International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.5, p. 469   | 1 | 2 |

## Section 4.5.2.3.3. Approximate helix symmetry

R. P. Millanea*

#### 4.5.2.3.3. Approximate helix symmetry

| top | pdf |

In some cases the nature of the subunits and their interactions results in a structure that is not exactly periodic. Consider a helical structure with subunits in v turns, where x is a small real number; i.e. the structure has approximate, but not exact, helix symmetry. Since the molecule has an approximate repeat distance c, only those layer planes close to those at show significant diffraction. Denoting by the Z coordinate of the nth Bessel order and its associated value of m, and using the selection rule shows that so that the positions of the Bessel orders are shifted by from their positions if the helix symmetry is exactly . At moderate resolution m is small so the shift is small. Hence Bessel orders that would have been coincident on a particular layer plane are now separated in reciprocal space. This is referred to as layer-plane splitting and was first observed in fibre diffraction patterns from tobacco mosaic virus (TMV) (Franklin & Klug, 1955). Splitting can be used to advantage in structure determination (Section 4.5.2.6.6).

As an example, TMV has approximately 493 helix symmetry with a c repeat of 69 Å. However, close inspection of diffraction patterns from TMV shows that there are actually about 49.02 subunits in three turns (Stubbs & Makowski, 1982). The virus is therefore more accurately described as a 2451150 helix with a c repeat of 3450 Å. The layer lines corresponding to this larger repeat distance are not observed, but the effects of layer-plane splitting are detectable (Stubbs & Makowski, 1982).

### References

Franklin, R. E. & Klug, A. (1955). The splitting of layer lines in X-ray fibre diagrams of helical structures: application to tobacco mosaic virus. Acta Cryst. 8, 777–780.Google Scholar
Stubbs, G. J. & Makowski, L. (1982). Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fibre diffraction data. Acta Cryst. A38, 417–425.Google Scholar