The Variance Map and the Analysis of Statistical Significance

We have seen in the foregoing that one of the uses of the variance map is to pinpoint image regions where the images in the set vary strongly. The possible sources of interimage variability are numerous:

(i) Presence versus absence of a molecule component, for example, in partial depletion experiments (Carazo et al., 1988)

(ii) Presence versus absence of a ligand, for example, in immunoelectron microscopy (Gogol et al., 1990; Boisset et al., 1993b), ribosome-factor binding complex (Agrawal et al., 1998; Valle et al., 2002), calcium release channel bound with a ligand (Wagenknecht et al., 1997; Sharma et al., 1998; Samso et al., 1999, 2000; Liu et al., 2002)

(iii) Conformational change, that is, movement of a mass (Carazo et al., 1988, 1989) or of many thin flexible masses (Wagenknecht et al., 1992)

(iv) (Compositional heterogeneity

(v) Variation in orientation, for example, rocking or flip/flop variation (van Heel and Frank, 1981; Bijlholt et al., 1982)

(vi) Variation in stain depth (Frank et al., 1981a, 1982; Boisset et al., 1990a)

(vii) Variation in magnification (Bijlholt et al., 1982)

Striking examples are found in many studies of negatively stained molecules, where the variance map often reveals that the stain depth at the boundary of the molecule is the strongest varying feature. The 40S ribosomal subunit of eukaryotes shows this behavior quite clearly (Frank et al., 1981a); see figure 3.17.

An example of a study in which structural information is gleaned from the variance map is found in the paper by Wagenknecht et al. (1992) (figure 3.18). Here, a core structure (E2 cores of pyruvate dehydrogenase) is surrounded by lipoyl domains which do not show up in the single particle average because they do not appear to assume fixed positions. Their presence at the periphery of the E2 domain is nevertheless reflected in the variance map by the appearance of a strong white halo of high variance. These early findings are particularly interesting in the light of the high-resolution structure emerging now by a combination of results from cryo-EM and X-ray crystallography, and indications of the dynamical behavior of this molecule (Zhou et al., 2001c).

Figure 3.17 Average and variance map obtained from an aligned set of macromolecules. (a) Sixteen of a total set of 77 images showing the 40S ribosomal subunit of HeLa in L-view orientation; (b) average image; (c) variance map; and (d) standard deviation map, showing prominent variations mainly at the particle border where the amount of stain fluctuates strongly (white areas indicate high variance). From Frank et al. (1981), with permission of the American Association for the Advancement of Science.

Figure 3.17 Average and variance map obtained from an aligned set of macromolecules. (a) Sixteen of a total set of 77 images showing the 40S ribosomal subunit of HeLa in L-view orientation; (b) average image; (c) variance map; and (d) standard deviation map, showing prominent variations mainly at the particle border where the amount of stain fluctuates strongly (white areas indicate high variance). From Frank et al. (1981), with permission of the American Association for the Advancement of Science.

However, this ''global'' variance analysis made possible by the variance map has some obvious shortcomings. While it alerts us to the presence of variations and inconsistencies among the images of a data set, and gives their location in the image field, it fails to characterize the different types of variation and to flag those images that have an outlier role. For a more specific analysis, the tools of multivariate data analysis and classification must be employed (see chapter 4).

Another important use of the variance map is the assessment of significance of local features in the average image (Frank et al., 1986), using standard methods of statistical inference (e.g., Cruickshank, 1959; Sachs, 1984): each pixel value in that image, regarded as an estimate of the mean, is accompanied by a confidence interval within which the true value of the mean is located with a given probability. In order to construct the confidence interval, we consider the random variable:

Figure 3.18 Example of the type of information contained in a variance map: visualization of a "corona" of highly flexible lipoyl domains surrounding the E2 core of puryvate dehydrogenase complex of Escherichia coli. (a) Electron micrograph of frozen-hydrated E2 cores presenting fourfold symmetric views. Each is surrounded by fuzzy structures believed to be lipoyl domains bound to the molecule. Due to their changing positions, the average map (b, left) depicts only the E2 core, but the variance map (b, right) shows a ring-shaped region of high variance. From Wagenknecht et al. (1992), reproduced with permission of Elsevier.

where s(rj) = [V(rj)/N]1/2 is the standard error of the mean, which can be computed from the measured variance map expressed by equation (3.48) (see figure 3.17). The true mean lies in the interval:

the confidence interval, with probability P, if t satisfies

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