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. 2.1, pp. 46-47   | 1 | 2 |

## Section 2.1.2. Symmetry

J. Drentha*

aLaboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: j.drenth@chem.rug.nl

### 2.1.2. Symmetry

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A symmetry operation can be defined as an operation which, when applied, results in a structure indistinguishable from the original one. According to this definition, the periodic repetition along a, b and c represents translational symmetry.

In addition, rotational symmetry exists, but only rotational angles of 60, 90, 120, 180 and are allowed (i.e. rotation over 360/n degrees, where n is an integer). These correspond to n-fold rotation axes, with and 1 (identity), respectively. Rotation axes with or are not found as crystallographic symmetry axes, because translations of unit cells containing these axes do not completely fill three-dimensional space. Another type of rotational symmetry axis is the screw axis. It combines a rotation with a translation. For a twofold screw axis, the translation is over 1/2 of the unit-cell length in the direction of the axis; for a threefold screw axis, it is 1/3 or 2/3 etc. In this way, the translational symmetry operators can be obeyed. The requirement that translations are 1/2, 1/3, 2/3 etc. of the unit-cell length does not exist for individual objects that are not related by crystallographic translational symmetry operators. For instance, an α-helix has 3.6 residues per turn.

Besides translational and rotational symmetry operators, mirror symmetry and inversion symmetry exist. Mathematically, it can be proven that not all combinations of symmetry elements are allowed, but that 230 different combinations can occur. They are the space groups which are discussed extensively in IT A (2005). The graphical and printed symbols for the symmetry elements are also found in IT A (Chapter 1.4 ).

Biological macromolecules consist of building blocks such as amino acids or sugars. In general, these building-block structures are not symmetrical and the mirror images of the macromolecules do not exist in nature. Space groups with mirror planes and/or inversion centres are not allowed for crystals of these molecules, because these symmetry operations interchange right and left hands. Biological macromolecules crystallize in one of the 65 enantiomorphic space groups. (Enantiomorphic means the structure is not superimposable on its mirror image.) Apparently, some of these space groups supply more favourable packing conditions for proteins than others. The most favoured space group is (Table 2.1.2.1). A consequence of symmetry is that multiple copies of particles exist in the unit cell. For instance, in space group (space group No. 4), one can always expect two exactly identical entities in the unit cell, and one half of the unit cell uniquely represents the structure. This unique part of the structure is called the asymmetric unit. Of course, the asymmetric unit does not necessarily contain one protein molecule. Sometimes the unit cell contains fewer molecules than anticipated from the number of asymmetric units. This happens when the molecules occupy a position on a crystallographic axis. This is called a special position. In this situation, the molecule itself obeys the axial symmetry. Otherwise, the molecules in an asymmetric unit are on general positions. There may also be two, three or more equal or nearly equal molecules in the asymmetric unit related by noncrystallographic symmetry.

 Table 2.1.2.1| top | pdf | The most common space groups for protein crystals
 Situation as of April 1997; data extracted from the Protein Data Bank and supplied by Rob Hooft, EMBL Heidelberg.
Space groupOccurrence (%)
23
11
8
6
6

### References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.