Pseudo translation and Twinning

Crystal peculiarities

• Pseudo translation • Twin • Order-disorder

b

Real space

a

Pseudo translation

Reciprocal space

Distance between spots: 1/a, 1/b Distance between spots: 1/(2a), 1/b Every second reflection is weak.

P0

Pseudo-translation

Cell Patterson 0.125 P0 Pst-vector

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: Constrain: Lattice symmetry *: (rotations only) Possible twinning: P3 P622 merohedral P2 β = 90º P222 pseudo -merohedral P2 P2 Domain 1 Twinning operator Domain 2

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 C222 1 single crystal C2 C222 1 Disordered OD-structure OD-twin C2 C2 Allotwin C2 C222 1

The whole crystal: twin or polysynthetic twin?

twin polysynthetic twin A single crystal can be cut out of the twin: yes no The shape of the crystal suggested that we dealt with polysynthetic OD-twin

Twins: Self-Rotation Function

Figures show sections of the self rotation function corresponding to two-fold axes Experimental data Model (single domain) • PDB code 1l2h • Spacegroup P4 3 • 1 molecule per AU • Merohedral twinning Crystallographic two-fold axis • PDB code 1igj • Spacegroup P2 1 • NCS (

Pseudosymmetry

): 2 monomers per AU • Pseudo-merohedral twinning Four equivalent twinning two-fold axes Crystallographic Pseudosymmetry and twinning Pseudosymmetry two-fold axis

non-twins

RvR-plot

  

R twin

 

h

|

I

(

h

) 

I

(

S twin h

) | 2 

h I

(

h

)

obs R twin

::

I



I obs calc R twin

::

I



I calc

A: B: C,C’: translational NCS mislabeling F  I mislabeling I  F Red: (potential) merohedral twins Black: (potential) pseudomerohedral twins

Symmetry environment of twinning

Merohedral 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

This test is implemented in TRUNCATE Untwinned + pseudosymmetry: test shows no twinning Twin + pseudosymmetry: Test shows only partial Twinning.

(decrease of contrast)

Partial twinning test

Non-linearity 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.

This test is implemented in SFCHECK

Electron density: 1rxf We will see occasionally this

“refmac” map “twin” map

Electron density: 1jrg More usual and boring case

“refmac” map “twin” map



Effect of twin on electron density: Noise level. Very, very approximate

|

F t

|

e i

  |

F R

|

e i

   (|

F w

|  |

F R

|)

e i

 F t F R F W - twinned structure factor structure factor from “correct” crystal 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.