Helical diffraction

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. Helical diffraction

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. Helical diffraction

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The theory of diffraction by helical structures (Cochran et al., 1952; Klug et al., 1958) has played an important part in the study of polypeptides, of nucleic acids and of tobacco mosaic virus.

1.3.4.5.1.1. Circular harmonic expansions in polar coordinates

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Let [f = f(x, y)] be a reasonably regular function in two-dimensional real space. Going over to polar coordinates $[x = r \cos \varphi\quad y = r \sin \varphi]$ and writing, by slight misuse of notation, $[f(r, \varphi)]$ for $[f(r \cos \varphi, r \sin \varphi)]$ we may use the periodicity of f with respect to φ to expand it as a Fourier series (Byerly, 1893): $[f(r, \varphi) = {\textstyle\sum\limits_{n \in {\bb Z}}} \;f_{n} (r) \exp (in \varphi)]$ with $[f_{n} (r) = {1 \over 2\pi} {\textstyle\int\limits_{0}^{2\pi}} f(r, \varphi) \exp (-in \varphi) \hbox{ d}\varphi.]$

Similarly, in reciprocal space, if $[F = F(\xi, \eta)]$ and if $[\xi = R \cos \psi\quad \eta = R \sin \psi]$ then $[F(R, \psi) = {\textstyle\sum\limits_{n \in {\bb Z}}} \;i^{n} F_{n} (R) \exp (in\psi)]$ with $[F_{n} (R) = {1 \over 2\pi i^{n}} {\textstyle\int\limits_{0}^{2\pi}} F(R, \psi) \exp (-in \psi) \hbox{ d}\psi,]$ where the phase factor $[i^{n}]$ has been introduced for convenience in the forthcoming step.

1.3.4.5.1.2. The Fourier transform in polar coordinates

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The Fourier transform relation between f and F may then be written in terms of $[f_{n}]$ 's and $[F_{n}]$ 's. Observing that $[\xi x + \eta y = Rr \cos (\varphi - \psi)]$ , and that (Watson, 1944) $[{\textstyle\int\limits_{0}^{2\pi}} \exp (iX \cos \theta + in\theta) \hbox{ d}\theta = 2 \pi i^{n} J_{n} (X),]$ we obtain: $[\eqalign{F(R, \psi) &= {\textstyle\int\limits_{0}^{\infty}} {\textstyle\int\limits_{0}^{2\pi}} \left[{\textstyle\sum\limits_{n \in {\bb Z}}} \;f_{n} (r) \exp (in\varphi)\right]\cr &\quad \times \exp [2 \pi i Rr \cos (\varphi - \psi)] r \hbox{ d}r \hbox{ d}\varphi\cr &= {\textstyle\sum\limits_{n \in {\bb Z}}} i^{n} \left[{\textstyle\int\limits_{0}^{\infty}} f_{n} (r) J_{n} (2 \pi Rr) 2 \pi r \hbox{ d}r\right] \exp (in\psi)\hbox{;}}]$ hence, by the uniqueness of the Fourier expansion of F: $[F_{n} (R) = {\textstyle\int\limits_{0}^{\infty}} f_{n} (r) J_{n} (2 \pi Rr) 2 \pi r \hbox{ d}r.]$ The inverse Fourier relationship leads to $[f_{n} (r) = {\textstyle\int\limits_{0}^{\infty}} F_{n} (R) J_{n} (2 \pi r R) 2 \pi R \hbox{ d}R.]$ The integral transform involved in the previous two equations is called the Hankel transform (see e.g. Titchmarsh, 1922; Sneddon, 1972) of order n.

1.3.4.5.1.3. The transform of an axially periodic fibre

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Let ρ be the electron-density distribution in a fibre, which is assumed to have translational periodicity with period 1 along z, and to have compact support with respect to the (x, y) coordinates. Thus ρ may be written $[\rho = \left[\delta_{x} \otimes \delta_{y} \otimes \left({\textstyle\sum\limits_{k \in {\bb Z}}} \delta_{(k)}\right)_{z}\right] * \rho^{0},]$ where $[\rho^{0} = \rho^{0} (x, y, z)]$ is the motif.

By the tensor product property, the inverse Fourier transform $[F = \bar{{\scr F}}_{xyz} [\rho]]$ may be written $[F = \left[1_{\xi} \otimes 1_{\eta} \otimes \left({\textstyle\sum\limits_{l \in {\bb Z}}} \delta_{(l)}\right)_{\zeta}\right] \times \bar{{\scr F}}[\rho^{0}]]$ and hence consists of `layers' labelled by l: $[F = {\textstyle\sum\limits_{l \in {\bb Z}}} F(\xi, \eta, l) (\delta_{(l)})_{\zeta}]$ with $[F(\xi, \eta, l) = {\textstyle\int\limits_{0}^{1}} \bar{{\scr F}}_{xy} [\rho^{0}] (\xi, \eta, z) \exp (2 \pi i l z) \hbox{ d}z.]$

Changing to polar coordinates in the (x, y) and $[(\xi, \eta)]$ planes decomposes the calculation of F from ρ into the following steps: $[\eqalign{g_{nl} (r) &= {1 \over 2\pi} {\textstyle\int\limits_{0}^{2\pi}} {\textstyle\int\limits_{0}^{1}} \rho (r, \varphi, z) \exp [i (-n \varphi + 2 \pi l z)] \hbox{ d}\varphi \hbox{ d}z \cr G_{nl} (R) &= {\textstyle\int\limits_{0}^{\infty}} g_{nl} (r) J_{n} (2 \pi Rr) 2 \pi r \hbox{ d}r\cr F (R, \psi, l) &= {\textstyle\sum\limits_{n \in {\bb Z}}} i^{n} G_{nl} (R) \exp (in\psi)}]$ and the calculation of ρ from F into: $[\eqalign{G_{nl} (R) &= {1 \over 2\pi i^{n}} {\textstyle\int\limits_{0}^{2\pi}} F(R, \psi, l) \exp (-in \psi) \hbox{ d}\psi\cr g_{nl} (r) &= {\textstyle\int\limits_{0}^{\infty}} G_{nl} (R) J_{n} (2 \pi rR) 2 \pi R \hbox{ d}R\cr \rho (r, \varphi, z) &= {\textstyle\sum\limits_{n \in {\bb Z}}}\; {\textstyle\sum\limits_{l \in {\bb Z}}} g_{nl} (r) \exp [i (n\varphi - 2 \pi l z)].}]$

These formulae are seen to involve a 2D Fourier series with respect to the two periodic coordinates φ and z, and Hankel transforms along the radial coordinates. The two periodicities in φ and z are independent, so that all combinations of indices (n, l) occur in the Fourier summations.

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

Byerly, W. E. (1893). An elementary treatise on Fourier's series and spherical, cylindrical and ellipsoidal harmonics. Boston: Ginn & Co. [Reprinted by Dover Publications, New York, 1959.]Google Scholar

Cochran, 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

Klug, A., Crick, F. H. C. & Wyckoff, H. W. (1958). Diffraction by helical structures. Acta Cryst. 11, 199–213.Google Scholar

Sneddon, I. N. (1972). The use of integral transforms. New York: McGraw-Hill.Google Scholar

Titchmarsh, E. C. (1922). Hankel transforms. Proc. Camb. Philos. Soc. 21, 463–473.Google Scholar

Watson, G. N. (1944). A treatise on the theory of Bessel functions, 2nd ed. Cambridge University Press.Google Scholar

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