Helical symmetry and associated selection rules

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 93-94 | 1 | 2 |

Section 1.3.4.5.1.4. Helical symmetry and associated selection rules

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.5.1.4. Helical symmetry and associated selection rules

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Helical symmetry involves a `clutching' between the two (hitherto independent) periodicities in φ (period 2π) and z (period 1) which causes a subdivision of the period lattice and hence a decimation (governed by `selection rules') of the Fourier coefficients.

Let i and j be the basis vectors along $[\varphi /2\pi]$ and z. The integer lattice with basis (i, j) is a period lattice for the $[(\varphi, z)]$ dependence of the electron density ρ of an axially periodic fibre considered in Section 1.3.4.5.1.3: $[\rho (r, \varphi + 2 \pi k_{1}, z + k_{2}) = \rho (r, \varphi, z).]$

Suppose the fibre now has helical symmetry, with u copies of the same molecule in t turns, where g.c.d. [(u, t) = 1] . Using the Euclidean algorithm, write $[u = \lambda t + \mu]$ with λ and μ positive integers and $[\mu \;\lt\; t]$ . The period lattice for the $[(\varphi, z)]$ dependence of ρ may be defined in terms of the new basis vectors:

I , joining subunit 0 to subunit l in the same turn;
J , joining subunit 0 to subunit λ after wrapping around.

In terms of the original basis $[{\bf I} = {t \over u} {\bf i} + {1 \over u} {\bf j},\quad {\bf J} = {-\mu \over u} {\bf i} + {\lambda \over u} {\bf j.}]$ If α and β are coordinates along I and J, respectively, $[\pmatrix{{\displaystyle{\varphi/2\pi}}\cr\noalign{\vskip 3pt} z\cr} = {1 \over u} \pmatrix{t &-\mu\cr 1 &\lambda\cr} \pmatrix{\alpha\cr \beta\cr}]$ or equivalently $[\pmatrix{\alpha\cr \beta\cr} = \pmatrix{\lambda &\mu\cr -1 &t\cr} \pmatrix{{\displaystyle{\varphi /2\pi}}\cr\noalign{\vskip 3pt} z\cr}.]$ By Fourier transformation, $[\eqalign{ \left({\varphi \over 2\pi}, z\right) &\Leftrightarrow (-n, l)\cr (\alpha, \beta) &\Leftrightarrow (m, p)}]$ with the transformations between indices given by the contragredients of those between coordinates, i.e. $[\pmatrix{n\cr l\cr} = \pmatrix{-\lambda &1\cr -\mu &t\cr} \pmatrix{m\cr p\cr}]$ and $[\pmatrix{m\cr p\cr} = {1 \over u} \pmatrix{-t &1\cr \mu &\lambda\cr} \pmatrix{n\cr l\cr}.]$ It follows that [l = tn + um,] or alternatively that $[\mu n = up - \lambda l,]$ which are two equivalent expressions of the selection rules describing the decimation of the transform. These rules imply that only certain orders n contribute to a given layer l.

The 2D Fourier analysis may now be performed by analysing a single subunit referred to coordinates α and β to obtain $[h_{m, \, p} (r) = {\textstyle\int\limits_{0}^{1}} {\textstyle\int\limits_{0}^{1}} \rho (r, \alpha, \beta) \exp [2 \pi i(m\alpha + p\beta)] \hbox{ d}\alpha \hbox{ d}\beta]$ and then reindexing to get only the allowed $[g_{nl}]$ 's by $[g_{nl} (r) = uh_{-\lambda m + p, \, \mu m + tp} (r).]$ This is u times faster than analysing u subunits with respect to the $[(\varphi, z)]$ coordinates.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 93-94