Refinement: small unit cells

Chandrasekaran, R.; Stubbs, G.

doi:10.1107/97809553602060000702

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

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International Tables for Crystallography (2006). Vol. F. ch. 19.5, pp. 447-448 | 1 | 2 |

Section 19.5.7.2. Refinement: small unit cells

R. Chandrasekaran^a ^* and G. Stubbs^b

^aWhistler Center for Carbohydrate Research, Purdue University, West Lafayette, IN 47907, USA, and ^bDepartment of Molecular Biology, Vanderbilt University, Nashville, TN 37235, USA
Correspondence e-mail: [email protected]

19.5.7.2. Refinement: small unit cells

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The widely used linked-atom least-squares (LALS) technique (Arnott & Wonacott, 1966; Smith & Arnott, 1978) and the variable virtual bond (PS79 ) method (Zugenmaier & Sarko, 1980) were developed for fibre structures. They are similar in principle to the least-squares refinement procedure for crystalline proteins (Hendrickson, 1985), although bond lengths and bond angles are usually kept fixed in the fibre refinements. The function minimized by the LALS program is of the form $[\Omega = \textstyle\sum\limits_{m}\displaystyle w_{m}\Delta F_{m}^{2} + \textstyle\sum\limits_{i}\displaystyle e_{i}\Delta \theta_{i}^{2} + \textstyle\sum\limits_{j}\displaystyle k_{j}\Delta c_{j}^{2} + \textstyle\sum\limits_{n}\displaystyle \lambda_{n}G_{n}. \eqno(19.5.7.1)]$ The first term on the right-hand side is the weighted sum of the squares of the differences, $[\Delta F_{m}]$ , between observed and calculated X-ray structure amplitudes of Bragg reflections or continuous diffraction. Either or both types of data can be used as necessary. The weights, $[w_{m}]$ , are inversely proportional to the estimated variance of the data. The second term minimizes the differences, $[\Delta\theta_{i}]$ , between the expected (standard) values of conformation and bond angles and those in the model; the weights, $[e_{i}]$ , are based on empirically determined variances. The third term is designed to take care of non-bonded interactions and thus keep the model free from steric compression. It includes the deviations from target values of both intra- and inter-chain hydrogen bonds and the differences between acceptable and calculated non-bonded distances for those contacts that are smaller than the acceptable limiting values. The weights, $[k_{j}]$ , are based on the Buckingham energy function for non-bonded contacts and empirical variances for hydrogen bonds. Finally, the fourth term imposes constraints ( $[G_{h}]$ , with Lagrange multipliers $[\lambda_{h}]$ ) for helix connectivity and ring closure, as in a furanose or pyranose, and it vanishes when all such constraints are satisfied. During the refinement, the structure factors are calculated with either the conventional atomic scattering factor f or with a solvent-corrected atomic scattering factor $[f_{w}]$ (Fraser et al., 1978; Chandrasekaran & Radha, 1992) given by the function $[f_{w}(D) = f(D) - v \sigma_{s} \exp (-\pi v^{2/3} D^{2}), \eqno(19.5.7.2)]$ where $[D = (R^{2} + Z^{2})^{1/2}]$ , $[\sigma_{s}]$ is the electron density of the solvent and v is the excluded volume of the atom. If the van der Waals radius of water is taken as 2 Å, $[\sigma_{s}]$ for water is 0.2984 e Å⁻³. Equation (19.5.7.2) allows for the solvent contribution to the diffracted intensity and is particularly useful in studying hydrated fibres in which structured and amorphous water can account for up to 50% of the total mass.

References

Arnott, S. & Wonacott, A. J. (1966). The refinement of the crystal and molecular structures of polymers using X-ray data and stereochemical constraints. Polymer, 7, 157–166.Google Scholar

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Hendrickson, W. A. (1985). Stereochemically restrained refinement of macromolecular structures. Methods Enzymol. 115, 252–270.Google Scholar

Smith, P. J. C. & Arnott, S. (1978). LALS: a linked-atom least-squares reciprocal-space refinement system incorporating stereochemical restraints to supplement sparse diffraction data. Acta Cryst. A34, 3–11.Google Scholar

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International Tables for Crystallography (2006). Vol. F. ch. 19.5, pp. 447-448