The sums are computed over Fourier components falling within rings defined by spatial frequency radii k ± Ak; k =|k| and plotted as a function of k (figure 3.21). In principle, as in the case of the Fourier ring correlation to be introduced below, the entire curve is needed to characterize the degree of consistency between the two averages. However, it is convenient to use a single

Figure 3.21 Resolution assessment by comparison of two subaverages (calcium release channel of skeletal fast twitch muscle) in Fourier space using two different criteria: differential phase residual (DPR, solid line, angular scale 0° ... 100°) and Fourier ring correlation (FRC, dashed line, scale 0... 1). The scale on the x-axis is in Fourier units, denoting the radius of the rings over which the expressions for DPR or FRC were evaluated. The DPR resolution limit (A<> = 45°; see arrowhead on y-axis) is 1/30 AA (arrowhead on x-axis). For FRC resolution analysis, the FRC curve was compared with twice the FRC for pure noise (dotted curve). In the current example, the two curves do not intersect within the Fourier band sampled, indicating an FRC resolution of better than 1/20 A_1. From Radermacher et al. (1992a), reproduced with permission of the Biophysical Society.

Figure 3.21 Resolution assessment by comparison of two subaverages (calcium release channel of skeletal fast twitch muscle) in Fourier space using two different criteria: differential phase residual (DPR, solid line, angular scale 0° ... 100°) and Fourier ring correlation (FRC, dashed line, scale 0... 1). The scale on the x-axis is in Fourier units, denoting the radius of the rings over which the expressions for DPR or FRC were evaluated. The DPR resolution limit (A<> = 45°; see arrowhead on y-axis) is 1/30 AA (arrowhead on x-axis). For FRC resolution analysis, the FRC curve was compared with twice the FRC for pure noise (dotted curve). In the current example, the two curves do not intersect within the Fourier band sampled, indicating an FRC resolution of better than 1/20 A_1. From Radermacher et al. (1992a), reproduced with permission of the Biophysical Society.

figure, k45, the spatial frequency for which A<(k, Ak) = 45°. As a conceptual justification for the choice of this value, one can consider the effect of superimposing two sine waves differing by A<. If A< is less than 45°, the waves tend to enforce each other, whereas for any A<>45°, the maximum of one wave already tends to fall in the vicinity of the zero of the other, and destructive interference starts to occur.

It is of crucial importance in the application to EM that the phase residual (and any other Fourier-based measures of consistency, to be described in the following) be computed "differentially," over successive rings or shells, rather than globally, over the entire Fourier domain with a circle of radius k. Such global computation is often used, for instance, to align particles with helical symmetry (Unwin and Klug, 1974). Since |F(k)| falls off rapidly as the phase difference A< increases, the figure k45 obtained with the global measure would not be very meaningful in our application; for instance, excellent agreement in the lower spatial frequency range can make up for poor agreement in the higher range and thus produce an overoptimistic value for k45. The differential form of the phase residual, equation (3.64), was first used by Crowther (1971) to assess the preservation of icosahedral symmetry as a function of spatial frequency. It was first used in the context of single-particle averaging by Frank et al. (1981a).

One can easily verify from equation (3.64) that the DPR is sensitive to changes in scaling between the two Fourier transforms. In computational implementations, equation (3.64) is therefore replaced by an expression in which |F2(k)| is dynamically scaled, that is, replaced by s|F2(k)|, where the scale factor s is allowed to run through a range from a value below 1 to a value above 1, through a range large enough to include the minimum of the function. The desired DPR then is the minimum of the curve formed by the computed residuals. One of the advantages of the DPR is that it relates to the measure frequently used in electron and X-ray crystallography to assess reproducibility and the preservation of symmetry.

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