Lecture 5: The Measurement and Processing of X-ray
diffraction data
5.1 Specimen preparation & the problem of radiation damage: “Freezing” crystals
5.2 The oscillation method for single crystal X-ray diffraction data collection
5.3 Estimation of Diffracted Intensities
5.4 The Merging and Scaling of data
5.5 The Determination of the crystal space group: Symmetry and Systematic
absences
5.6 A common pathology: Merohedral twinning (Omitted in 2015)
5.7 Basic characterization of a novel X-ray data set
5.7.1 Estimating the number of molecules in the asymmetric unit
5.7.3 Detecting rotational NCS
5.7.4 Detecting translational NCS (omitted in 2015)
Wednesday, 18 March 15
1
The paper that
founded the field
of protein
crystallography
The essential observation:
Protein crystals contain a lot
of water which is essential to
their integrity.
Wednesday, 18 March 15
2
Another innovative idea from the brain of J.D.
Bernal: The Bernal sphere …
Wednesday, 18 March 15
3
… inhabitable space stations orbiting the
earth
Wednesday, 18 March 15
4
Capillary-mounting of protein crystals …
formerly the standard method
From Blundell and Johnson (1976)
From Blow (2002)
Wednesday, 18 March 15
5
Problems with capillary-mounting of protein
crystals
1. It’s fiddly, and hard to do well, especially for small crystals.
2. Protein crystals at room temperature suffer severe radiation damage from
exposure to X-rays. This process can be dramatically slowed (but not
eliminated) by cooling the crystals to liquid Nitrogen temperatures.
Wednesday, 18 March 15
6
Cryo-crystallography
•Most X-ray diffraction data on protein crystals is now collected with the crystal
maintained near the temperature of liquid nitrogen.
‣Liquid
nitrogen boils at 77 K (-196 °C)
‣Crystals typically maintained at 110 K (-163 °C) for collection of diffraction data
•Crystals are prepared for data collection by
suspending them in a thin film of liquid,
in a fiber loop, and then rapidly immersing them in liquid nitrogen.
•Cryoprotectants (essentially antifreeze) are usually added to the buffer, to help
suppress crystalline ice formation and prevent physical damage to crystals.
•It’s quite common for crystals to get damaged during cooling and optimization of
the cryoprotectant, and its concentration, is usually required.
Wednesday, 18 March 15
7
Mounting protein crystals in fiber loops
Harvesting and the Flash
cooling process: The crystal is
scooped up from the drop in a
loop, and quickly immersed in liquid
Nitrogen. A vial with a magnetic rim
is used to protect and transport the
Crystal/Loop/Pin/Base assembly
Text
Magnetic “wand”, used for handling
From Rupp (2010)
Magnetic base
Pin
Loop (invisible at this magnification)
Cryovial with magnetic rim
The basic setup
Image courtesy Phil Jeffrey:
http://xray0.princeton.edu/~phil/Facility/Guides/XrayDataCollection.html
Wednesday, 18 March 15
8
8-+
,-%=
1-&
-($
E$
8/
$&(
%$0
$&6
&-(
G$&
-!*
1&'
1*
1&'
0$*
"+$
E$
$&(
()$
-&=
$*(
/%$
-2
8*(
'$+
.$
("%=
& -2
$$&
"7*
1&
7-6
-&*
*("%
("%=
!"#"
F$*
J8"&(1(7 -2 0"("= !)"( 1* 0$*1+"E%$a
Maintaining loop-mounted
crystals close to liquid Nitrogen
temperature
;&0-:1 <
4 (7/1,"% $I/$+1.$&("% "++"&'$.$&( 2-+ " ,+7-,+7*("%%-'+"/)1, 0"("
,-%%$,(1-&3 4 ."'&$(1, +8EE$+ 01*, 1* /1$+,$0 !1() " *("1&%$** *($$% /1&= ()$
(-/ -2 !)1,) 1* +-8&0$0 "&0 ()$ E-((-. -2 !)1,) M(* 1&(- ()$ )-%$ 1& ()$
'-&1-.$($+ )$"03 D)$ ."'&$( .8*( E$ *(+-&' $&-8') (- ."G$ " +1'10
,-&&$,(1-&= E8( !$"G $&-8') (- "%%-! ()$ $I/$+1.$&($+ M&$ ,-&(+-% -2
()$ (-/ )"(= !)1,) 1* 2"E+1,"($0 2+-. " ."'&$(1, .$("% *8,) "* *("1&%$**
*($$% -+ &1,G$%3 9T1'8+$ +$/+-08,$0 2+-. :"+."& ; <,)&$10$+= >??@3C
Wednesday, 18 March 15
From Garman. Cool data: quantity AND quality. Acta Crystallogr. D Biol.
Crystallogr. (1999) vol. 55 (Pt 10) pp. 1641-53
9
Role of cryoprotectants
A primary role of the cryoprotectant is to suppress formation of crystalline ice.
X-ray diffraction image from a poorly cooled
crystal
Garman and Schneider (1997) J. Appl. Cryst. 30, 211-237
Wednesday, 18 March 15
10
Role of cryoprotectants
Diffraction patterns obtained from flash cooled
water-glycerol mixtures
Because the tiny ice crystals are randomly oriented, they
generally give rise to a powder diffraction pattern
The resolution of the powder diffraction rings arising from
crystalline ice are indicated (hexagonal ice has unit cell
dimensions a=b= 4.51 Å, c= 7.35 Å)
If ice rings are present in diffraction patterns collected from
protein crystals, the data in the immediate vicinity of the ice ring
cannot be reliably estimated.
Garman and Doublié. Cryocooling of macromolecular crystals: optimization methods. Meth. Enzymol. (2003) vol. 368 pp. 188-216
Wednesday, 18 March 15
11
Role of cryoprotectants
A secondary role of cryoprotectants is to prevent physical damage
to the crystal resulting from the differential contraction of solvent
and protein.
a
b
c
d
Juers and Matthews (2004) Quart. Rev. Biophys. 37, 105-119
Schematic showing three possible responses of solvent to cooling. (a) The room-temperature crystal, with the portion in blue illustrating the
solvent occupying the interstitial space between the four macromolecules in the unit cell. (b)–(d ) The crystal with its low-temperature packing
arrangement, and three possibilities for solvent contraction. (b) The solvent shrinks less than the interstices, resulting in the extrusion of solvent
into the neighboring interstices. (c) The solvent shrinks by the same amount as the interstitial space. Some solvent rearrangement is required
to accommodate the lattice repacking. (d ) The solvent shrinks more than the interstices, resulting in import of solvent into the interstice from
neighboring ones.
Wednesday, 18 March 15
12
Radiation damage
The problem of radiation damage is particularly acute at high intensity
synchrotron radiation sources, where protein crystals can get toasted … very
quickly. Freezing helps, but it’s not a universal panacea.
Garman and Owen(2006) Acta Cryst D62, 32-47.
Crystal of bacteriorhodopsin exposed (at 100K) to a 30 μM diameter X-ray
beam at the ESRF, Grenoble
Wednesday, 18 March 15
13
Radiation damage
As we learned in Lecture 1, X-rays can interact with the crystal in 3 possible ways:
1. They can be elastically scattered, without loss of energy (Thomson scattering).
2. They can be inelastically scattered, and lose energy to the molecules in the crystal (Compton
scattering).
3. They can be absorbed due to the Photoelectric effect. This generates free “photoelectrons”
and results in emission of secondary radiation.
•Only the first of these contributes to the useful part of the observed diffraction pattern.
•However most (~90%) of the interactions of X-rays with the crystal are of types 2 and 3.
•Therefore most of the X-rays interacting with the crystal deposit their energy into it, which causes
radiation damage. Photo-absorption (mechanism 3) is the main contributor to radiation damage in
protein crystallography.
Wednesday, 18 March 15
14
Radiation damage
Conceptually, radiation damage can be divided into two
components.
1. Primary radiation damage. The initial deposition of
energy generates photoelectrons, leading to rupture
of covalent bonds and production of free radicals
(highly reactive chemical species with unpaired
electrons). Primary radiation damage is
unavoidable and not dependent on temperature.
Incident X-ray
2. Secondary radiation damage. Free radicals produced by
the incident radiation can diffuse through the crystal
and cause further damage. Secondary radiation
damage is time and temperature dependent. Excepting
free electrons, most radicals are immobilized at liquid
Nitrogen temperatures. This is why cooling is good.
But cooling does not eliminate secondary radiation
damage.
Wednesday, 18 March 15
15
Radiation damage
Radiation damage in proteins localizes at particular sites. Common manifestations
of radiation damage:
• Disulfide bond rupture
• Decarboxylation of Aspartatic and Glutamatic acids
• Loss of hydroxyl group from Tyrosine
• Carbon-sulfur bond cleavage in Methionine.
• Disulfide bond rupture
Schulze-Briese, Wagner, Tomizaki & Oetiker (2005) J. Synch. Rad. 12, 261-267.
Increasing X-ray exposure
The evolution of radiation damage in a disulfide bond (protein = insulin)
Wednesday, 18 March 15
16
i.e. the ‘noise’) normalized to the intensity I1/! 1 of the first
data set is not a robust metric since the noise ! D increases with
dose and thus ID/! D reduces by an amount that more than
represents the true loss of diffracting power.
(ii) Rd, the pairwise R factor between identical and
symmetry-related reflections occurring on different diffraction
images, plotted against the difference in dose, !D, between
Radiation damage
Decarboxylation of
Glutamic acid
Figure 3
Photograph of a 400 mm neuraminidase crystal (subtype N9 fro
influenza isolated from a noddy tern), space group I432, that h
irradiated on ID14-4 at the ESRF at 100 K and then allowed to w
to RT. The three black marks are from the 100 # 100 mm be
discolouration is an indication of radiation damage.
Carbon-Sulfur bond
rupture in methionine
Garman, E. F. Radiation damage in macromolecular crystallography: what
is it and why should we care? Acta Crystallographica Section D 66, 339–
351 (2010).
Specific structural damage inflicted on a cryocooled crystal of apoferritin during sequential data sets collected on beamline ID14-4 at
Figure 4
ESRF. Figure 2
3 after An
Specificdensity
structural
inflicted
on a cryocooled
of apoferritin
idealized
Rd(b)
, the
pairwise
R factor between identi
(a) Electron
mapdamage
surrounding
Glu63
contoured crystal
at 0.2 electrons/Å
a dose
of 2.5 plot
MGyofand
after
50 MGy.
3 after symmetry-related
during density
sequential
sets collected
on contoured
beamline ID14-4
at ESRF. (a)
occurring
on different diffraction
(c) Electron
mapdata
surrounding
Met96
at 0.2 electrons/Å
a dose of 2.5 MGyreflections
and (d) after
50 MGy
"3
2Fo " Fc map of Glu63 contoured at 0.2 e Å after a dose of 2.5 MGy
Wednesday,
March
15 50 MGy. (c) 2F " F map of Met96 contoured at 0.2 e Å"3
and18(b)
after
plotted against the difference in dose, !D, between the images o
17
the reflections were collected (Diederichs, 2006). The plot is a stra
The Screenless Oscillation Method
The crystal is rotated through a small angle about an axis perpendicular to the X-ray beam, while
the diffraction pattern is recorded on a suitable detector (generally a CCD or an imaging plate). The
total angular range that has to be swept out to collect a complete dataset depends on the crystal
symmetry and crystal orientation (also note that for some orientations of a crystal it is
physically impossible to collect all the data by simply rotating around a single
axis).
Typically we might sweep through 180° in 1° increments, resulting in 180 diffraction images which
must be further analyzed.
From Outline of Crystallography for Biologists, Blow
Wednesday, 18 March 15
18
Visualizing the Laue conditions geometrically
We aren’t going to attempt a systematic treatment of diffraction geometry. But
you should be aware of the following “geometric” interpretation of the Laue
conditions
Recall that these are the Laue conditions - the conditions for observing diffraction
from a 3D crystal:
s.a = h
s.b = k
s.c = l
a, b, and c are the vectors which define the unit cell. h,k and l are integers.
Diffraction only occurs when s - the scattering vector - satisfies these conditions.
The points at which all 3 Laue conditions are satisfied are the points of the
reciprocal lattice.
Wednesday, 18 March 15
19
Visualizing the Laue conditions geometrically
Imagine that we have the a axis of the
cr ystal aligned with the axis of
oscillation. Geometrically, the first of the
Laue conditions specifies that diffraction
is restricted to a series of cones
arranged symmetrically about the
oscillation axis.
s.a = h
s.b = k
s.c = l
(To understand this remember that s.a is
proportional to the projection of s in the
direction of a )
a
From Woolfson. An Introduction to X-ray crystallography (1978)
So
2θ
s
Wednesday, 18 March 15
S
20
Visualizing the Laue conditions geometrically
If we record the diffraction pattern from
this crystal on a flat detector we see that
the fir st Laue condition restricts
diffraction to arcs on the detector
surface.
The diffraction from a 3D
cr ystal is restricted to
points which lie at the
intersection of the arcs
arising from all 3 Laue
conditions.
The main point to take away - with a flat
detector, the oscillation method gives
rise to a “distorted” picture of the
reciprocal lattice.
Adapted from van Holde, Johnson & Ho. Principles of Physical Biochemistry. (2006)
Wednesday, 18 March 15
21
What oscillation data looks like
A movie made by compiling actual X-ray diffraction data from a crystal
of GCN4-N16A peptide in space group P3121. Each frame is a 1°
oscillation..
James Holton, Berkeleyo
Wednesday, 18 March 15
22
Some practicalities
Must select an oscillation
angle, and a crystaldetector distance, sufficient
to resolve the diffraction
maxima and avoid overlaps.
This crystal is
orthorhombic: space group
P212121
Cell dimensions:
a= 76 Å b=77 Å c= 297 Å
α = 90° β= 90° γ= 90°
To resolve the closely
spaced reflections along c*
the detector could not be
moved much closer.
From Biomolecular Crystallography, Rupp
Wednesday, 18 March 15
23
Some practicalities
Must choose an appropriate exposure time.
The same oscillation image, recorded with different exposure times
30 ms exposure
Too noisy, high resolution
data not reliably recorded
From Biomolecular Crystallography, Rupp
Wednesday, 18 March 15
1 s exposure
About right
60 s exposure
Overdone. Detector is
saturated at low
resolution, Little gain in
high resolution region
24
The processing of oscillation data
The basic steps to be carried out:
•Autoindexing: Determine a lattice and crystal orientation that
can predict the positions of the diffracted intensities.
•Data integration: Get a numerical estimate of the diffracted
intensity
•Data Scaling and Merging: Put the intensity estimates from
different images on a common scale and combine multiple
measurements (“Reducing the data”). The user must make an
assumption about the symmetry of the crystal. The assumptions
about crystal symmetry need to be carefully examined. X-ray data
processing is now highly automated. When it goes wrong it’s
usually because of incorrect symmetry assignment
by the user.
Wednesday, 18 March 15
25
Autoindexing and data integration
Modern autoindexing algorithms are based on the Fourier transform, and are very robust. They will
output the crystal parameters (the unit cell dimensions, the centering operations, and the orientation of
the crystal with respect to the laboratory coordinate system). This is required to predict the diffraction
pattern, and assign each spot on the image an index hkl. We will skip the details of data integration,
through which a numerical intensity estimate is derived from the images. It works well !!
A typical macromolecular diffraction pattern for a strongly diffracting crystal. The original image is shown on the left, with the predicted
reflections shown superposed on the right. Each reflection is shown as a box, colour-coded blue and yellow for fully recorded and partially
recorded reflections, respectively. Leslie (2005) Acta Cryst D62, 48-57.
Wednesday, 18 March 15
26
The processing of oscillation data
What we want to end up with is a list of indices h k l and an
estimate I(hkl) for the diffracted intensity.
Note once again that experimentally we measure I(hkl) - the
intensity - which is proportional to the square of the structure
factor amplitudes |F(hkl)| which appear in the expression for the
electron density.
I(hkl) ∝
Wednesday, 18 March 15
2
|F(hkl)|
27
The assignment of symmetry
•The big problem in data processing, is to correctly understand the symmetry
of the crystal ... i.e. assign the crystal space group.
• While this step has been automated, it’s good to have a strong
understanding of the problem, because things still go wrong here.
•To assign the symmetry of the crystal we have to work backwards from the
diffraction data - by consideration of the geometry of the reciprocal lattice
and the symmetry present in the diffraction pattern.
•Recall that there are 230 space groups. However because protein molecules
are chiral (they have handedness) we need consider only the 65
enantiomorphic space groups.
At the end of the symmetry lectures we listed these “biological” space
groups.
Wednesday, 18 March 15
28
The 65 “Biological” Space Groups
Bravais Lattice
Possible space groups
Associated
point group
Primitive Cubic
P23 (195), P213 (198)
P432 (207), P4132 (213), P4232 (208), P4332 (212)
23
432
I centered Cubic
I23 (197), I213 (199)
I432 (211), I4132 (214)
23
432
F centered Cubic
F23 (196)
F432 (209),F4132 (210)
23
432
Rhombohedral
R3 (146)
R32 (155)
3
32
Primitive Hexagonal
P3 (143), P31 (144), P32 (145)
P312 (149), P3112 (151), P3212 (153), P321 (150), P3121 (152), P3221 (154)
P6 (168), P61 (169), P62 (171), P63 (173), P64 (172), P65 (170)
P622 (177), P6122 (178), P6222 (180), P6322 (182), P6422 (181), P6522 (179)
3
32
6
622
Primitive Tetragonal
P4 (75), P41 (76), P42 (77), P43 (78)
P422 (89), P4212 (90), P4122 (91), P41212 (92), P4222 (93), P42212 (94), P4322,
(95), P43212 (96)
4
422
I centred Tetragonal
I4 (79), I41 (80)
I422 (97), I4122 (98)
4
422
Primitive Orthorhombic
P222 (16), P2221 (17), P21212 (18), P212121 (19)
222
C Centered Orthorhombic
C222 (21), C2221 (20)
222
I Centered Orthorhombic
I222 (23), I212121 (24)
222
F Centered Orthorhombic
F222 (22)
222
Primitive Monoclinic
P2 (3), P21 (4)
2
C Centered Monoclinic
C2 (5)
2
Triclinic
P1 (1)
1
Wednesday, 18 March 15
29
The assignment of space group symmetry ...
overview
1. Collect diffraction data
& successfully index the
diffraction pattern.
2. Figure out which of
the Crystal Systems /
Bravais lattices we
appear to be dealing
with
4. Narrow the list of
space group possibilities
by consideration of
systematically absent
observations
3. Analyze the symmetry
present in the diffraction
pattern - deduce
possible space groups
Wednesday, 18 March 15
30
Bravais Lattices and Crystal Systems
•Auto-indexing algorithms will produce a list of possible lattices, and the cell
dimensions, and indicate the degree to which they can predict the diffraction
pattern. This gives us our first indications of the likely symmetry of the of the
crystal.
•This is because symmetry places constraints on the angles and lengths of a
conventionally chosen unit cell. Consult the table on the seven crystal systems
(repeated on the following slide)
•Initially we want to try and figure out which of the Bravais
Lattices we are working with, since that narrows the space group
possibilities. Remember that there are a total of 14 Bravais lattices, split
among 7 Crystal Systems ...
•However the cell dimensions are only a guide to the crystal symmetry. While
they tell you the highest possible symmetry of the crystal, you can always
have less symmetry than the cell dimensions might suggest.
Wednesday, 18 March 15
31
The seven crystal systems
System
Essential rotational
symmetry
Conventional choice of
axes
Unit Cell restrictions
Possible Lattices
Triclinic
None
No constraints
None
P
Monoclinic
Two-fold axis
b parallel to 2-fold
α = γ = 90°
P,C
Orthorhombic
Three perpendicular
2-fold axes
a, b, c parallel to 2fold axes
α = β = γ = 90°
P,C,I,F
Trigonal/Hexagonal
3-fold or 6-fold axis
c parallel to 3-fold or
6-fold
a=b
α = β = 90°, γ = 120°
P
Rhombohedral
3-fold axis
a, b, c related by
three fold axis
a=b=c
α=β=γ
R
Tetragonal
4-fold axis
c parallel to 4-fold
a=b
α = β = γ = 90°
P,I
Cubic
4 3-fold axes
a, b, c related by
three fold axis
a=b=c
α = β = γ = 90°
P,I,F
Wednesday, 18 March 15
32
A reminder about centered cells ...
For 7 of the 14 Bravais lattices, crystallographers select non-primitive cells, with more than 1 lattice point
in the cell, because these better reflect the space group symmetry. Here they are ...
Monoclinic crystal system
C centering
Tetragonal crystal system
I centering
Orthorhombic crystal system
I centering
C centering
F centering
Cubic crystal system
I centering
F centering
Images from Ron Stenkamp, University of Washington
Wednesday, 18 March 15
33
Bravais Lattices and Crystal Systems
Let’s look at some examples …
Primitive Lattice: Cell dimensions a= 23 Å b=34 Å c= 55 Å, Cell angles α = 87° β= 93° γ= 98°
C Centered Lattice: Cell dimensions a= 23 Å b=34 Å c= 55 Å, Cell angles α = 90° β= 90° γ= 90°
Primitive Lattice: Cell dimensions a= 60 Å b=60 Å c= 94 Å, Cell angles α = 90° β= 90° γ= 120°
I Centered Lattice: Cell dimensions a= 60 Å b=60 Å c= 94 Å, Cell angles α = 90° β= 90° γ= 90°
From Drenth (2002)
Wednesday, 18 March 15
34
The seven crystal systems
System
Essential rotational
symmetry
Conventional choice of
axes
Unit Cell restrictions
Possible Lattices
Triclinic
None
No constraints
None
P
Monoclinic
Two-fold axis
b parallel to 2-fold
α = γ = 90°
P,C
Orthorhombic
Three perpendicular
2-fold axes
a, b, c parallel to 2fold axes
α = β = γ = 90°
P,C,I,F
Trigonal/Hexagonal
3-fold or 6-fold axis
c parallel to 3-fold or
6-fold
a=b
α = β = 90°, γ = 120°
P
Rhombohedral
3-fold axis
a, b, c related by
three fold axis
a=b=c
α=β=γ
R
Tetragonal
4-fold axis
c parallel to 4-fold
a=b
α = β = γ = 90°
P,I
Cubic
4 3-fold axes
a, b, c related by
three fold axis
a=b=c
α = β = γ = 90°
P,I,F
Wednesday, 18 March 15
35
The Laue symmetry - The symmetry of the
diffraction pattern
To go further, we need to consider not just the unit cell dimensions, and the Bravais lattice, but
the actual symmetry of the diffraction pattern. The symmetry of the diffraction pattern
is just the rotational symmetry of the Space group, plus inversion, and is termed
the Laue symmetry. Here’s an example, for Space group P2.
Here is space group P2 …
The rotational symmetry is just a
2-fold axis
(formally: the cyclic point group 2)
And the symmetry of the
diffraction pattern is 2/m …
2
From International Tables for X-ray Crystallography.
From Fundamentals of Crystals,Vainshtein.
Even-fold axes in the space group generate mirror planes in the diffraction data, and it is their
presence or absence that allows us to discriminate many of the space groups from their diffraction
patterns
Wednesday, 18 March 15
36
The Laue symmetry - The symmetry of the
diffraction pattern
Here’s a reminder of how coupling inversion with two-fold rotation
generates a mirror plane, in case you’re not getting it …
From Crystal Structure Analysis, Glusker and TrueBlood.
Wednesday, 18 March 15
37
The Laue symmetry - The symmetry of the
diffraction pattern
Here’s another example
The rotational symmetry is just a
Here is space group P61 …
6-fold axis
(formally: the cyclic point group 6)
From International Tables for X-ray Crystallography.
Wednesday, 18 March 15
And the symmetry of the
diffraction pattern is 6/m …
From Fundamentals of Crystals,Vainshtein.
38
The Laue symmetry - The symmetry of the
diffraction pattern
And a final example
Here is spacegroup P6122…
From International Tables for X-ray Crystallography.
Wednesday, 18 March 15
The associated rotational
symmetry is the dihedral
point group 622
And the symmetry of the
diffraction pattern is 6/mmm
…
From Fundamentals of Crystals,Vainshtein.
39
Here are the Laue groups which describe the point group
symmetry of X-ray diffraction patterns
Crystal point group 4
↓
Crystal point group 422
↓
Crystal point group 1
↓
Crystal point group 222
↓
Crystal point group 6
↓
Crystal point group 2
↓
Crystal point group 3
↓
Crystal point group 32
↓
Crystal point group 23
↓
Crystal point group 622
↓
Crystal point group 432
↓
Adapted from Fundamentals of Crystals, Vainshtein.
Wednesday, 18 March 15
40
The Laue symmetry defines the asymmetric unit in
reciprocal space.
Crystal point group 222
↓
Crystal point group 3
↓
Crystal point group 32
↓
Laue
Symmetry
Asymmetric
unit
Adapted from Fundamentals of Crystals, Vainshtein and Biomolecular Crystallography, Rupp
Wednesday, 18 March 15
41
The assignment of symmetry
Here’s an heuristic example of how we can use symmetry in the diffraction pattern of a protein
crystal, to work backwards and assign the space group
This image was generated by Precession
photography, an old technique which gives a
very straightforward picture of the reciprocal
lattice. We can now easily generate and
inspect similar plots using a computer.
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
42
The assignment of symmetry
2. Figure out which of the
Bravais lattices & crystal
systems we appear to be
dealing with
This diffraction pattern can be indexed on a
primitive hexagonal lattice. That already
tells us this crystal probably belongs to the
trigonal/hexagonal crystal system.
This is the hk0 plane of the reciprocal lattice (we
are looking straight down the unique axis c*)
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
43
The assignment of symmetry
3. Analyze the symmetry
present in the diffraction
pattern - deduce possible space
groups
Now we need to examine the symmetry
present in the diffraction pattern:
We can see clear six fold rotational
symmetry. That means we are dealing with
either Laue group 6/m or 6/mmm
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
44
The assignment of symmetry
3. Analyze the symmetry
present in the diffraction
pattern - deduce possible space
groups
We can see the tell-tale mirror planes bisecting
the diffraction pattern
The Laue group is therefore 6/mmm.
The crystal point group is therefore 622.
That narrows the space group possibilities to six
P622 (#177), P6122 (#178), P6222 (#180),
P6322 (#182), P6422 (#181), P6522 (#179)
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
45
The assignment of symmetry
3. Analyze the symmetry
present in the diffraction
pattern - deduce possible space
groups
These days we don’t generate precession
photographs but we can use statistics measuring
the agreement between symmetry-related
observations to accomplish the same job.
Following indexing we would know that this
crystal likely belonged to the trigonal/hexagonal
crystal system.
We could then merge the data 4 times, each
time assuming one of the Laue symmetries
consistent with this crystal system
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
46
The assignment of symmetry
3. Analyze the symmetry
present in the diffraction
pattern - deduce possible space
groups
If an assumed symmetry isn’t present you will be
merging intensities which aren’t truly equivalent
and your agreement statistics will start to look
very much worse.
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
47
The assignment of symmetry
4. Narrow the list of space
group possibilities by
consideration of systematically
absent observations
However we do it, identifying the correct Laue
symmetry for this crystal (6/mmm) narrows the
space group possibilities to six:
P622 (#177), P6122 (#178), P6222 (#180), P6322
(#182), P6422 (#181), P6522 (#179)
Consideration of the systematic absences
would narrow the possibilities further, leading to
assignment of the space group as P6122 or
P6522.
Wei et al (1979) J Biol Chem,254, 4892-4894
Wednesday, 18 March 15
48
Systematic absences
Screw rotation axes generate systematically absent reflections
(structure factor amplitudes always zero) along certain lines in the reciprocal
lattice. This is very important for space group determination, since it’s the only
way to discriminate between many space groups.
For example
the four Orthorhombic space groups:
P222, P212121, P21212 and P2221
which all have mmm Laue symmetry
and the six Hexagonal space groups:
P622, P6122, P6222, P6322, P6422, P6522
which all have 6/mmm Laue symmetry
Let’s illustrate the meaning of systematic absences with an example …
Wednesday, 18 March 15
49
Systematic absences
00l reciprocal lattice line
Here’s an oscillation image from
crystals of a protein called nmrA.
The crystal space group is P6122
Cell dimensions:
a= 82 Å b=82 Å c= 363 Å
α = 90° β= 90° γ= 120°
Because of the 61 screw axis we
the 00l reflections will be absent if
l is not a multiple of six
And that’s what we see
Nichols et al (2001) Acta Cryst D57, 1722-1725
Wednesday, 18 March 15
50
Crystallographic screw-rotations and
systematic absences.
Screw-rotation
Class of reflection
affected
Reflections are present
only if
21axis in direction of a
h00
h even
21axis in direction of b
0k0
k even
21axis in direction of c
00l
l even
31or 32 axis in direction of c
00l
l = 3n
41or 43 axis in direction of c
00l
l = 4n
42 axis in direction of c
00l
l = 2n
61or 65 axis in direction of c
00l
l = 6n
62or 64 axis in direction of c
00l
l = 3n
63 axis in direction of c
00l
l = 2n
Wednesday, 18 March 15
51
Systematic
absences, another
example ...
From McPherson (2003)
Wednesday, 18 March 15
52
Centered cells and systematic absences.
Finally ... note that choice of a centered cell also leads to systematically absent observations. This
additional complexity is one of the “costs” of selecting a centered unit cell. These absences aren’t useful
for space group discrimination per se, but you should be aware of this.
Cell type
Class of reflection
affected
Condition for
presence
Primitive
hkl
None
Body-centered (I)
hkl
h+k+l = n
(where n is an even number)
Centered on the C face (C)
hkl
h+k = n
(where n is an even number)
Centered on all faces (F)
hkl
h,k,l all = n
(where n is an even number)
or
h,k,l all = n
(where n is an odd number)
i.e. h,k,l are permutable
Wednesday, 18 March 15
53
A final note on the assignment of symmetry
As you may have already noticed, some space groups cannot be
discriminated on the basis of the diffraction pattern. That is, the
Laue symmetry and the systematic absences are identical, yet the underlying
space group is different.
Examples …
P61 and P65.
P62 and P64
etc etc
These space group pairs differ in the hand of their screw axis, and can only be
discriminated during the process of phase determination (Lecture 6). In practice
you have to do parallel calculations in both space groups, until the correct
answer emerges.
Wednesday, 18 March 15
54
Scaling and merging
Determining the symmetry isn’t the only problem to be faced.
Estimates of the intensities on individual images will not all be on the same scale, so we have to take
care of that in data processing
.
Here’s a couple of physical reasons why the scale varies (we will not be exhaustive) …
1. The incident beam intensity may change with time. At a synchrotron this is certain to happen …
Wednesday, 18 March 15
55
Scaling and merging
2. Absorption by the crystal and surrounding material may attenuate the primary and secondary
beam. For a crystal of very unequal dimensions, absorption can vary greatly from reflection to
reflection. Additional the loop and surrounding vitrified solvent can also absorb significantly attenuating
thepapers
X-ray scattering in some directions.
research
Figure 3
Composite
of the mounted
sample
in orientations
corresponding
the three reference
positions
to Cryst
assess(2008)
the effect
of pp.
sample
orientation
Lealimages
et al. Absorption
correction
based
on a three-dimensional
modeltoreconstruction
from visual
images.used
J Appl
vol. 41
729-737
on the absorption correction. Schematics of the side-on view are also shown for clarity. (a) Starting position, batch 1. (b) Mounted sample perpendicular
to the detector, batch 40. (c) Mounted sample parallel to the detector, batch 130.
is special:about
here there
is little
distinguish
(corresponding
image batches
40 and
aroundWeOrientation
Scalingdatatakes
care ofto these,
andaround
other
issues.
will notCworry
how
wetoscale.
Just note that
between the four scaling protocols.
where the scale factors show higher deviation from unity.
every230)
image
will
usually
have
several
scale
factors
associated with it. Some scaling packages will also
In order to compare the performance of this method with an
empirical
correction
(as implemented
SCALA),
the effecexplicitly
model
absorption
by inthe
crystal.
tiveness of the algorithm was tested over the same three zones
of the mounted sample. The tests used increasing amounts of
data
(in terms
of number of image batches). The results (Fig. 7)
Wednesday, 18
March
15
5. Discussion
The final three-dimensional model of this crystal is of good
56
Scaling and merging:
The usual measure of data agreement.
Once we have put all the data on a common scale we must merge the
redundant (symmetry-equivalent) observations to arrive at the final output. The
.
usual statistic that’s calculated is Rmerge (sometimes also called Rsymm)
In words, this is nothing more than the sum of the differences of all
measurements from the mean value of the measurement, divided by the sum of
all measurements. Or in the form of an equation
∑ ∑
Rmerge =
hkl i
I i (hkl) − I (hkl)
∑ ∑ I i (hkl)
hkl i
€
Wednesday, 18 March 15
57
Agreement statistics and resolution
Because agreement statistics are resolution-dependent, in addition to overall
statistics we usually report agreement as a function of either
.
|S| = 2sinθ/λ (Å-1)
or
1/|S| = λ/2sinθ (Å) ... this is the resolution
In the last lecture we discussed how these quantities could be readily calculated
from the Miller indices h,k,l and the lattice parameters (see lecture 4, slide 25)
Wednesday, 18 March 15
58
Scaling and merging:
The usual measure of data agreement.
Rmerge increases at higher scattering angles (high resolution) as the data becomes
weaker and less well measured. Like this …
.
|s| (Å-1)
1/|s| (Å) = resolution
Adapted From Blow (2002).
Wednesday, 18 March 15
59
Scaling and merging:
The usual measure of data agreement.
•A low overall Rmerge is an indication of good quality data.
•However, a major problem with Rmerge as a quality indicator is that it is inherently
dependent on the redundancy of the data.
•The more often a given reflection is observed, the higher the Rmerge will be, even
though by simple statistical reasoning the average value of the measurements
becomes more precise.
This is, a priori, flawed
Wednesday, 18 March 15
60
Scaling and merging:
A better measure of data agreement.
The problems with Rmerge can be remedied by including a corrective term for the
multiplicity,
N, of the measurement. This statistic is known as Rmeasure
.
∑
Rmeasure = hkl
N
∑ I i (hkl) − I (hkl)
N −1 i
∑ ∑ I i (hkl)
hkl i
Diederichs and Karplus. Improved R-factors for diffraction data analysis in macromolecular crystallography. Nat Struct Biol (1997) vol. 4 (4) pp. 269-75
€
Wednesday, 18 March 15
61
Basic characterization of a novel X-ray data set
Once you’ve collected a novel X-ray diffraction data set, there’s a few things you
can (and should) do, even without phases ...
1.Have a look at the crystal packing density, and estimate the probable number
of molecules in the asymmetric unit.
2. If there’s more than one molecule in the asymmetric unit, try and learn
something about the non-crystallographic symmetry.
Wednesday, 18 March 15
62
The Matthews coefficient and protein solvent
content
The standard way of describing protein crystal packing density, and by
derivation, the solvent content, was introduced by Brian Matthews in 1968 (J.
Mol. Biol. 33, 491-497).
The Matthews coefficient (Vm ) is given by
Volume of the unit cell
Vm =
No of molecules in the unit cell x Molecular weight
Wednesday, 18 March 15
63
The Matthews coefficient and protein solvent
content
Volume of the unit cell
Vm =
No of molecules in the unit cell x Molecular weight
3/Da.
The
mean
V
for
protein
crystals
is
about
2.7
Å
•
m
•To obtain an estimate for the %protein and %solvent in the cell,
we need additional information … the density of the protein. On
average this is 1.35 g/cm3.
•Given this, it’s easy to show that the %protein in the unit cell is
123 / Vm ( and the %solvent is, of course, 100 - %protein).
•Hence the average protein crystal is 123 / 2.7 = 45% protein and
1 - 45 = 55%Solvent (!!).
Wednesday, 18 March 15
64
The Matthews coefficient and protein solvent
content
Frequency distribution of values observed
for VM. Data taken from Matthews (1968)
and from 10,471 non-redundant protein
crystal forms from the November 2002
release of the Protein Data Bank.
Kantardjieff and Rupp (2003) Prot. Sci 12,1865-1871
The principal utility of the Matthews coefficient is that it allows us to estimate the
number of molecules there are likely to be in the unit cell of a uncharacterized
crystal, and hence in the asymmetric unit.
Wednesday, 18 March 15
65
Characterization of non-crystallographic
symmetry
If there is more than one copy of a molecule in the asymmetric unit, and these molecules have the
same basic conformation, then there will be some non-crystallographic symmetry operations
which relate them. We can often detect the rotation which relates them using the self-rotation
function. We will not discuss this in any detail, but note that no phases are required - it can be
calculated from the measured intensities alone.
Here’s a very clear example for a heptameric molecule - part of
the 20S proteosome - that has a 7-fold rotational symmetry. In this
case there were 7 molecules in the asymmetric unit of the crystal.
But in what direction is the 7-fold rotation axis pointing with
respect to the unit cell axes ?
A contour map of the self-rotation function is shown, for all
possible directions of the 7-fold rotation axis.
The “correct”
orientation for the axis (i.e. the orientation observed in the
crystal), corresponds to the peak in the self-rotation function.
Johnston et al. The proteasome 11S regulator subunit REG alpha (PA28 alpha) is a heptamer. Protein Sci. (1997) vol. 6 (11) pp. 2469-73
Wednesday, 18 March 15
66