Chapter 18.5 Coordinate uncertainty

Crystallography of biological macromolecules

Second Online Edition (2012)

Part 18. Refinement

  1. D. W. J. Cruickshank

Published Online: 14 APR 2012

DOI: 10.1107/97809553602060000859

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Cruickshank, D. W. J. 2012. Coordinate uncertainty. International Tables for Crystallography. F:18:18.5:499–511.

Author Information

  1. Chemistry Department, UMIST, Manchester M60 1QD, England

Publication History

  1. Published Online: 14 APR 2012


Full-matrix least-squares is taken as the basis for an examination of protein-structure precision. A two-atom 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. As a protein example, data with 0.94 Å resolution for concanavalin A are used in unrestrained and restrained full-matrix inversions to provide e.s.d.’s 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 also between σ(r) and σ(l). An unrestrained full-matrix inversion for an immunoglobulin with 1.7 Å data is also discussed. Several approximate methods are examined critically. These include Luzzati plots and the diffraction-component precision index (DPI). The DPI estimate of σ(r, Bavg) is given by a simple formula, which uses R or Rfree and is based on a very rough approximation to the least-squares method. Examples show its usefulness as a precision comparator for high- and low-resolution structures.


  • R factors;
  • Rfree;
  • accuracy;
  • atomic displacement parameters;
  • block-matrix approximation;
  • concanavalin A;
  • coordinate uncertainty;
  • DPI;
  • diffraction-component precision index;
  • errors;
  • free R factor;
  • full-matrix inversion;
  • goodness of fit;
  • least-squares methods;
  • low-resolution structures;
  • Luzzati plot;
  • modified Fourier method for estimating coordinate uncertainty;
  • normal equations;
  • position error;
  • precision;
  • refinement;
  • residual function;
  • restrained full-matrix inversion for concanavalin A;
  • restrained refinement;
  • restraints;
  • temperature factors;
  • unrestrained full-matrix inversion;
  • weighting