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Full-matrix least squares is taken as the basis for an examination of protein structure precision. A two-atom protein model is used to compare the precisions of unre­strained and restrained refinements. In this model, restrained refinement determines a bond length which is the weighted mean of the unrestrained diffraction-only length and the geometric dictionary length. Data of 0.94 Å resolution for the 237-residue protein concanavalin A are used in unrestrained and restrained full-matrix inversions to provide standard uncertainties σ(r) for positions and σ(l) for bond lengths. σ(r) is as small as 0.01 Å for atoms with low Debye B values but increases strongly with B. The results emphasize the distinction between unrestrained and restrained refinements and between σ(r) and σ(l). Other full-matrix inversions are reported. Such inversions require massive calculations. Several approximate methods are examined and compared critically. These include a Fourier map formula [Cruickshank (1949). Acta Cryst. 2, 65–82], Luzzati plots [Luzzati (1952). Acta Cryst. 5, 802–810] and a new diffraction-component precision index (DPI). The DPI estimate of σ(r, Bavg) is given by a simple formula. It uses R or Rfree and is based on a very rough approximation to the least-squares method. Many examples show its usefulness as a precision comparator for high- and low-resolution structures. The effect of restraints as resolution varies is examined. More regular use of full-matrix inversion is urged to establish positional precision and hence the precision of non-dictionary distances in both high- and low-resolution structures. Failing this, parameter blocks for representative residues and their neighbours should be inverted to gain a general idea of σ(r) as a function of B. The whole discussion is subject to some caveats about the effects of disordered regions in the crystal.
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