Quaternary Structure - Symmetry
Symmetry is the concept of repetitive arrangements of similar objects
in space. In three dimensions, objects may be arranged in a large number
of ways - many of these are exhibited in crystal forms. Each possible arrangement
is achieved by a combination of simple, basic operations. These include
translation, rotation, screw-rotation, mirror, inversion, inversion-rotation,
and glide-reflection. The total number of possible ways of arranging copies
of the same object to form a repeating lattice in 3D-space is 230. These
are the `Space Groups' familiar to crystallographers. They are enumerated
and detailed in The International Tables of Crystallography.
Proteins are chiral objects, and cannot be mirror-inverted whilst remaining the same. Their mirror reflection is different. Thus, many of these arrangements are actually precluded. In fact, proteins may only adopt 65 of the 230 possible 3D space groups. Many of these are observed when we crystallise proteins.
In the case of naturally occurring multimers of proteins, other constraints occur which limit the possible arrangements. We are speaking here of individual assemblies of monomer units, creating (usually soluble) complexes which exhibit internal symmetry.
Thus, the monomers must associate with van der Waals contact interfaces between the sub-units of the assembly. That is, the sub-units can touch but not intersect. They may interpenetrate only in so far as there exist corresponding `holes' into which `knobs' can be fit. Furthermore, axes of rotational symmetry may not actually pass `through' a protein monomer.
This leads to the limited possibilities listed below.
(Diagrams adapted from Voet and Voet, 1990; after Irving Geis)
A cyclic symmetry, designated C_{N} has N identical units related by a single N-fold rotational axis. That is, successive rotations of a single unit through (360/N)° result in the positions of the other units. Therefore, cyclic symmetries of N = 2 to infinity are permitted. Examples presented below range from C_{2} to C_{11}.
Note: the biological unit of this protein is an undecamer (11-mer), exhibiting C_{11} symmetry; however in this particular crystal (1wap) the rings pack back-to-back, and there are two rings in the asymmetric unit. Therefore the appearance is of D_{11} symmetry (see below). Diagram courtesy of The Protein Structure Group, Chemistry Department, University of York; if you would like to read more about this protein, follow this link: Trp attenuation protein of Bacillus subtilis.
A dihedral symmetry, designated D_{N} has 2xN identical units related by a single N-fold rotational axis and N 2-fold rotational axes.
Fibrous proteins exhibit helical symmetry, although in some cases there are several entwined helical strands (e.g. 2 in the case of actin filaments, 3 in collagen). Single helical structures are the protein coats of rod-shaped viruses, and microtubules. Click for a diagram of a microtubule (from the page on larger assemblies in this Section).
Birkbeck are distributing a Symmetry Teaching programme that runs under Microsoft Windows. Let us know if you find it useful.
Alan Mills, John Kenney, John Walshaw
Last updated 14th April '97