Highlighted Articles from IUCr Journals
John Helliwell, Editor-in-Chief of Acta Cryst and Chairman of
the IUCr's Commission on Journals has arranged for reviews of
articles in current issues of IUCr journals to appear in this
and future issues of the IUCr Newsletter.
The Foundations of Macromolecular Structure Refinement
Techniques for macromolecular refinement continue to advance
apace. The quality of data is improving with easier access to
synchrotron sources, computing power is increasing while the
cost is declining, and new algorithms are helping to put the
calculations on a firmer statistical basis. In the past six months,
three papers have appeared in Acta Cryst. A and D discussing
the role of the normal matrix, the second-derivative matrix of
the refinement target function.
The first problem to address is simply one of size; the normal
matrix has the dimension of the number of parameters squared,
and the time taken to generate it by classical techniques is
proportional to the square of the number of atoms multiplied
by the number of observations. The papers by Tronrud ["Efficient
calculation of normal matrix in least-squares refinement of macromolecular
structures", Acta Cryst. A55 (1999), 700-703]; Murshudov,
Vagin, Lebedev, Wilson and Dodson ["Efficient anisotropic
refinement of macromolecular structures using FFT", Acta
Cryst. D55 (1999), 247-255]; and Templeton ["Faster calculation
of the full matrix for least-squares refinement", Acta Cryst.
A55 (1999), 695-699] all discuss ways of speeding up the calculations.
These methods can be traced back first to Cruickshank who described
an approach to refinement using Fourier methods in the 1950s
[Acta Cryst. 5 (1952), 511-518; Acta Cryst. 9 (1956), 747-753],
and then to Agarwal who showed how to exploit fast Fourier transforms
(FFTs) [Acta Cryst. A34 (1978), 791-809]. He outlined a method
for the fast calculation of second-derivative matrices, which
is extended by Tronrud and generalized by Murshudov et al. Templeton
extends another aspect of Agarwal's results, i.e., he makes approximations
by replacing summations with integration assuming that the reciprocal
space sampling is dense enough to justify this.
Most experience in exploiting the full normal matrix is in the
field of small-molecule crystallography; here, it is feasible
to generate the derivatives analytically, and to perform the
inversion. When there is a high ratio of observations to parameters,
its use has been shown to speed up the rate of convergence, and
to reduce the likelihood of reaching a false minimum. When the
refinement has converged, the inverse of the normal matrix reveals
both the precision of the model parameters and the correlations
between them. The eigenvectors and eigenvalues of the normal
matrix provide information about parameters or parameter combinations
that are not determined by the original data.
For large macromolecules, even if the full normal matrix is generated,
there will be considerable problems in the inversion of such
a large array. However, there are now many numerical tools developed
within a variety of disciplines addressing such problems. The
stability of the inversion procedure can be analysed using eigenvalues
and -vectors, and this was discussed by Cowtan and
Ten Eyck at the 18th European Crystallographic Meeting in Prague,
Czech Republic in August 1998. They have performed such an eigensystem
analysis on the normal matrices resulting from the least-squares
refinement of a small metalloprotein using two datasets and models
determined at different resolutions (all performed by classical
techniques using the program SHELXL at the San Diego Supercomputer
Center) [Sheldrick (1995), SHELXL93, a Program for the Refinement
of Crystal Structures from Diffraction Data. Institut für
Anorganische Chemie, Göttingen, Germany]. As a protein refinement
is usually underdetermined without the application of geometric
restraints, and these contributions are routinely included in
the minimization residual, they have repeated their analysis
including the contributions from such restraints. They show that
the eigenvalue spectra reveal considerable information about
the conditioning of the problem as the resolution varies. In
the case of a restrained refinement, the spectra also provide
information about the impact of various restraints on the refinement.
The established procedure used in macromolecular refinement programs
such as SFALL/PROLSQ, XPLOR, and REFMAC has been to generate
only the diagonal elements of the normal matrix for the X-ray
data. Although this limits the rate of convergence, it does mean
that each cycle is completed quickly because structure factors
and gradients can both be generated using FFTs. All such programs
incorporate the restraint derivative terms into the matrix, with
some rather arbitrary weighting relative to the X-ray elements.
The derivatives for these contributions can be generated simply,
and provide off-diagonal terms for linked atoms only. Templeton's
approximations for the maximum contribution for the X-ray off-diagonal
terms show that these fall off rapidly with the distance between
the atom pairs. Hence the geometric restraint contributions may
well swamp any X-ray contribution to off-diagonal elements, especially
when the resolution of the X-ray data is limited. But as Cowtan
and Ten Eyck show, there are special situations where unexpected
interactions link distant atoms, and these can degrade the course
of refinement.
The FFT techniques for estimating the X-ray off-diagonal elements
will make it possible to derive a much improved form for the
full normal matrix in a realistic time, and we can look forward
to further analyses, and a better understanding of the proper
parameterization and accuracy of macromolecular structures.
Eleanor Dodson, U. of York, UK
Area Detectors - Current Trends and Why You Can't Have
the Perfect Bucket
There are three factors that prevent you from owning the ideal
area detector -the one with infinite count rate, zero dead time
and unity detection efficiency out to 25 keV, namely:
* The "Laws of Physics" - don't ask for zero response
time and zero noise floor.
* Available materials - it would be nice to have high-quality
semiconductor wafers with absorbance coefficients of a few tens
of micrometres for Eh = 3-30 keV and a direct bandgap so that
only about a dozen electron-hole pairs are created per incident
photon; but you've got silicon, gallium arsenide, indium phosphide
and a few other esoteric materials with suspect mechanical properties.
