Wilson and Sim structure-factor distributions in P1

Read, R. J.

doi:10.1107/97809553602060000688

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 15.2, pp. 325-326 | 1 | 2 |

Section 15.2.3.1. Wilson and Sim structure-factor distributions in P1

R. J. Read^a ^*

^a Department of Haematology, University of Cambridge, Wellcome Trust Centre for Molecular Mechanisms in Disease, CIMR, Wellcome Trust/MRC Building, Hills Road, Cambridge CB2 2XY, England
Correspondence e-mail: rjr27@cam.ac.uk

15.2.3.1. Wilson and Sim structure-factor distributions in P1

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For the Wilson distribution (Wilson, 1949), it is assumed that the atoms in a crystal structure in space group P1 are scattered randomly and independently through the unit cell. In fact, it is sufficient to make the much less restrictive assumption that the atoms are placed randomly with respect to the Bragg planes defined by the Miller indices. The assumption of independence is somewhat more problematic, since there are restrictions on the distances between atoms, large volumes of protein crystals are occupied by disordered solvent and many protein crystals display noncrystallographic symmetry; as discussed elsewhere (Vellieux & Read, 1997), the resulting relationships among structure factors are exploited implicitly in averaging and solvent-flattening procedures. The higher-order relationships among structure factors are used explicitly in direct methods for solving small-molecule structures and are being developed for use in protein structures (Bricogne, 1993). For the purposes of simpler relationships between the calculated and true structure factors for a single hkl, however, the lack of complete independence does not seem to create serious problems.

When atoms are placed randomly relative to the Bragg planes, the contribution of each atom to the structure factor will have a phase varying randomly from 0 to 2π. The overall structure factor can then be considered to be the result of a random walk in the complex plane, which can be treated as an application of the central limit theorem. The structure factor is the sum of the independent atomic scattering contributions, each of which has a probability distribution defined as a circle in the complex plane centred on the origin, with a radius of $[f_{j}]$ . The centroid of this atomic distribution is at the origin, and the variance for each of the real and imaginary parts is $[{1 \over 2} f_{j}^{2}]$ . The probability distribution of the structure factor that is the sum of these contributions is a two-dimensional Gaussian, the product of the one-dimensional Gaussians for the real and imaginary parts. Because the variances are equal in the real and imaginary directions, it can be simplified, as shown below, and expressed in terms of a single distribution parameter, $[\Sigma_{N}]$ . $[\eqalign{{\bf F} &= \textstyle\sum\limits_{j = 1}^{N}\displaystyle f_{j} \exp (2 \pi i {\bf h} \cdot {\bf x}_{j}) = A + iB\hbox{;}\quad \langle A\rangle = \langle B\rangle = 0\hbox{;}\hfill \cr \sigma^{2} (A) &= \sigma^{2} (B) = \textstyle{1 \over 2}\displaystyle \textstyle\sum\limits_{j = 1}^{N}\displaystyle f_{j}^{2} = {\textstyle{1 \over 2}} \Sigma_{N}, \hbox{so} \hfill\cr p(A) &= [1/(\pi \Sigma_{N})^{1/2}] \exp \left(-A^{2}/\Sigma_{N}\right),\hfill\cr p(B) &= [1/(\pi \Sigma_{N})^{1/2}] \exp \left(-B^{2}/\Sigma_{N}\right), \hfill\cr p({\bf F}) &= p(A, B) = (1/\pi \Sigma_{N}) \exp \left(-|{\bf F}|^{2}/\Sigma_{N}\right). \hfill\cr}]$

The Sim distribution (Sim, 1959), which is relevant when the positions of some of the atoms are known, has a very similar basis, except that the structure factor is now considered to arise from a random walk starting from the position of the structure factor corresponding to the known part, $[{\bf F}_{P}]$ . Atoms with known positions do not contribute to the variance, while each of the atoms with unknown positions (the `Q' atoms) contributes $[{1 \over 2} f_{j}^{2}]$ to each of the real and imaginary parts, as in the Wilson distribution. The distribution parameter in this case is referred to as $[\Sigma_{Q}]$ . The Sim distribution is a conditional probability distribution, depending on the value of $[{\bf F}_{P}]$ , $[p({\bf F}\hbox{;}\ {\bf F}_{P}) = (1/\pi \Sigma_{Q}) \exp \left(-|{\bf F} - {\bf F}_{P}|^{2}/\Sigma_{Q}\right).]$

The Wilson (1949) and Woolfson (1956) distributions for space group $[P\bar{1}]$ are obtained similarly, except that the random walks are along a line and the resulting Gaussian distributions are one-dimensional. (The Woolfson distribution is the centric equivalent of the Sim distribution.) For more complicated space groups, it is reasonable to assume that acentric reflections follow the P1 distribution and that centric reflections follow the $[P\bar{1}]$ distribution. However, for any zone of the reciprocal lattice in which symmetry-related atoms are constrained to scatter in phase, the variances must be multiplied by the expected intensity factor, ɛ, for the zone, because the symmetry-related contributions are no longer independent.

References

Bricogne, G. (1993). Direct phase determination by entropy maximization and likelihood ranking: status report and perspectives. Acta Cryst. D49, 37–60.Google Scholar

Sim, G. A. (1959). The distribution of phase angles for structures containing heavy atoms. II. A modification of the normal heavy-atom method for non-centrosymmetrical structures. Acta Cryst. 12, 813–815.Google Scholar

Vellieux, F. M. D. & Read, R. J. (1997). Non-crystallographic symmetry averaging in phase refinement and extension. Methods Enzymol. 277, 18–53.Google Scholar

Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–321.Google Scholar

Woolfson, M. M. (1956). An improvement of the `heavy-atom' method of solving crystal structures. Acta Cryst. 9, 804–810.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.2, pp. 325-326