Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing

Tong, L.; Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000765

International Tables for Crystallography (2010). Vol. B. ch. 2.3, pp. 244-281 | 1 | 2 |
https://doi.org/10.1107/97809553602060000765

Chapter 2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing

2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing (pp. 244-281) | html | pdf | chapter contents |
L. Tong, M. G. Rossmann and E. Arnold
- 2.3.1. Introduction (pp. 244-247) | html | pdf |
  - 2.3.1.1. Background (pp. 244-245) | html | pdf |
  - 2.3.1.2. Limits to the number of resolved vectors (p. 245) | html | pdf |
  - 2.3.1.3. Modifications: origin removal, sharpening etc. (pp. 245-246) | html | pdf |
  - 2.3.1.4. Homometric structures and the uniqueness of structure solutions; enantiomorphic solutions (pp. 246-247) | html | pdf |
  - 2.3.1.5. The Patterson synthesis of the second kind (p. 247) | html | pdf |
- 2.3.2. Interpretation of Patterson maps (pp. 247-251) | html | pdf |
  - 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin (pp. 247-248) | html | pdf |
  - 2.3.2.2. Harker sections (p. 248) | html | pdf |
  - 2.3.2.3. Finding heavy atoms (p. 249) | html | pdf |
  - 2.3.2.4. Superposition methods. Image detection (pp. 249-250) | html | pdf |
  - 2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies (p. 251) | html | pdf |
- 2.3.3. Isomorphous replacement difference Pattersons (pp. 251-255) | html | pdf |
  - 2.3.3.1. Introduction (p. 251) | html | pdf |
  - 2.3.3.2. Finding heavy atoms with centrosymmetric projections (pp. 251-252) | html | pdf |
  - 2.3.3.3. Finding heavy atoms with three-dimensional methods (p. 252) | html | pdf |
  - 2.3.3.4. Correlation functions (pp. 252-253) | html | pdf |
  - 2.3.3.5. Interpretation of isomorphous difference Pattersons (pp. 253-254) | html | pdf |
  - 2.3.3.6. Direct structure determination from difference Pattersons (pp. 254-255) | html | pdf |
  - 2.3.3.7. Isomorphism and size of the heavy-atom substitution (p. 255) | html | pdf |
- 2.3.4. Anomalous dispersion (pp. 255-257) | html | pdf |
  - 2.3.4.1. Introduction (pp. 255-256) | html | pdf |
  - 2.3.4.2. The $[P_{\rm s}({\bf u})]$ function (p. 256) | html | pdf |
  - 2.3.4.3. The position of anomalous scatterers (pp. 256-257) | html | pdf |
  - 2.3.4.4. Computer programs for automated location of atomic positions from Patterson maps (p. 257) | html | pdf |
- 2.3.5. Noncrystallographic symmetry (pp. 258-260) | html | pdf |
  - 2.3.5.1. Definitions (pp. 258-259) | html | pdf |
  - 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry (pp. 259-260) | html | pdf |
- 2.3.6. Rotation functions (pp. 260-269) | html | pdf |
  - 2.3.6.1. Introduction (pp. 260-261) | html | pdf |
  - 2.3.6.2. Matrix algebra (pp. 262-263) | html | pdf |
  - 2.3.6.3. Symmetry (pp. 263-264) | html | pdf |
  - 2.3.6.4. Sampling, background and interpretation (pp. 264-267) | html | pdf |
  - 2.3.6.5. The fast rotation function (p. 268) | html | pdf |
  - 2.3.6.6. Locked rotation functions (pp. 268-269) | html | pdf |
- 2.3.7. Translation functions (pp. 269-272) | html | pdf |
  - 2.3.7.1. Introduction (p. 269) | html | pdf |
  - 2.3.7.2. Position of a noncrystallographic element relating two unknown structures (p. 270) | html | pdf |
  - 2.3.7.3. Position of a known molecular structure in an unknown unit cell (pp. 270-271) | html | pdf |
  - 2.3.7.4. Position of a noncrystallographic symmetry element in a poorly defined electron-density map (p. 271) | html | pdf |
  - 2.3.7.5. Locked translation function (pp. 271-272) | html | pdf |
  - 2.3.7.6. Computer programs for rotation and translation function calculations (p. 272) | html | pdf |
- 2.3.8. Molecular replacement (pp. 272-275) | html | pdf |
  - 2.3.8.1. Using a known molecular fragment (pp. 272-273) | html | pdf |
  - 2.3.8.2. Using noncrystallographic symmetry for phase improvement (pp. 273-274) | html | pdf |
  - 2.3.8.3. Update on noncrystallographic averaging and density-modification methods (pp. 274-275) | html | pdf |
  - 2.3.8.4. Equivalence of real- and reciprocal-space molecular replacement (p. 275) | html | pdf |
- 2.3.9. Conclusions (p. 275) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.3.1.1. Effect of `sharpening' Patterson coefficients (p. 246) | html | pdf |
  - Fig. 2.3.1.2. (c) The point Patterson of the two homometric structures in (a) and (b) (p. 246) | html | pdf |
  - Fig. 2.3.2.1. Origin selection in the interpretation of a Patterson of a one-dimensional centrosymmetric structure (p. 247) | html | pdf |
  - Fig. 2.3.2.2. The c-axis projection of cuprous chloride azomethane complex (C₂H₆Cl₂Cu₂N₂) (p. 247) | html | pdf |
  - Fig. 2.3.2.3. Atoms ABCD, arranged as a quadrilateral, generate a Patterson which is the sum of the images of the quadrilateral when each atom is placed on the origin in turn (p. 250) | html | pdf |
  - Fig. 2.3.3.1. Three different cases which can occur in the relation of the native, $[{\bf F}_{\rm N}]$ , and heavy-atom derivative, $[{\bf F}_{\rm NH}]$ , structure factors for centrosymmetric reflections (p. 251) | html | pdf |
  - Fig. 2.3.3.2. Vector triangle showing the relationship between $[{\bf F}_{\rm N}]$ , $[{\bf F}_{\rm NH}]$ and $[{\bf F}_{\rm H}]$ , where $[{\bf F}_{\rm NH} = {\bf F}_{\rm N} + {\bf F}_{\rm H}]$ (p. 252) | html | pdf |
  - Fig. 2.3.3.3. A Patterson with coefficients $[({\bf F}_{\rm NH1} - {\bf F}_{\rm NH2})^{2}]$ will be equivalent to a Patterson whose coefficients are $[(AB)^{2}]$ (p. 253) | html | pdf |
  - Fig. 2.3.3.4. The phase α of the native compound (structure factor $[{\bf F}_{\rm N}]$ ) is determined either as being equal to, or 180° out of phase with, the presumed heavy-atom contribution when only a single isomorphous compound is available (p. 254) | html | pdf |
  - Fig. 2.3.3.5. Let (a) be the original structure which contains three heavy atoms ABC in a noncentrosymmetric configuration (p. 254) | html | pdf |
  - Fig. 2.3.3.6. A plot of mean isomorphous differences as a function of resolution (p. 255) | html | pdf |
  - Fig. 2.3.4.1. (a) A model structure with an anomalous scatterer at A (p. 256) | html | pdf |
  - Fig. 2.3.4.2. Anomalous-dispersion effect for a molecule (p. 257) | html | pdf |
  - Fig. 2.3.5.1. The two-dimensional periodic design shows crystallographic twofold axes perpendicular to the page and local noncrystallographic rotation axes in the plane of the paper (p. 258) | html | pdf |
  - Fig. 2.3.5.2. The objects $[A_{1}B_{1}]$ and $[A_{2}B_{2}]$ are related by an improper rotation θ (p. 258) | html | pdf |
  - Fig. 2.3.5.3. The position of the twofold rotation axis which relates the two piglets is completely arbitrary (p. 259) | html | pdf |
  - Fig. 2.3.6.1. Shape of the interference function G for a spherical envelope of radius R at a distance H from the reciprocal-space origin (p. 261) | html | pdf |
  - Fig. 2.3.6.2. Relationships of the orthogonal axes $[X_{1}, X_{2}, X_{3}]$ to the crystallographic axes $[a_{1}, a_{2}, a_{3}]$ (p. 262) | html | pdf |
  - Fig. 2.3.6.3. Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ relating the rotated axes $[X'_{1}, X'_{2}, X'_{3}]$ to the original unrotated orthogonal axes $[X_{1}, X_{2}, X_{3}]$ (p. 262) | html | pdf |
  - Fig. 2.3.6.4. Variables ψ and φ are polar coordinates which specify a direction about which the axes may be rotated through an angle κ (p. 262) | html | pdf |
  - Fig. 2.3.6.5. Rotation space group diagram for the rotation function of a Pmmm Patterson function $[(P_{1})]$ against a Patterson function $[(P_{2})]$ (p. 264) | html | pdf |
  - Fig. 2.3.6.6. The locked rotation function, L, applied to the determination of the orientation of the common cold virus (p. 267) | html | pdf |
  - Fig. 2.3.7.1. Crosses represent atoms in a two-dimensional model structure (p. 270) | html | pdf |
  - Fig. 2.3.7.2. Vectors arising from the structure in Fig. 2.3.7.1 (p. 270) | html | pdf |
- Tables
  - Table 2.3.1.1. Matrix representation of Patterson peaks (p. 245) | html | pdf |
  - Table 2.3.2.1. Coordinates of Patterson peaks for C₂H₆Cl₂Cu₂N₂ projection (p. 248) | html | pdf |
  - Table 2.3.2.2. Square matrix representation of vector interactions in a Patterson of a crystal with M crystallographic asymmetric units each containing N atoms (p. 248) | html | pdf |
  - Table 2.3.2.3. Position of Harker sections within a Patterson (p. 249) | html | pdf |
  - Table 2.3.5.1. Possible types of vector searches (p. 259) | html | pdf |
  - Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell (p. 259) | html | pdf |
  - Table 2.3.6.1. Different types of uses for the rotation function (p. 260) | html | pdf |
  - Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations (p. 264) | html | pdf |
  - Table 2.3.6.3. Numbering of the rotation-function space groups (p. 264) | html | pdf |
  - Table 2.3.6.4. Rotation-function Eulerian space groups (pp. 265-266) | html | pdf |
  - Table 2.3.8.1. Molecular replacement: phase refinement as an iterative process (p. 273) | html | pdf |

Chapter 2.3. Patterson and molecular replacement techniques, and the use of noncrystallographic symmetry in phasing

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