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Space group determination (view source)

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Space group determination entails the following steps:
Space group determination entails the following steps:
# determine the '''Laue group''': this is the symmetry of the intensity-weighted point lattice (diffraction pattern). 1,2,3,4,6=n-fold rotation axis; -n means inversion centre; m means mirror.
# determine the '''Laue class''': this is the symmetry of the intensity-weighted point lattice (diffraction pattern). 1,2,3,4,6=n-fold rotation axis; -n means inversion centre (normally the - is written over the n); m means mirror.
# find out about '''Bravais type''': the second letter signifies centering (P=primitive; C=centered on C-face; F=centered on all faces; I=body-centered; R=rhombohedral). The first letter (a=triclinic; m=monoclinic; o=orthorhombic; h=trigonal or hexagonal; c=cubic) is redundant since it can be inferred from the Laue group.
# find out about '''Bravais type''': the second letter signifies centering (P=primitive; C=centered on C-face; F=centered on all faces; I=body-centered; R=rhombohedral). The first letter (a=triclinic; m=monoclinic; o=orthorhombic; h=trigonal or hexagonal; c=cubic) is redundant since it can be inferred from the Laue group.
# possible spacegroups are now given in '''columns 3 and 4'''; CORRECT always suggests the one given in column 3 but this is no more likely than those in column 4.
# possible spacegroups are now given in '''columns 3 and 4'''; CORRECT always suggests the one given in column 3 but this is no more likely than those in column 4.
# choose according to screw axis; this may result in two possibilities (enantiomorphs).
# choose according to screw axis (according to the table "REFLECTIONS OF TYPE H,0,0  0,K,0  0,0,L OR EXPECTED TO BE ABSENT (*)" in [[CORRECT.LP]]); this may result in two possibilities (enantiomorphs).
# determine correct enantiomorph - this usually means that one tries to solve the structure in both spacegroups, and only one gives a sensible result (like helices that are right-handed, amino acids of the L type).
# determine correct enantiomorph - this usually means that one tries to solve the structure in both spacegroups, and only one gives a sensible result (like helices that are right-handed, amino acids of the L type).




== Table of space groups by Laue class and Bravais type ==
{| cellpadding="10" cellspacing="0" border="1"  
{| cellpadding="10" cellspacing="0" border="1"  
! Laue group
! Laue class
! Bravais type
! Bravais type
! spacegroup <br> number <br> suggested by <br> CORRECT
! spacegroup <br> number <br> suggested by <br> CORRECT
! other possibilities
! other possibilities (with screw axes)
! alternative indexing <br> possible?
! alternative indexing <br> possible?
! choosing among all spacegroup possibilities
! choosing among all spacegroup possibilities
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| -1 || aP || 1 ||-||||  
| -1 || aP || 1 ||-||||  
|-   
|-   
| 2m || mP || 3 || 4 |||| screw axis extinctions let you decide
| 2/m || mP || 3 || 4 |||| screw axis extinctions let you decide
|-   
|-   
| 2m || mC ||  5 || - |||| The [[pointless]] article discusses why I2 (mI) should not be used.
| 2/m || mC ||  5 || - |||| The [[pointless]] article discusses why I2 (mI) should not be used.
|-   
|-   
| mmm || oP || 16 || 17, 18, 19 |||| screw axis extinctions let you decide. The [[pointless]] article discusses why CCP4's 1017 (P2<sub>1</sub>22), 2017 (P22<sub>1</sub>2), 2018 (P2<sub>1</sub>22<sub>1</sub>) , 3018 (P22<sub>1</sub>2<sub>1</sub>) are not needed and should not be used.
| mmm || oP || 16 || 17, 18, 19 |||| screw axis extinctions let you decide. The [[pointless]] article discusses why CCP4's 1017 (P2<sub>1</sub>22), 2017 (P22<sub>1</sub>2), 2018 (P2<sub>1</sub>22<sub>1</sub>) , 3018 (P22<sub>1</sub>2<sub>1</sub>) are not needed and should not be used.
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| mmm || oI || 23 || 24 |||| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 23/24 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition).
| mmm || oI || 23 || 24 |||| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 23/24 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition).
|-
|-
| 4/m || tP || 75 || 76, 77, 78 ||yes|| screw axis extinctions let you decide, except between 76/78 enantiomorphs
| 4/m || tP || 75 || 76, 77, 78 ||k,h,-l|| screw axis extinctions let you decide, except between 76/78 enantiomorphs
|-
|-
| 4/m || tI || 79 || 80 ||yes|| screw axis extinctions let you decide
| 4/m || tI || 79 || 80 ||k,h,-l|| screw axis extinctions let you decide
|-
|-
| 4/mmm || tP || 89 || 90, 91, 92, 93, 94, 95, 96 |||| screw axis extinctions let you decide, except between 91/95 and 92/96 enantiomorphs
| 4/mmm || tP || 89 || 90, 91, 92, 93, 94, 95, 96 |||| screw axis extinctions let you decide, except between 91/95 and 92/96 enantiomorphs
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| 4/mmm || tI || 97 || 98 |||| screw axis extinctions let you decide
| 4/mmm || tI || 97 || 98 |||| screw axis extinctions let you decide
|-
|-
| -3 || hP || 143 || 144, 145 ||4 possibilities|| screw axis extinctions let you decide, except between 144/145 enantiomorphs
| -3 || hP || 143 || 144, 145 ||-h,-k,l; k,h,-l; -k,-h,-l|| screw axis extinctions let you decide, except between 144/145 enantiomorphs
|-
|-
| -3 || hR || 146 || - ||yes, and obverse/reverse||  
| -3 || hR || 146 || - ||k,h,-l, and obverse (-h+k+l=3n) / reverse (h-k+l=3n)||  
|-
|-
| -3/m || hP || 149 || 151, 153 ||yes|| screw axis extinctions let you decide, except between 151/153 enantiomorphs. Note: the twofold goes along the diagonal between a and b.
| -3/m || hP || 149 || 151, 153 ||k,h,-l|| screw axis extinctions let you decide, except between 151/153 enantiomorphs. Note: the twofold goes along the diagonal between a and b.
|-
|-
| -3/m || hP || 150 || 152, 154 ||yes|| screw axis extinctions let you decide, except between 152/154 enantiomorphs. Note: compared to previous line, the twofold goes along a.
| -3/m || hP || 150 || 152, 154 ||-h,-k,l|| screw axis extinctions let you decide, except between 152/154 enantiomorphs. Note: compared to previous line, the twofold goes along a.
|-
|-
| -3/m || hR || 155 || - ||obverse/reverse||  
| -3/m || hR || 155 || - ||obverse/reverse||  
|-
|-
| 6/m || hP || 168 || 169, 170, 171, 172, 173 ||yes|| screw axis extinctions let you decide, except between 169/170 and 171/172 enantiomorphs
| 6/m || hP || 168 || 169, 170, 171, 172, 173 ||k,h,-l|| screw axis extinctions let you decide, except between 169/170 and 171/172 enantiomorphs
|-
|-
| 6/mmm || hP || 177 || 178, 179, 180, 181, 182 |||| screw axis extinctions let you decide, except between 178/179 and 180/181 enantiomorphs
| 6/mmm || hP || 177 || 178, 179, 180, 181, 182 |||| screw axis extinctions let you decide, except between 178/179 and 180/181 enantiomorphs
|-
|-
| m-3 || cP || 195 || 198 ||yes|| screw axis extinctions let you decide
| m-3 || cP || 195 || 198 ||k,h,-l|| screw axis extinctions let you decide
|-
|-
| m-3 || cF || 196 ||-||yes ||  
| m-3 || cF || 196 ||-||k,h,-l||  
|-
|-
| m-3 || cI || 197 || 199 ||yes|| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 197/199 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition).
| m-3 || cI || 197 || 199 ||k,h,-l|| screw axis extinctions do '''not''' let you decide because the I centering results in h+k+l=2n and the screw axis extinction 00l=2n is just a special case of that. 197/199 do '''not''' form an enantiomorphic, but a ''special'' pair (ITC A §3.5, p. 46 in the 1995 edition).
|-
|-
| m-3m || cP || 207 || 208, 212, 213 |||| screw axis extinctions let you decide, except between 212/213 enantiomorphs
| m-3m || cP || 207 || 208, 212, 213 |||| screw axis extinctions let you decide, except between 212/213 enantiomorphs
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|}
|}


 
Alternative indexing possibilities taken from http://www.ccp4.ac.uk/html/reindexing.html (for R3 and R32, obverse/reverse are specified).
If you find an error in the table please send an email to kay dot diederichs at uni-konstanz dot de !
If you find an error in the table please send an email to kay dot diederichs at uni-konstanz dot de !


== The 65 space groups in which proteins composed of L-amino acids can crystallize ==
== The 65 Sohncke space groups in which proteins composed of L-amino acids can crystallize ==




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=== Subgroup and supergroup relations of these space groups ===
Compiled from [https://onlinelibrary.wiley.com/doi/book/10.1107/97809553602060000001 International Tables for Crystallography (2006) Vol. A1 (Wiley)]. Simply put, for each space group, a maximum ''translationengleiche'' subgroup has lost a single type of symmetry, and a minimum ''translationengleiche'' supergroup has gained a single symmetry type. Example: P222 is a supergroup of P2, and a subgroup of P422 (and P4222 and P23). Of course the sub-/supergroup relation is recursive, which is why P1 is also a (sub-)subgroup of P222 (but not a maximum ''translationengleiche'' subgroup). The table below does not show other types of relations, e.g. non-isomorphic ''klassengleiche'' supergroups which may result e.g. from centring translations, because I find them less relevant in space group determination.
The table is relevant because in particular (perfect) twinning adds a symmetry type, and leads to an apparent space group which is the supergroup of the true space group.
{| cellpadding="0" cellspacing="0" border="1"
! spacegroup number
! maximum ''translationengleiche'' subgroup
! minimum ''translationengleiche'' supergroup
! spacegroup name
|-
|  1 ||-|| 3, 4, 5, 143, 144, 145, 146 || P 1
|- 
|  3 || 1 || 16, 17, 18, 21, 75, 77, 168, 171, 172 || P 2
|- 
|  4 || 1 || 17, 18, 19, 20, 76, 78, 169, 170, 173 || P 21
|- 
|  5 || 1 || 20, 21, 22, 23, 24, 79, 80, 149, 150, 151, 152, 153, 154, 155 || C2
|- 
|  16 || 3 || 89, 93, 195 || P 2 2 2
|-
|  17 || 3, 4 || 91, 95 || P 2 2 21
|-
|  18 || 3, 4 || 90, 94 || P 21 21 2
|-
|  19 || 4 || 92, 96, 198 || P 21 21 21
|-
|  20 || 4, 5 || 91, 92, 95, 96, 178, 179, 182 || C 2 2 21
|-
|  21 || 3, 5 || 89, 90, 93, 94, 177, 180, 181 || C 2 2 2
|-
|  22 || 5 || 97, 98, 196 || F 2 2 2
|-
|  23 || 5 || 97, 197 || I 2 2 2
|-
|  24 || 5 || 98, 199 || I 21 21 21
|-
|  75 || 3 || 89, 90 || P 4
|-
|  76 || 4 || 91, 92 || P 41
|-
|  77 || 3 || 93, 94 || P 42
|-
|  78 || 4 || 95, 96 || P 43
|-
|  79 || 5 || 97 || I 4
|-
|  80 || 5 || 98 || I 41
|-
|  89 || 16, 21, 75 || 207 || P 4 2 2
|-
|  90 || 18, 21, 75 || - || P 4 21 2
|-
|  91 || 17, 20, 76 || - || P 41 2 2
|-
|  92 || 19, 20, 76 || 213 || P 41 21 2
|-
|  93 || 16, 21, 77 || 208 || P 42 2 2
|-
|  94 || 18, 21, 77 || 93, 97 || P 42 21 2
|-
|  95 || 17, 20, 78 || - || P 43 2 2
|-
|  96 || 19, 20, 78 || 212 || P 43 21 2
|-
|  97 || 22, 23, 79 || 209, 211 || I 4 2 2
|-
|  98 || 22, 24, 80 || 210, 214 || I 41 2 2
|-
| 143 || 1 || 149, 150, 168, 173 || P 3
|-
| 144 || 1 || 151, 152, 169, 172 || P 31
|-
| 145 || 1 || 153, 154, 170, 171 || P 32
|-
| 146 || 1 || 155, 195, 196, 197, 198, 199 || R 3
|-
| 149 || 5, 143 || 177, 182 || P 3 1 2
|-
| 150 || 5, 143 || 177, 182 || P 3 2 1
|-
| 151 || 5, 144 || 178, 181 || P 31 1 2
|-
| 152 || 5, 144 || 178, 181 || P 31 2 1
|-
| 153 || 5, 145 || 179, 180 || P 32 1 2
|-
| 154 || 5, 145 || 179, 180 || P 32 2 1
|-
| 155 || 5, 146 || 207, 208, 209, 210, 211, 212, 213, 214 || R 3 2
|-
| 168 || 3, 143 || 177 || P 6
|-
| 169 || 4, 144 || 178 || P 61
|-
| 170 || 4, 145 || 179 || P 65
|-
| 171 || 3, 145 || 180 || P 62
|-
| 172 || 3, 144 || 181 || P 64
|-
| 173 || 4, 143 || 182 || P 63
|-
| 177 || 21, 149, 150, 168 || - || P 6 2 2
|-
| 178 || 20, 151, 152, 169 || - || P 61 2 2
|-
| 179 || 20, 153, 154, 170 || - || P 65 2 2
|-
| 180 || 21, 153, 154, 171 || - || P 62 2 2
|-
| 181 || 21, 151, 152, 172 || - || P 64 2 2
|-
| 182 || 20, 149, 150, 173 || - || P 63 2 2
|-
| 195 || 16, 146 || 207, 208 || P 2 3
|-
| 196 || 22, 146 || 209, 210 || F 2 3
|-
| 197 || 23, 146 || 211 || I 2 3
|-
| 198 || 19, 146 || 212, 213 || P 21 3
|-
| 199 || 24, 146 || 214 || I 21 3
|-
| 207 || 89, 155, 195 || - || P 4 3 2
|-
| 208 || 93, 155, 195 || - || P 42 3 2
|-
| 209 || 97, 155, 196 || - || F 4 3 2
|-
| 210 || 98, 155, 196 || - || F 41 3 2
|-
| 211 || 97, 155, 197 || - || I 4 3 2
|-
| 212 || 96, 155, 198 || - || P 43 3 2
|-
| 213 || 92, 155, 198 || - || P 41 3 2
|-
| 214 || 98, 155, 199 || - || I 41 3 2
|-
|}


== Space group selected by XDS: ambiguous with respect to enantiomorph and screw axes ==
== Space group selected by XDS: ambiguous with respect to enantiomorph and screw axes ==


In case of a crystal with an unknown space group (SPACE_GROUP_NUMBER=0 in [[XDS.INP]]), XDS (since [http://www.mpimf-heidelberg.mpg.de/~kabsch/xds/html_doc/Release_Notes.html version June 2008]) helps the user in determination of the correct space group, by suggesting possible space groups compatible with the Laue symmetry and Bravais type of the data, and by calculating the R<sub>meas</sub> for these space groups.
In case of a crystal with an unknown space group (SPACE_GROUP_NUMBER=0 in [[XDS.INP]]), XDS (since [http://xds.mpimf-heidelberg.mpg.de/html_doc/Release_Notes.html version June 2008]) helps the user in determination of the correct space group, by suggesting possible space groups compatible with the Laue symmetry and Bravais type of the data, and by calculating the R<sub>meas</sub> for these space groups.


XDS (or rather, the [[CORRECT.LP|CORRECT]] step) makes an attempt to pick the correct space group automatically: it chooses that space group which has the highest symmetry (thus yielding the lowest number of unique reflections) and still a tolerable R<sub>meas</sub> compared to the R<sub>meas</sub> the data have in any space group (which is most likely a low-symmetry space group - often P1).
XDS (or rather, the [[CORRECT.LP|CORRECT]] step) makes an attempt to pick the correct space group automatically: it chooses that space group which has the highest symmetry (thus yielding the lowest number of unique reflections) and still a tolerable R<sub>meas</sub> compared to the R<sub>meas</sub> the data have in any space group (which is most likely a low-symmetry space group - often P1).


In some cases the automatic choice is the correct one, and re-running the [[CORRECT.LP|CORRECT]] step is then not necessary. However, neither the correct enantiomorph nor screw axes (see below) are determined automatically by XDS.  
In some cases the automatic choice is the correct one, and re-running the [[CORRECT.LP|CORRECT]] step is then not necessary. However, neither the correct enantiomorph nor [[Space_group_determination#Screw_axes|screw axes]] (see below) are determined automatically by XDS.
 


== Space group selected by user ==
== Space group selected by user ==
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There are two ways to have XDS choose an indexing consistent with some other dataset:
There are two ways to have XDS choose an indexing consistent with some other dataset:
* using REFERENCE_DATA_SET=    ! see also [[REFERENCE_DATA_SET]]
* using REFERENCE_DATA_SET=    ! see also [[REFERENCE_DATA_SET]]
* using [http://www.mpimf-heidelberg.mpg.de/~kabsch/xds/html_doc/xds_parameters.html#UNIT_CELL_A-AXIS= UNIT_CELL_A-AXIS=], UNIT_CELL_B-AXIS=, UNIT_CELL_C-AXIS= from a previous data collection run with the same crystal
* using [http://xds.mpimf-heidelberg.mpg.de/html_doc/xds_parameters.html#UNIT_CELL_A-AXIS= UNIT_CELL_A-AXIS=], UNIT_CELL_B-AXIS=, UNIT_CELL_C-AXIS= from a previous data collection run with the same crystal


One can also manually force a specific indexing, using the REIDX= keyword, but this is error-prone.
One can also manually force a specific indexing, using the REIDX= keyword, but this is error-prone.
Line 185: Line 326:
</pre>
</pre>
The automatic choice gave 23 as a result. This is because the associated R<sub>meas</sub> (39.5) is lower than MAX_FAC_RMEAS * (lowest R<sub>meas</sub>) = 2 * 22.5 = 45.0.  
The automatic choice gave 23 as a result. This is because the associated R<sub>meas</sub> (39.5) is lower than MAX_FAC_RMEAS * (lowest R<sub>meas</sub>) = 2 * 22.5 = 45.0.  
However, the C2 values of 30.7 and 31.0 are significantly lower still, and the question needs further investigation. <code>[[pointless]] XDS_ASCII.HKL</code> gives:
However, the C2 values of 30.7 and 31.0 are significantly lower still, and the question needs further investigation.  
 
=== checking the CORRECT assignment with [[pointless]] ===
 
<code>[[pointless]] XDS_ASCII.HKL</code> gives:
<pre>
<pre>
Analysing rotational symmetry in lattice group I 4/m m m
Analysing rotational symmetry in lattice group I 4/m m m
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In this particular case, CORRECT chose the wrong symmetry, whereas pointless identified the correct symmetry elements. I have also seen cases where pointless mis-identified the symmetry, usually on the side of too high symmetry. To avoid mistakes in space group identification, it is absolutely crucial to read and understand the tables that the programs print.
In this particular case, CORRECT chose the wrong symmetry, whereas pointless identified the correct symmetry elements. I have also seen cases where pointless mis-identified the symmetry, usually on the side of too high symmetry. To avoid mistakes in space group identification, it is absolutely crucial to read and understand the tables that the programs print.


Important note: pointless should really be run with the "SETTING SYMMETRY-BASED" option. When doing that, the output changes to  
=== a more sensible way to run pointless ===
 
pointless should really be run with the "SETTING SYMMETRY-BASED" option. When doing that, the output changes to  
<pre>
<pre>
Best Solution:    space group C 1 2 1
Best Solution:    space group C 1 2 1
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so the spacegroup given is the more normal C2 setting (instead of A2 or I2). Unfortunately, pointless does not seem to print out the table of "alternative indexing possibilities" in this mode - but possibly the table is only printed in the I2 case because the beta angle comes out as 90.0°.
so the spacegroup given is the more normal C2 setting (instead of A2 or I2). Unfortunately, pointless does not seem to print out the table of "alternative indexing possibilities" in this mode - but possibly the table is only printed in the I2 case because the beta angle comes out as 90.0°.


Another note: CORRECT can easily be forced ''not'' to assign a spacegroup, and consequently will not reject outliers based on a too high symmetry assignment. To this end one simply supplies space group P1 and correct unit cell:
=== not biasing the cell parameters, and avoiding premature outlier rejection ===
 
CORRECT can easily be forced ''not'' to assign a spacegroup, and consequently will not reject outliers based on a too high symmetry assignment. To this end one simply supplies space group P1 and its unit cell:
<pre>
<pre>
SPACE_GROUP_NUMBER=1  
SPACE_GROUP_NUMBER=1  
UNIT_CELL_CONSTANTS= 74.1  78.6  124.0 108.2 105.2  90.4
UNIT_CELL_CONSTANTS= 74.1  78.6  124.0 108.2 105.2  90.4
</pre>
</pre>
in XDS.INP and re-runs CORRECT. That gives an unbiased XDS_ASCII.HKL. In this case though, the output of pointless is practically unchanged, whether XDS_ASCII.HKL was scaled in oI (=22 or 23) or P1..
in XDS.INP and re-runs CORRECT. That gives an unbiased XDS_ASCII.HKL, with no angles set to 90°. pointless ''with'' SETTING SYMMETRY_BASED then gives
<pre>
Best Solution:    space group C 1 2 1
 
  Reindex operator:                [-k-2l,-k,-h]             
  Laue group probability:            0.849
  Systematic absence probability:    1.000
  Total probability:                  0.849
  Space group confidence:            0.770
  Laue group confidence              0.770
 
  Unit cell:  233.70  78.42  73.22    90.31 105.34  89.77
</pre>
and ''without'' SETTING SYMMETRY_BASED
<pre>
Best Solution:    space group I 1 2 1
 
  Reindex operator:                [-h,-k,h+k+2l]           
  Laue group probability:            0.838
  Systematic absence probability:    1.000
  Total probability:                  0.838
  Space group confidence:            0.753
  Laue group confidence              0.753
 
  Unit cell:  233.70  78.42  73.22    90.31 105.34  89.77
 
</pre>
the dreadful I2 space group nomenclature (and I have no idea why the probability and confidence values are worse than in C2). But anyway this shows that there are two non-equivalent ways to index the same lattice!
 
=== final steps ===
 
Finally, XDS (but only JOB=DEFPIX INTEGRATE CORRECT) should be re-run with
<pre>
SPACE_GROUP_NUMBER=5
UNIT_CELL_CONSTANTS= 233.70  78.42  73.22     90 105.34  90
</pre>
because this enforces just the correct cell constraints.
 
== See also ==
 
[http://pd.chem.ucl.ac.uk/pdnn/symm3/allsgp.htm The 230  3-Dimensional Space Groups]
Kay
Bureaucrats
2,652

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