Phase Extension Using Noncrystallographic Symmetry

If an asymmetric unit possesses high symmetry, as is often the case for large macromolecu-lar complexes such as the pyruvate dehydrogenase enzyme complex, or icosahedral viruses, then another approach to solving the phase problem becomes available. Viruses, in particular, are amenable because their symmetry operators are very precise and their orientations are well defined. Asymmetric units also occassionally contain redundant copies of a protein that are related by noncrystallographic symmetry through proper or improper rotations and translations. 

on the other hand, the electron density map calculated at some resolution r is somehow improved in quality in real space, then if it is transformed to produce structure factors, phases at somewhat higher resolution, r + Ar, can be computed that do have some measure of validity. Improvement of an electron density map thus allows gradual extension of phases in reciprocal space to higher resolution, and ultimately to an electron density map of sufficient quality and detail that a model can be constructed. This is another example of those bootstrap, incremental procedures so common to X-ray crystallography. 

The means for improving the real space, electron density map may involve a combination of ideas, but by far the most powerful is application of noncrystallographic symmetry. If the asymmetric unit contains noncrystallographic symmetry, then its electron density map does as well. If the dispositions of the noncrystallographic symmetry operators are known, then the electron density map can be self-averaged using these operators. 

The low-resolution phases initially required may be obtained in a variety of ways, but frequently these depend on other imaging techniques outside of X-ray crystallography. These may include transmission electron microscopy, cryo-electron microscopy, or atomic force microscopy. Low-resolution phases are even more often obtained by placing the known structure of a closely related virus, or complex, in the correct disposition in the unit cell (determined by rotation and translation functions) and using its low resolution calculated phases. 

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