The Order-disorder (OD) theory

The order-disorder (OD) terminology was developed to describe disorder phenomena observed in crystal lattices. The basic concept of this theory introduced by Dornberger-Schiff in the 1950s is the existence of layer slabs through a crystal forming the overall crystal lattice (Dornberger-Schiff, 1956; Dornberger-Schiff & Grell-Niemann, 1961), with a set of recurring symmetry operations converting one layer into the adjacent layer (layer group). This special transformation of adjacent layers by this symmetry operation is only existent for a subspace of the whole crystal and can therefore be denoted as local symmetry. This symmetry is not part of the structure symmetry and does not appear in its space group. The periodicity of this set of local symmetry operations indicate whether the crystal is ordered (periodic) or disordered (non-periodic). Slabs in such

crystals are referred to as OD-layers forming the crystal. Symmetry operations of the whole crystal are summarized in the space group, whereas the addition of local symmetry to the set of crystal symmetry operations describes the space groupoid (Brandt, 1927; Dornberger-Schiff, 1964).

Most important for a clear definition of layer slabs in this issue is the fulfilling of so called vicinity conditions (VC), which define geometrically equivalent sets of paired layers. Three vicinity conditions can be postulated (Dornberger-Schiff & Fichtner, 1972; Dornberger-Schiff, 1979):

1. Vicinity condition : VC layers are either geometrically equivalent or very few in kind.

2. Vicinity condition Sets of all translation operations (translation group) of all VC layers are either equal or have a common subgroup.

3. Vicinity condition Equivalent layers face equivalent adjacent layers by a equivalent interface, so that the resulting pair is equivalent.

The position of a layer only defined by the positions of the adjacent layer and fulfilling the VC are termed fully ordered. If more than one position is allowed fulfilling the VC conditions, the structure is termed an OD structure and the layers are OD layers.

Symmetry in OD structures is either referring to a layer itself or it builds the adjacent layer and is classified as follows (Dornberger-Schiff, 1956; Ďurovič, 1997):

-symmetry: Is defined as the symmetry proper of an individual layer.

-symmetry: Is the symmetry of a layer pair.

Symmetry operations arising from these symmetry conditions depending on the kind of symmetry operation are:

-operations: Symmetry operations transforming a layer into itself, the set of -operations add to a diperiodic group (layer group).

-operations: Coincidence operations transforming a layer into the adjacent one.

-operations: Symmetry or coincidence operations which do not invert the sign of adjacent layer orientation.

-operations: Symmetry or coincidence operations which do invert the sign of the coordinate stacking direction of a layer and thus turn the adjacent layer upside down.

Choice of the layers is crucial for the correct description of the symmetry in an OD-structure as the concept of OD-layer choice is not an absolute one, but helps only to describe the observed architecture of a crystal.

Classification of OD-structures

Classification of OD-structures helps to compare reasons and consequences of OD arrangement in a more distinct manner (Dornberger-Schiff, 1956).

OD structures of the first kind are consisting of layers with planes being parallel to each other, which either form a set of translatable layers or belong to several such layers. Further classification can be made on the set of layers by:

Type A: All layers belong to a set of translatable layers.

Type B: All layers are substantially equal, but belong to two or more different interleaved sets of translatable layers. Adjacent layers can belong to different sets and are transformed by -operations.

OD structures of the second kind consist of rod-like structures with one dimensional periodicity and can exhibit disorder effects as well. Further classification has not been introduced so far.

OD twinning

The presence of OD-structures can explain the occurrence of twinning, as symmetry operations that are not part of the holohedral point group of specialized metric or the symmetry of the whole crystal lattice are only valid for a subspace of the crystal. OD

twins therefore consist of periodic OD-layers and the twin operation relating twin individuals is isomorphous to - and -operations described by the OD groupoid (Nespolo et al., 2004). The process leading to twinning is the ambiguity in layer stacking described by the OD theory, as boundary interactions of adjacent layers are equal in energy for two or more different orientations.

4.2 Aim of the present work

Biochemical and structural characterization of the diheme c-type cytochrome DHC2 from Geobacter sulfurreducens have been carried out, obtaining structural data indicating the presence of a conserved heme packing motif and several structural features for fine tuning of the observed mid-point potentials of -139 mV and - 289 mV.

Indicators for data and model quality indicate reliable experimental and structural model data. Furthermore, the model seems to be in line with other multi-heme c-type cytochrome structures and no evidence can be drawn, that the current model does not represent the state of DHC2 in its crystalline form. However, R factors indicated a problem in structure refinement, as the process converged at disagreeably high R factors for data of this quality and resolution, pointing to additional features of DHC2 crystals that were not detected during the process of structure solution and refinement in the first place. Reassessing diffraction data from the point of data merging will be described in the following, looking for signs revealing the presence of twinning, using a variety of intensity statistics methods, with a final refinement using the information gained by these methods. Structural and thermodynamic reasons for the presence of pseudo-merohedral twinning in crystals of DHC2 will be discussed on the basis of non crystallographic symmetry relations and protein crystal packing behavior as described by the order-disorder (OD) terminology.