Cross Correlation Search

A global correlation search, to find the exact docking position, can be realized in a straightforward way by the use of a CCF that incorporates both translational and rotational search. The density representing the small ligand structure is placed in the origin of a 3D array that has the same size as the array in which the cryo-EM map is housed. For different rotations of the ligand structure, the translational CCF (chapter 3, section 3.3) has to be repeatedly computed via the fast Fourier transform algorithm.

Experience has shown that for the success of such a search, the method of normalization is of critical importance. Global normalization, using the global variance of the larger structure as in the definition of the cross-correlation coefficient (chapter 3, section 3.3), will often fail since a correlation peak arising from the correct match may be dwarfed by peaks caused by the occurrence of local density maxima in the cryo-EM map of the complex. The solution to this problem is provided by a locally variable normalization [Volkmann and Hanein, 1999; Rossmann, 2000 (incorporated in the program EMFIT); Roseman, 2000]. In Roseman's notation:

where Si and Ti are the samples of the X-ray search object and the target map, respectively, P is the number of points in the search object, and Mt is a mask function that defines the boundary of the search object. The product MiTi+x

defines the 3D region of the target map as the search object assumes all possible positions symbolically indicated by x. Finally, SS, T are the respective means, and aS, oMT (x) are the local variances of the two densities within the exact region of intersection in the current relative position.

Chacon and Wriggers (2002) introduced a modified cross-correlation search to deal with the problem of low accuracy encountered with low-resolution (20-30 A) density maps. The modification lies in the application of a Laplacian filter to the target structure, with the purpose of enhancing the contributions from the molecular boundaries. A modification of a different kind was proposed by Wu et al. (2003) in the core-weighted fitting method. These authors introduce the concept of the core of a structure within a low-resolution density map, as a region whose density is unlikely to be affected by the presence of an adjacent structure. A ''core index'' is defined, which quantifies the depth of a grid point within the core. With the help of this index, a weight is defined, which, when applied to the cross-correlation function, has the property of exerting a "pull" with the tendency to make the core regions of search structure and target structure coincide. Test computations indicate that this fitting method allows correct fitting at resolutions where unweighted cross-correlation fails.

In the form in which it is stated in equation (6.13), the correlation search is computationally expensive since it does not lend itself to the use of the fast Fourier transformation in the calculation of the local normalization. Of particular interest is Roseman's (2003) modified version of the algorithm, which makes it possible to perform the computation of the local variances in Fourier space. To understand this approach, one has to realize that the local mean and local variance of the target density in equation (6.13) can be written as convolution products of the mask with the target density or its square, respectively (see also van Heel, 1982). This means, according to the convolution theorem (appendix 1), that they can be computed in Fourier space by performing a simple scalar multiplication. An implementation of Roseman's fast algorithm also exists in the form of a SPIDER script (Rath et al., 2003). Here, the method was demonstrated by a search of several proteins and an RNA motif within the cryo-EM map of the E. coli ribosome. Also, as mentioned earlier in the context of volume matching (chapter 5, section 12.3.3), a spherical harmonics expansion can be used to greatly accelerate the orientation part of the search.

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