Patterson and molecular-replacement techniques

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 235-263 | 1 | 2 |
https://doi.org/10.1107/97809553602060000556

Chapter 2.3. Patterson and molecular-replacement techniques

2.3. Patterson and molecular-replacement techniques (pp. 235-263) | html | pdf | chapter contents |
M. G. Rossmann and E. Arnold
- 2.3.1. Introduction (pp. 235-238) | html | pdf |
  - 2.3.1.1. Background (p. 235) | html | pdf |
  - 2.3.1.2. Limits to the number of resolved vectors (pp. 235-236) | html | pdf |
  - 2.3.1.3. Modifications: origin removal, sharpening etc. (pp. 236-237) | html | pdf |
  - 2.3.1.4. Homometric structures and the uniqueness of structure solutions; enantiomorphic solutions (p. 237) | html | pdf |
  - 2.3.1.5. The Patterson synthesis of the second kind (p. 238) | html | pdf |
- 2.3.2. Interpretation of Patterson maps (pp. 238-242) | html | pdf |
  - 2.3.2.1. Simple solutions in the triclinic cell. Selection of the origin (pp. 238-239) | html | pdf |
  - 2.3.2.2. Harker sections (p. 239) | html | pdf |
  - 2.3.2.3. Finding heavy atoms (pp. 239-240) | html | pdf |
  - 2.3.2.4. Superposition methods. Image detection (pp. 240-241) | html | pdf |
  - 2.3.2.5. Systematic computerized Patterson vector-search procedures. Looking for rigid bodies (pp. 241-242) | html | pdf |
- 2.3.3. Isomorphous replacement difference Pattersons (pp. 242-246) | html | pdf |
  - 2.3.3.1. Introduction (p. 242) | html | pdf |
  - 2.3.3.2. Finding heavy atoms with centrosymmetric projections (pp. 242-243) | html | pdf |
  - 2.3.3.3. Finding heavy atoms with three-dimensional methods (p. 243) | html | pdf |
  - 2.3.3.4. Correlation functions (pp. 243-244) | html | pdf |
  - 2.3.3.5. Interpretation of isomorphous difference Pattersons (pp. 244-245) | html | pdf |
  - 2.3.3.6. Direct structure determination from difference Pattersons (p. 245) | html | pdf |
  - 2.3.3.7. Isomorphism and size of the heavy-atom substitution (pp. 245-246) | html | pdf |
- 2.3.4. Anomalous dispersion (pp. 246-248) | html | pdf |
  - 2.3.4.1. Introduction (p. 246) | html | pdf |
  - 2.3.4.2. The $[P_s({\bf u})]$ function (pp. 246-247) | html | pdf |
  - 2.3.4.3. The position of anomalous scatterers (pp. 247-248) | html | pdf |
- 2.3.5. Noncrystallographic symmetry (pp. 248-250) | html | pdf |
  - 2.3.5.1. Definitions (pp. 248-249) | html | pdf |
  - 2.3.5.2. Interpretation of Pattersons in the presence of noncrystallographic symmetry (pp. 249-250) | html | pdf |
- 2.3.6. Rotation functions (pp. 250-258) | html | pdf |
  - 2.3.6.1. Introduction (pp. 250-252) | html | pdf |
  - 2.3.6.2. Matrix algebra (pp. 252-253) | html | pdf |
  - 2.3.6.3. Symmetry (pp. 253-254) | html | pdf |
  - 2.3.6.4. Sampling, background and interpretation (pp. 254-255) | html | pdf |
  - 2.3.6.5. The fast rotation function (pp. 255-258) | html | pdf |
- 2.3.7. Translation functions (pp. 258-260) | html | pdf |
  - 2.3.7.1. Introduction (pp. 258-259) | html | pdf |
  - 2.3.7.2. Position of a noncrystallographic element relating two unknown structures (p. 259) | html | pdf |
  - 2.3.7.3. Position of a known molecular structure in an unknown unit cell (pp. 259-260) | html | pdf |
  - 2.3.7.4. Position of a noncrystallographic symmetry element in a poorly defined electron-density map (p. 260) | html | pdf |
- 2.3.8. Molecular replacement (pp. 260-262) | html | pdf |
  - 2.3.8.1. Using a known molecular fragment (pp. 260-261) | html | pdf |
  - 2.3.8.2. Using noncrystallographic symmetry for phase improvement (pp. 261-262) | html | pdf |
  - 2.3.8.3. Equivalence of real- and reciprocal-space molecular replacement (p. 262) | html | pdf |
- 2.3.9. Conclusions (pp. 262-263) | html | pdf |
  - 2.3.9.1. Update (pp. 262-263) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 2.3.1.1. Effect of `sharpening' Patterson coefficients (p. 237) | html | pdf |
  - Fig. 2.3.1.2. (c) The point Patterson of the two homometric structures in (a) and (b) (p. 237) | html | pdf |
  - Fig. 2.3.2.1. Origin selection in the interpretation of a Patterson of a one-dimensional centrosymmetric structure (p. 238) | html | pdf |
  - Fig. 2.3.2.2. The c-axis projection of cuprous chloride azomethane complex (C₂H₆Cl₂Cu₂N₂) (p. 238) | 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. 241) | html | pdf |
  - Fig. 2.3.3.1. Three different cases which can occur in the relation of the native, $[{\bf F}_{N}]$ , and heavy-atom derivative, $[{\bf F}_{NH}]$ , structure factors for centrosymmetric reflections (p. 242) | html | pdf |
  - Fig. 2.3.3.2. Vector triangle showing the relationship between $[{\bf F}_{N}]$ , $[{\bf F}_{NH}]$ and $[{\bf F}_{H}]$ , where $[{\bf F}_{NH} = {\bf F}_{N} + {\bf F}_{H}]$ (p. 243) | html | pdf |
  - Fig. 2.3.3.3. A Patterson with coefficients $[({\bf F}_{NH1} - {\bf F}_{NH2})^{2}]$ will be equivalent to a Patterson whose coefficients are $[(AB)^{2}]$ (p. 243) | html | pdf |
  - Fig. 2.3.3.4. The phase α of the native compound (structure factor $[{\bf F}_{N}]$ ) 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. 244) | html | pdf |
  - Fig. 2.3.3.5. Let (a) be the original structure which contains three heavy atoms ABC in a noncentrosymmetric configuration (p. 245) | html | pdf |
  - Fig. 2.3.3.6. A plot of mean isomorphous differences as a function of resolution (p. 246) | html | pdf |
  - Fig. 2.3.4.1. (a) A model structure with an anomalous scatterer at A (p. 247) | html | pdf |
  - Fig. 2.3.4.2. Anomalous-dispersion effect for a molecule (p. 247) | 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. 248) | 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. 249) | html | pdf |
  - Fig. 2.3.5.3. The position of the twofold rotation axis which relates the two piglets is completely arbitrary (p. 249) | 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. 252) | 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. 252) | 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. 253) | 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. 253) | 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. 254) | 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. 255) | html | pdf |
  - Fig. 2.3.7.1. Crosses represent atoms in a two-dimensional model structure (p. 259) | html | pdf |
  - Fig. 2.3.7.2. Vectors arising from the structure in Fig. 2.3.7.1 (p. 259) | html | pdf |
- Tables
  - Table 2.3.1.1. Matrix representation of Patterson peaks (p. 236) | html | pdf |
  - Table 2.3.2.1. Coordinates of Patterson peaks for C₂H₆Cl₂Cu₂N₂ projection (p. 239) | 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. 239) | html | pdf |
  - Table 2.3.2.3. Position of Harker sections within a Patterson (p. 240) | html | pdf |
  - Table 2.3.5.1. Possible types of vector searches (p. 250) | html | pdf |
  - Table 2.3.5.2. Orientation of the glyceraldehyde-3-phosphate dehydrogenase molecular twofold axis in the orthorhombic cell (p. 250) | html | pdf |
  - Table 2.3.6.1. Different types of uses for the rotation function (p. 251) | html | pdf |
  - Table 2.3.6.2. Eulerian symmetry elements for all possible types of space-group rotations (p. 254) | html | pdf |
  - Table 2.3.6.3. Numbering of the rotation function space groups (p. 254) | html | pdf |
  - Table 2.3.6.4. Rotation function Eulerian space groups (pp. 256-257) | html | pdf |
  - Table 2.3.8.1. Molecular replacement: phase refinement as an iterative process (p. 261) | html | pdf |

Chapter 2.3. Patterson and molecular-replacement techniques

Contents