InternationalCrystallography of biological macromoleculesTables for Crystallography Volume F Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F, ch. 13.4, pp. 279-280
## Section 13.4.2. Noncrystallographic symmetry (NCS) |

Crystallographic symmetry is valid for the infinite crystal lattice. Any crystallographic symmetry element relates all points within the crystal to equivalent points elsewhere. In contrast, an NCS operator is valid only locally within a finite volume (Fig. 13.4.2.1); if a periodic structure is superimposed on itself after operation with an NCS operator, it will superimpose only within the envelope^{1} defining the limits of the local symmetry.

A product of superimposed periodic structures will be non-periodic, containing only the point symmetry of the noncrystallographic operators (Fig. 13.4.2.2). This fact can frequently be used to select a molecular envelope that was not obvious prior to noncrystallographic averaging [see *e.g.* Buehner *et al.* (1974) or Lin *et al.* (1986)]. Although no knowledge of the crystallographic envelope is needed for this first averaging, it is necessary to determine it for the averaged molecular structure within the crystallographic cell to permit Fourier back-transformation.

There must be space between the confining envelopes governed by the local symmetry. Only the crystallographic symmetry is valid within this space. In the limit, when this space has diminished to zero, the local symmetry will have become a true crystallographic operator.

The definition of NCS can be extended to symmetry that relates similar objects in different crystal lattices. An operation that relates an object in one lattice to an equivalent object in another lattice will apply only to the chosen objects in each lattice. Beyond the confines of the chosen objects, there will be no coincidence of pattern.

Two kinds of NCS elements may be defined: proper and improper. The former satisfies a closed point group [*e.g.* a 17-fold rotation as occurs in tobacco mosaic virus disk protein (Champness *et al.*, 1976)]. Here, it does not matter whether a rotation axis is applied right- or left-handedly; the result is indistinguishable. On the other hand, the relationship between different molecules in a crystallographic asymmetric unit is unlikely to be a closed point group. Thus, a rotation in one direction (followed by a translation) might achieve superposition of the two molecules, while a rotation in the opposite direction would not. This is called an improper NCS operator. An operation which takes a molecule in one unit cell to that in another unit cell (initially, the cells are lined up with, say, their orthogonalized *a*, *b* and *c* axes parallel) must equally be an improper rotation.

The position in space of a noncrystallographic rotation symmetry operator can be arbitrarily assigned. The rotation operation will orient the two molecules similarly. A subsequent translation, whose magnitude depends upon the location of the NCS operator, will always be able to superimpose the molecules (Fig. 13.4.2.3). Nevertheless, it is possible to select the position of the NCS axis such that the translation is a minimum, and that will occur when the translation is entirely parallel to the noncrystallographic rotation axis.

The position of an NCS axis, like everything else in the unit cell, must be defined with respect to a selected origin. Consider the noncrystallographic rotation defined by the matrix [C]. Then, if the point **x** is rotated to **x**′ (both defined with respect to the selected origin and axial system), where **d** is a three-dimensional vector which expresses the translational component of the NCS operation. The magnitude of the components of **d** is quite arbitrary unless the position of the rotation axis is defined. If the rotation axis represents a proper NCS element, there will exist a point **x** on the rotation axis, when positioned to eliminate translation, such that it is rotated onto **x**′. It follows that for such a point from which **d** can be determined if the position of the molecular centre is known. Note that if, and only if, the noncrystallographic rotation axis passes through the crystallographic origin.

The presence of proper NCS in a crystal can help phase determination considerably. Consider, for example, a tetramer with 222 symmetry. It is not necessary to define the chemical limits of any one polypeptide chain as the NCS is true everywhere within the molecular envelope and the boundaries of the polypeptide chain are irrelevant to the geometrical considerations. The electron density at every point within the molecular envelope (which itself must have 222 symmetry) can be averaged among all four 222-related points without any chemical knowledge of the configuration of the monomer polypeptide. On the other hand, if there is only improper NCS, then the envelope must define the limits of one noncrystallographic asymmetric unit, although the crystallographic asymmetric unit contains two or more such units.

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