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.

# Protein Multimer Symmetry

(Diagrams adapted from Voet and Voet, 1990; after Irving Geis)

## Cyclic symmetries

A cyclic symmetry, designated CN 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 C2 to C11.

C2
diagram VERY common - examples include horse-liver alcohol dehydrogenase , growth factors (e.g. nerve growth factors, by Judith Murray-Rust)

C3
diagram Examples include chloramphenicol acetyltransferase (pdb file of trimer in PPS hypertree, 400Kb, generated from structure 3cla, gif1, gif2, gif3, gif4), and glucagon 1gcn (25Kb) [Bbk|BNL|ExP|Waw|Hal] (Exercise: Create the trimer)

C5
diagram Examples include serum amyloid P-component(GIF) ( 1sac (703Kb) [Bbk|BNL|ExP|Waw|Hal] gif1, gif2)

C6
eg C-reactive protein from the horse-shoe crab (Limulus) 1lim (864Kb) [Bbk|BNL|ExP|Waw|Hal]

C9
eg Light harvesting complex

C11
eg TRAP Trp attenuation protein of Bacillus subtilis (GIF) 1wap (1.2Mb) [Bbk|BNL|ExP|Waw|Hal]

Note: the biological unit of this protein is an undecamer (11-mer), exhibiting C11 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 D11 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.

## Dihedral symmetries

A dihedral symmetry, designated DN has 2xN identical units related by a single N-fold rotational axis and N 2-fold rotational axes.

D2
diagram Examples include lactate dehydrogenase ( pdb file,790Kb, of complete biological unit at Brookhaven, gif), glyceraldehyde dehydrogenase ( 1gd1 (966Kb) [Bbk|BNL|ExP|Waw|Hal], gif1, gif2), Fe or Mn superoxide dismutase ( pdb, gif1, gif2)

D3
diagram

D4
diagram Examples include hemerythrin ( pdb file of octamer in PPS hypertree, 549Kb, generated from structure 1hmo gif1, gif2)

D6
eg glutamine synthetase ( 2gls (3.6Mb)[Bbk|BNL|ExP|Waw|Hal], gif1, gif2, gif3)

D7
eg GroEL

## Higher symmetries

### Octahedral

diagram An octohedral symmetry, designated O has 24 identical units related by three 4-, four 3- and six 2-fold rotational axes.

### Tetrahedral (Tetramer of Trimers)

diagram A tetrahedral symmetry, designated T has 12 identical units related by four 3- and three 2-fold rotational axes.

### Icosahedral symmetry

diagram An icosahedral symmetry, designated I/532 has 60 identical units related by six 5-, ten 3- and fifteen 2-fold rotational axes.

## Helical Symmetry

diagram The units in a helical symmetry are related by a screw axis (a translation and rotation operation).

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).

## Tutorial Software

Birkbeck are distributing a Symmetry Teaching programme that runs under Microsoft Windows. Let us know if you find it useful.

## References

• Voet, D. and Voet, J.G. (1990) Biochemistry, John Wiley and Sons, New York
• Klotz, L.M., Klippenstein, G.L., & Hendrickson, W.A. (1976) Science 192, 335-344
• Cooper, J., McIntyre, K., Badasso, M., Wood, S., Zhang, Y., Garbe, T. & Young, D. (1995) J.Mol.Biol. 246, 531-544

Alan Mills, John Kenney, John Walshaw
Last updated 14th April '97