Chapter 13.4 Noncrystallographic symmetry averaging of electron density for molecular-replacement phase refinement and extension

Crystallography of biological macromolecules

Second Online Edition (2012)

Part 13. Molecular replacement

  1. M. G. Rossmann1,
  2. E. Arnold2

Published Online: 14 APR 2012

DOI: 10.1107/97809553602060000842

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Rossmann, M. G. and Arnold, E. 2012. Noncrystallographic symmetry averaging of electron density for molecular-replacement phase refinement and extension. International Tables for Crystallography. F:13:13.4:352–363.

Author Information

  1. 1

    Department of Biological Sciences, Purdue University, West Lafayette, IN 47907–1392, USA

  2. 2

    Biomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854–5638, USA

Publication History

  1. Published Online: 14 APR 2012


Noncrystallographic symmetry (NCS) occurs when symmetry operations are true only within a confined envelope, as opposed to being valid throughout the essentially infinite crystal lattice. Computationally, it is useful to define the molecular symmetry with reference to an arbitrary cell (the ‘h-cell’) with the relationship Xn = [Rn]X1 (n = 1, N). Then the assembly of N NCS equivalent objects can be moved into the actual crystal (the ‘p-cell’) using the relationship Y = [E]X. Hence each of the N units can be referred to the reference unit by Yn = [E][Rn]X1. In turn, the N units in the p-cell asymmetric unit can be multiplied by the crystal symmetry to produce the whole unit cell from the reference subunit in the h-cell. Procedures of averaging electron density will require a definition of the envelope either for the reference subunit or the whole of the molecular assembly if the NCS represents a closed point group (‘proper’ NCS). Averaging beyond the range of the NCS operators means that averaging is between non-equivalent densities. This causes the mean height of the average density to diminish and thus accurately indicates the limits of the NCS envelope. Various symmetry situations are examined, such as averaging subunits within the same crystal lattice (maybe proper symmetry) or between different crystal forms (necessarily improper symmetry). Phase extension is shown to be possible only by small defined increments of resolution after each cycle of averaging and solvent flattening.


  • ab initio phasing;
  • computer programs;
  • electron-density averaging;
  • h-cell;
  • molecular envelopes;
  • molecular replacement;
  • multidomain averaging;
  • multiple-crystal-form averaging;
  • noncrystallographic symmetry;
  • p-cell;
  • phase extension;
  • phase refinement;
  • phasing