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∑

€

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.