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Phases can be determined for geometrically redundant amplitudes by iteration of the following procedure: compute an electron density map from the currently available phases; average the electron densities of all the crystallographically independent molecules; rebuild the crystal(s) from this averaged subunit, setting the density outside the molecular boundaries to its average value; obtain phase information from the resulting structure, and combine it with that given by isomorphous replacement to produce the phases to be used in the next iteration. This algorithm converges very rapidly, and has proved to be a powerful tool in the solution of two large unknown protein structures. This paper describes the computational techniques developed to implement it, which include a swift and general method for real-space averaging of electron density maps.
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