 STARANISO
Anisotropy of the Diffraction Limit and
Bayesian Estimation of Structure Amplitudes


 Perform an anisotropic
diffraction cutoff of the merged intensities, instead of the
traditional isotropic cutoff, using a locallyaveraged mean
I/σ(I) as the cutoff threshold. The
local average is calculated within a sphere of reciprocal space centered
on each reflection (default radius r* = 0.15Å^{1}),
and the contributions to the average are weighted by the exponential
function w = exp(4(s*/r*)^{2}) of the
reciprocalspace distance s* from the center.
 Determine the anisotropy of the observed intensity
distribution, corrected where necessary by the systematic absence factor
(Wilson [1987]), using either the errorfree likelihood function
proposed by Popov & Bourenkov [2003], or the Bayesian likelihood
function (default) which uses the FrenchWilson formalism and which
takes experimental errors into account. In either case, by default
a precalculated expected intensity profile is used (thanks to Alexander
Popov for furnishing this); this assumes an average solvent content so
in cases where the actual solvent content is much higher than normal it
may not provide an accurate estimate of the contribution of bulk water.
Maximum likelihood optimization of the overall scale and the elements
of the overall anisotropic displacement tensor that are not constrained
by the pointgroup symmetry is performed. Note that the default P &
B profile was obtained by averaging the observed profiles of a number of
proteinonly structures so is not strictly applicable to structures
containing a substantial proportion of nucleic acid.
 Optionally, renormalize the intensity profile by applying a
d*dependent scale factor determined such that the mean
normalized intensity (Z) is 1 in all d* bins. This
may help to average out the effect of differences from the averaged
profile.
 Use the anisotropy
from step #13 to compute an anisotropic prior of the expected intensity,
i.e. divide the expected intensity obtained from the profile by
the scale/anisotropy correction.
 Perform Bayesian estimation of structure amplitudes by the method of
French & Wilson [1978], but using the anisotropic prior in place of the
traditional isotropic prior originally suggested by F & W.
STARANISO incorporates subroutines from the Netlib repository, in place
of the approximate lookup tables used in TRUNCATE, to compute
highaccuracy
parabolic cylinder functions (scaled to avoid numerical
under/overflow issues: Gil et al. [2006]) and thereby obtain all
the required moments.
 Input anomalous data are treated differently from nonanomalous data
in the Bayesian estimation. If anomalous data are present on input
it is naturally assumed that the anomalous differences are statistically
significant (otherwise what is the point of keeping the Bijvoet pairs
separate?). If this is not the case then the correct course of
action is to rerun the merging step, this time also merging the Bijvoet
pairs, since this will deal with outliers correctly. Otherwise the
Bayesian estimation is performed twice per unique reflection on the
separately merged means of I[+] and I[] (where these are
observed), not on the overall merged mean including all
I[+] and I[].
This is because the Bayesian estimation assumes a centric or acentric
Wilson distribution as appropriate, but the average of two random
variates each with an acentric distribution with different expected
values does not necessarily itself have an acentric distribution.
Hence it is not correct to perform the Bayesian estimation as currently
implemented on the average of two Wilson intensity variates with
different expected values. Rather I[+] and I[]
should be separately converted to Fs, and then averaged.
There are further
issues concerning the optimal procedure for averaging F[+]
and F[] when they have different standard uncertainties.
 Optionally correct the amplitudes for anisotropy.
 Finally, create a new MTZ file containing F and
σ(F) columns (and also anomalous F and
σ(F) columns if anomalous I columns were read
in). Note that it is formally invalid to take Fs from
the Bayesian estimation and square them in a misguided attempt to
recover the Is! (needed for example by some twinning tests).
Rather, the posterior Is should be estimated by the same
procedure as for the posterior Fs. For this reason there is
an option to output the posterior intensities (MTZ column labels
Ipost, SIGIpost etc.).
REFERENCES
French, S. & Wilson, K.S. (1978) "On the treatment of negative
intensity observations." Acta Cryst. A34, 517525.
See also: "Bayesian
treatment of negative intensity measurements in crystallography" .
Gil, A., Segura, J. & Temme, N.M. (2006) "Algorithm 850: Real
parabolic cylinder functions U(a,x),
V(a,x)." ACM Transactions on Mathematical Software
(TOMS). 32, 10212. See also: "Computing
the real parabolic cylinder functions U(a,x),
V(a,x)".
Morris, R.J., Blanc, E. & Bricogne, G. (2003) "On the interpretation
and use of <E^{2}>(d*) profiles."
Acta Cryst. D60, 22740.
Popov, A.N. & Bourenkov, G.P. (2003) "Choice of datacollection
parameters based on statistical modelling." Acta Cryst. D59,
114553.
Wilson, A.J.C. (1987) "Treatment of enhanced zones and rows in
normalizing intensities." Acta Cryst. A43, 2502.