Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 99   | 1 | 2 |

Section Discussion of macromolecular refinement techniques

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Discussion of macromolecular refinement techniques

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The impossibility of carrying out a full-matrix least-squares refinement of a macromolecular crystal structure, caused by excessive computational cost and by the paucity of observations, led Diamond (1971)[link] to propose a real-space refinement method in which stereochemical knowledge was used to keep the number of free parameters to a minimum. Refinement took place by a least-squares fit between the `observed' electron-density map and a model density consisting of Gaussian atoms. This procedure, coupled to iterative recalculation of the phases, led to the first highly refined protein structures obtained without using full-matrix least squares (Huber et al., 1974[link]; Bode & Schwager, 1975[link]; Deisenhofer & Steigemann, 1975[link]; Takano, 1977a[link],b[link]).

Real-space refinement takes advantage of the localization of atoms (each parameter interacts only with the density near the atom to which it belongs) and gives the most immediate description of stereochemical constraints. A disadvantage is that fitting the `observed' electron density amounts to treating the phases of the structure factors as observed quantities, and to ignoring the experimental error estimates on their moduli. The method is also much more vulnerable to series-termination errors and accidentally missing data than the least-squares method. These objections led to the progressive disuse of Diamond's method, and to a switch towards reciprocal-space least squares following Agarwal's work.

The connection established above between the Cruickshank–Agarwal modified Fourier method and the simple use of the chain rule affords a partial refutation to both the premises of Diamond's method and to the objections made against it:

  • (i) it shows that refinement can be performed through localized computations in real space without having to treat the phases as observed quantities;

  • (ii) at the same time, it shows that measurement errors on the moduli can be fully utilized in real space, via the Fourier synthesis of the functional derivative [\partial R/\partial \rho\llap{$-\!$}^{\rm calc}({\bf x})] or by means of the coefficients of a quadratic model of R in a search subspace.


Bode, W. & Schwager, P. (1975). The refined crystal structure of bovine β-trypsin at 1.8Å resolution. II. Crystallographic refinement, calcium-binding site, benzamidine binding site and active site at pH 7.0. J. Mol. Biol. 98, 693–717.
Deisenhofer, J. & Steigemann, W. (1975). Crystallographic refinement of the structure of bovine pancreatic trypsin inhibitor at 1.5 Å resolution. Acta Cryst. B31, 238–250.
Diamond, R. (1971). A real-space refinement procedure for proteins. Acta Cryst. A27, 436–452.
Huber, R., Kulka, D., Bode, W., Schwager, P., Bartels, K., Deisenhofer, J. & Steigemann, W. (1974). Structure of the complex formed by bovine trypsin and bovine pancreatic trypsin inhibitor. II. Crystallographic refinement at 1.9 Å resolution. J. Mol. Biol. 89, 73–101.
Takano, T. (1977a). Structure of myoglobin refined at 2.0 Å resolution. I. Crystallographic refinement of metmyoglobin from sperm whale. J. Mol. Biol. 110, 537–568.
Takano, T. (1977b). Structure of myoglobin refined at 2.0 Å resolution. II. Structure of deoxymyoglobin from sperm whale. J. Mol. Biol. 110, 569–584.

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