\<!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\>General Pathologies\</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\>
\<snm\>Lebedev\</snm\>\</au\>,\<orf id=\"^b^\"\>
\<au\>\<fnm\>Garib\</fnm\> \<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 Laboratories,
Crystallographic Computational 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; phase transitions; 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, 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 relationship between groups is visualised via
space group graphs. We also provide a number of practical examples.

# 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, 1983). 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 paragraph
XXX), 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 Shmueli et al. (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 figure 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 et al. (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 (XXX), due to
the introduction of Seleno-methionine residues (XXX), halide or heavy
metal soaking (XXX), or under different crystallisation conditions
(XXX).

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 (XXX
correct?).

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 close to a
perfect two-fold crystallographic rotation. This means, a
crystallographic symmetry operator was transformed into a NCS operator.
The 'remaining operators' in Tab. 1 can thus be interpreted as potential
"ideal" NCS 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
(XXX). 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 equal
to

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

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 $( - y + 1/2, - x + 1/2, - z + 0\text{.}\text{31})$ 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% (XXXX). In this
case it is therefore 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, and 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. For example, the structure YYY (XXX) has
an NCS operator approximately equal to $(x,y,z + 1/5)$. 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 (XXX), where an NCS operator
$(x + 1/2,y + 1/2,z)$ 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 (Read *et al.* in this
issue).

An interesting crystallographic pathology can arise when pseudo centring
is present. If the pseudo translational symmetry operators are mistaken
for a lattice translation and the lattice translations are interpreted
as pseudo translational symmetry operators during structure solution via
molecular replacement, further refinement can be stalled. For example,
if one has space group P21 with a pseudo translation equal to (x+½,y,z),
the approximate symmetry is equal to two P21 cells, stacked side by side
on the (**b**,**c**) face of the unit cell. This collection of symmetry
operators is nothing other than a regular space group in an unusual
setting. This particular set of operators (P21 with and added lattice
translation equal to (x+½,y,z)) is fully described by the symbol 'P 1
2~1~ 1 (2a,b,c)'. A full list of symmetry operators in this group is
shown in Table 2. From this set of operators, a number of groups can be
constructed. Operators not used in the construction of the group, can be
regarded as NCS operators. If operators A and B are designated as
crystallographic symmetry, the space group is P2~1~ and operators C and
D are NCS operators. If however operators A and D are designated to be
crystallographic, than operators B and C are non-crystallographic, while
the space group is an origin shifted P2~1~. The origin shift relating
the two groups is equal to (¼,0,0). Further details of this phenomenon
in relation to structure solution via molecular replacement and
refinement can be found elsewhere (XXXX). It is however noteworthy, that
if this particular type of pseudo centring is present, the structure can
be translated by the origin shift relating the two groups subgroups. For
both structures reasonable R-values will be obtained, as will be show in
paragraph XXXX, although only one solution is correct.

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

Twinning is a phenomenon where multiple reciprocal lattices partially or
fully overlap, giving rise to recorded intensities being equal to the
sum of the intensities of the individual domains with different
orientations that build up the crystal. The presence of twinning in an
X-ray data reveals itself usually by intensity statistics deviating from
well-described theoretical distributions. The presence of pseudo
rotational symmetry (especially when parallel to the twin law) or pseudo
translational symmetry can obscure the presence of twinning however.
Basic intensity statistics elucidating the problems of pseudo symmetry
in combination with twinning are explained thoroughly in Lebedev *et
al.* (XXXX).

### 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) unit cell 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 belong
to symmetry of the point group of the intensities, are the so-called
twin laws.

If two (or more) crystal domains related by a certain twin law are grown
together and are used in a diffraction experiment, the recorded
diffraction pattern is then equal to the sum of the diffraction patterns
of the individual domains. When the reciprocal lattice is invariant
under the twin law (merohedral twinning), it is not obvious from the
diffraction pattern, that the crystal is not a *single crystal*.

In some cases however, the reciprocal lattice is only approximately
invariant under a given twin law. In this case, called pseudo merohedral
twinning, twin related intensities may be identified as individual
reflections if the diffraction pattern is clear enough. Examples of a
number of (pseudo)merohedrally twinned structures are given in Table 3.

The presence of a non crystallographic 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.

The breaking of symmetry due to a temperature dependent phase transition
can introduce twinning in a similar manner (Helliwell *et .al* 2006;
Herbst-Irmer & Sheldrick 1998). This phase-transition induced twinning
can in principle be introduced by ligand or heavy atom soaks. An example
of such a phase transition is described by Dauter et al (2001). In that
particular case however, the symmetry of the crystals before a halide
soak had a lower symmetry than the after the halide 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.

\<These paragraphs have to be merged I think somehow\>

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

Twinning by reticular merohedral is slightly more complicated than
merohedral twinning. In brief, reticular merohedral twinning can be
described as merohedral twinning on an assembly of unit cells, a so
called sublattice (XXX). 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 intensities that have
contributions from either 1 or from multiple twin domains.

A well known example of twinning by reticular merohedry is the
obverse-reverse twinning in rhombohedral space groups, where systematic
absences of one domain overlap with systematic absences of another
domain.

An excellent introduction into twinning by reticular merohedry is given
by Parsons (XXXX). Examples of diffraction patters are found in Dauter
(XXXX).

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

Order disorder twins (XXXX, YYY, ZZZZ) are a less common type of twin,
but have been observed for protein structures in a number of cases
(XXXX, YYYY). 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 (XXXX, YYYY). A correction for this effect can be vital for
structure solution (XXXX) and result in lower R-values during refinement
(YYYY)

## Avoidable (user-introduced) pathologies {#avoidable-user-introduced-pathologies .IUCr-heading-2}

A number of user introduced pathologies can make life difficult. A few
examples are listed here.

*Misindexing.* When the beam centre has not been defined accurately
enough, auto indexing programs can return an indexing solution for 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 (XXXX).

*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 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.

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 a correct unit cell has been obtained, the
appropriate Bravais lattice and intensity symmetry have to be chosen.
The presence of pseudo symmetry can make this choice difficult but can
be done automatically by programs such as Xtriage () , Xprep (),
Pointless () or Labelit (). 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 by
increasing the 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 (XXXX),
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}

## Pathological cases from the PDB {#pathological-cases-from-the-pdb .IUCr-heading-2}

A number of datasets in the PDB (XXXX) 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 (XXXX) was indexed in C2 with
unit cell parameters equal to (74.58, 48.69, 67.55, 90, 102.3, 90). The
Patterson function reveals a peak at (0, 0.5, 0) with a height
approximately equal to that of the origin (99%). The rmsd between the
C-alphas of the two molecules given the translational NCS operator
specified by the Patterson function is equal to 0.08 Å. The intensities
of the reflections with miller indices that should be equal to zero if
the NCS operator were crystallographic, all barely rise above the noise
as judged from their associated standard deviation.

*2A8Y*. The unit cell constants for this structure are equal to (96.60,
96.56, 96.63, 91.57, 91.23, 91.52). The space group is P1 (XXXX).
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 (XXX) reveals that the intensity
symmetry corresponds to the space group C2, 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 (XXXX) in various point groups (including
C2), but the data only scaled well in space group P1. Given the pseudo
symmetric nature of the lattice (pseudo rhombohedral), three possible C2
groups are possible (see figure XXX) corresponding to the three
orientations of the two fold axis in space group R32:R. The integration
suite used to initially process the data only gave a single indexing
choice for C2, which was unfortunately incorrect. Currently, the
structure is being re-refined in the higher symmetry C2 space group
(Ealick, private communication).

*1UPP.* The structure of a spinach rubisco complex (XXXX) has associated
unit cell parameters equal to (155.9 156.3 199.8 90.00 90.00 90.00) and
space group C222~1~. As can be seen from the unit cell parameters, a is
approximately equal to b resulting in the presence of a twin law equal
to (k,h,-l). Furthermore, the Patterson function indicates a
translational NCS vector equal to (0.5, 0, 0.5) at a height of 40% of
the origin. The presence of pseudo translational symmetry can make the
detection of twinning difficult (XXXX), but the results of the L-test is
quite clear (Table 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. A free R set was
not deposited.

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

It has previously been shown that structure solution using experimental
phasing techniques is feasible with twinned data (XXXX). Therefore, it
is not surprising that molecular replacement on twinned data is usually
successful. Using artificially twinned data, it is quite clear that the
contrast of the rotation function decreases in proportion to the twin
fraction, Figure 4. A similar observation is made for the translation
function. It is noteworthy that when the data is twinned perfectly and
merged in a too high space group, one is not able to solve the
structure, as the ASU is typically too small to contain the true
contents of the crystal. Reduction of the space group, even in the
presence of twinned data, does result in a structure solution.

More details about molecular replacement on twinned data can be found
elsewhere in this issue (XXXX).

## 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 (XXXX). An example
structure solution utilising group-subgroup relations is presented here.

The crystals of 1EIX and 1JJK have been grown under similar conditions,
albeit that one structure is a seleno-methionine derivative, while the
other is a native protein. The cell parameters are listed in Table 5.
The ratio of the unit cell volumes is 2.12, indicating a possible
relation between the two unit cells. Another clue that the two unit
cells are related, is found in the Patterson function of 1JJK: a large
peak is found at (0.5,0,0.5). If this NCS operator would be a
crystallographic operator, the unit cell dimensions of 1JJK would be
equal (apart from a permutation) 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 (P2~1~ versus P2~1~2~1~2~1~).

A relation between the two unit cells was identified with the tool
*iotbx.explore_metric_symmetry* (XXXX) and is depicted in Figure 6. The
procedure used to solve the structure of 1JJK with the model of 1EIK via
group-subgroup relations is described in (XXXX). First, the appropriate
asymmetric unit is constructed by apply a two fold screw axis on 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 (Table 6).

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

There are numerous pathologies possible in macromolecular
crystallographic data, resulting from X, Y and Z. Fortunately there are
now a number of tools that make it possible to identify many of the
cases. 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.

\<TO BE EXPANDED\>

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.
    The algorithms described here are available in the PHENIX software
    suite
    ([http://www.phenix-online.org](http://www.phenix-online.org/)) and
    XXX (CCP4?).

\<XXXX a word about how the sg graphs are made maybe? XXXX\>

References

Adams, P. D., Gopal, K., Grosse-Kunstleve, R. W., Hung, L.-W., Ioerger,
T. R., McCoy, A. J., Moriarty, N. W., Pai, R. K., Read, R. J., Romo, T.
D., Sacchettini, J. C., Sauter, N. K., Storoni, L. C. & Terwilliger, T.
C. (2004). *Journal of Synchrotron Radiation* **11**, 53-55.

Adams, P. D., Grosse-Kunstleve, R. W., Hung, L.-W., Ioerger, T. R.,
McCoy, A. J., Moriarty, N. W., Read, R. J., Sacchettini, J. C., Sauter,
N. K. & Terwilliger, T. C. (2002). *Acta Crystallographica Section D*
**58**, 1948-1954.

Brunzelle, J. S., Shafaee, P., Yang, X., Weigand, S., Ren, Z. &
Anderson, W. F. (2003). *Acta Crystallogr D Biol Crystallogr* **59**,
1138-1144.

Dauter, Z., Li, M. & Wlodawer, A. (2001). *Acta Crystallogr D Biol
Crystallogr* **57**, 239-249.

Emsley, P. & Cowtan, K. (2004). *Acta Crystallogr D Biol Crystallogr*
**60**, 2126-2132.

Giacovazzo, C. (1992). Editor. Fundamentals of Crystallography.
IUCr/Oxford University Press.

Hahn, T. (1983). International Tables for Crystallography, Vol. A.
Dordrecht: Kluwer.

Holton, J. & Alber, T. (2004). *Proc Natl Acad Sci U S A* **101**,
1537-1542.

Holton, T., Ioerger, T. R., Christopher, J. A. & Sacchettini, J. C.
(2000). *Acta Crystallogr D Biol Crystallogr* **56**, 722-734.

Ioerger, T. R., Holton, T., Christopher, J. A. & Sacchettini, J. C.
(1999). *Proc Int Conf Intell Syst Mol Biol* 130-137.

Lamzin, V. S. & Perrakis, A. (2000). *Nat Struct Biol* **7 Suppl**,
978-981.

Lamzin, V. S., Perrakis, A., Bricogne, G., Jiang, J., Swaminathan, S. &
Sussman, J. L. (2000). *Acta Crystallogr D Biol Crystallogr* **56**,
1510-1511.

McRee, D. E. (1999). *J Struct Biol* **125**, 156-165.

Morris, R. J., Perrakis, A. & Lamzin, V. S. (2003). *Methods Enzymol*
**374**, 229-244.

Morris, R. J., Zwart, P. H., Cohen, S., Fernandez, F. J., Kakaris, M.,
Kirillova, O., Vonrhein, C., Perrakis, A. & Lamzin, V. S. (2004).
*Journal of Synchrotron Radiation* **11**, 56-59.

Navaza, J. (1994). *Acta Crystallographica Section A* **50**, 157-163.

Perrakis, A., Morris, R. & Lamzin, V. S. (1999). *Nat Struct Biol*
**6**, 458-463.

Read, R. (2001). *Acta Crystallographica Section D* **57**, 1373-1382.

Snell, G., Cork, C., Nordmeyer, R., Cornell, E., Meigs, G., Yegian, D.,
Jaklevic, J., Jin, J., Stevens, R. C. & Earnest, T. (2004). *Structure*
**12**, 537-545.

Shmueli, U., Hall S.R., Grosse-Kunstleve, R.W. (2001). In International
Tables for Crystallography, Volume B: Reciprocal space, U. Shmueli, Ed.,
Kluwer Academic Publishers (Dordrecht), 107-119.

Terwilliger, T. (2002a). *Acta Crystallographica Section A* **58**, C57.

Terwilliger, T. (2002b). *Acta Crystallographica Section D* **58**,
1937-1940.

Vagin, A. & Teplyakov, A. (2000). *Acta Crystallogr D Biol Crystallogr*
**56**, 1622-1624.

Winter, G. M. (2007). *To be published*.

1.  A graphical representation of all group-subgroup relations for P222
    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
    (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/image2.png){width="4.625694444444444in"
    height="1.7694444444444444in"}

3.  A space group graph showing all subgroups of space group R32:R.
    Specifically, three distinct C2 subgroups are available,
    corresponding to three possible directions the two fold axes are
    oriented in R32.

    ![](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"}

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1.  Subgroups of P222. For each subgroup, the symmetry operators are
    specified, together with the operators that are element of P222, but
    not of the subgroup. Although P222 is not a subgroup of itself, its
    details are shown for completeness.

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

  P222            (x,y,z), (-x,y,-z), (x,-y,-z), None
                  (-x,-y,z)                      

  P211            (x,y,z), (x,-y,-z)             (-x,y,-z), (-x,-y,z)

  P121            (x,y,z), (-x,y,-z)             (x,-y,-z), (-x,-y,z)

  P112            (x,y,z), (-x,-y,z)             (-x,y,-z), (x,-y,-z)

  P1              (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 P21 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  I4           (-k,-h,-l)    8%         M

  1EYX     180, 180, 36, 90, 90,    R3: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       P2~1~

  1JJK                 61, 96, 145, 90, 90, 90          P2~1~2~1~2~1~
  -------------------- -------------------------------- -----------------

6.  Results of the manual molecular replacement

  --------------- --------------- --------------- ------------------------
  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)
  --------------- --------------- --------------- ------------------------