InternationalReciprocal spaceTables for Crystallography Volume B Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 4.5, p. 570
## Section 4.5.2.3.3. Approximate helix symmetry R. P. Millane
^{a} |

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 *n*th 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 49_{3} 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 2451_{150} 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.*

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Stubbs, G. J. & Makowski, L. (1982).

*Coordinated use of isomorphous replacement and layer-line splitting in the phasing of fibre diffraction data. Acta Cryst.*A

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