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

International Tables for Crystallography (2006). Vol. F. ch. 21.1, pp. 501-502   | 1 | 2 |

Section 21.1.7.2.1. Geometry and stereochemistry

G. J. Kleywegta*

aDepartment of Cell and Molecular Biology, Uppsala University, Biomedical Centre, Box 596, SE-751 24 Uppsala, Sweden
Correspondence e-mail: gerard@xray.bmc.uu.se

21.1.7.2.1. Geometry and stereochemistry

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The covalent geometry of a model can be assessed by comparing bond lengths and angles to a library of `ideal' values. In the past, every refinement and modelling program had its own set of `ideal' values. This even made it possible to detect (with 95% accuracy) with which program a model had been refined, simply by inspecting its covalent geometry (Laskowski, Moss & Thornton, 1993[link]). Nowadays, standard sets of ideal bond lengths and bond angles derived from an analysis of small-molecule crystal structures from the CSD (Allen et al., 1979[link]) are available for proteins (Engh & Huber, 1991[link]; Priestle, 1994[link]) and nucleic acids (Parkinson et al., 1996[link]). For other entities, typical bond lengths and bond angles can be taken from tables of standard values (Allen et al., 1987[link]) or derived by other means (Kleywegt & Jones, 1998[link]; Greaves et al., 1999[link]).

For bond lengths, the r.m.s. deviation from ideal values is invariably quoted. Deviations from ideality of bond angles can be expressed directly as an angular r.m.s. deviation or in terms of angle distances (i.e. the angle ∠ABC is measured by the 1–3 distance |AC|; note that this distance is also implicitly dependent on the bond lengths |AB| and |BC|). There are some indications that protein geometry cannot always be captured by assuming unimodal distributions (i.e. geometric features with only a single `ideal' value). For example, Karplus (1996[link]) found that the main-chain bond angle τ3 (∠N—Cα—C) varies as a function of the main-chain torsion angles ϕ and ψ.

Chirality is another important criterion in the case of biomacromolecules: most amino-acid residues will have the L configuration for their Cα atom. Also, the Cβ atoms of threonine and isoleucine residues are chiral centres (IUPAC–IUB Commission on Biochemical Nomenclature, 1970[link]; Morris et al., 1992[link]). Chirality can be assessed in terms of improper torsion angles or chiral volumes. For example, to check if the Cα atom of any residue other than glycine has the L configuration, the improper (or virtual) torsion angle Cα—N—C—Cβ should have a value of about +34° (a value near −34° would indicate a D-amino acid). The torsion angle is called improper or virtual because it measures a torsion around something other than a covalent bond, in this case the N—C `virtual bond'. The chiral volume is defined as the triple scalar product of the vectors from a central atom to three attached atoms (Hendrickson, 1985[link]). For instance, the chiral volume of a Cα atom is defined as[V_{{\rm C}^{\alpha}} = ({\bf r}_{\rm N} - {\bf r}_{\rm C^{\alpha}})\cdot [({\bf r}_{\rm C} - {\bf r}_{\rm C^{\alpha}}) \times ({\bf r}_{\rm C^{\beta}} - {\bf r}_{\rm C^{\alpha}})],] where rX is the position vector of atom X. It should be noted that the chiral volume also implicitly depends on the bond lengths and angles involving the four atoms.

Another issue to consider is that of moieties that are necessarily planar (e.g. carboxylate groups, phenyl rings; Hooft et al., 1996a[link]). Again, planarity can be assessed in two different ways: by inspecting a set of (possibly improper) torsion angles and calculating their r.m.s. deviation from ideal values (e.g. all ring torsions in a perfectly flat phenyl ring should be 0°) or by fitting a least-squares plane through each set of atoms and calculating the r.m.s. distance of the atoms to that plane. Note that for double bonds, cis and trans configurations cannot be distinguished by deviations from a least-squares plane, but they can be distinguished by an appropriately defined torsion angle.

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