Generating symmetry: observed macromolecular crystal contacts explain space group frequencies

Jennifer E. Padilla,^a and Todd O. Yeates^a

^aDepartment of Chemistry & Biochemistry, University of California, Los Angeles, CA 90095-1569, USA (padilla@mbi.ucla.edu)

Biological macromolecules have been crystallized in all of the 65 biological space groups, however, these 65 space group symmetries are unevenly represented. An entropic model has been invoked to explain the observed space group frequencies [1]. According to this model, favored space groups, such as P2₁2₁2₁, allow more rigid body degrees of freedom, found in the number of independent unit cell parameters, and the number of meaningful ways to orient and position a molecule in the unit cell. Conversely, degrees of freedom are lost for each unique contact that is required in order to achieve connectivity in the crystal. In the 65 biological space groups, this minimum contact number ranges from 2 to 5, depending on the space group.

Here we present a study of the number of unique contacts actually found within crystals reported in the PDB. We are observing the frequency with which the minimum number of contacts actually occurs. Where the minimum does not occur, degrees of freedom have been sacrificed for other gains, such as closer packing within the crystal, or overcoming geometric hindrances to making certain contacts. Each set of unique contacts within a crystal corresponds to a set of symmetry elements that generate the space group symmetry of the crystal. When this set is not a minimal set, it may contain non-essential elements. These correspond to unique contacts that are unnecessary in order to maintain the connectivity of the crystal.

We find that minimal sets of unique contacts do not make up the majority of generating sets in real crystals. Sometimes, the generating set that is used is larger than the minimum, but still does not contain unnecessary elements. In certain space groups, this is because the minimal set of generators cannot be achieved without an unusual geometry of the molecule being crystallized. However, there are still are large number of cases in which nonessential crystal contacts are present. We are reinterpreting the observed space group frequencies in light of this new information on the actual number of unique contacts found to generate each space group. This study reveals the interplay between the degrees of freedom available in the nucleation of a crystal versus the advantage of achieving close packing within the crystal, and better explains the space group frequencies observed in the PDB.

References

1 Wukovitz, S. W., and Yeates, T. O. (1995) Nat. Struct. Bio. 2, 1062-1067.