot of the CTF in diffraction space shows that it periodically passes through zero at points determined by the wavelength of the electron wave, the spherical aberration of the objective lens, and by the focus of the objective lens. The zeroes appear as concentric circles in the diffraction pattern (Figure 3c), and the radii of the circles can be used to calculate accurately the focus of the objective lens. If the objective lens has residual astigmatism, the focus is different in orthogonal directions, and the zeroes produce concentric ellipses rather than circles.

### 4.2. Digitizing

In order to process electron micrographs by computer, they must first be converted to digital form. Although it is now possible to purchase sensitive high resolution digital cameras for transmission electron microscopes, film is still the best media on which to record low dose high resolution images. The most accurate scanners have been mechanical, based upon wrapping the micrograph around a rotating drum or on precise movement in two dimensions, and these are generally quite expensive. More recently, high resolution digital cameras based upon charge-coupled device (CCD) technology have become available, and these represent a suitable lower cost alternative to mechanical scanners for many applications. Whatever the device used, there are several factors to consider in digitizing an image, and these have been covered in earlier publications (19,26). The first is the resolution one requires or expects in the digitized image. In digital sampling of a continuous function (analog signal), one must sample the function at an interval that is one half the interval or resolution one wants to obtain in the digitized image; this is called the Nyquist sampling rate. Thus, if one requires 10 A information in a digital image, it must be sampled (digitized) at 5 A or smaller intervals. In order to obtain very high resolution information from a low dose image, one must digitize the image of a large 2D crystal, at very small intervals, producing a very large data file. Electron micrographs of 2D crys tals are typically recorded at about 40 000x, and 1 A spacing in the specimen corresponds to 4 pm on the film. Thus, in order to record 10 A information in the digitized electron micrograph, one would have to sample it at 20-pm (5 A) intervals, and for 4 A resolution, the sampling would have to be at 8-pm intervals. In practice, one actually samples more finely than strictly required by the Nyquist limit, since there is some fall off in the transfer of high resolution information that depends on the sampling aperture size.

### 4.3. Fourier Filtering

One of the principal goals of image processing applied to electron micrographs of 2D crystals is to extract an image with high S/N from a micrograph with a very low S/N; this is accomplished by signal averaging. The Fourier transform of a perfect 2D crystal of infinite extent would be nonzero only at the points of a 2D reciprocal lattice, but as seen in Figure 3, the Fourier transform of an actual image of a 2D protein crystal contains nonperiodic noise, and the peaks at the points of the reciprocal lattice are spread out somewhat as the crystal is finite and not perfectly ordered. If the image is reconstructed using only the information lying at the reciprocal lattice points of the Fourier transform, the result is the average of all of the unit cells within the digitized image, and the S/N is increased by the square root of the number of unit cells averaged. This is shown in Figure 3b compared with the original image in Figure 3 a.

### 4.4. Correlation Alignment

The resolution of the averaged image depends upon the inherent resolution of the original electron micrograph (defined by the CTF) and upon the order of the crystal being averaged. As techniques of electron microscopy and computational averaging improved, leading to improvements in image resolution, it became apparent that disorder in membrane protein crystals was limiting the resolution that could be achieved after averaging the unit cells contained within an image. This problem was first recognized by Crowther and Sleytr who developed the first computer programs to attempt to correct for crystalline disorder (15). Henderson et al. later developed a method and software to correct the lattice disorder in images of very large 2D protein crystals in order to improve the S/Ns of their images at higher resolution, and their method is described in Procedure 5 (41). Although this may seem like a laborious process, the improvements in data can be dramatic, and most steps of the procedure are automated.

A similar approach to this problem developed out of efforts that began in Joachim Frank's group to create software tools to align electron microscope images of individual particles using correlation methods. The single particle correlation methods can also be used to align individual unit cells if the S/N is high enough to permit accurate alignment, or it can be applied to patches of unit cells in the case of low S/N images of unstained specimens. Once the patches of unit cells are aligned, they can be averaged to increase dramatically the S/N of the resulting images (27). The single particle averaging software can also be used for structural study of molecules that are not crystalline and have yielded dramatic results when applied to images of individual ribosomes and ribosomal subunits (58).

❖ Procedure 5. Resolution of the

Electron Micrograph

1. Apply a mask to the Fourier transform that passes only the information that lies within a specified radius of each reciprocal lattice point.

2. Calculate the inverse Fourier transform of the "masked" transform to produce a "coarsely" filtered image in which each unit cell is averaged with its nearest neighbors.

3. Select a reference image from the coarsely filtered image and calculate the cross-correlation function of this reference image with the entire filtered image.

Identify the positions of each unit cell by searching for the peaks in the crosscor-relation function.

Reinterpolate the sampling of the original image based upon the positions of all of the unit cells identified in the crosscorrela-tion function in order to remove crystal lattice disorder.

### 4.5. Correcting the CTF

The performance of the objective lens of an electron microscope is defined by the CTF, which is the Fourier transform of the point-spread-function that describes how a point on the object (specimen) appears in the image (23,26). Figure 3c shows the effect of the CTF in modulating the intensities of the Fourier transform (displayed in the optical diffraction pattern) of the image in Figure 3 a, and the inset graph shows the phase contrast component of the CTF for this defocus, demonstrating that it causes periodic phase reversals (phase shifts of 180°) within concentric bands of spatial frequencies

Note: When the CTF is plotted in this manner, correct transfer of contrast is indicated when sin c(a) = -1.

Not shown in the inset is the effect of amplitude contrast generated when electrons are scattered into the objective lens aperture removing them from the image; amplitude contrast is important at low spatial frequencies contributing contrast to low resolution features (approximately 50

A or greater). Correction of the CTF is not absolutely required if all of the information contained in a micrograph lies within the first zero of the CTF; this is the region around the origin of the Fourier transform and within the first ring of low noise where the CTF goes through zero on the inset graph. However, if one wishes to obtain an accurate representation of the object, even at low resolution, the amplitudes of the Fourier transform must be increased by varying amounts to compensate for the fact that the CTF is not -1.0 across this frequency spectrum. The most significant correction is for those bands of the frequency spectrum of the Fourier transform where the CTF has caused a phase shift of 180°. In Figure 3c, the circled lattice points contain information about the crystal structure whose phases have been shifted by 180°, and these will contribute incorrect information to the image unless they are corrected. Once the defocus and residual astigmatism of a particular micrograph has been defined, the CTF can be calculated, and the phase changes are easily corrected. Correction of the amplitudes is more complicated because: (i) the amount by which amplitudes must be increased can be difficult to determine, since one must include contributions from amplitude contrast; and (ii) regions of the Fourier transform near the zeroes of the CTF require very large corrections, which can greatly magnify the contribution of noise in the image. The proper methods for making this correction are beyond the scope of this article, but more detailed information can be found in the literature (73,89).

### 4.6. 3D Reconstruction

The ultimate goal is to calculate a 3D structure of the protein under study. Electron micrographs are 2D projections of the 3D electron density. Most people are intuitively aware that one can gain a better knowledge of a complex 3D object's structure by viewing it from several angles. This intuitive approach is quantitatively achieved by a number 3D reconstruction algorithms that make use of 2D projections along different directions. In the case of 2D crystals, the most common of these algorithms make use of a property of Fourier transforms described by the central section theorem; this states that the Fourier transform of a 2D projection of a 3D object is a central section (a section that passes through the origin) of the 3D Fourier transform of the object. Thus, as one collects 2D projections along different angles, one can fill in the 3D Fourier transform and estimate its value at a resolution limited by:

1. The number of 2D projections and the angle between them. This is so because higher resolution information is contained further from the origin of the Fourier transform where the 2D central sections are further apart; eventually they diverge enough that the values of the 3D Fourier transform can no longer be estimated from the values on the 2D sections.

2. The size of the object. The reason for this limitation is more subtle and arises from the fact that larger objects have Fourier transforms that vary more rapidly than smaller objects with the same level of detail. Thus, the 2D sections for larger objects must be more closely spaced to achieve comparable resolution compared to smaller objects. Another way of viewing this is to recognize that defining the structure of a larger object requires more data (e.g., more projections).

The semiquantitative relationship between resolution, object size, and number of tilts was expressed by Crowther et al. in the equation:

m s nD/d where m equals the number of views, D is the particle diameter, and d is the desired resolution (14).

Fourier transforms of 2D crystals have a special property that renders the use of Fourier transforms computationally efficient; they are sampled on a 2D lattice parallel to the plane of the crystal and are continuous along "lattice lines" perpendicular to the crystal plane at the points of the 2D reciprocal lattice as shown in Figure 4. For example, in the Fourier transform of the cytochrome oxidase crystal in Figure 3, the information about the crystalline structure is contained at the points of the reciprocal lattice defined by the lattice vectors a* and b*, while nonperiodic noise is distributed over the entire transform. If one could view the 3D Fourier transform, one would see that the information intersected by this central section varies continuously along lines, lattice lines, parallel to one another, and perpendicular to the plane of the transform in Figure 3c as diagrammed in Figure 4. The Fourier transforms of images of tilted crystals sample these lattice lines at the positions where the central section intersects them as shown in Figure 4, and collecting the information required for a complete 3D reconstruction of the crystal requires collecting central sections at different tilt angles in order to sample these lattice lines finely enough to estimate their value continuously out to the desired resolution. The higher the resolution and the larger the unit cell, the more projections required and the finer the angular intervals between them. Once the lattice lines have been measured from central sections, they can be sampled at appropriate regular intervals, and the 3D structure calculated by an inverse 3D Fourier transformation (4).

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