International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. F. ch. 18.5, pp. 406408
Section 18.5.4.1. Unrestrained and restrained inversions for concanavalin A^{a}Chemistry Department, UMIST, Manchester M60 1QD, England 
G. M. Sheldrick extended his SHELXL96 program (Sheldrick & Schneider, 1997) to provide extra information about protein precision through the inversion of leastsquares full matrices. His programs have been used by Deacon et al. (1997) for the highresolution refinement of native concanavalin A with 237 residues, using data at 110 K to 0.94 Å refined anisotropically. After the convergence and completion of fullmatrix restrained refinement for the structure, the unrestrained full matrix (coordinates only) was computed and then inverted in a massive calculation. This led to s.u's , , and for all atoms, and to and for all bond lengths and angles. is defined as . For concanavalin A the restrained full matrix was also inverted, thus allowing the comparison of restrained and unrestrained s.u.'s.
The results for concanavalin A from the inversion of the coordinate matrices of order 6402 (= 2134 × 3) are plotted in Figs. 18.5.4.1 and 18.5.4.2. Fig. 18.5.4.1 shows versus for the fully occupied atoms of the protein (a few atoms with B > 60 Å^{2} are offscale). The points are colourcoded black for carbon, blue for nitrogen and red for oxygen. Fig. 18.5.4.1(a) shows the restrained results, and Fig. 18.5.4.1(b) shows the unrestrained diffractiondataonly results. Superposed on both sets of data points are leastsquares quadratic fits determined with weights . At high B, the unrestrained can be at least double the restrained , e.g., for carbon at B = 50 Å^{2}, the unrestrained is about 0.25 Å, whereas the restrained is about 0.11 Å. For B < 10 Å^{2}, both 's fall below 0.02 Å and are around 0.01 Å at B = 6 Å^{2}.

Plots of versus for concanavalin A with 0.94 Å data, (a) restrained fullmatrix , (b) unrestrained fullmatrix . Carbon black, nitrogen blue, oxygen red. 

Plots of versus average for concanavalin A with 0.94 Å data, (a) restrained fullmatrix , (b) unrestrained fullmatrix . C—C black, C—N blue, C—O red. 
For B < 10 Å^{2}, the better precision of oxygen as compared with nitrogen, and of nitrogen as compared with carbon, can be clearly seen. At the lowest B, the unrestrained in Fig. 18.5.4.1(b) are almost as small as the restrained in Fig. 18.5.4.1(a). [The quadratic fits of the restrained results in Fig. 18.5.4.1(a) are evidently slightly imperfect in making tend almost to 0 as B tends to 0.]
Fig. 18.5.4.2 shows versus for the bond lengths in the protein. The points are colourcoded black for C—C, blue for C—N and red for C—O. The restrained and unrestrained distributions are very different for high B. The restrained distribution in Fig. 18.5.4.2(a) tends to about 0.02 Å, which is the standard uncertainty of the applied restraint for 1–2 bond lengths, whereas the unrestrained distribution in Fig. 18.5.4.2(b) goes off the scale of the diagram. But for B < 10 Å^{2}, both distributions fall to around 0.01 Å.
The differences between the restrained and unrestrained and can be understood through the twoatom model for restrained refinement described in Section 18.5.3. For that model, the equation relates the bondlength s.u. in the restrained refinement, , to the of the unrestrained refinement and the s.u. assigned to the length in the stereochemical dictionary. In the refinements, was 0.02 Å for all bond lengths. When this is combined in (18.5.3.16) with the unrestrained of any bond, the predicted restrained is close to that found in the restrained full matrix.
It can be seen from Fig. 18.5.4.2(b) that many bond lengths with average B < 10 Å^{2} have Å. For these bonds the diffraction data have greater weight than the stereochemical dictionary. Some bonds have as low as 0.0080 Å, with around 0.0074 Å. This situation is one consequence of the availability of diffraction data to the high resolution of 0.94 Å. For large (i.e., high B), equation (18.5.3.16) predicts that Å, as is found in Fig. 18.5.4.2(a).
In an isotropic approximation, . Equation (18.5.3.12) of the twoatom model can be recast to give For low B, say in concanavalin, (18.5.4.1) gives quite good predictions of from . For instance, for a carbon atom with B = 15 Å^{2}, the quadratic curve for carbon in Fig. 18.5.4.1(b) shows Å, and Fig. 18.5.4.1(a) shows Å. While if Å is used with (18.5.4.1), the resulting prediction for is 0.028 Å.
However, for high B, say B = 50 Å^{2}, the quadratic curve for carbon in Fig. 18.5.4.1(b) shows Å, and Fig. 18.5.4.1(a) shows Å, whereas (18.5.4.1) leads to the poor estimate Å.
Thus at high B, equation (18.5.4.1) from the twoatom model does not give a good description of the relationship between the restrained and unrestrained . The reason is obvious. Most atoms are linked by 1–2 bond restraints to two or three other atoms. Even a carbonyl oxygen atom linked to its carbon atom by a 0.02 Å restraint is also subject to 0.04 Å 1–3 restraints to chain and N atoms. Consequently, for a highB atom, when the restraints are applied it is coupled to several other atoms in a group, and its is lower, compared with the diffractiondataonly , by a greater amount than would be expected from the twoatom model.
References
Deacon, A., Gleichmann, T., Kalb (Gilboa), A. J., Price, H., Raftery, J., Bradbrook, G., Yariv, J. & Helliwell, J. R. (1997). The structure of concanavalin A and its bound solvent determined with smallmolecule accuracy at 0.94 Å resolution. J. Chem. Soc. Faraday Trans. 93, 4305–4312.Google ScholarSheldrick, G. M. & Schneider, T. R. (1997). SHELXL: high resolution refinement. Methods Enzymol. 277, 319–343.Google Scholar