Twinning in PDB and REFMAC Garib N Murshudov Chemistry Department, University of York, UK
Contents Crystal peculiarities Problem of twinning Occurrences of twinning in PDB Recognition of potential operators Likelihood function and data organisation Map coefficients Some test results
Pseudo translation Twin Order disorder Crystal peculiarities
Pseudo translation Real space a b Reciprocal space Distance between spots: 1/a, 1/b Distance between spots: 1/(2a), 1/b Every second reflection is weak.
Pseudo translation Cell P0 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: Non merohedral twin Helices, DNA Order disorder
Twinning
Programs that can currently handle twinning: SHELXL CNS phenix.refine But you need to know twinning operators and sometimes initial twin fraction estimation. As far as I know current implementations use least squares approach. phenix.refine may have an automatic approach
merohedral and pseudo merohedral twinning Crystal symmetry: P3 P2 P2 Constrain: = 90º Lattice symmetry *: P622 P222 P2 (rotations only) Possible twinning: merohedral pseudo merohedral 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 Allotwin C2 C2 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 selfrotation function corresponding to two fold axes Experimental data Model (single domain) PDB code 1l2h Spacegroup P4 3 1 molecule per AU Merohedral twinning PDB code 1igj Spacegroup P2 1 NCS (Pseudosymmetry): 2 monomers per AU Pseudo merohedral twinning Four equivalent twinning two fold axes Pseudosymmetry and twinning Pseudosymmetry Crystallographic two fold axis Crystallographic two fold axis
RvR plot non twins R twin = R obs twin :: I R calc twin :: I I(h) - I(S twin h) h 2 I(h) h I obs I calc A: translational NCS B: mislabeling F I C,C : 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
Detection of potential twin operators Take cell parameters and find reduced cell Find symmetry of the cell Take symmetry of the crystal Use coset decomposition and find symmetry of the cell that are not symmetry of the crystal These are potential (pseudo)merohedral twin operators
Likelihood for twin refinement L = blocks log AB P(I block ;I twin )P(F true ;F c )dadb I block - all reflections related with each other via twin I twin = a ik I true contributors F c - calculated structure factors I true = F true 2 To estimate this likelihood and its derivatives we use Laplace approximation
List of observations linked together Data organisation List of reflections contributing to each observation asymmetric unit List of references to the asymmetric unit Reasons why observations are linked: unmerged data, twinned crystals, split crystals Asymmetric unit has its own structure. E.g. Friedel pairs together
Problem with Rfree When using twinning refinement there is a problem with free reflections. Reflections are related via twinning operators. Current implementation makes free reflections property of block of reflections.
Problems with R factors When there is a twin then statistics may be lower than in the absence of it. R factor: random R factor in the absence of twin is around 58.5%. In the presence of twin it could be around 40% Rmerge: Rmerger for random structure factors is around 50%. If there is twin and merging happens on the axis independent from twin operator then random Rmerge could go as low as 37.5 So: One must be careful in the presence of twin. Structures may not be comparable, merging may not be reasonable.
Map coefficients < F >= F P(F;observations)dAdB F is structure factor of interest Observations are all available data. They could related by twin operators. Unmerged data can also be used. This equation defines expected structure factors for a single crystal.
Test cases: Preliminary results PDB ID R in pdb R after refmac** R after twin Twin fractions Comments 1rxf 11.9 21.5 12.0 0.69 0.31 Refined with twin 1ap9* 25.8 31.7 27.6 0.65 0.35 Data between 5 2.35 were used 1gwy 21.6 22.1 18.4 0.74 0.26 Refined without twin 1jrg 21.1 23.5 16.7 0.73 0.27 Refined without twin *Data could have been detwinned (bad idea) **Zero cycle of refinement in REFMAC was used
Electron density: 1ap9 What we would like to see refmac map twin map Data may have been detwinned (very bad idea!!!!)
Electron density: 1gwy What we will see refmac map Twin map differences between electron densities are marginal. That is usual case especially when twin and NCS are almost parallel
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 if F R e if + a ( F w - F R )e if F t twinned structure factor F R structure factor from correct crystal F W 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.
Conclusion Twinning occurs more often than we would like Twinning and rotational NCS occur very often together Twin refinement improves statistics and occasionally electron density PDB is a fantastic resource for testing and development
Current and future work Complete automation of MR Automatic twin refinement (merohedral, nonmerohedral, split crystals etc) Replacement for free R Dealing with OD with and without twin General modulated crystals
Acknowledgement Andrey Lebedev Dan Zhou Alexei Vagin CCP4 PDB depositor and maintainers: a fantastic resource Wellcome Trust, NIH, BBSRC for funding