Although the different resolution criteria emphasize different aspects of the signal, it is possible to characterize the resolution limit qualitatively as the limit, in Fourier space, beyond which the signal becomes so small that it "drowns" in noise. Beyond a certain limit, no amount of averaging will retrieve a trace of the signal. This ''practical resolution limit" is the result of many adversary effects that diminish the contrast at high spatial frequencies. The effects of various factors on the contrast of specimens embedded in ice have been discussed in detail (Henderson and Glaeser, 1985; Henderson, 1992). Useful in this context is the so-called relative Wilson plot, defined in regions where CTF = 0 by image Fourier amplitudes 75^
electron diffraction amplitudes x CTF '
where both the expressions in numerator and denominator are derived by averaging over a ring in Fourier space with radius k = |k|. This plot is useful because it allows those effects to be gauged that are not caused by the CTF but nevertheless affect the image only, not the diffraction pattern of the same specimen recorded by the same instrument. In the parlance of crystallography, these would be called phase effects since they come to bear only because diffracted rays are recombined in the image, so that their phase relationship is important. The most important effects in this category are specimen drift, stage instability, and charging. [Here, we consider effects expressed by classical envelope terms associated with finite illumination convergence and finite energy spread (see chapter 2, section 3.3.2) as part of the CTF.] For instance, drift will leave the electron diffraction pattern unchanged, while it leads to serious deterioration in the quality of the image since it causes an integration over images with different shifts during the exposure time.
Therefore, an ideal image would have a Wilson plot of w(k) = const. Instead, a plot of w(k) for tobacco mosaic virus, as an example of a widely used biological test specimen, shows a falloff in amplitude by more than an order of magnitude as we go from 1/i to 1/10 A_1. This cannot be accounted for by illumination divergence (Frank, 1973a) or energy spread (Hanszen, 1971; Wade and Frank, 1977), but must be due to other factors. Henderson (1992) considered the relative importance of contributions from four physical effects: radiation damage, inelastic scattering, specimen movement, and charging. Of these, only the last two, which are difficult to control experimentally, were thought to be most important. Both effects are locally varying, and might cause some of the spatial variation in the power spectrum across the micrograph described by Gao et al.
Concern about beam-induced specimen movement led to the development of spot scanning (Downing and Glaeser, 1986; Bullough and Henderson, 1987; see section 3.2 in chapter 2). Charging is more difficult to control, especially at temperatures close to liquid helium where carbon becomes an insulator. Following Miyazawa et al. (1999), objective apertures coated with gold are used as a remedy, since their high yield of back-scattered electrons neutralizes the predominantly positive charging of the specimen.
In addition to the factors discussed by Henderson (1992), we have to consider the effects of conformational variability that is intrinsically larger in single molecules than in those bound together in a crystal. Such variability reduces the resolution not only directly, by washing out variable features in a 2D average or 3D reconstruction, but it also decreases the accuracy of alignment since it leads to reduced similarity among same-view images, causing the correlation signal to drop (see Fourier computation of the CCF, section 3.3.2).
5.5. Statistical Requirements following the Physics of Scattering
So far, we have looked at signal and noise in the image of a molecule, and the resolution achievable without regard to the physics of image formation. If we had the luxury of being able to use arbitrarily high doses, as is the case for many materials science applications of EM, then we could adjust the SNR to the needs of the numerical evaluation. In contrast, biological specimens are extremely radiation sensitive. Resolution is limited both on the sides of high and low doses. For high doses, radiation damage produces a structural deterioration that is by its nature stochastic and irreproducible; hence, the average of many damaged molecules is not a high-resolution rendition of a damaged molecule, but rather a low-resolution, uninformative image that cannot be sharpened by "deblurring." For low doses, the resulting statistical fluctuations in the image limit the accuracy of alignment; hence, the average is a blurred version of the signal.
The size of the molecule is a factor critical for the ability to align noisy images of macromolecules (Saxton and Frank, 1977) and bring single-particle reconstruction to fruition. The reason for this is that for a given resolution, the size determines the amount of structural information that builds up in the correlation peak ("structural content'' in terms of Linfoot's criteria; see section 3.3 in chapter 2). Simply put, a particle with 200 A diameter imaged at 5 A resolution covers 2002/52 = 1600 resolution elements, while a particle with 100 A diameter imaged at the same resolution covers only 400.
By comparing the size of the expected CCF peak with the fluctuations in the background of the CCF for images with Poisson noise, Saxton and Frank (1977) arrived at a formula that links the particle diameter D to the contrast c, the critical dose pcrit, and the sampling distance d. Accordingly, the minimum particle diameter Dmin allowing significant detection by cross-correlation is
c2 dp crit
Henderson (1995) investigated the limitations of structure determination using single-particle methods in EM of unstained molecules, taking into account the yield ratio of inelastic to elastic scattering, which determines the damage sustained by the molecule, versus the amount of useful information per scattering event in bright-field microscopy. He asked the question "What is the smallest size of molecule for which it is possible to determine from images of unstained molecules the five parameters needed to define accurately its orientation (three parameters) and position (two parameters) so that averaging can be performed?'' The answer he obtained was that it should be possible, in principle, to reconstruct unstained protein molecules with molecular mass in the region of 100 kDa to ~3A resolution.
However, by a long shot, this theoretical limit has not been reached in practice, for two main reasons: there is likely a portion of noise not covered in these calculations, and secondly, the signal amplitude in Fourier space falls off quite rapidly on account of experimental factors not considered in the contrast transfer theory, a discrepancy observed and documented earlier on (see Henderson, 1992). Experience has shown that proteins should be ~400 kDa or above in molecular mass for reconstructions at resolutions of 10 A to be obtained. The spliceosomal U1 snRP (molecular mass of ~200kD) reconstructed by Stark et al. (2001) is one of the smallest particles investigated with single-particle methods without the help of stain, but it does have increased contrast over protein on account of its RNA.
Another conclusion of the Henderson study that may be hard to reconcile with experiments is that the number of images required for a reconstruction at 3 A resolution is ~ 19,000, independent of particle size. The reconstruction of the ribosome to 11.5 A already required 73,000 images (Gabashvili et al., 2000), though later, due to improvements in EM imaging and image processing, the same resolution could be achieved with half that number (e.g., Valle et al., 2003a). Still, this means that 19,000 images is just half the number necessary to achieve a resolution that is lower by a factor of 3 than predicted. Only when the problems of specimen and stage instability and charging are under better control will the estimations by Henderson (1995) match with the reality of the experiment.
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