Tables for
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

International Tables for Crystallography (2006). Vol. F. ch. 18.5, pp. 406-408   | 1 | 2 |

Section Unrestrained and restrained inversions for concanavalin A

D. W. J. Cruickshanka*

aChemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: Unrestrained and restrained inversions for concanavalin A

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G. M. Sheldrick extended his SHELXL96 program (Sheldrick & Schneider, 1997[link]) to provide extra information about protein precision through the inversion of least-squares full matrices. His programs have been used by Deacon et al. (1997)[link] for the high-resolution 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 full-matrix 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 [\sigma (x)], [\sigma (y)], [\sigma (z)] and [\sigma (r)] for all atoms, and to [\sigma (l)] and [\sigma (\theta)] for all bond lengths and angles. [\sigma (r)] is defined as [[\sigma^{2}(x) + \sigma^{2}(y) + \sigma^{2}(z)]^{1/2}]. 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.[link] and[link]. Fig.[link] shows [\sigma (r)] versus [B_{\rm eq}] for the fully occupied atoms of the protein (a few atoms with B > 60 Å2 are off-scale). The points are colour-coded black for carbon, blue for nitrogen and red for oxygen. Fig.[link] shows the restrained results, and Fig.[link] shows the unrestrained diffraction-data-only results. Superposed on both sets of data points are least-squares quadratic fits determined with weights [1/B^{2}]. At high B, the unrestrained [\sigma_{\rm diff}(r)] can be at least double the restrained [\sigma_{\rm res}(r)], e.g., for carbon at B = 50 Å2, the unrestrained [\sigma_{\rm diff}(r)] is about 0.25 Å, whereas the restrained [\sigma_{\rm res}(r)] is about 0.11 Å. For B < 10 Å2, both [\sigma (r)]'s fall below 0.02 Å and are around 0.01 Å at B = 6 Å2.


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Plots of [\sigma (r)] versus [B_{\rm eq}] for concanavalin A with 0.94 Å data, (a) restrained full-matrix [\sigma_{\rm res}(r)], (b) unrestrained full-matrix [\sigma_{\rm diff}(r)]. Carbon black, nitrogen blue, oxygen red.


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Plots of [\sigma (l)] versus average [B_{\rm eq}] for concanavalin A with 0.94 Å data, (a) restrained full-matrix [\sigma_{\rm res}(l)], (b) unrestrained full-matrix [\sigma_{\rm diff}(l)]. C—C black, C—N blue, C—O red.

For B < 10 Å2, 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 [\sigma_{\rm diff}(r)] in Fig.[link] are almost as small as the restrained [\sigma_{\rm res}(r)] in Fig.[link]. [The quadratic fits of the restrained results in Fig.[link] are evidently slightly imperfect in making [\sigma_{\rm res}(r)] tend almost to 0 as B tends to 0.]

Fig.[link] shows [\sigma (l)] versus [B_{\rm eq}] for the bond lengths in the protein. The points are colour-coded 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.[link] 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.[link] goes off the scale of the diagram. But for B < 10 Å2, both distributions fall to around 0.01 Å.

The differences between the restrained and unrestrained [\sigma (r)] and [\sigma (l)] can be understood through the two-atom model for restrained refinement described in Section 18.5.3[link]. For that model, the equation [1 / \sigma_{\rm res}^{2} (l) = 1 / \sigma_{\rm diff}^{2} (l) + 1 / \sigma_{\rm geom}^{2} (l) \eqno(] relates the bond-length s.u. in the restrained refinement, [\sigma_{\rm res}(l)], to the [\sigma_{\rm diff}(l)] of the unrestrained refinement and the s.u. [\sigma_{\rm geom}(l)] assigned to the length in the stereochemical dictionary. In the refinements, [\sigma_{\rm geom}(l)] was 0.02 Å for all bond lengths. When this is combined in ([link] with the unrestrained [\sigma_{\rm diff}(l)] of any bond, the predicted restrained [\sigma_{\rm res}(l)] is close to that found in the restrained full matrix.

It can be seen from Fig.[link] that many bond lengths with average B < 10 Å2 have [\sigma_{\rm diff}(l)\lt 0.014] Å. For these bonds the diffraction data have greater weight than the stereochemical dictionary. Some bonds have [\sigma_{\rm diff}(l)] as low as 0.0080 Å, with [\sigma_{\rm res}(l)] 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 [\sigma_{\rm diff}(l)] (i.e., high B), equation ([link] predicts that [\sigma_{\rm res}(l) = \sigma_{\rm geom}(l) = 0.02] Å, as is found in Fig.[link].

In an isotropic approximation, [\sigma (r) = 3^{1/2}\sigma (x)]. Equation ([link] of the two-atom model can be recast to give [\sigma_{\rm res}^{2} (r) = \sigma_{\rm diff}^{2} (r) \left\{\left[\sigma_{\rm diff}^{2} (r) + 3(0.02)^{2}\right]\bigg/\left[2\sigma_{\rm diff}^{2} (r) + 3(0.02)^{2}\right]\right\}. \eqno(] For low B, say [B \leq 15\ \hbox{\AA}^{2}] in concanavalin, ([link] gives quite good predictions of [\sigma_{\rm res}(r)] from [\sigma_{\rm diff}(r)]. For instance, for a carbon atom with B = 15 Å2, the quadratic curve for carbon in Fig.[link] shows [\sigma_{\rm diff}(r) = 0.034] Å, and Fig.[link] shows [\sigma_{\rm res}(r) = 0.029] Å. While if [\sigma_{\rm diff}(r) = 0.034] Å is used with ([link], the resulting prediction for [\sigma_{\rm res}(r)] is 0.028 Å.

However, for high B, say B = 50 Å2, the quadratic curve for carbon in Fig.[link] shows [\sigma_{\rm diff}(r) = 0.25] Å, and Fig.[link] shows [\sigma_{\rm res}(r) = 0.11] Å, whereas ([link] leads to the poor estimate [\sigma_{\rm res}(r) = 0.18] Å.

Thus at high B, equation ([link] from the two-atom model does not give a good description of the relationship between the restrained and unrestrained [\sigma (r)]. 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 [\hbox{C}_{\alpha}] and N atoms. Consequently, for a high-B atom, when the restraints are applied it is coupled to several other atoms in a group, and its [\sigma_{\rm res}(r)] is lower, compared with the diffraction-data-only [\sigma_{\rm diff}(r)], by a greater amount than would be expected from the two-atom model.


First citationDeacon, 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 small-molecule accuracy at 0.94 Å resolution. J. Chem. Soc. Faraday Trans. 93, 4305–4312.Google Scholar
First citationSheldrick, G. M. & Schneider, T. R. (1997). SHELXL: high resolution refinement. Methods Enzymol. 277, 319–343.Google Scholar

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