Figure 5.32 Cross-validation of refinement. A shell is removed from the 3D reference at the indicated place in the Fourier transform. In the refinement, the Fourier shell correlation (FSC) behaves differently inside from outside the removed shell, and the difference is most dramatic for pure noise. (a) Real data (ribosome particles from cryo-EM micrographs). The FSC within the shell decreases, reflecting the existence of a certain amount of model dependence. (b) Pure noise. In this case, the FSC falls to a value that is indeed expected for pure noise. From Shaikh et al. (2003), reproduced with permission of Elsevier Science.

Figure 5.32 Cross-validation of refinement. A shell is removed from the 3D reference at the indicated place in the Fourier transform. In the refinement, the Fourier shell correlation (FSC) behaves differently inside from outside the removed shell, and the difference is most dramatic for pure noise. (a) Real data (ribosome particles from cryo-EM micrographs). The FSC within the shell decreases, reflecting the existence of a certain amount of model dependence. (b) Pure noise. In this case, the FSC falls to a value that is indeed expected for pure noise. From Shaikh et al. (2003), reproduced with permission of Elsevier Science.

As the SNR of the data decreases, the reference increasingly "takes over,'' up to the point, reported by Grigorieff (2000), where data representing pure noise give rise to a reconstruction that is a perfect copy of the reference. The cross-validation of Shaikh et al. (2003) makes it possible to spot those cases by the large drop in the FSC at the radius of the selected shell.

It is clear from the properties of the Fourier representation of a bounded object (see section 2.2) that the region in Fourier space from which data is to be excised must have a minimum width, which is given by the lateral extent of the object's shape transform. (We recall that each Fourier coefficient is surrounded by a "region of influence'' within which neighboring Fourier coefficients are correlated with it. For the cross-validation to be true, we can only look at the behavior of Fourier coefficients that lie outside that correlation range from its data-carrying neighbors.) This extent roughly corresponds to the inverse of the object's diameter. For example, the ribosome has a globular shape with ~250 A diameter, so its shape transform has a width of 1/250 A_1. If the ribosome is contained in a box of size

500 A, then the width of the shell should be at least 4 Fourier pixels (or 4/500 A—1) so that the center of the shell is safely away from influence on both sides.

Excision of Fourier information over a shell may seem risky, as it could conceivably produce artifacts that affect the outcome of the 3D projection matching and the course of the refinement. A careful check (T. Shaikh, unpublished results) revealed that these effects are insignificant if the shell is thin enough, as chosen in Shaikh et al. (2003).

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