Rigid body -...
Transcript of Rigid body -...
![Page 1: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/1.jpg)
Rigid body Rigid body approach
“Model molecule” is a fragment, which is placed arbitrarily and does not contribute to structure factors
Model molecule is transformed to “Actual positions” by translation vector and three rotation angles:
xm is position in the model molecule xo is a reference point
Reference point determines the symmetry
Rotations can be combined with an inversion
Molecule can have its own local symmetry
( ) 101 txxRx +−⋅= mi
![Page 2: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/2.jpg)
Model molecule
Actual position #1
Actual position #2
The parameters of atoms of the model molecule (coordinates, ADP …) can be refined. Thus the model molecule is less rigid then an object fixed by “Keep geometry rigid”
![Page 3: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/3.jpg)
Model molecule
Actual position #1
Actual position #2
![Page 4: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/4.jpg)
Model molecule
Actual position #1
Actual position #2
![Page 5: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/5.jpg)
Model molecule
Actual position #1
Actual position #2
![Page 6: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/6.jpg)
Model molecule
Actual position #1
Actual position #2
![Page 7: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/7.jpg)
Model molecule
Actual position #1
Actual position #2
TLS tensors
Instead of individual ADP of the model molecule we can assume that all atoms have amplitudes appropriate to a rigid body and that all atoms are moving in phase. Then we can refine TLS tensors (20 parameters) for each actual position. Symmetry restrictions will be done according to the reference point.
![Page 8: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/8.jpg)
Rigid body approach applied to structure of cyclodextrin
β-Cyclodextrin duplex connected by two disulfide bonds Chem. Eur. J. 2012, in print, DOI: 10.1002/chem.201201239
a=b=21.04 Å, c=26.69 Å, α=β=γ=90° V=11833 Å3
Symmetry I4
![Page 9: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/9.jpg)
Some of glucose units are disordered, the others not. The disordered part could not be described reliably by splitting of atoms because of low data quality. Fixing geometry by restraints would be extremely laborious.
![Page 10: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/10.jpg)
One of not disordered glucose units was taken as a model molecule. The model molecule is indicated in yellow and suffix “a”. O1, O2 and C1a were refined freely.
![Page 11: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/11.jpg)
The model molecule enters to ten actual positions. The pairs g-f, h-I, e-d describe disordered glucose units, with total occupancy 1. Because all actual position have common geometry refined only for the model molecule, disorder can be reliably refined. Introduction of rigid bodies did not affect R value. This is a prove that the glucose unit is the same in all position within our data precision. Advantages for visualization: minor actual position can be easily removed.
![Page 12: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/12.jpg)
Modulated structures very short explanation what does it mean
![Page 13: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/13.jpg)
Sodium carbonate View along c z from 0 to 1
![Page 14: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/14.jpg)
Sodium carbonate View along c z from 1 to 2
![Page 15: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/15.jpg)
Sodium carbonate View along c z from 2 to 3
![Page 16: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/16.jpg)
Is there some translation periodicity? Probably yes because the diffraction pattern looks like usually
![Page 17: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/17.jpg)
c*
a*
…. but we cannot describe all peak positions with a lattice
![Page 18: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/18.jpg)
c*
a*
![Page 19: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/19.jpg)
c*
a*
* * *1 2 3q q q= + +q a b c
The remaining spots can be described with q-vector, which has irrational components. Indices h k l m
![Page 20: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/20.jpg)
Four-dimensional reciprocal lattice Modulated and composite crystals can be described in a (3+d) dimensional superspace. The theory has developed by P.M.DeWolff, A.Janner and T.Janssen (Aminoff prize 1998). The basic idea: real diffraction pattern is a projection from the (3+d) dimensional superspace. The basic assumption: all satellites are clearly separated. (the intensities diminish for large satellite index).
e q
*4A
*3
*3 Aa =*3a
*3R
![Page 21: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/21.jpg)
Modulation function
The atom is displaced from its basic position by a periodic modulation function that can be expressed as a Fourier expansion. In the first approximation intensities of satellites reflections up to order m are determined by modulation waves of the same order.
( ) ( ) ( )4 , 4 , 41 1
sin 2 cos 2m m
s n c nn n
x nx nxπ π= =
= +
= +∑ ∑
r r u
u A A
e=A4
Basic position
incommensurate
![Page 22: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/22.jpg)
Example 5.2: Anhydrous Sodium carbonate Structure with strong harmonic modulation of positions and ADP. Chemical formula: Na2CO3
3d → incommensurate → commensurate
Alpha 9.02, 5.21, 6.50 90, 90, 90 P63/mmm
Beta 8.98, 5.25, 6.21 90, 90.33, 90 C2/m
Gamma 8.92, 5.25, 6.05 90, 101.35, 90 C2/m(α0γ) q=(0.182,0,0.322)
Delta 8.90, 5.24, 6.00 90, 101.87, 90 C2/m(α0γ) q=(1/6,0,1/3)
757K 628K 170K
Change of rotation symmetry Change of translation symmetry
The main change of the structure occurs during the alpha → beta transition which changes rotation symmetry. Although modulations in gamma and delta phase are very strong they only modify the beta structure keeping the rotation symmetry unchanged.
Alpha Beta Gamma Delta
![Page 23: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/23.jpg)
Data for the phase gamma Single crystal data measured with KUMA four-circle diffractometer Input files: naco.hkl, naco.cif_od Frame scaling, absorption correction: done with software of diffractometer Unit cell: a=8.9338, b=5.2678, c=6.0565, α=90.1018. β=101.551, γ=89.7404 q vector: (0.1776, 0, 0.3252)
a*
c*
![Page 24: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/24.jpg)
Import wizard
1 2
3
![Page 25: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/25.jpg)
Symmetry wizard
1
2
![Page 26: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/26.jpg)
3
Data processing was done with C centering.
![Page 27: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/27.jpg)
Symbol C2/m(0b0)s0 means: - 3d symmetry C2/c - q vector (0 β 0) - shift by ½ in x4 M50: symmetry x1 x2 x3 x4 symmetry -x1 x2 -x3 -x4+1/2 symmetry -x1 -x2 -x3 -x4 symmetry x1 -x2 x3 x4+1/2
Superspace group from satellite reflections
5
6
Space group from main reflections 4
![Page 28: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/28.jpg)
Structure solution wizard
The only reason for the random seed is compatibility with the text of Jana Cookbook. Convergence with R~21%
![Page 29: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/29.jpg)
Output of Superflip confirms the symmetry
![Page 30: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/30.jpg)
1 2
3
The initial structure model
![Page 31: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/31.jpg)
Individual atoms
![Page 32: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/32.jpg)
File M40
![Page 33: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/33.jpg)
Structure plots
or
![Page 34: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/34.jpg)
The plot helps to identify oxygens C1 and C2 will be changed to oxygens with “Edit atoms” tool of Jana
![Page 35: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/35.jpg)
Structure refinement
Refinement of the initial structure model
Adding position modulation waves
![Page 36: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/36.jpg)
Refinement with harmonic ADP and four position modulation waves for all atoms
![Page 37: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/37.jpg)
Fourier maps and Contour plots
Coordinates of Na1: x,y,z Sections to plot: x2-x4 Range of x2: from (y – 2Å/b) to (y + 2Å/b) step 0.02Å Range of x4: from 0 to 2, step 0.02 Stacking of sections: x2-x4 x3(1) x1(1) x2-x4 x3(2) x1(1) …. x2-x4 x3(n) x1(1) x2-x4 x3(1) x1(2) x2-x4 x3(2) x1(2) …. x2-x4 x3(n) x1(m) Range of x1: from (x-1Å/a) to (x+1Å/a) step 0.02Å Range of x3: from (z-1Å/c) to (z+1Å/c) step 0.02Å
![Page 38: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/38.jpg)
Observed and difference Fourier map for Na1 indicates modulation of ADP Similar features can be seen in difference maps of other atoms except carbon
![Page 39: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/39.jpg)
Adding ADP modulation waves
Final refinement
Atom Position waves ADP waves
Na1 4 3
Na2 4 2
Na3 4 2
O1 4 2
O2 4 4
C1 3 0
![Page 40: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/40.jpg)
Final difference maps
Contour step 0.1
![Page 41: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/41.jpg)
Interpretation of structure model using Grapht
t=0
t=1
Internal coordinate t
![Page 42: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/42.jpg)
Interpretation of structure model using Dist
![Page 43: Rigid body - crm2.univ-lorraine.frcrm2.univ-lorraine.fr/mathcryst/pdf/uberlandia/Dusek_Rigid_body_an… · Rigid body . Rigid body approach “Model molecule” is a fragment, which](https://reader033.fdocuments.in/reader033/viewer/2022042804/5f5277d00553c23f59200604/html5/thumbnails/43.jpg)
Interpretation of structure model using plots