International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. F, ch. 25.2, pp. 697698

Several methods are used for structurefactor and phasing calculations depending on the nature of the model and how the results will be used. The methods available in the package are described below.
Phasing by heavyatombased methods (isomorphous replacement and/or anomalous scattering) begins when one or more `scaled' data sets are input to the program PHASIT (batch). Userspecified rejection criteria are first applied to each data set, and structure factors corresponding to the heavyatom or anomalousscatterer substructure are computed from where is the occupancy, is the (possibly complex) scattering factor, is the isotropic temperature parameter and , , are the fractional coordinates of the jth atom. Sc is a scale factor relating the calculated structure factor (absolute scale) to the scale of the observed data. The summation is taken over all heavy atoms or anomalous scatterers in the unit cell. Alternatively, anisotropic temperature parameters can be used for each atom if desired. A subset of reflections comprising all centric data (plus the largest 25% of the isomorphous or anomalous differences if there are insufficient centric data) is selected and used to estimate Sc by a leastsquares fit to the observed differences. Initial estimates of the `standard error' E (expected lack of closure) are determined from this subset as a function of F magnitude, treating centric and acentric data separately. SIR (single isomorphous replacement) or SAS (singlewavelength anomalous scattering) phase probability distributions are given by where the lack of closure is defined by for isomorphousreplacement data and for anomalousscattering data, with the and − superscripts denoting members of a Bijvoet pair, and with ϕ denoting the protein phase, and and denoting the heavyatom structurefactor amplitude and phase, respectively. The distributions, however, are cast in the A, B, C, D form (Hendrickson & Lattman, 1970). After all input data sets are processed in this manner, the individual phase probability distributions for common reflections are combined via with k as a normalization constant and the sums taken over all contributing data sets. The resulting combined distributions are then integrated to yield a centroid phase and figure of merit for each reflection. The standard error estimates, E, as a function of structurefactor magnitude are then updated for each data set, this time using all reflections and a probabilityweighted average over all possible phase values for the contribution from each reflection (Terwilliger & Eisenberg, 1987a,b). With these updated standard error estimates, the individual SIR and/or SAS phase probability distributions are recomputed for all reflections and combined again to yield an improved centroid phase and figure of merit for each reflection. The resulting phases, figures of merit and probability distribution information are then available for use in map calculations or for further parameter or phase refinement. This method is used to produce MIR (multiple isomorphous replacement), SIRAS (single isomorphous replacement with anomalous scattering) MIRAS (multiple isomorphous replacement with anomalous scattering) and MAD phases as well as other possible phase combinations.
Structurefactor amplitudes and phases for a macromolecular structure can be computed directly from atomic coordinates corresponding to a tentative model with the programs PHASIT and GREF (both run as batch processes). This allows one to obtain structurefactor information from an input model typically derived from a partial chain trace or from a molecularreplacement solution. Equation (25.2.1.5) is used, but this time the sum is taken over all known atoms in the cell, and the scale factor is refined by least squares against the native amplitudes rather than against the magnitudes of isomorphous or anomalous differences. The computed structure factors may be used directly for map calculations, including `omit' maps, or for combination with other sources of phase information. One can output probability distribution information for the calculated phases, if desired, as well as coefficients for various Fourier syntheses, including those using sigma_A weighting (Read, 1986) for the generation of reducedbias native or difference maps.
For the purpose of improving phases by densitymodification methods, such as solvent flattening, negativedensity truncation and/or NC symmetry averaging, one must compute structure factors by Fourier inversion of an electrondensity map rather than from atomic coordinates. The program MAPINV (batch) is a companion program to FSFOUR and carries out this inverse Fourier transform. It accepts a fullcell map in FSFOUR format and inverts it to produce amplitudes and phases for a selected set of reflections when given the target range of Miller indices. A variableradix 3D fast Fourier transform algorithm is used. Optionally, the program can modify the density prior to inversion by truncation below a cutoff and/or by squaring the density values. Other types of density modification are handled by different programs in the package and are carried out prior to running MAPINV. The indices, calculated amplitude and phase are written to a file for each target reflection.
References
Hendrickson, W. A. & Lattman, E. E. (1970). Representation of phase probability distributions for simplified combination of independent phase information. Acta Cryst. B26, 136–143.Google ScholarRead, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.Google Scholar
Terwilliger, T. C. & Eisenberg, D. (1987a). Isomorphous replacement: effects of errors on the phase probability distribution. Acta Cryst. A43, 6–13.Google Scholar
Terwilliger, T. C. & Eisenberg, D. (1987b). Isomorphous replacement: effects of errors on the phase probability distribution. Erratum. Acta Cryst. A43, 286.Google Scholar