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

International Tables for Crystallography (2006). Vol. B. ch. 4.5, pp. 467-468   | 1 | 2 |

Section 4.5.2.3.1. Helix symmetry

R. P. Millanea*

4.5.2.3.1. Helix symmetry

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The presence of a unique axis about which there is rotational disorder means that it is convenient to use cylindrical polar coordinate systems in fibre diffraction. We denote by [(r, \varphi, z)] a cylindrical polar coordinate system in real space, in which the z axis is parallel to the molecular axes. The molecule is said to have [u_{v}] helix symmetry, where u and v are integers, if the electron density [f(r, \varphi, z)] satisfies [f\big(r, \varphi + (2\pi mv/u), z + (mc/u)\big) = f(r, \varphi, z), \eqno(4.5.2.2)] where m is any integer. The constant c is the period along the z direction, which is referred to variously as the molecular repeat distance, the crystallographic repeat, or the c repeat. The helix pitch P is equal to [c/v]. Helix symmetry is easily interpreted as follows. There are u subunits, or helix repeat units, in one c repeat of the molecule. The helix repeat units are repeated by integral rotations of [2\pi v/u] about, and translations of [c/u] along, the molecular (or helix) axis. The helix repeat units may therefore be referenced to a helical lattice that consists of points at a fixed radius, with relative rotations and translations as described above. These points lie on a helix of pitch P, there are v turns (or pitch-lengths) of the helix in one c repeat, and there are u helical lattice points in one c repeat. A [u_{v}] helix is said to have `u residues in v turns'.

Since the electron density is periodic in ϕ and z, it can be decomposed into a Fourier series as [f(r, \varphi, z) = {\textstyle\sum\limits_{l=-\infty}^{\infty}}\; {\textstyle\sum\limits_{n=-\infty}^{\infty}} g_{nl}(r) \exp \big(i[n\varphi - (2\pi lz/c)]\big), \eqno(4.5.2.3)] where the coefficients [g_{nl} (r)] are given by [{g_{nl} (r) = (c/2\pi) {\textstyle\int\limits_{0}^{c}} {\textstyle\int\limits_{0}^{2\pi}} f(r, \varphi, z) \exp \big(i[-n \varphi + (2\pi lz/c)]\big) \;\hbox{d}\varphi\;\hbox{d}z.} \eqno(4.5.2.4)] Assume now that the electron density has helical symmetry. Denote by [g(r, \varphi, z)] the electron density in the region [0  \lt  z  \lt  c/u]; the electron density being zero outside this region, i.e. [g(r, \varphi, z)] is the electron density of a single helix repeat unit. It follows that [f (r, \varphi, z) = {\textstyle\sum\limits_{m = -\infty}^{\infty}} g [r, \varphi + (2\pi mv/u), z + (mc/u)]. \eqno(4.5.2.5)] Substituting equation (4.5.2.5)[link] into equation (4.5.2.4)[link] shows that [g_{nl} (r)] vanishes unless [(l - nv)] is a multiple of u, i.e. unless [l = um + vn \eqno(4.5.2.6)] for any integer m. Equation (4.5.2.6)[link] is called the helix selection rule. The electron density in the helix repeat unit is therefore given by [g (r, \varphi, z) = {\textstyle\sum\limits_{l}} {\textstyle\sum\limits_{n}} g_{nl} (r) \exp \big(i[n \varphi - (2\pi lz/c)]\big), \eqno(4.5.2.7)] where [{g_{nl} (r) = (c/2\pi) {\textstyle\int} {\textstyle\int} g (r, \varphi, z) \exp \big(i [-n\varphi + (2\pi l/c)]\big) \;\hbox{d}\varphi\;\hbox{d}z,} \eqno(4.5.2.8)] and where in equation (4.5.2.7)[link] (and in the remainder of this section) the sum over l is over all integers, the sum over n is over all integers satisfying the helix selection rule and the integral in equation (4.5.2.8)[link] is over one helix repeat unit. The effect of helix symmetry, therefore, is to restrict the number of Fourier coefficients [g_{nl} (r)] required to represent the electron density to those whose index n satisfies the selection rule. Note that the selection rule is usually derived using a rather more complicated argument by considering the convolution of the Fourier transform of a continuous filamentary helix with a set of planes in reciprocal space (Cochran et al., 1952[link]). The approach described above, which follows that of Millane (1991)[link], is much more straightforward.

References

First citationCochran, W., Crick, F. H. C. & Vand, V. (1952). The structure of synthetic polypeptides. I. The transform of atoms on a helix. Acta Cryst. 5, 581–586.Google Scholar
First citationMillane, R. P. (1991). An alternative approach to helical diffraction. Acta Cryst. A47, 449–451.Google Scholar








































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