\<!DOCTYPE IUCR-ART PUBLIC \"IUCr//DTD IUCr article dtd
V1.1//EN\"\>\<iucr-art jid=\"D000000\" aid=\"xx0000\" access=\"pay\"
docsubty=\"FA\" crt=\"International Union of Crystallography\"
language=\"0\"\>\<jnlinfo name=\"Acta Crystallographica Section D\"
yr=\"2004\" issue=\"1\" volume=\"60\" abbrtitle=\"Acta Cryst. D\"
coden=\"ABCRE6\" editor=\"J. P. Glusker\" issn=\"0907-4449\" fpage=\"0\"
lpage=\"0\"\>\<fm\>\<atl\>Surprises and pitfalls due to (approximate)
symmetry\</atl\>

\<aug\>\<au\>\<fnm\>Peter H.\</fnm\> \<snm\>Zwart\</snm\>\</au\>,\<orf
id=\"^a^\"\>\<cor email=\"PHZwart@lbl.gov\"\>\</cor\>^\*^ Ralf. W.
Grosse-Kunstleve^a^\"\>, \<au\>\<fnm\>Andrey\</fnm\> A.
\<snm\>Lebedev\</snm\>\</au\>,\<orf id=\"^b^\"\>
\<au\>\<fnm\>Garib\</fnm\> N. \<snm\>Murshudov\</snm\>\</au\>\<orf
id=\"^b^\"\> and \<au\>\<fnm\>Paul D. \</fnm\>
\<snm\>Adams\</snm\>\</au\>\<orf id=\"^a^\"\>

\<aff\>\<oid id=\"^a^\"\>Lawrence Berkeley National Laboratory,
Computational Crystallography Initiative, \<cny\>USA\</cny\>\</aff\>,
and \<aff\>\<oid id=\"^b^\"\>York Structural Biology Laboratories,
\<cny\>UK\</cny\>\</aff\>\</aug\>. E-mail: PHZwart@lbl.gov

1.  It is not uncommon for protein crystals to crystallise with more
    than a single molecule per asymmetric unit. The possibility of
    multiple favourable inter molecular contacts often forms the
    structural basis for polymorphisms that can result in various
    pathological situations such as twinning, modulated crystals and
    pseudo translational or rotational symmetry. The presence of pseudo
    symmetry can lead to uncertainties about the correct space group,
    especially in the presence of twinning. We present the background to
    certain common pathologies and introduce a new notation for space
    groups in unusual settings. Main concepts are illustrated with
    various examples from the literature and PDB.

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1.  pathology; twinning; pseudo symmetry

# Introduction {#introduction .IUCr-heading-1}

With the advent of automated methods in crystallography (Ada*ms et a*l.,
2002; Ada*ms et a*l., 2004; Brunzel*le et a*l., 2003; Lamzin & Perrakis,
2000; Lamz*in et a*l., 2000; Sne*ll et a*l., 2004), it is possible to
solve a structure without visual inspection of the diffraction images
(Winter, 2007; Holton & Alber, 2004), interpretation of the output of a
molecular replacement program (Read, 2001; Navaza, 1994; Vagin &
Teplyakov, 2000) or, in extreme cases, manually building a model or even
looking at the electron density map (Emsley & Cowtan, 2004; Terwilliger,
2002b; Morr*is et a*l., 2004; Morr*is et a*l., 2003; Terwilliger, 2002a;
Holt*on et a*l., 2000; Ioerg*er et a*l., 1999; McRee, 1999; Perrak*is et
a*l., 1999). Although automated methods often handle many routine
structure solution scenarios, pitfalls due to certain pathologies are
still outside the scope of most automated methods and often require
human intervention to ensure a smooth progress of structure solution or
refinement.

This manuscript studies situations that arise when non-crystallographic
symmetry (NCS) operators are close to true crystallographic symmetry.
Pathologies of this type are often seen in protein crystallography (e.g.
Daut*er et a*l., 2005), since a large number of proteins crystallise
with more than a single copy in the asymmetric unit or in various space
groups.

The distinction between "simple" NCS and pseudo symmetry can be be made
in a number of ways. One way of defining pseudo symmetry is by
idealising NCS operators to crystallographic operators and by
determining the *root mean square displacement* (RMSD) between the
structural assembly in the true low-symmetry space group and the
structure with the idealised symmetry. If the resulting RMSD is below a
certain threshold value (say 3Å), the structure can be called pseudo
symmetric. Using this definition, we find that about 6% of the
structures deposited in the PDB exhibit pseudo symmetry. This is in line
with the observations of Wang & Janin (XXXX) who conclude that the
alignment of NCS axes is biased towards crystallographic symmetry axes.
On a year-to-year basis, there is a slow increase in the fraction of new
structures that exhibit pseudo symmetry (Fig 1). This small increase is
most likely due to improvements in hard and software that allow for a
more routine detection, solution and refinement of structures with
pseudo symmetry, as well as a general tendency to focus on more
challenging proteins or protein complexes.

In order to develop a better understanding of the consequences of
pseudo-symmetry, we review some basic concepts and introduce an
efficient way of describing space groups in unusual settings. We
furthermore \'visualise\' relations between space groups *via* graphs,
similar to those generated by the Bilbao crystallographic server (XXXX).
On contrast to these, the graphs presented here include the point groups
or space groups in all orientations in which they occur in the
supergroups, rather than just one representative per point or space
group type. This results in a more informative and complete overview of
the relations between different groups.

A number of examples from the PDB (Berm*an et a*l., 2000; Bernste*in et
a*l., 1977) and the literature are provided to illustrate common
surprises and pitfalls. We point out structures with suspected incorrect
symmetry, give an example of molecular replacement with twinned data and
ambiguous space group choices, and we illustrate the uses of
group-subgroup relations.

# Space groups, symmetry and approximate symmetry {#space-groups-symmetry-and-approximate-symmetry .IUCr-heading-1}

##  {#section .IUCr-heading-2}

##  {#section-1 .IUCr-heading-2}

##  {#section-2 .IUCr-heading-2}

## Space groups in unusual settings {#space-groups-in-unusual-settings .IUCr-heading-2}

The standard reference for crystallographic space group symmetry is
International Tables for Crystallography, Volume A (Hahn, 2002). In the
following we will use *ITVA* to refer to this work. *ITVA* Table 4.3.1
defines Hermann-Mauguin space group symbols for 530 conventional
settings of the 230 space group types. This means, in general there are
multiple settings for a given space group type. For example, assume we
are given an X-ray dataset that can be integrated and scaled in space
group *P*222. Further analysis of the data reveals systematic absences
for (*0,k,0*) (*k* odd). This suggests the space group is *P*22~1~2. It
may be useful or necessary (e.g. for compatibility with older software)
to reindex the dataset so that the two-fold screw axis is perpendicular
to a new **a** axis, to obtain space group *P*2~1~22. The space groups
and unit cells before and after reindexing are said to be in different
settings.

In the context of group-subgroup analysis with respect to a given metric
(unit cell parameters), unusual settings not tabulated in *ITVA* arise
frequently. To be able to represent these with concise symbols, we have
introduced *Universal Hermann-Mauguin Symbols*, by borrowing an idea
introduced in (Shmue*li et a*l., 2001): a change-of-basis symbol is
appended to the conventional Hermann-Mauguin symbol. To obtain short
symbols, two notations are used. For example (compare with Fig. 4
below):

*C* 1 2 1 (*x*-*y*, *x*+*y*, *z*)

*C* 1 2 1 (1/2\**a*-1/2\**b*, 1/2\**a*+1/2\**b*, *c*)

These two symbols are equivalent, i.e. encode the same unconventional
setting of space group No. 5. The change-of-basis matrix encoded with
the *x*,*y*,*z* notation is the inverse-transpose of the matrix encoded
with the *a*,*b*,*c* notation. Often, for a given change-of-basis, one
notation is significantly shorter than the other. The shortest symbol is
used when composing the universal Hermann-Mauguin symbol.

Note that both change-of-basis notations have precedence in *ITVA*. The
*x*,*y*,*z* notation is used to symbolise symmetry operators which act
on coordinates. Similarly, the *x*,*y*,*z* change-of-basis symbol
encodes a matrix that transforms coordinates from the reference setting
to the unconventional setting. The *a*,*b*,*c* notation appears in
*ITVA* section 4.3, where it encodes basis-vector transformations. Our
*a*,*b*,*c* notation is compatible with this convention. The *a*,*b*,*c*
change-of-basis symbol encodes a matrix that transforms basis-vectors
from the reference setting to the unconventional setting. A
comprehensive overview of transformation relationships is given in and
around Table 2.E.1 of (Giacovazzo, 1992).

## Relations between groups {#relations-between-groups .IUCr-heading-2}

A subgroup $H$ of a group $G$ ($H \neq G$) is a subset of the elements
of $G$ which also forms a (smaller) group. For instance, the symmetry
operators of space group *P*222 can be described by {(*x*,*y*,*z*),
(-*x*,*y*,-*z*), (*x*,-*y*,-*z*), (-*x*,-*y*,*z*)}. Subgroups of *P*222
can be constructed by selecting only certain operators. The full list of
subgroups of *P2*22 and the set of 'remaining operators' for each
subgroup with respect to *P*222 are given in Tab. 1.

Note that if the operators of *P*211 are combined with one of the
'remaining' operators (-*x*,*y*,-*z*) or (-*x*,-*y*,*z*), the other
operator is generated by group multiplication, leading to *P*222. A
depiction of the relations between all subgroups of *P*222 is shown in
Fig. 1. In this figure, nodes representing space groups are linked with
arrows. The arrows between the space groups indicate that the
multiplication of a single symmetry operator into a group results in the
other group. For example, the arrow in Fig. 1 from *P*1 to *P*211
indicates that a single symmetry element (in this case (*x,-y*,-*z*) )
combined with *P*1 results in the space group *P*211.

## Pseudo symmetry {#pseudo-symmetry .IUCr-heading-2}

As mentioned before, it is not uncommon that non-crystallographic
symmetry approximates crystallographic symmetry. A change of the space
group symmetry of a known crystal form, either a reduction or an
increase of the symmetry, is often associated with ligand binding, the
introduction of Seleno-methionine residues, halide or heavy metal
soaking, or crystal growth under different conditions (Daut*er et a*l.,
2001; Pouls*en et a*l., 2001; Parsons, 2003).

The presence of an approximate symmetry can be a structural basis for
twinning or pseudo-rotational or pseudo-translational symmetry.
Group-subgroup relations and their graphical representations as outlined
in section 2.2, are a useful tool for understanding approximate symmetry
and the resulting relations between the space groups of different
crystal forms. The graphical representations can often provide an easy
way of enumerating and illustrating all possible subgroups of a space
group. This enumeration of possible space or point groups, can be useful
in the case of perfect merohedral twinning.

Constructing artificial structures with pseudo symmetry is
straightforward. For example, given the asymmetric unit of a protein in
*P*222, generate a symmetry-equivalent copy using the operator
(-*x*,*y*,-*z*) or (-*x*,-*y*,*z*). If small random perturbations are
applied to this new copy (e.g. a small overall rotation or small random
shifts), the resulting symmetry is *P*211, with *P*222 pseudo symmetry.
The two protein molecules in the *P*211 asymmetric unit are related by a
non-crystallographic symmetry (NCS) operator that is very similar to a
perfect two-fold crystallographic rotation.

Note that in the previous example, crystallographic symmetry operators
were transformed into an NCS operator by the application of a small
perturbation of the coordinates. The 'remaining operators' in Tab. 1 can
be seen as NCS operators that are approximately equal to the listed
operators.

# Common pathologies {#common-pathologies .IUCr-heading-1}

## Rotational pseudo symmetry {#rotational-pseudo-symmetry .IUCr-heading-2}

Rotational pseudo symmetry (RPS) can arise if the point-group symmetry
of the lattice is higher than the point-group symmetry of the crystal.
RPS is generated by an NCS operator parallel to a symmetry operator of
the lattice that is *not* also a symmetry operator of the crystal space
group. A prime example of such a case can be found in PDB entry 1Q43
(Zagot*ta et a*l., 2003). The structure crystallises in space group
*I*4, with two molecules per *asymmetric unit* (ASU). The RMSD between
the two copies in the ASU is 0.27 Å. The NCS operator (in fractional
coordinates) that relates one molecule to the other is:

$$R = \begin{pmatrix}
 + 0\text{.}\text{056} & - 0\text{.}\text{998} & - 0\text{.}\text{003} \\
 - 0\text{.}\text{998} & - 0\text{.}\text{056} & - 0\text{.}\text{003} \\
 + 0\text{.}\text{002} & + 0\text{.}\text{003} & - 1\text{.}\text{000}
\end{pmatrix}\ T = \begin{pmatrix}
0\text{.}\text{50} \\
0\text{.}\text{50} \\
0\text{.}\text{31}
\end{pmatrix}$$

The rotational part $R$of the NCS operator can be recognised as being
almost identical to a two-fold axis in the *xy*-plane. If the idealised
operator $( - y + 12, - x + 12, - z + 0\text{.}\text{31})$ is multiplied
into space group *I*4 we obtain space group *I*422 with an arbitrary
origin shift along z, which is a polar axis in *I*4.

The R-value between pseudo-symmetry related intensities as calculated
from the coordinates is equal to 44%. For unrelated (independent)
intensities, the R-value is expected to be equal to 50% (Lebed*ev et
a*l., 2006). In this case it is clear that the correct symmetry is *I*4
rather than *I*422. However, there is a grey area where it may be
possible to merge the data with reasonable statistics in the higher
symmetry. While this has the advantage of reducing the number of model
parameters, over-idealisation of the symmetry may lead to problems in
structure solution and particularly refinement. Furthermore, information
about biologically significant differences may be lost. In case of doubt
the best approach is to process and refine in both the lower and the
higher symmetry, and to compare the resulting R-free values and model
quality indicators.

## Translational pseudo symmetry and pseudo centring {#translational-pseudo-symmetry-and-pseudo-centring .IUCr-heading-2}

Translational pseudo symmetry (TPS) is an NCS operator whose rotational
part is close to a unit matrix. If a TPS operator, or a combination of
TPS operators, is very similar to a group of lattice-centring operators,
it can be denoted as pseudo centring. An example is PDB entry 1SCT
(Roy*er et a*l., 1995) where an NCS operator $(x + 12,y + 12,z)$ mimics
a C-centring operator. In this particular case, the true space group is
*P*2~1~2~1~2~1~, but pseudo symmetric *C*222~1~.

In reciprocal space, the presence of pseudo centring operators
translates into a systematic modulation of the observed intensities
(e.g. Chook et al., XXXX) and is most easily detect by inspection of the
Patterson function (XXXX). The subset of reflections that would be
systematically absent given idealised centring operators will have
systematically low intensities. If these intensities are sufficiently
low, data processing programs may index and reduce the diffraction
images in a unit cell that is too small. This situation is very similar
to the case of higher rotational symmetry as discussed in the previous
section. A smaller unit cell is also a higher symmetry leading to a
reduction of the number of model parameters. The "grey area"
considerations of the previous section apply also to TPS. In addition,
the efficiency of likelihood-based approaches that rely on specific
assumptions about the distribution of the observed amplitudes can be
impeded by the presence of TPS (see Read *et al.* in this issue).

An interesting crystallographic pathology can arise when pseudo centring
is present. An example is given by (Lebed*ev et a*l., 2007). In this
case the space group is *P*2~1~ with a pseudo translation (*x*+½, *y*,
*z*). The approximate symmetry is equal to two *P*2~1~ cells stacked
side by side on the (**b**,**c**)-face of the unit cell. The resulting
symmetry is described by the universal Hermann-Mauguin symbol *P*12~1~1
(2*a*, *b*, *c*). A full list of symmetry operators in this setting is
shown in Tab. 2. From this set of operators, a number of subgroups can
be constructed. Operators not used in the construction of the subgroup
can be regarded as NCS operators. If operators A and B are designated as
crystallographic symmetry, the space group is *P*2~1~ and operators C
and D are NCS operators. If however operators A and D are designated to
be crystallographic, the space group is *P*2~1~ with an origin shift of
(¼,0,0) and B and C are NCS operators. Both choices produce initially
reasonable R-values, but only one choice is correct and eventually leads
to the best model. Further details of problems encountered due to this
ambiguity are reported by Lebed*ev et a*l. (2007).

## Twinning {#twinning .IUCr-heading-2}

Twinning is the partial or full overlap of multiple reciprocal lattices.
Resulting recorded intensities are therefore equal to the sum of the
intensities of the individual domains with different orientations. The
presence of twinning in an X-ray data set reveals itself usually by
intensity statistics deviating from theoretical distributions. However,
the presence of pseudo-rotational symmetry (especially when parallel to
the twin law) or pseudo-translational symmetry can offset the effects of
twinning on the intensity statistics, making it more difficult to detect
the twinning. Basic intensity statistics elucidating the problems of
pseudo symmetry in combination with twinning are explained thoroughly by
Lebed*ev et a*l. (2006). Prime examples of problems with space-group
assignment due to the presence pseudo symmetry and twinning are
described by Abrscica (XXXX) , Lee (XXXX), Rudino-pinera, XXXX and Dicer
(XXXX).

The relative sizes of the twin domains building up the crystal are the
*twin fractions*. The sum of the twin fractions is one. The situation
where all twin fractions are all equal is called *perfect twinning*. A
twin with an arbitrary ratio of twin fractions is denoted as a partial
twin. A number of high-quality papers are available from the literature
that deal with basic introduction to twinning (XXXX), as well as case
studies of particular proteins (Yang et al (2000), Rudolph collection
XXXX, Lehtio 2005, Barends 2003, 2005).

### Merohedral and pseudo merohedral twins {#merohedral-and-pseudo-merohedral-twins .IUCr-heading-3}

Merohederal or pseudo-merohedral twinning is a form of twinning in which
the (primitive) lattice has a higher symmetry than the symmetry of the
unit cell content. If this occurs, the arrangement of reciprocal lattice
points will have a higher symmetry than the symmetry of the intensities
associated with the reciprocal lattice points. The symmetry operators
that belong to the point group of the reciprocal lattice, but not to the
symmetry of the point group of the intensities, are potential *twin
laws*.

If the reciprocal lattice is perfectly invariant under a given twin law
(merohedral twinning), the presence of twinning can be detected only by
inspection of the intensity statistics or model-based techniques.
However, if the reciprocal lattice is only approximately invariant under
a given twin law (pseudo-merohedral twinning), twin related intensities
may be identified as individual reflections in the diffraction pattern.
Examples of a number of (pseudo) merohedrally twinned structures are
given in Tab. 3.

The presence of an NCS operator that is an approximate crystallographic
operator provides a structural basis for the presence of twinning.
Twin-domain interfaces have molecular contacts that are very similar to
interfaces seen in non-twinned domains which allows or promotes the
growth of twinned crystals in general. In a similar manner, twinning can
be introduced by the breaking of symmetry due to a temperature dependent
phase transition (Helliwe*ll et a*l., 2006; Herbst-Irmer & Sheldrick,
1998; Parsons, 2003), or by other external influences such as inclusion
of a ligand or heavy atom soaks. An example of such a phase transition
is described by (Daut*er et a*l., 2001). In that particular case
however, the symmetry of the crystals before a halide soak had a lower
symmetry than after the soak, eliminating the possibility of twinning.

Note that when a crystal is perfectly twinned or almost perfectly
twinned, the data will scale well in a space group that is incorrect.
The use of an incorrect space group often impedes a successful structure
solution procedure. A general theme in most cases studies involving
difficulties with twinning (see references in paragraph 3.3) is that
structure solution is possible once the correct space group has been
found. Incorrect assignment of the space group for datasets with close
to perfect twinning seems to be the most important factor hampering
structure solution.

### Reticular merohedral twinning {#reticular-merohedral-twinning .IUCr-heading-3}

Reticular merohedral twinning can be understood as merohedral twinning
on an assembly of unit cells, a so called sublattice (Rutherford, 2006).
In this type of twinning, only a certain part of the reciprocal lattices
of the twin domains will overlap, resulting in a diffraction pattern
that consists of intensity sums with contributions from a variable
number of twin domains. A well known example of twinning by reticular
merohedry is the obverse-reverse twinning in rhombohedral space groups.
An excellent introduction into twinning by reticular merohedry is given
by (Parsons, 2003). Examples of diffraction patterns are found in
(Dauter, 2003).

### Order-disorder twinning {#order-disorder-twinning .IUCr-heading-3}

Order-disorder twinning (Dornberger-Schiff & Dunitz, 1965;
Dornberger-Schiff & Grell-Niemann, 1961; Dornberger-Schiff, 1966, 1956)
is a less well-classified type of twinnig, but has been observed for
protein structures in a number of cases (Trame & McKay, 2001; Wa*ng et
a*l., 2005). Order-disorder twinning can occur when a crystal lattice is
build up of successive layers of molecules, while the crystal contacts
between the layers allow the molecules to shift relative to each other.
If one of the layers has a random perturbation in the plane of this
layer of molecules, only a subset of the crystal lattice is ordered. The
presence of disordered layers of molecules in the lattice introduces a
modulation on the intensities of specific reflections. A correction for
this effect can be vital for structure solution (Wa*ng et a*l., 2005)
and can result in lower R-values during refinement (Trame & McKay,
2001). As noted by (Nespo*lo et a*l., 2004), order-disorder phenomena in
combination with twinning may easily go unnoticed during structure
solution and refinement.

## Common pitfalls {#common-pitfalls .IUCr-heading-2}

*Misindexing.* If the beam centre has not been defined accurately
enough, auto indexing programs can return an indexing solution in which
the (0,0,0) reflection (the direct beam) is, for instance, indexed as
(0,0,1). Subsequent merging of the data will fail if the Miller indices
are not corrected. Misindexing can be avoided by obtaining the position
of the direct beam on the detector using powder methods or by using more
robust auto-indexing routines (Saut*er et a*l., 2004).

*Incorrect unit cell.* When more than one single crystal is present or
when the diffraction images are noisy in general, it is possible that
auto-indexing procedures will result in a unit cell that is too large.
In the integrated and merged data this issue can reveal itself as a
prominent peak in a Patterson function. In contrast, if the structure
under investigation has a strong pseudo translation, it can occur that
the indexing solution corresponds to a unit cell that is too small. In
such a case, reflections that are systematically weak due to the pseudo
translation are ignored and the pseudo translation is treated as a
lattice translation.

*Incorrect space group.* If an approximately correct unit cell has been
obtained, the space group has to be determined based on the intensities.
The presence of pseudo symmetry can make this choice difficult, but it
can often be made automatically by programs such as phenix.xtriage
(Zwa*rt et a*l., 2005), Xprep (Sheldrick, 2000), Pointless (Evans, 2006)
or Labelit (Saut*er et a*l., 2006). Assigning an incorrect space group
can result in a number of difficulties. If the assigned space group is
too low, structure solution and refinement is made artificially
difficult because of the larger number of molecules in the asymmetric
unit. Furthermore, differences between molecules can subsequently be
over-interpreted resulting in incorrect biological conclusions.

If the data is twinned and as a result the assigned space group is too
high, it may not be possible to solve the structure. An excellent
example that illustrates this (and other) pitfalls is given by L*ee et
a*l. (2003), where the presence of pseudo translational symmetry and
perfect twinning resulted in an incorrect choice of both the unit cell
and space group.

# Examples {#examples .IUCr-heading-1}

## Interesting cases from the PDB {#interesting-cases-from-the-pdb .IUCr-heading-2}

A number of datasets in the PDB show interesting pathologies, such as
twinning and pseudo-rotational and or pseudo-translational symmetry. A
few examples are highlighted here.

### 2BD1: Incorrect symmetry*.*  {#bd1-incorrect-symmetry. .IUCr-heading-3}

The structure of phospholipase A2 (Sek*ar et a*l., 2006) was indexed in
*C*2 with unit cell parameters (74.58, 48.69, 67.55, 90, 102.3, 90). The
Patterson function reveals a peak at (0, ½, 0) with a height
approximately equal to that of the origin (99%). Correspondingly, the
intensities of the reflections with Miller indices that would be equal
to zero if the NCS operator was crystallographic, barely rise above the
noise as judged from their associated standard deviations. The RMSD
between the C~α~ of the two molecules related by the translational NCS
operator obtained from the Patterson function is very small, 0.08 Å. In
comparison, the cross-validated estimate of the coordinate error is 0.19
Å, which strongly suggests that the unit cell is in fact too large.

### 2A8Y: Incorrect symmetry {#a8y-incorrect-symmetry .IUCr-heading-3}

The unit cell constants for this structure are (96.60, 96.56, 96.63,
91.57, 91.23, 91.52). The deposited space group is *P*1 (Zha*ng et a*l.,
2006). Cursory analysis of the unit cell parameters suggests that the
highest possible symmetry is rhombohedral. An analysis of the merged
intensities with phenix.xtriage reveals that the intensity symmetry
corresponds to the space group *C*2, with unit cell parameters (135.2,
138.1, 96.6, 90, 92.2, 90). In this particular case, the authors did
attempt to merge the data in various point groups (including *C*2), but
the data only scaled well in space group *P*1 (Zha*ng et a*l., 2006).
Given the pseudo symmetric nature of the lattice (pseudo rhombohedral),
*C*2 can be embedded in the higher-symmetry lattice in three different
ways (see Fig. 4), corresponding to the three orientations of the
two-fold axis in space group *R*32. The integration suite used to
initially process the data only gave a single indexing choice for *C*2,
which was unfortunately incorrect. Currently, the structure is being
re-refined in the higher symmetry *C*2 space group (Ealick, private
communication).

### *1UPP: Pseudo translational symmetry and twinning*  {#upp-pseudo-translational-symmetry-and-twinning .IUCr-heading-3}

The structure of a spinach rubisco complex (Karkehaba*di et a*l., 2003)
has associated unit cell parameters (155.9, 156.3, 199.8, 90, 90, 90)
and space group *C*222~1~. Obviously, ***a*** is approximately equal to
***b***, resulting in the presence of the twin law (k,h,-l).
Furthermore, the Patterson function indicates a translational NCS vector
(½, 0, ½) with a height of 40% of the origin. The presence of
pseudo-translational symmetry can make the detection of twinning
difficult, but the results of the L-test (Padilla & Yeates, 2003) are
quite clear (Tab. 4). Refinement of the twin fraction given the
deposited structure indicates that the twin fraction is approximately
45%. Including twinning in the R-value calculations (while keeping the
model fixed), reduces the R-value from 0.25 to 0.17.

## Molecular replacement using twinned data {#molecular-replacement-using-twinned-data .IUCr-heading-2}

Using artificially twinned data, it can be demonstrated that the
contrast of the rotation function decreases in proportion to the twin
fraction (Fig. 4). A similar observation is made for the translation
function (Fig. 5). However, from practical experience we know that
molecular replacement based on twinned data is often successful if the
quality of the search model is reasonable.

In the case of perfect twinning, a data reduction program may pick a
space group that is too high (see section 3.3.1). In this situation it
is unlikely that molecular replacement will produce a solution, as the
ASU is typically too small to contain the true contents of the crystal.
Working with the data re-processed in the lower symmetry may be
successful, even though the data is perfectly twinned. This is
illustrated by the following example.

The X-ray dataset of Dicer (XXXX) crystals were initially processed in
point group *P*422, but failed to give interpretable maps using
experimental phasing methods in *P*4~x~2~y~2 groups. Intensity
statistics revealed that the data are twinned, and reprocessing the data
in point group *P*4, resulted in partially interpretable SAD maps in
space group *P*4~1~, assuming a two fold twin law along **b**. A
complete and refinable model could however not be obtained in any
tetragonal space group. Collecting data from a new specimen revealed
that the point group symmetry was *P*222 with almost perfect twinning,
leading to a pseudo tetragonal system. A successful structure solution
via SAD methods was obtained in space group *P*2~1~22~1~. Although the
Dicer structure was solved via experimental phasing methods, it is
instructive to re-solve the structure via molecular replacement under
various space group hypotheses. To this end, we reindexed data submitted
to the PDB with accession code 2QWV from *P*2~1~2~1~2 to *P2~1~22~1~*
with operator (-a,c,b), to obtain a setting that corresponds to the
standard setting if the data were merged in point group *P*422. The data
with intensity symmetry *P*222 was subsequently merged to *P*422. After
this, the merged data were expanded out to point group *P*4 and *P*222,
the two point groups directly 'below' point group *P*422. Molecular
replacement with chain A of the deposited model was used to determine
the structure in all possible space groups for the given point groups.
The rotation function gave 2 clear solutions in point group *P*422, and
4 clear solutions in point groups *P*4 and *P*222 (Tab 5). Subsequent
translation functions and refinement of the twin fraction, resulted in
three likely possible solutions in space groups *P*4~1~, *P*2~1~22~1~
and *P*22~1~2~1~ (Tab. 6). Further rigid-body and ADP group refinement
lowered the R-values of the space group candidates in the orthorhombic
system to 25%, while the model in *P*4~1~ had an R-value of 29%.

Note that it is not surprising that *P*2~1~22~1~ and *P*22~1~2~1~ are
both possible solutions since the data are perfectly twinned in point
group *P*222 with twin law (-k,-h,-l). Data with a lower twin fraction
or the presence of anomalous differences can be used to determine the
space group.

## Manual molecular replacement using group-subgroup relations {#manual-molecular-replacement-using-group-subgroup-relations .IUCr-heading-2}

It is not uncommon that a protein molecule crystallises in various space
groups (polymorphs). In some cases, the polymorphs are related and one
can use the structure of one polymorph to solve the other without the
aid of automated molecular replacement software (Di Costan*zo et a*l.,
2003). An example structure solution utilising group-subgroup relations
is presented here.

The crystals of 1EIX and 1JJK (Pouls*en et a*l., 2001) have been grown
under similar conditions, but 1EIX is a native protein structure, while
1JJK is a seleno-methionine derivative. The cell parameters are listed
in Tab. 6. The ratio of the unit cell volumes is 2.12, suggesting the
possibility of a relation between the two unit cells. Another piece of
evidence suggesting a relation is found in the Patterson function of
1JJK: a large peak is located at (½, 0, ½), which can be interpreted as
pseudo centring (translational NCS). If this NCS operator is idealised
to a crystallographic operator, the unit cell dimensions of 1JJK become
equal to the unit cell constants of 1EIX (apart from a permutation of
the basis vectors). It is thus clear that 1JJK is related to 1EIX via
pseudo translational symmetry (as seen from the Patterson peak) and
pseudo rotational/screw symmetry (*P*2~1~ versus *P*2~1~2~1~2~1~).

A relation between the two unit cells was identified with the tool
*iotbx.explore_metric_symmetry* (Zwa*rt et a*l., 2006) and is depicted
in Fig. 6. The procedure used to solve the structure of 1JJK with the
model of 1EIK via group-subgroup relations is described in (Di Costan*zo
et a*l., 2003). First, the appropriate ASU is constructed by applying a
two-fold screw axis to the ASU of 1EIX. Subsequently, a lattice
translation along **a** is applied to these two molecules. An
appropriate change of basis to bring the model in the correct
orientation and a subsequent origin shift generates a possible solution.
In this particular case, a group-theoretical analysis reveals that two
origin shifts are possible (see section 3.2). Rigid-body refinement of
the two possible solutions, taking into account the presence of
twinning, resulted in a single, clear solution (Tab. 7).

Using the same group-subgroup relations, one can test the presence of a
relation between crystal forms by computing intensity correlations
between reindexed datasets. If the crystal form with the smaller unit
cell is reindexed to a unit cell that is related to the larger cell, the
intensities can be compared relatively straightforwardly (XXXX). This
allows one to verify a possible relation between two crystal forms
before attempting manual molecular replacement.

# Discussion and Conclusions {#discussion-and-conclusions .IUCr-heading-1}

There are numerous special cases and pitfalls arising from the interplay
of crystallographic and non-crystallographic symmetry in macromolecular
crystals. Clearly this paper only touches the tip of the iceberg.
Fortunately there are now a number of tools that make it possible to
identify many of the most common problems (XXXX). In some situations it
is possible to correct for the problem, in others use of the appropriate
algorithms in subsequent structure solution and refinement can lead to
accurate final models suitable for biological interpretation. Experience
suggests that it is initially best to treat all experimental data with
suspicion and apply all available tests to identify possible pathologies
as soon as possible. In an ideal world data would be stored in an
unmerged form in space group *P*1 and decisions are made automatically
as more information becomes available. In the case of (close to) perfect
twinning, knowledge of the proper space group can for instance only be
available when a partial model is build. A similar argument can be made
for the detection and for dealing with order-disorder twinning. Note
that this scheme assumes that the correct unit cell has been found by
the auto-indexing software. Incorporating decision-making schemes that
include changes in the primitive unit cell parameters will most likely
require access to the raw data.

Although access to the raw data is preferred for decision-making
procedures, the Dicer (XXXX) example illustrates that integrating the
expansion to a lower symmetry into the molecular replacement procedure
could in some cases solve the structure. Similar arguments can probably
be made for structure solution routes via experimental phasing
techniques.

1.  The authors would like to thank the NIH for generous support of the
    PHENIX project (1P01 GM063210). This work was partially supported by
    the US Department of Energy under Contract No. DE-AC02-05CH11231
    (PHZ, RWGK & PDA), the Wellcome Trust (064405/Z/01/A, GNM), and NIH
    (1-RO1-GM069758-03, AAL). The programs mentioned here are available
    in the PHENIX
    ([http://www.phenix-online.org](http://www.phenix-online.org/)) and
    CCP4 ([http://www.ccp4.ac.uk](http://www.ccp4.ac.uk/)) software
    suites.

2.  

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1.  Prevalence of structures with pseudo symmetry in the PDB since 1990.
    ![](media/image1.wmf){width="4.016666666666667in"
    height="2.729861111111111in"}

2.  A graphical representation of all group-subgroup relations for
    *P*222 and it subgroups. The arrows connecting two space groups
    represent the addition of a single operator to the parent space
    groups and its result.

    ![](media/image2.png){width="3.125in" height="1.6868055555555554in"}

3.  A space group graph showing all subgroups of space group *P* 1 21 1
    (2a,b,c). Specifically, two distinct P21 subgroups are available
    with equal unit cell dimensions, related by an origin shift of ¼.
    See text and Table 2 for details.

    ![](media/image3.png){width="4.625694444444444in"
    height="1.7694444444444444in"}

4.  A space group graph showing all subgroups of space group R32:R.
    Specifically, three distinct *C*2 subgroups are available,
    corresponding to three possible directions the two fold axes are
    oriented in *R*32.

    ![](media/image4.png){width="3.497916666666667in" height="2.025in"}

5.  The value of the largest peak in the rotation function of the
    program MOLREP for synthetically twinned data. The blue line shows
    the value for a model with 100% sequence identity, whereas the
    purple line indicates the results of the rotation function when the
    model has 27% sequence identity. In both runs, the search model
    compromised only one quarter of the total ASU content.

    ![](media/image5.wmf){width="3.125in" height="2.7041666666666666in"}

6.  The value of the contrast of the translation function of the program
    MOLREP for synthetically twinned data, given a correct orientation
    of the model. The search model was identical to model used to
    compute the artificially twinned data.

    ![](media/image6.wmf){width="3.125in" height="2.279166666666667in"}

7.  The relations between projections of the unit cells of 1EIX and
    1JJK. The basis vectors of the orthorhombic unit cell of 1EIX are
    shown in black. The blue and red basis vectors are two equivalent
    choices for a unit cell of 1JJK. The purple vectors indicate that
    1JJK is approximately C-centred orthorhombic.

    ![](media/image7.png){width="2.214583333333333in"
    height="2.2597222222222224in"}

```{=html}
<!-- -->
```
1.  Subgroups of *P*222. For each subgroup, the symmetry operators are
    specified, together with the operators that are element of *P*222,
    but not of the subgroup. The parent space group *P*222 is included
    for completeness.

  --------------- ------------------------------ -------------------------
  Space group     Operators                      Remaining operators

  *P*222          (x,y,z), (-x,y,-z), (x,-y,-z), None
                  (-x,-y,z)                      

  *P*211          (x,y,z), (x,-y,-z)             (-x,y,-z), (-x,-y,z)

  *P*121          (x,y,z), (-x,y,-z)             (x,-y,-z), (-x,-y,z)

  *P*112          (x,y,z), (-x,-y,z)             (-x,y,-z), (x,-y,-z)

  *P*1            (x,y,z)                        (-x,y,-z), (x,-y,-z),
                                                 (-x,-y,z)
  --------------- ------------------------------ -------------------------

2.  The presence of a pseudo-centring operator (x+½,y ,z) in *P*21 can
    lead to an interesting pathology. If all operators are
    crystallographic operators, and (x+½,y ,z) is designated to be a
    lattice translation (LT), the a number of groups can be formed. See
    figure 2 and the main text for details.

  -------------- -------------------------- ------------------------------
  Name           Operators                  Description

  A              (x,y,z)                    E

  B              (-x,y+½ ,-z)               2~1~ through (0,0,0)

  C              (x+½, y,z)                 LT

  D              (-x+½,y+½ ,-z)             2~1~ through (¼,0,0)
  -------------- -------------------------- ------------------------------

3.  Examples of (pseudo-) merohedrally twinned structures.

  -------- ------------------------ ------------ ------------- ---------- ------
  PDBID    Unit cell constants      Space group  Twin law      Fraction   Type

  1Q43     95, 95, 125, 90, 90, 90  *I*4         (-k,-h,-l)    8%         M

  1EYX     180, 180, 36, 90, 90,    *R*3:H       (h,-h-k,-l)   45%        M
           120                                                            

  1UPP     155.8, 156.2, 199.7, 90, *C* 2 2 2~1~ (h,k,-l)      45%        PM
           90, 90                                                         

  1L2H     53.9, 53.9, 77.4, 90,    *P* 4~3~     (-h,k,-l)     37%        M
           90, 90                                                         
  -------- ------------------------ ------------ ------------- ---------- ------

4.  Intensity statistics of 1UPP. Only the L-test indicates that the
    data might be twinned.

  ------------------------------------------------------------------------------- ----------- -------------------- ----------------------
  Statistic                                                                       Observed    Theory (untwinned)   Theory (perfect twin)

  $$\frac{\left\langle I^{2} \right\rangle}{\left\langle I \right\rangle^{2}}$$   2.09        2                    1.5

  $$\frac{\left\langle F \right\rangle^{2}}{\left\langle F^{2} \right\rangle}$$   0.80        0.785                0.885

  $$\left\langle |E^{2} - 1| \right\rangle$$                                      0.731       0.736                0.541

  $$\left\langle L \right\rangle$$                                                0.43        0.50                 0.375
  ------------------------------------------------------------------------------- ----------- -------------------- ----------------------

5.  Rotation function peaks of Dicer (XXXX) data.

  ------------------- ---------------------------------------------------
  Point group         Top rotation function peaks (Rf/sigma)

  *P*422              **8.29 7.55** 4.61

  *P*4                **8.32 8.32 7.58 7.58** 4.62

  *P*222              **8.27 8.27 7.55 7.55** 4.61
  ------------------- ---------------------------------------------------

6.  Translation function and refinement results for Dicer (XXXX).

  --------------------- ------------- ------------- ----------- -----------
  Space group           Copies        R~molrep~     R~twin~     R~group~

  *P*4~1~22             2             47%           *NA*        *NA*

  *P*4~1~2~1~2          2             48%           *NA*        *NA*

  *P*4~1~               4             43%           38%         29%

  *P*222~1~             4             49%           39%         *NA*

  *P*2~1~22~1~          4             43%           36%         25%

  *P*22~1~2~1~          4             43%           37%         25%

  *P*2~1~2~1~2~1~       4             49%           43%         *NA*
  --------------------- ------------- ------------- ----------- -----------

Only molecular replacement solutions with initial R-values (R~molrep~)
below 50% are shown. All other space groups results in R~molrep~ values
above 50%. R~twin~ is the R-value of the model to the data as obtained
directly after molrep, but with taking into account the effects of
twinning. R~group~ is the R-value of the model to the data after
rigid-body refinement of individual chains and refinement of grouped
ADP\'s. Twin fractions refined to 50% for all cases.

7.  Unit cell parameters for structures 1EIX and 1JJK.

  -------------------- -------------------------------- -----------------
  Code                 Unit cell parameters             Space group

  1EIX                 115, 149, 116, 90, 115, 90       *P*2~1~

  1JJK                 61, 96, 145, 90, 90, 90          *P*2~1~2~1~2~1~
  -------------------- -------------------------------- -----------------

8.  Results of the manual molecular replacement and subsequent
    rigid-body refinement.

  --------------- --------------- --------------- ------------------------
  Origin shift    R value (start) R-value (rigid) R-value (twin)

  Choice 1        0.54            0.34            0.30 (twin fraction:
                                                  0.43)

  Choice 2        0.54            0.30            0.25 (twin fraction:
                                                  0.37)
  --------------- --------------- --------------- ------------------------