The account given thus far relates to the theoretical resolution, or the maximum resolution expected for a particular data collection geometry. There are, however, many effects that prevent this resolution from being realized. Some of these have already been mentioned in passing in the relevant sections on instrumentation and 2D data analysis. It is useful to have a list of these effects in one place:

(i) Instrumental limitation: the projections themselves may be limited in resolution, due to (i) partial spatial and energy coherence (Wade and Frank, 1977; see also chapter 2, section 3.3), and (ii) effects due to charging and mechanical instability (Henderson and Glaeser, 1985; Henderson, 1992; Gao et al., 2002). While concerns about partial coherence have been put to rest in recent years by the ready availability of instruments with field emission guns and high stability of voltage and lens currents, the effects in the other category (ii) are determined by the exact set-up and operating conditions in the laboratory.

(ii) Heterogeneity: the coexistence of particles with different conformations invalidates the entire single-particle reconstruction approach, as it leads to density maps in which features of different structures are intermixed. One of the noticeable effects is resolution loss, either in the entire density map or in parts of it. In fortuitous situations, this problem may be addressed by classification (section 9).

(iii) Numerical errors: rounding errors are possible in the course of the data analysis, and they may accumulate unless precautions are taken (chapter 3, section 3.4.3). A critical factor is the ratio between the resolution of the structural information and the resolution of the digital representation— scanning should be with a sampling increment that is half or less of the critical distance required by the sampling theorem (see chapter 3, section 2.2).

(iv) Alignment errors: due to the low SNR of the raw data, the parameters describing the geometric position of a particle in the molecule's coordinate frame may be incorrect. Even magnification errors that occur when data are merged from different instruments can have a substantial

Figure 5.31 Predicted effect of angular errors, in terms of an envelope function affecting the Fourier transform of the 3D density map. Abscissa: sR, the product of spatial frequency and the radius of the particle. The curves are labeled by the standard deviation of the rotational error imposed. From Jensen (2001), reproduced with permission of Elsevier.

Figure 5.31 Predicted effect of angular errors, in terms of an envelope function affecting the Fourier transform of the 3D density map. Abscissa: sR, the product of spatial frequency and the radius of the particle. The curves are labeled by the standard deviation of the rotational error imposed. From Jensen (2001), reproduced with permission of Elsevier.

influence. These effects have been analyzed by Jensen (2001) in a series of tests with modeled data. A result of particular interest is the dependence of resolution on the error of angle assignment (figure 5.31). Similar errors of angular assignments affect the resolution if, following classification, particles within a finite angular neighborhood are assigned the same angles for the 3D reconstruction (Stewart et al., 1999), as in some reconstruction schemes following the angular reconstitution route (sections 3.4 and 6).

Discussions of the relative importance of these effects are found in many reviews of cryo-EM of single particles (see references in appendix 3). For articles specifically addressing the topic of resolution and resolution loss, see Glaeser and Downing (1992), Stewart and coworkers (1999, 2000), and Jensen (2001).

For all these reasons, the significant resolution of a reconstruction (i.e., the resolution up to which true, object-related features are actually represented in the 3D image) can differ substantially from the theoretical resolution deduced from the geometry, and needs to be independently assessed in each project.

As in the practical assessment of 2D resolution (section 5.2 in chapter 3), there are two different approaches; one goes through the evaluation of reconstructions done from randomly drawn halfsets of the projection data, the other a multiple comparison first devised by Unser and coworkers (1987, 1989), the spectral signal-to-noise ratio (SSNR).

Two reconstructions are calculated from two randomly drawn subsets of the projection set, and these reconstructions are compared in Fourier space using differential phase residual (DPR) or Fourier ring correlation (FRC) criteria. Obviously, in the 3D case, the summation spelled out in the defining formulas [chapter 3, equations (3.64) and (3.65)] now has to go over shells of k = |k| = constant. This extension of the differential resolution criterion from two to three dimensions is straightforward, and was first implemented (under the name of Fourier shell correlation, abbreviated to FSC) by Harauz and van Heel (1986a) for the FRC and by Radermacher et al. (1987a, b) for the DPR.

One of the consistency tests for a random-conical reconstruction is the ability to predict the 0° projection from the reconstruction, a projection that does not enter the reconstruction procedure yet is available for comparison from the analysis of the 0° data. Mismatch of these two projections is an indication that something has gone wrong in the reconstruction, while the reverse line of reasoning is incorrect: an excellent match, up to a resolution R, is no guarantee that this resolution is realized in all directions. This is easy to see by invoking the projection theorem as applied to the conical projection geometry (see figure 5.13): provided that the azimuths 'i are correct, the individual tilted planes representing tilted projections furnish correct data for the 0° plane, irrespective of their angle of tilt. It is, therefore, possible to have excellent resolution in directions defined by the equatorial plane, as evidenced by the comparison between the 0° projections, while the resolution might be severely restricted in all other directions. The reason that this might happen is that the actual tilt angles of the particles could differ from the nominal tilt angle assumed (see Penczek et al., 1994).

Most resolution tests are now based on the FSC curve (typical curves are shown in figure 5.24, where it is used to monitor the progress of refinement), although the value quoted can differ widely depending on the criterion used. In the following, the different criteria are listed with a brief discussion of their merits.

(i) 3-a criterion (Orlova et al., 1997): In this and the following criteria, the signal FSC on a given shell is compared with the correlation of pure noise, which is obtained by 1 /vN, where N is the number of independent points in the shell (Saxton and Baumeister, 1982). Initially, 2a had been proposed (Saxton and Baumeister, 1982; van Heel, 1987a). Experience shows that the 3-a criterion is still too lenient—reconstructions limited to this resolution show a substantial amount of noise. Indeed, the number of points on the Fourier grid is always substantially larger than the number of independent points, because of the mutual dependence of Fourier coefficients within the range of the shape transform. The problem is also evident when one pays attention to the behavior of the signal FSC in the vicinity of the 3-a cutoff: it usually fluctuates significantly in this region, which indicates that the cutoff resolution itself becomes a matter of chance. Objections against the use of the noise curve have been voiced by Penczek (appendix in Malhotra et al., 1998) on theoretical grounds. A brief account of the controversy is given in Frank (2002). An insightful critique is also found in the appendix of the article by Rosenthal et al. (2003).

(ii) 5-a criterion (e.g., Radermacher, 1988; Radermacher et al., 2001; Ruiz et al., 2003): This criterion is tied to the same noise curve as the previous one, but attempts to address the problems by the use of a larger multiple of the noise.

(iii) 0.5 cutoff: This cutoff was introduced by Crowther's group (Bottcher et al., 1997) as it became clear in their reconstruction of the hepatitis B virus that the 3-a cutoff, even when corrected for the effect of the icosahedral symmetry, reported a resolution that proved untenable. Later, Penczek pointed out (appendix in Malhotra et al., 1998) that this choice of cutoff will limit the SNR in the Fourier shells to values equal to, or larger than, one. On the other hand, van Heel (presentation at the ICEM in Durban, 2002) has cautioned against the blind application of this criterion since at very low resolutions it can yield a value that is too optimistic. This problem is quite real for low-resolution maps of viruses but has no bearing on resolution assessments of single-particle reconstructions with resolutions better than 30 A.

(iv) 0.143 cutoff: Rosenthal et al. (2003) pointed out that, judging from experience, the 0.5 cutoff is too conservative. These authors suggest a rationale for the use of an FSC cutoff at 0.143 on the basis of a relationship (condition for improvement of the density map by including additional shells of data) that is known to be valid for data in X-ray crystallography. As yet, however, it is unclear whether this relationship holds for EM data, as well.

In trying to choose among the different measures, we can go by comparisons between X-ray and cryo-EM maps of the same structure (see chapter 6 for two examples). Such comparisons are now available for an increasing number of structures. Saibil's group (Roseman et al., 2001) argued, from a comparison of the appearance of the cryo-EM map with that obtained by filtering the X-ray structure of GroEL, that the cutoff should be placed somewhere between the 0.5 and 3-ct values. In its tendency, this would support Rosenthal et al. contention [point (iv) above]. On the other hand, a recent reconstruction of the ribosome from 130,000 particles (Spahn et al., 2005) shows the best agreement with the X-ray map when the latter is presented at the FSC = 0.5 resolution, namely at 7.8 A (chapter 6, section 3.3). It seems prudent, therefore, to hold on to this criterion until a strong case can be built for an alternative one.

8.2.2. Truly Independent Versus Partially Dependent Halfset Reconstructions

The method of assessing resolution based on halfset reconstructions is not without problems: in the usual refinement procedure, the alignment of both halfsets is derived with the help of the same 3D reference. To the extent that each of the final halfset reconstruction is influenced by the choice of reference (''model dependence''), it also has similarity to its counterpart which is reflected by the FSC. True independence of the two halfset reconstructions, which would be a desirable feature of the test, cannot be achived under these circumstances. The statistical dependence of the halfset reconstructions results in a systematic increase of the FSC curve, which leads to an overestimation of resolution when any of the criteria (i)-(iv) are applied. (Fortuitously, the resolution overestima-tion is to some extent counteracted by its underestimation, resulting from the fact that the resolution measured pertains to the reconstruction that is obtained from one-half of the data only).

The problem of statistical dependence of halfset reconstructions originating from the same reference and possible remedies were extensively discussed by Grigorieff (2000), Penczek (2002a), and Yang et al. (2003). To circumvent the influence of model dependence, Grigorieff (2000) proposed to use, in a departure from the usual 3D projection matching procedure (Penczek et al., 1994), two distinctly different references, and to carry out the refinement of both halfsets totally independently. Evidently, convergence to a highly similar structure in such independent processing routes will attest to the validity and self-consistency of the result. What is unclear, however, is the nature of the competing references—how much should they differ to make the results acceptable?

The recently developed cross-validation method (''free FSC'', Shaikh et al., 2003) provides a way to establish the degree of the model dependence and obtain a model-independent resolution estimate (see section 8.3).

8.2.3. Three-Dimensional Spectral Signal-to-Noise Ratio

Considerations along the lines of a definition of a 3D SSNR were initially developed by Liu (1993) and Grigorieff (1998). We recall that the 2D SSNR as originally defined by Unser and coworkers (1987, 1989; see also Grigorieff, 2000) is based on a comparison between the signal and the noise components of an image set in Fourier space that contribute to the average at a given spatial frequency, and is expressed as a ratio. Summation over subdomains (such as rings in two dimensions) of the spatial frequency range will give specific SSNR dependencies on spatial frequency radius, or angles specifying a direction. Penczek (2002a) considered how this concept can be generalized in three dimensions. The necessity to trace contributions from the 2D Fourier transforms of the projections to the 3D Fourier transform of the reconstruction singles out reconstruction algorithms based on Fourier interpolation, and practically excludes other algorithms such as weighted back-projection or iterative algebraic reconstruction in the generalization of the SSNR concept.

In Fourier interpolation (see section 4.3 in this chapter), for any given point of the 3D Fourier grid, contributions are summed that originate from all 2D Fourier transforms in its vicinity. These contributions are derived by centering the 3D shape transform—a convolution kernel reflecting the approximate shape of the 3D object or the volume it is contained in (see section 2.2)—on each 2D grid point of a central section. The fact that the 3D shape transform has, for practical purposes, a limited extent limits the number of projections and the number of 2D grid points that need to be considered in the summation.

Penczek (2002a), in derivations that cannot be properly paraphrased in this limited space, obtained approximate expressions for the 3D SSNR in two situations: in one, he considered the interpolation scheme with multiple overlapping contributions described above, and in the other, a poor nearest-neighbor interpolation scheme. It is noteworthy that the numerical application of the 3D SSNR (computed according to the more sophisticated interpolation scheme) yielded results that agreed very well with the FSC-based estimations in the range between SSNR = 0 and 100, or between FSC = 0 and 0.99. It is also important to note that the 3D SSNR provides a way to measure resolution for tomographic reconstructions (Penczek, 2002a; see also application in the study by Hsieh et al., 2002). Furthermore, it provides a very useful way for mapping out the direction dependence of resolution, to be discussed in section 8.2.4.

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