\<!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 (broken)
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. We present the
    background to certain common pathologies with examples from the
    literature.

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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 not impossible
to solve a structure without a 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.

The pathologies dealt with in this manuscript are related to the
breaking of symmetry elements or the interplay between
non-crystallographic symmetry and 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 as well as in a different space
groups.

In order to have a better understanding of the issues at hand, we review
basic group theory. The relationships between groups are visualised in
via space group graphs. We also provide a number of examples from the
PDB (Berm*an et a*l., 2000; Bernste*in et a*l., 1977) and literature.

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

## Background and notation {#background-and-notation .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.

A mathematical group $G$ is a set of elements $g$ with special
properties under a binary operation $g_{i}g_{j}$:

1.  $G$ is closed under the binary operation: if $g_{i}$ and $g_{j}$ are
    elements of the group $G$, than so are $g_{i}g_{j}$ and $g_{j}g_{i}$

2.  Multiplication of group elements is associative:
    $\left( g_{i}g_{j} \right)g_{k} = g_{i}\left( g_{j}g_{k} \right)$

3.  There is an element $E$ (german: *Einheit*) in the group that
    \"doesn\'t do anything\": $\text{gE} = g$ and $\text{Eg} = g$. $E$is
    known as the *identity element* or also *unit element*.

4.  For each element $g$ in $G$, there is an inverse element
    $g^{- 1}$defined by $\text{gg}^{- 1} = E$.

Of the 230 crystallographic space group types, only 65 are compatible
with chiral compounds, such as proteins or nucleic acids. The elements
$g$ of these 65 groups are restricted to:

1.  Rotations (2, 3, 4 and 6-fold)

2.  Screw rotations (2~1~, 3~1~, 3~2~, 4~1~, 4~2~, 4~3~, 6~1~, 6~2~,
    6~3~, 6~4~ and 6~5~ axes)

3.  Lattice translations (unit translations and centring translations)

For example, the space group *P*21 according to *ITVA* contains the
elements:

1.  The identity operator : (*x*, *y*, *z*)

2.  A two fold screw axis (2~1~) along y: (-*x* , *y*+½ , -*z*)

3.  Lattice translations: (*x*+*n*, *y*+*m*, *z*+*o*), where (*n*, *m*,
    *o*) are integers

The lattice translations are often not mentioned explicitly. However, in
the context of pathologies caused by broken symmetry (see below),
explicit consideration of the lattice translations is crucial.

*ITVA* Table 4.3.1 defines Hermann-Mauguin space group symbols for 530
conventional settings. In the context of group-subgroup analysis with
respect to a given metric (unit cell parameters), other unusual settings
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 operations 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 has all the group properties as listed in paragraph 2.1.

For instance, the symmetry elements of space group *P*222 are
{(*x*,*y*,*z*), (-*x*,*y*,-*z*), (*x*,-*y*,-*z*), (-*x*,-*y*,*z*)}.
Subgroups of *P*222 can be constructed by selecting only certain
elements. The full list of subgroups of *P*222 and the set of
\'remaining elements\' for each subgroup with respect to *P*222 are
given in Tab. 1.

Note that if the elements of *P*211 are combined with one of the
'remaining' elements (-*x*,*y*,-*z*) or (-*x*,-*y*,*z*), the other
element 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 element 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.

## Broken symmetry {#broken-symmetry .IUCr-heading-2}

It is not uncommon that non-crystallographic symmetry is close to being
crystallographic symmetry. The transition of the arrangement of
symmetrically arranged molecules to approximately symmetrically arranged
molecules can be seen as a case of "broken symmetry". Breaking of
crystallographic symmetry can occur upon ligand binding, due to the
introduction of Seleno-methionine residues, halide or heavy metal
soaking, or crystal growth under different crystallisation conditions
(Daut*er et a*l., 2001; Pouls*en et a*l., 2001; Parsons, 2003).

The presence of broken symmetry can be a structural basis for twinning
or pseudo rotational or pseudo translational symmetry. Group-subgroup
relations and their graphical representation as outlined in section 2.2
are a useful tool for understanding broken symmetry and the resulting
relations between the space groups of different crystal forms.

Constructing examples of broken 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}

A number of pathologies will be discussed based on the symmetry and
group tools just described.

## 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 *root mean
square distance* (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 equal to a two-fold axis in the *xy*-plane. If the idealized
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. The
subset of reflections that would be systematically absent given
idealized 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* 1 2~1~
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 is correct and eventually leads to the
best model. Further details of problems encountered due to this
ambiguity are reported in Lebed*ev et a*l. (2007).

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

Twinning is a phenomenon where multiple reciprocal lattices partially or
fully overlap. 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).

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 equal is called *perfect twinning*. A twin
with an arbitrary ratio of twin fractions is denoted as a partial twin.

### 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 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
(exact 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 argument 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 the halide soak was lower
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.

### 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 twins {#order-disorder-twins .IUCr-heading-3}

Order-disorder twins (Dornberger-Schiff & Dunitz, 1965;
Dornberger-Schiff & Grell-Niemann, 1961; Dornberger-Schiff, 1966, 1956)
are a less well-classified type of twins, but have been observed for
protein structures in a number of cases (Trame & McKay, 2001; Wa*ng et
a*l., 2005). An order-disorder twin 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 in such a crystal 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 result in lower R-values during
refinement (Trame & McKay, 2001). As it is 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.* When 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 autoindexing routines (Saut*er et a*l., 2004).

*Incorrect unit cell.* When more than a single crystal is present or
when the diffraction images are noisy in general, it is possible that
auto indexing proceduces 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 the
choice can 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 are 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.* This 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~ atoms 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*. 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 may be 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.* 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~. As can be seen from the unit cell
parameters, ***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) at 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.

## Manual molecular replacement using group-sub group relations {#manual-molecular-replacement-using-group-sub-group-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. 5. 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 would be a
crystallographic operator, the unit cell dimensions of 1JJK would be
equal (apart from a permutation of the basis vectors) to the unit cell
constants of 1EIX. It is thus clear that 1JJK is related to 1EIX via
broken translational symmetry (as seen from the Patterson peak) and
broken 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 while taking into account the presence of
twinning resulted in a single, clear solution (Tab. 6).

# 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. 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 after data collection and processing. In an ideal
world data would be stored in an unmerged form in space group *P*1 and
certain 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
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.

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 algorithms described 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.

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1.  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/image1.png){width="3.125in" height="1.6868055555555554in"}

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

    ![](media/image2.png){width="4.625694444444444in"
    height="1.7694444444444444in"}

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

    ![](media/image3.png){width="3.497916666666667in" height="2.025in"}

4.  The value of the largest peak in the rotation 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/image4.wmf){width="3.125in" height="2.7041666666666666in"}

5.  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/image5.wmf){width="3.125in" height="2.279166666666667in"}

6.  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/image6.png){width="2.214583333333333in"
    height="2.2597222222222224in"}

```{=html}
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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. Although *P*222 is not a subgroup of
    itself, its details are shown 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
    Fig. 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.  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~
  -------------------- -------------------------------- -----------------

6.  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)
  --------------- --------------- --------------- ------------------------