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

International Tables for Crystallography (2010). Vol. B, ch. 4.5, p. 570   | 1 | 2 |

Section 4.5.2.3.3. Approximate helix symmetry

R. P. Millanea

4.5.2.3.3. Approximate helix symmetry

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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 [u + x] subunits in v turns, where x is a small [(x \ll 1)] real number; i.e. the structure has approximate, but not exact, [u_{v}] helix symmetry. Since the molecule has an approximate repeat distance c, only those layer planes close to those at [Z = l/c] show significant diffraction. Denoting by [Z_{mn}] the Z coordinate of the nth Bessel order and its associated value of m, and using the selection rule shows that [Z_{mn} = [(um + vn)/c] + (mx/c) = (l/c) + (mx/c), \eqno(4.5.2.15)]so that the positions of the Bessel orders are shifted by [mx/c] from their positions if the helix symmetry is exactly [u_{v}]. 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[link]). Splitting can be used to advantage in structure determination (Section 4.5.2.6.6[link]).

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[link]). 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[link]).

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








































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