Experimental phasing
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In order to calculate an electron density map, we require both the structure factor magnitudes and their phases. However the experiment only provides intensities, from which the magnitudes can be determined but not the phases. Therefore, additional experiments are required to obtain phase estimates.
This process involves several steps:
- Collect multiple data sets by some means.
- Determine the differences in scattering which lead to differences in the observations - usually a heavy atom substructure.
- Refine the parameters of the substructure.
- Use the observations and the substructure to estimate phases.
In the simplest case, two data sets are collected from crystals which are similar except for the insertion of one or more heavy atoms in the derivative crystal (the original being called the native). The scattering from the molecule of interest remains the same, with the difference in the structure factor being accounted for by the scattering from the heavy atoms. This leads to a difference in the observed magnitude. From the experiment, we do only know the magnitude of the scattering from each structure, so our knowledge is represented by the circles on the diagram. If we can determine the heavy atom substructure, then we can calculate the magnitude and phase of Fh(h).
Since the native part of the structure factor is the same, we can superimpose the diagrams so that these vectors are aligned. The possible values of the native phase are given by the intersections of the two circles. There are two possible values. This is referred to as a Harker construction.
If we have more than one derivative, we obtain multiple estimates of the possible phases. In the absence of errors, this allows the phase to be determined unambiguously.
It is more common these days to use anomalous dispersion to obtain phase information. When an anomalous scatterer is present in the structure, an additional component is introduced to the scattering of that atom, at 90 degrees to the normal contribution. Most significantly, this shifted component does not obey the usual phase relationship for Friedel opposites that φ(-h) = -φ(h). As a result, the Friedel opposites may now differ in both magnitude and phase.
To clarify the situation, the Argand diagram for the (-h) reflection is flipped to superimpose the conserved components of the structure factors.
By moving the contribution of the anomalous scatterers to the other end of the structure factor, and recentering the circles, we can obtain an estimate of the phase of the non-anomalous scattering from the remainder of the structure, in a similar manner to the SIR case.
In real experiments, there are errors in the observations, and also in the substructure model used to describe the isomorphous or anomalous differences. Errors in the data can be represented as a "blurring" of the phase circles. Errors in the substructure model lead to a Gaussian uncertainty in the contribution of the substructure to the total scattering. This uncertainty may be convoluted with the phase circles to further increase the width of the phase circles.
As a result, a clear phase indication is no longer obtained, instead we obtain a phase probability distribution which may indicate that some phases are more probable than others.
The final result of the experimental phasing calculation is generally represented in terms of Hendrickson Lattman coefficients. Modern software may also output a set of synthetic structure factors which represent the best combination of the different experimental observations, which lead to reduced errors.
--Kevin Cowtan 02:19, 23 April 2008 (CDT)
