where s(Rj) = ^J V(Rj)/N is again the standard error of the mean. s(Rj) can be estimated either from the variance estimate V(Rj) [equation (6.2)]—which is normally not available, unless we can draw from many independent 3D reconstructions (see section 2.2 for the detailed argument)—or from the estimate V(Rj) [equation (6.6)], which is always available when weighted back-projection or one of the linear iterative reconstruction algorithms is used as the method of reconstruction.
Having constructed the random variable t(Rj), we can then infer the confidence interval for each element of the 3D map, following the considerations outlined in the 2D case earlier on (section 4.2, chapter 3). On this basis, the local features within the 3D reconstruction of a macromolecule can now be accepted or rejected [see, for instance, the use of t-maps by Trachtenberg and DeRosier (1987) to study the significance of features in reconstructions of frozen-hydrated filaments].
The most important use of the variance map is in the assessment of the statistical significance of features in a 3D difference map. Due to experimental errors, such a difference map contains many features that are unrelated to the physical phenomenon studied (e.g., addition of antibodies, deletion of a protein, or conforma-tional change). Following the 2D formulation, equation (3.52) in chapter 3, we can write down the standard error of the difference between two corresponding elements of the two maps, j?i(R/) and p2(R/) as follows:
sd[ P1(R/), P2(R/)] = [V1(Rj)/N1 + V2(R/)/N2]1/2 (6.9)
where V1(R/) and V2(Rj) are the 3D variances. In studies that allow the estimation of the variance from repeated reconstructions according to equation (6.2) (see, e.g., Milligan and Flicker, 1987; Trachtenberg and DeRosier, 1987), N1 and N2 are the numbers of reconstructions that go into the computation of the averages jp1(R/) and p2(Rj). In the other methods of estimation, from projections, N1 = N2 = 1 must be used.
On the basis of the standard error of the difference, equation (6.9), it is now possible to reject or accept the hypothesis that the two elements are different. As a rule of thumb, differences between the reconstructions are deemed significant in those regions where they exceed the standard error by a factor of three or more (see section 4.2 in chapter 3).
Analyses of this kind have been used by Milligan and Flicker (1987) to pinpoint the position of tropomyosin in difference maps of decorated actin filaments. In the area of single-particle reconstruction, they are contained in the work by Liu (1993), Frank et al. (1992), and Boisset et al. (1994a). In the former two studies, the 50S-L7/L12 depletion study of Carazo et al. (1988) was re-evaluated with the tools of the variance estimation. The other application concerned the binding of an Fab fragment to the Androctonus australis hemocyanin molecule in three dimensions (Liu, 1993; Boisset et al., 1994a; Liu et al., 1995). In the case of the hemocyanin study (figure 6.3), the appearance of the Fab mass on the corners of the complex is plain enough, and hardly requires statistical confirmation. However, interestingly, the t-map (figure 6.3f) also makes it possible to delineate the epitope sites on the surface of the molecule with high precision.
Among the groups that have most consistently used formal significance tests in assessing the validity of structural differences are DeRosier's (Trachtenberg and DeRosier, 1987; Thomas et al., 2001), Wagenknecht's (Sharma et al., 1998, 2000; Samso et al., 1999), and Saibil's (Roseman et al., 1996).
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