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Twinning and other pathologies Andrey Lebedev University of York

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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Crystal disorder Single crystal Twinned crystal Partially disordered crystal Sizes of ordered domains decrease 1- 2- and 3-dimensional disorder OD-structures Diffraction:Twinning & Disorder = Missing global periodicity

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S1S1 S1S1 S1S1 S2S2 OD-structures - two-dimensional periodicity - a potential for disorder in the third dimension - identical layers - identical interfaces between the layers - but: two or more ways of packing adjacent layers * ) MX: "identical" means Ca r.m.s.d. < 1 A * ) S 1 and S 2. are called stacking vectors

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Single crystal OD-twin C2 Single crystal Allotwin P2 1 P2 1 2 1 2 1 Examples Partially disordered OD-structure P2 1 Example 1Example 2Example 3

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The diffraction images can be indexed in C2 with two different orientation of the crystal The problem with the data is that some of the reflections from two lattices overlap. The presence of layers of overlapping reflections is the reason of non-origin peaks in the Patterson map. C2 Indexing in C2 Rye et al. (2007) Acta Cryst. D67 L-2-haloacid dehalogenase from Sulfolobus tokodaii Example 1: OD-twin (twin by lattice pseudomerohedy)

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a c C2 OD-twin: real and reciprocal lattices Example1

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Original data Demodulated data v = w = 0 R / R-free 0.21 / 0.27 0.16 / 0.23 I T1 = q'(h) I 1 q'(h) = p 0 + p 1 cos(2 th) + p 2 cos(4 th) +... q'(h) C2 Example1 OD-twin: demodulation

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Dauter et al. (2005). Acta Cryst. D61, 967-975. P2 1 P2 1 2 1 2 1 P2 1 R / R-free = 0.21 / 0.31 Crystals of Lon protease Resolution 3Å P2 1 2 1 2 1 R / R-free = 0.19 / 0.35 Example 2: allotwin

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Wang et al. (2005). Acta Cryst. D61, 67-74. a* The translation symmetry is perturbed in the direction a*. The diffraction pattern is characterised by the presence of the diffuse streaks along a*. Crystals of Phi29 DNA polymerase Resolution 2.2Å The structure was solved using demodulated data and experimental phasing Refinement against corrected data: R=0.28 P2 1 Example 3: partially disordered OD-structure

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Example 4: Four types of domains Patterson maps at Z=0 Interpretation of the Patterson map for 1k7v: four types of domains - P2 1 (orientation 1) - P2 1 (orientation 2) - C2 (orientation 1) contribute to some of the P2 1 spots, - C2 (orientation 2) hence non-origin Patterson peaks P2 1 structure (1k7u, 1k7v) Putative C2 structure 1k7u 1k7v Twinned P2 1 data

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crystal soaking twinned orthorhombic crystaltwinned tetragonal crystal Example 5: Conserved one-dimensional substructures Roberto Steiner, Kings college, University of London

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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Important special case: twinning by (pseudo)merohedry - All spots overlap with related spots from another individual crystal - Detection requires analysis of intensity statistics - Significant effect on model if ignored Twins by reticular merohedry (inc some OD-twins), allotwins, disordered structures - Can be readily seen in images with predictions

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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Single crystalPartial twin Perfect twin Acentric reflections Centric reflections s sss Theoretical intensity statistics

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C-terminal domain of gp2 protein from phage SPP1 (unpublished) perfect twin PDB entry 1i1j single crystal Two good, two bad

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PDB code 1l2h partial twin Bad example 1

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human deoxycytidine kinase single crystal Bad example 2

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Twinning tests in CCP4I (ctruncate) 1 2 3 4 6 5

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Cumulative intensity distribution in Ctruncate To compare: Red: Acentric theoretical, Blue: Acentric observed Z ≈ |E| 2 > Cumulative intensity distribution > Cumulative... (Centric and acentric) Untwinned dataTwinned data

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Second moments of Z (fourth moments of |E|) in Ctruncate To compare with the line = 2 Untwinned dataTwinned data > Acentric moments of E for k=1,3,4 > 4th moments of E...

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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H-test and L-test J1 J2 J1 J2 H = | J1 – J2 | / ( J1 + J2 )L = | J1 – J2 | / ( J1 + J2 ) twin axes sublattices with strong and weak reflections (pseudotranslation)

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Theoretical distribution of H Single crystal H P(H) Partial twin H P(H) Perfect twin H P(H)

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Distribution of H can be perturbed by NCS and weak observations H P(H) Blue: ideal distribution for partial twin Green: blue + effect of NCS axis || twin axis Red: green + effect of intensities with small I/ sig(I)

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Examples of experimental P(H) Despite NCS and effect of weak observations correct interpretation is possible

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0.00.51.0 0.0 0.5 1.0 L P(L) Theoretical distribution of L Partial twin P(L) 0.00.51.0 0.0 0.5 1.0 L Single crystal L 0.00.51.0 0.0 0.5 1.0 P(L) Perfect twin

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Distribution of L can be strongly perturbed by weak observations Resolution, A / 0.00.51.0 0.0 0.5 1.0 P(L) L All data: as if a perfect twin 0.00.51.0 0.0 0.5 1.0 P(L) L Data below 3A: untwinned Cell: 64.2 109.2 100.2 90 93.8 90 Spacegroup: P2 1 No pseudo symmetry Pseudomerohedral twinning is impossible Nevertheless the L-test is very useful when performed with right resolution range (or with several ranges)

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> Acentric moments of E for k=1,3,4 > 4th moments of E... Statistics of one intensity are strongly affected by pseudotranslation PDB:1jjk: Pseudotranslation results in clearly seen alteration of strong and weak reflections

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L-test and H-test are not affected by pseudotranslation > H test for twinning (operator...) > cumulative distribution function for |H| > L test for twinning > cumulative distribution function for |L|

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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Why so many tests? Statistics of one observation Statistics of two observations P(Z) H-testL-test Specific for a given operation ––+– Insensitive to pseudotranslation ––+ / –+ Insensitive to anisotropy ––+ / –+ Specific for a given resolution shell –+–– Insensitive to weak reflections at high resolution –(–)––

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Are these tests always sufficient? Resolution Weak observations may obscure twinning Pseudosymmetry may behave as exact symmetry (and may obscure twinning) HighLow Validation of crystallographic symmetry instead of twinning tests: refinement in space groups compatible with - unit cell - current model (considered as at least approximately correct) How to handle the cases with strong pseudosymmetry?

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Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation

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http://www.ysbl.york.ac.uk/YSBLPrograms/index.jsp Zanuda

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Submitting Zanuda job 1yup.mtz 1yup.pdb

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Zanuda output Download of output pdb- and mtz-files Symmetry analysis

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An example of symmetry correction PDB code:1yup spacegroup (PDB):P18 molecules per a.u. spacegroup (true):P2 1 4 molecules per a.u. Pseudo-symmetry spacegroup:C22 molecules per a.u. (because of pseudo-translation)

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Monoclinic structures related to 1yup False spacegroupP2 P2 1 True spacegroup C2Pseudo-symmetry spacegroup Spacegroup and its relation to the structure 1yup Crystallographic axes NCS axes Positions of molecules

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Data processing ( 2/m ) Molecular replacement ( P2 ) Refinement ( P2 ) R-free ≈ 0.37 Data processing ( -1 ) Molecular replacement ( P1 ) Refinement ( P1 ) R / R-free = 0.24 / 0.31 PDB: 1yup ( P1 ) PDB: 1yup Zanuda ( P2 1 ) R-free = 0.33 Structure solution and symmetry validation

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Spacegroup validation: step 1 if this value is too big value (>1.5), then convergency is unlikely, and the results will almost certainly be unreliable

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Spacegroup validation: step 2 2-axis2 1 -axis C2crystallographic crystallographic P2crystallographic NCS P2 1 NCS crystallographic

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spacegroup validation: step 3 Output (P2 1 )

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Zanuda protocol is not perfect Assumptions: - The pseudosymmetry is very strong (r.m.s.d. from exact symmetry ≈ 1A) - The structure is almost correct (although it might have been refined / rebuilt in an incorrect spacegroup) If it is not so, the results will likely to be wrong. Things go wrong way

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CCP4I interface Refinement > Symmetry validation This is not jet in the ccp4i distribution

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