Pseudo translation and Twinning
Crystal peculiarities
• Pseudo translation• Twin• Order-disorder
Pseudo translation
Real space Reciprocal space
a
b
Distance between spots: 1/a, 1/b
Distance between spots: 1/(2a), 1/b
Every second reflection is weak.
Pseudo-translation
P0
0.125 P0
Patterson
Pst-vector
Cell
Pseudo translation (PST) may cause problems in molecular replacement. Refinement usually does not have much problem. However in the presence of PST the solution may be in wrong origin.
There may be other sources of pseudotranslation:
1) Non-merohedral twin
2) Helices, DNA
3) Order-disorder
Twinning
merohedral and pseudo-merohedral twinning
Crystal symmetry: P3 P2 P2Constrain: - β = 90º -Lattice symmetry *: P622 P222 P2(rotations only)Possible twinning: merohedral pseudo-merohedral -
Domain 1
Domain 2
Twinning operator
-
Crystal lattice is invariant with respect to twinning operator.
The crystal is NOT invariant with respect to twinning operator.
More than three layers, but less than the whole crystal.
C2 single crystal
C2
OD-twin
C2
C2
C2221 single crystal
C2221
Allotwin
C2
C2221
Disordered OD-structure
The whole crystal: twin or polysynthetic twin?
A single crystal can be cut out of the twin:
twin
yes
polysynthetic twin
no
The shape of the crystal suggested that we dealt with polysynthetic OD-twin
Experimental data
Model(single domain)
Twins: Self-Rotation Function
• PDB code 1l2h• Spacegroup P43
• 1 molecule per AU• Merohedral twinning
Crystallographic two-fold axis
Four equivalent twinning two-fold
axes
Figures show sections of the self-rotation function corresponding to two-fold axes
• PDB code 1igj• Spacegroup P21
• NCS (Pseudosymmetry): 2 monomers per AU• Pseudo-merohedral twinning
Crystallographic two-fold axis
Pseudosymmetry and twinning Pseudosymmetry
RvR-plot
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Rtwin =| I(h) − I(Stwinh) |h
∑2 I(h)
h∑
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Rtwinobs :: I→ Iobs
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Rtwincalc :: I→ I calc
A: translational NCSB: mislabeling FIC,C’: mislabeling IF
Red: (potential) merohedral twinsBlack: (potential) pseudomerohedral twins
non-twins
Symmetry environment of twinningMerohedral twinning:
– crystal symmetry assumes more symmetric lattice– twinning would not require extra constraints on unit cell dimensions
Conclusions:
– Cases with pseudosymmetry are more frequent in general, and dominate for pseudomerohedral twins.
– Among solved structures, pseudomerohedral twinning is less frequent than merohedral. It is likely, that this is partially because of the problems with diagnostic.
Perfect twinning test
Untwinned + pseudosymmetry: test shows no twinning
Twin + pseudosymmetry:Test shows only partialTwinning.
(decrease of contrast)
This test is implemented in TRUNCATE
Partial twinning test
No pseudosymmetry: linear for both twins and non-twins.Tilt shows twinning fraction.
The test is useless for perfect twins (cannot distinguish it from higher symmetry)
Pseudosymmetry causes non-linearity.
Experimental errors + this non-linearity makes the test hardly interpretable in some cases.
Non-linearity
This test is implemented in SFCHECK
Electron density: 1rxf
We will see occasionally this “refmac” map “twin” map
Electron density: 1jrgMore usual and boring case
“refmac” map “twin” map
Effect of twin on electron density: Noise level. Very, very approximate
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|Ft | eiφ ≈|FR | e
iφ +α (|Fw | − |FR |)eiφ
Ft - twinned structure factorFR - structure factor from “correct” crystalFW - structure factor from “wrong” crystal
The first term is correct electron density the second term corresponds to noise. When twin and NCS are parallel then the second term is even smaller.
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