This subproject is one of many research subprojects utilizing theresources provided by a Center grant funded by NIH/NCRR. The subproject andinvestigator (PI) may have received primary funding from another NIH source,and thus could be represented in other CRISP entries. The institution listed isfor the Center, which is not necessarily the institution for the investigator.Electron cryo-tomography has advantages for studying large objects having pleomorphic structures. However, tomography suffers from the 'missing wedge' problem due to the limited tilts of specimens in the microscope column. Methods are being developed to sort out objects of the same symmetry, conformation or size by cross-correlation and then to average the subgroups separately in order to enhance the signal to noise ratio and thus the resolution.It is desirable to cross-correlate and average 3D volumes corresponding to particles isolated from a tomographic reconstruction. This is useful either to characterize the contents of a cell, or to develop an unbiased 3D model or models if the symmetry, size or conformation of the particles is uncertain.Determining the relative orientations of the particles is crucial to the success of this operation. However, we encountered a difficulty in these comparisons having to do with the geometry of data collection for tomographic reconstructions. Because a tilt angle range of 90 cannot be achieved in the electron microscope, central sections in Fourier space for tilts higher than 60-70 are missing, leading to the problem commonly referred to as the missing wedge in Fourier space. After extraction from the tomogram and before applying any rotations, the orientation of the missing wedge in Fourier space is identical for each particle/volume. However, the orientation of the particles themselves is random with respect to each other and must be searched for in 3D rotation space. Because of the strong signal from the pattern of the missing wedge, the cross-correlation map may merely lead to the alignment of the missing wedges, which is certainly not correct. This problem does not occur to the same extent in cases where an isotropic starting model is available and used for 3-D alignment.We solved this problem by understanding the relevant issues in reciprocal space. As two 3-D volumes in Fourier space, each containing a missing wedge, are rotated against each other in the cross-correlation search, and one volume is multiplied by the complex conjugate of the other, zeros are generated. They occur when the missing-data region for one of the particles is multiplied by data in the other particle, and the number of such zeros changes at each rotation angle increment in the search. To minimize the effect of the missing wedge on the overall scale of the 3-D cross-correlation map, we computed the number of non-zero terms after the complex multiplication in Fourier space for each relative 3-D orientation in the space searched for the two particles, and scaled the 3-D cross-correlation map at that orientation search angle by the reciprocal of this number. This correction factor is approximately right because the total power in the Fourier transform of a cross-correlation function influences the values of the subsequent cross-correlation map. This normalization was good enough to allow the true proper relative orientation of the particles to be determined as demonstrated from our simulated data with missing wedge (M.F. Schmid, unpublished). We also developed methods for accounting for the missing wedge when averaging particles together. Our methods lead to more accurate averages.
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