Twinning and other pathologies

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Twinning and other pathologies Andrey Lebedev CCP4

Twinning by (pseudo)merohedry Statistics of one observation OD-structures Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne OD-structures - identical layers - identical interfaces between the layers - but: two or more ways of packing three adjacent layers *) MX: "identical" means Ca r.m.s.d. < 1 A S1 S2 *) S1 and S2. are called stacking vectors - two-dimensional periodicity - a potential for disorder in the third dimension June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Example 1: OD-twin C2 Indexing in C2 Indexing in C2 L-2-haloacid dehalogenase from Sulfolobus tokodaii Rye et al. (2007) Acta Cryst. D67 The diffraction images can be indexed in C2 with two different orientation of the crystal Some reflections from two lattices overlap. June 27, 2014 APS/CCP4 School, Argonne

Real and reciprocal lattices Maximum overlap is not at integer h. Twinning by reticular merohedry with twin index 10 and obliquity 0.1° Integration of a single lattice: in effect, twinning coefficient depends on h June 27, 2014 APS/CCP4 School, Argonne

Intensities of the overlapping reflections J J  L1 J  L2 Fourier transform of the tetramer Diffraction pattern of domain 1 Diffraction pattern of domain 2 Tetramers in different twin domains are in the same orientation Therefore, if reflections of the two lattices overlap, they have close intensities. The stronger the overlap, the closer the intensities are. June 27, 2014 APS/CCP4 School, Argonne

Demodulation Original data: R / R-free = 0.21 / 0.27 Modulation function q'(h) = p0 + p1 cos(2th) + p2 cos(4th) + ... Corrected data: R / R-free = 0.16 / 0.23 June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Patterson Map Indexed lattice The second lattice z x Non-origin peaks in the Patterson map: • contribution from the second lattice • because of the overlapping spots v=0 w u June 27, 2014 APS/CCP4 School, Argonne

Improvement in the electron density Visually, improvement occurred only for the electron density for solvent molecules (Poor density for solvent was the original reason for data revision) The electron density maps (2-1 at 1.5σ and 1-1 at 3σ) around the pyruvate molecule before and after demodulation June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne OD-structures Single crystal Single crystal Partially disordered OD-structure OD-twin Allotwin C2 P21 P212121 P21 Examples 1 & 2 Example 3 & 4 Examples 4 & 5 June 27, 2014 APS/CCP4 School, Argonne

Classification: OD-structures vs. twins Single crystals allotwin OD-twin (partially) disordered OD-structure This is structure based classification of a specific class of structures Twinning: by (pseudo)merohedry by reticular (pseudo)merohedry ... This is geometry based classification accounting for crystal and lattice symmetries. June 27, 2014 APS/CCP4 School, Argonne

Symbols for groupoid symmetry In 2P, P is a non-integer subscript. Special values of P correspond to space group symmetry or specialised groupoid symmetry The following types are possible (I) two surfaces of a single layer are identical; (II) two surfaces of a single layer are different and contacts are made by different surfaces. (III) two surfaces of a single layer are different but contacts are made by identical surfaces. An example of symbol for groupoid of type (III): June 27, 2014 APS/CCP4 School, Argonne

Example 2: OD-twin with zero obliquity Space group: C2 a = 95.9 Å, b = 95.6 Å, c = 81.8 Å β =122.2° OD layer: P(2)2121 • The data were processed in C2 but in the twin lattice (twin index = 3) a'=229.5 Å, c'=86.8Å, β =90° • non-overlapping reflections from the minor twin component were removed • overlapping reflections were detwinned a' a c c' Uppenberg et al. (1995). Biochemistry 34, 16838-51. Molecule: Lipase B from Candida antarctica PDB code 1lbs June 27, 2014 APS/CCP4 School, Argonne

Example 2: OD-twin with zero obliquity This packing could be assumed by similarity with the previous example This packing is more likely to occur as it explains the exactly orthorhombic twin lattice The previous example: twin index 10 obliquity 0.1° This example: twin index 3 obliquity 0° In general, protein OD-twins frequently have zero obliquity (twins by metric merohedry) June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Example 3: allotwin P21 P212121 Crystals of Lon protease Resolution 3Å Dauter et al. (2005). Acta Cryst. D61, 967-975. P21 a = 48.5 Å b = 86.3 Å c = 138.0 Å β = 92.3° P212121 a = 86.3 Å b = 90.6 Å c = 148.0 Å June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Example 3: allotwin PDB code 1z0t P212121 0.19 / 0.35 PDB code 1z0v P21 0.21 / 0.31 Structures of both crystal forms were solved R / R-free Dauter et al. (2005). Acta Cryst. D61, 967-975. Crystals of Lon protease Resolution 3Å June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Four types of domains Patterson maps at W=0 P21 structure (1k7v) Putative C2 structure ? Interpretation of the Patterson map for: four types of domains - P21 (orientation 1) - P21 (orientation 2) - C2 (orientation 1) contribute to some of the P21 spots, - C2 (orientation 2) hence non-origin Patterson peaks Twinned P21 data set PDB 1k7v R 23.2 R-free 31.0 June 27, 2014 APS/CCP4 School, Argonne

Crystal disorder Twinning, partial disorder: Missing global periodicity size of ordered domains Single crystal (Single ordered domain) Twinned crystal (Two or more ordered domains) Partially disordered crystal (Many ordered domains) Coherence length of X-rays June 27, 2014 APS/CCP4 School, Argonne

Example 4: partially disordered OD-structure Crystals of Phi29 DNA polymerase Resolution 2.2Å The translation symmetry is not global in the direction a*. The diffraction pattern is characterized by the presence of the diffuse streaks along a*. The structure was solved using demodulated data and experimental phasing Refinement against corrected data: R=0.28 P21 Wang et al. (2005). Acta Cryst. D61, 67-74. a* June 27, 2014 APS/CCP4 School, Argonne

Example 5: Partial disorder with several stacking vectors Trame, C. B. & McKay, D. B. (2001). ActaCryst. D57, 1079–1090. Heat-shock locus U protein from Haemophilus influenzae and its complexes Several crystal forms, all partially disordered OD belonging to different OD-families. Data: Resolution 2.3Å Processed in P622 a = 110.6, c = 335.8 OD layer: P(6)22 model of P2221 single crystal model of disordered crystal June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry Statistics of one observation OD-structures Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry Twins by reticular merohedry (inc some OD-twins), allotwins, disordered structures - Can be readily seen in diffraction images (with spot predictions shown) Important special case: twinning by (pseudo)merohedry - All spots overlap with related spots from another individual crystal - Detection requires analysis of intensity statistics - More significant effect on model if ignored - Point group and, consequently, space group determination may be a problem June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry lattice in orientation 1 lattice in orientation 2 = ≠ crystal in orientation 1 crystal in orientation 2 Twinned crystal + Example: P2,  = 90 Twinned crystal Individual crystal 1 Individual crystal 2 Twin two-fold axis Crystallographic Unit cell June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry lattice in orientation 1 lattice in orientation 2 = ≠ crystal in orientation 1 crystal in orientation 2 Twinned crystal + Intensities from individual crystal 1 Intensities from individual crystal 2 Partial twin (individual crystal of different size) Perfect twin (individual crystal of equal size) h l k=0 P121,  = 90 some weak reflections vanish Twin two-fold axis Crystallographic two-fold axis June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry 432 622 23 422 6 32 3 222 4 2 1 (Pseudo)merohedral twining is impossible supergroup subgroup Twinning by merohedry – higher lattice symmetry is determined by crystal symmetry Twinning by pseudomerohedry – specialised unit cell parameters June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Twinning analysis 432 622 23 422 6 32 3 222 4 2 1 1/2 molecule per AU: Impossible! 222 One molecule per AU 2 Two reasons to show this slide Crystal content analysis may help recognising twinned crystal Deterministic case: structure cannot be "solved" in wrong space group Not always that easy June 27, 2014 APS/CCP4 School, Argonne

Monoclinic OD-twin (twin by pseudomerohedry) Au et al. (2006). Acta Cryst. D62, 1267-1275. Ferrochelatase-1 from B. anthracis PDB code 2c8j Space group: P21 Resolution 2.2Å a = 49.9, b = 109.9, c = 59.4 Å α = β = γ = 90° OD layer: P2(1)1 The only reflection with h = 2n, k=l=0 and I > 3 sig(I) June 27, 2014 APS/CCP4 School, Argonne

Monoclinic OD-twin (twin by pseudomerohedry) P21212 symmetrised structure Molecules shifted along c by 2.5Å R-free = 40% P21 true structure The lattice is exactly orthorhombic R-free = 27% Twinning was suspected only after several unsuccessful attempts at solving structure in an orthorhombic space group June 27, 2014 APS/CCP4 School, Argonne

Enantiomorphic stacking vectors Structures (1) and (2) • belong to different space groups: (1) P31 (2) P32 • are not necessarily related by inversion • but have the same structure amplitudes: F(1) = F(2) • and belong to the same OD family (1) (2) June 27, 2014 APS/CCP4 School, Argonne

Enantiomorphic stacking vectors Gulbis et al. (1996). Structure of the C-terminal region of p21WAF1/CIP1 complexed with human PCNA. Cell 87, 297–306. PDB code 1axc Space group: P3221 a = 83.5 Å, c = 233.9Å OD layer: P(3)21 Structure: from PDB generated Space group: P3221 P3121 R (%): 22.09 22.35 R-free (%): 29.15 30.02 Asymmetry of OD layer is within 0.2Å, but it helps choosing the right space group June 27, 2014 APS/CCP4 School, Argonne

Examples of crystal pathologies Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

Resolution bins (resolution shells) s < s(h,k,l) < s + ds <I>(s) = mean( I(h,k,l) ) <I>(s) ≈ C* exp( - 2* B* s2 ) B - Overall temperature factor s Wilson Plot: log( <I> ) against s June 27, 2014 APS/CCP4 School, Argonne

Normalised intensity: Second moment I(hkl) Z(hkl) = ---------- <I>(s) <Z>(s) = 1 (by definition of Z) many weak reflections Z(hkl): 1 1 1 1 <Z>(s) = 1 Z2(hkl): 1 1 1 1 <Z2>(s) = 1 Z(hkl): 0 0 2 2 <Z>(s) = 1 Z2(hkl): 0 0 4 4 <Z2>(s) = 2 Z(hkl): 0 0 0 4 <Z>(s) = 1 Z2(hkl): 0 0 0 16 <Z2>(s) = 4 <Z2> 1 s a few weak reflections June 27, 2014 APS/CCP4 School, Argonne

Normalised intensity: Cumulative distribution { Z(hkl) } --> sort --> { Zi } Z1 < Z2 < Z3 < ... < ZN many weak reflections 1 1/N Z1 Z2 Z3 ... ZN 1 3/N 2/N ... P(Z) Z a few weak reflections June 27, 2014 APS/CCP4 School, Argonne

Theoretical distribution of intensities Single crystal Partial twin Partial twin Perfect twin P(Z) P(Z) P(Z) P(Z) Z Z Z Z <Z2> s (works for incomplete data set) Acentric reflections Centric reflections June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Pseudosymmetry twin axis Twinning + Generic NCS Twinning axis || NCS axis Single crystal June 27, 2014 APS/CCP4 School, Argonne

Theoretical distribution of intensities Single crystal Partial twin Partial twin Perfect twin P(Z) P(Z) P(Z) P(Z) Z Z Z Z <Z2> s (works for incomplete data set) Acentric reflections Centric reflections June 27, 2014 APS/CCP4 School, Argonne

Theoretical distribution of intensities Single crystal Partial twin OR Perfect twin + pseudosymmetry axis || twin axis Perfect twin P(Z) P(Z) P(Z) P(Z) Z Z Z Z <Z2> s Acentric reflections Centric reflections June 27, 2014 APS/CCP4 School, Argonne

Monoclinic OD-twin (twin by pseudomerohedry) P21212 symmetrised structure Molecules shifted along c by 2.5Å R-free = 40% P21 true structure The lattice is exactly orthorhombic R-free = 27% Twinning was suspected only after several unsuccessful attempts at solving structure in an orthorhombic space group June 27, 2014 APS/CCP4 School, Argonne

Perfect twinning tests: proper behaviour PDB entry 1i1j single crystal C-terminal domain of gp2 protein from phage SPP1 perfect twin June 27, 2014 APS/CCP4 School, Argonne

Perfect twinning tests: resolution cutoff PDB code 1l2h partial twin June 27, 2014 APS/CCP4 School, Argonne

Perfect twinning tests: resolution cutoff human deoxycytidine kinase single crystal June 27, 2014 APS/CCP4 School, Argonne

Twinning tests in CCP4I (ctruncate) 1 5 6 2 3 4 June 27, 2014 APS/CCP4 School, Argonne

Cumulative intensity distribution To compare: Red: Acentric theoretical, Blue: Acentric observed Z ≈ |E|2 Untwinned data Twinned data > Cumulative intensity distribution > Cumulative ... (Centric and acentric) June 27, 2014 APS/CCP4 School, Argonne

Second moments of Z (fourth moments of |E|) Compare the experimental curve with the line <E4> = 2 Untwinned data Twinned data > Acentric moments of E for k=1,3,4 > 4th moments of E ... June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry Statistics of one observation OD-structures Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne H-test and L-test H = | J1 – J2 | / ( J1 + J2 ) L = | J1 – J2 | / ( J1 + J2 ) J1 J1 J2 J2 sublattices with strong and weak reflections (pseudotranslation) twin axes June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne H-test and L-test H = | J1 – J2 | / ( J1 + J2 ) L = | J1 – J2 | / ( J1 + J2 ) J1 J1 J2 J2 sublattices with strong and weak reflections (pseudotranslation) twin axes June 27, 2014 APS/CCP4 School, Argonne

Theoretical distribution of H Single crystal Partial twin Perfect twin P(H) P(H) P(H) H H H June 27, 2014 APS/CCP4 School, Argonne

Distribution of H can be perturbed by NCS and weak observations 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) June 27, 2014 APS/CCP4 School, Argonne

Examples of experimental P(H) An almost ideal case + effect of NCS axis || twin axis + effect of intensities with small I/ sig(I) June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne H-test and L-test H = | J1 – J2 | / ( J1 + J2 ) L = | J1 – J2 | / ( J1 + J2 ) J1 J1 J2 J2 sublattices with strong and weak reflections (pseudotranslation) twin axes June 27, 2014 APS/CCP4 School, Argonne

Theoretical distribution of L Single crystal Partial twin Perfect twin 1.0 1.0 1.0 P(L) 0.5 P(L) 0.5 P(L) 0.5 0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 L L L June 27, 2014 APS/CCP4 School, Argonne

Distribution of L can be strongly perturbed by weak observations Cell: 64.2 109.2 100.2 90 93.8 90 Space group: P21 No pseudo symmetry Pseudomerohedral twinning is impossible Resolution, A <sig(F)> / <F> 0.0 0.5 1.0 P(L) L All data: as if a perfect twin 0.0 0.5 1.0 P(L) L Data below 3A: untwinned Nevertheless the L-test is very useful when performed with right resolution range (or with several ranges) June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Pseudotranslation Crystallographic translation Pseudo-translation Pseudotranslation C/2 Planes 2L+1 contain weak reflections Limiting case, C' = C/2 • Weak reflections vanish Two times larger reciprocal lattice spacing Crystallographic translation June 27, 2014 APS/CCP4 School, Argonne

Statistics of one intensity are strongly affected by pseudotranslation 1jjk: Pseudotranslation results in alteration of strong and weak reflections > Acentric moments of E for k=1,3,4 > 4th moments of E ... June 27, 2014 APS/CCP4 School, Argonne

L-test and H-test are not affected by pseudotranslation > L test for twinning > cumulative distribution function for |L| > H test for twinning (operator ...) > cumulative distribution function for |H| June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry Statistics of one observation OD-structures Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

Statistics of one observation Statistics of two observations Why so many tests? Statistics of one observation Statistics of two observations P(Z) <Z^2> H-test L-test Specific for a given resolution shell No Yes Specific for a given twin operation Can detect perfect twinning + – Works for incomplete data Insensitive to pseudotranslation + / – Insensitive to anisotropy Insensitive to weak reflections at high resolution (–) June 27, 2014 APS/CCP4 School, Argonne

Are these tests always sufficient? Resolution Weak observations may obscure twinning Pseudosymmetry may behave as exact symmetry (and may obscure twinning) High Low How to handle the cases with strong pseudosymmetry? Validation of crystallographic symmetry: refinement in space groups compatible with unit cell current model (considered as at least approximately correct) June 27, 2014 APS/CCP4 School, Argonne

Twinning by (pseudo)merohedry Statistics of one observation OD-structures Twinning by (pseudo)merohedry Statistics of one observation Statistics of two observations Twinning tests summary Space group validation June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne YSBL server http://www.ysbl.york.ac.uk/YSBLPrograms/index.jsp Zanuda June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne CCP4I interface CCP4I > Validation & Deposition > Validate space group Starting from ccp4-6.3.0 (forthcoming release) June 27, 2014 APS/CCP4 School, Argonne

Pseudotranslation: what else can go wrong? Cell and H-M symbol are the same P2 (a,b,c) P2 (a,b,c) Crystallographic and pseudosymmetry axes are confused True structure False structure Molecular Replacement: Two structures are globally very similar (e.g. rmsd = 0.5A) MR can in some cases pick up a wrong solution June 27, 2014 APS/CCP4 School, Argonne

An example of symmetry correction PDB code: 1yup space group (PDB): P1 8 molecules per a.u. space group (true): P21 4 molecules per a.u. Pseudo-symmetry space group: C2 2 molecules per a.u. (because of pseudo-translation) June 27, 2014 APS/CCP4 School, Argonne

Monoclinic structures related to 1yup Space group and its relation to the structure 1yup Crystallographic axes Pseudosymmetry axes Positions of molecules C2 Pseudo-symmetry space group False space group P2 P21 True space group June 27, 2014 APS/CCP4 School, Argonne

Structure solution and symmetry validation Data processing ( 2/m ) Molecular replacement ( P2 ) Refinement R-free ≈ 0.37 ( -1 ) ( P1 ) R / R-free = 0.24 / 0.31 PDB: 1yup ( P1 ) PDB: 1yup Zanuda ( P21 ) R-free = 0.33 June 27, 2014 APS/CCP4 School, Argonne

APS/CCP4 School, Argonne Zanuda: limitations Assumptions: The pseudosymmetry is very strong (r.m.s.d. from exact symmetry ≈ 1A) The structures of individual molecules are almost correct although they might have been refined / rebuilt in an incorrect space group If assumptions are not satisfied, the results will likely to be wrong. June 27, 2014 APS/CCP4 School, Argonne

Four alternative solutions in two space groups GAF (N-terminal) domain of CodY protein from Bacillus subtilis Levdikov, V. M. et al. (2006). J Biol Chem 281, 11366-73. MR solution June 27, 2014 APS/CCP4 School, Argonne

Twinned crystal with pseudo-symmetric substructure Human macrophage receptor CLEC5A for dengue virus Watson, A. A. et al. (2011). J Biol Chem 286, 24208-18. pseudo-P3121 (a' b' c) P31 (a b c) a' b' a b a b 3-fold axes with respect to the true structure: crystallographic pseudosymmetry for (A) + (A) (B) Substructure (A) is common for twin individuals Substructure (B) is not even approximately symmetric relative to and The choice of correct origin was essential for structure completion June 27, 2014 APS/CCP4 School, Argonne

Acknowledgements Eleanor Dodson University of York Roberto Steiner King's College, London Zbigniew Dauter APS, Chicago Gleb Bournkov EMBL-Hamburg Michail Isupov University of Exeter Garib Murshudov MRC LMB Cambridge All the authors of cited papers June 27, 2014 APS/CCP4 School, Argonne