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We have
now published an analysis of the role played by convolutional structure
factor equations in the determination of phases by extrapolation from
a confidently known subset, and in the refinement of all phases in accordance
with an objective function based on these equations. Our conclusions establish
more clearly than ever before that the Sayre relation provides very powerful
extrapolation, but nevertheless cannot drive refinement effectively because
of the multi-modality of the objective function landscape over the range
of possible phases. The complex-valued implementation of the objective
function does, however, provide an important advantage: in at least one
example, it prevents over-fitting of the objective function, with concurrent
degradation of the phase information, which is almost invariably observed
using the real part of the objective function. The chief obstacle to using
Sayre-type equations effectively in refinement algorithms therefore appears
to be that we cannot yet generate good estimates for the variances of
the phases used to evaluate the objective function, which could then be
used to formulate a weighted least squares or maximum likelihood refinement
algorithm. To overcome the limitation imposed by the lack of phase variances,
we have implemented localized versions of the Sayre equations, together
with evaluation of the local mean value and local variance of the current
electron density function. This process, which is somewhat analogous to
the "Baking" portion of the successful "Shake and Bake" phasing algorithm,
potentially provides such information on phase accuracy, and can potentially
be used as the basis for an automated interpretation of electron density
maps for which atomic resolution diffraction data have been obtained.
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