Chapter 18.5 Coordinate uncertainty

Crystallography of biological macromolecules

First Online Edition (2006)

Part 18. Refinement

  1. D. W. J. Cruickshank

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000697

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Cruickshank, D. W. J. 2006. Coordinate uncertainty. International Tables for Crystallography. F:18:18.5:403–418.

Author Information

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

Publication History

  1. Published Online: 1 JAN 2006



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