Michael Sawaya
ACA Meeting
Thursday, July 27,
2006, 4:35 PM
Honolulu/Kahuku
Diffraction Anisotropy
diffraction strength differs with cell direction
mean |F| vs.
resolution
c*
a*
b*
c*
c*
a*
b*
ANISOTROPIC
a*
b*
c*
ISOTROPIC
c*
c*
b*
a*
Diffraction anisotropy arises when then number of
lattice contacts is less in one cell direction than another
Myohemerythrin (PDB ID 2MHR) crystal packing viewed from two orthogonal directions
The crystal diffracts to 1.3 Å along b*, 1.7 Å along a* and c*
Sheriff & Hendrickson, (1987) Description of overall Anisotropy in Diffraction from Macromolecular Crystals. Acta A43, 118-121.
View perpendicular to b
View parallel to b
Diffraction anisotropy presents two major
problems to crystallographers
Problem 1: choice of resolution
boundaries of the data set
•Clearly, one would like to chose an
ellipsoidal boundary for anisotropic data.
•a) Concentric ellipsoids more accurately
describe the intensity contours of anisotropic
data sets than do concentric spheres.
•b) Reflection bounded by ellipsoidal shells
will have the similar I/s and Rsym.
•Currently available programs provide only
spherical shells for selecting a resolution
cutoff and reporting diffraction statistics. (An
•Anisotropic data is best contoured
using ellipsoidal shells
anisotropic B is allowed in Scala, but is not recommended
because parameters for this option are likely to be poorly
determined.)
The inadequacy of spherical shells in
reporting diffraction statistics
Problem 1 (continued)
•1) Anisotropic data quality varies not only
with resolution, but also with direction.
•2) Within a spherical shell, data quality
(I/s, Rsym) will be highly varied depending on
direction. For example…
3) If one wishes to keep the strong data at
high resolution, one is forced to accept the
weak, poorly measured data bounded by
the same spherical shell.
Accept bad I/s, Rsym in high shell
Must justify bad stats to peers
•Same data set as previous slide,
but bordered by spherical shells
•4) If you discard the high resolution data,
you discard the details of the electron
density map. Which will it be?
Problem 2: The need for an anisotropic
scale factor for comparing Fcalc & Fobs
c*
 Refinement of a structure
c*
Areas of
poor
agreement
b*
b*
against anisotropic data will
stall at a high R-factor
the agreement between
Fobs and Fcalc will be very
poor
|Fobs| has a directional
dependence and |Fcalc|
does not;
 An anisotropic scale factor
must be applied to either
|Fobs| or |Fcalc| to make them
comparable.
|Fobs |
| Fcalc |
plane h=0
plane h=0
 Anisotropic diffraction is not
modeled by TLS disorder
parameters nor individual
isotropic B-factors.
How the anisotropic scale factor
works.
•
Scale factor
B=12 Å2
Same for all
lattice
directions
(a*,b*,c*)
•
resolution
•
•
•
b11= +2
along a*
Å2
Scale factor
Scale factor
Scale factor
resolution
resolution
b22= -5
along b*
Å2
•
resolution
b33= +12
along c*
Å2
An anisotropic scale factor is a
multiplicative factor like the
overall B-factor.
Like the overall B-factor, its value
varies with resolution.
But, unlike the overall B-factor,
its value also varies with
direction.
It has three principle
components, b11, b22, and b33
acting as B-factors along a*,b*,c*
directions, respectively.
An anisotropic data set can be
made isotropic by applying the
appropriate scale factor that
increases |F| in weak diffracting
direction or decrease |F| in the
strong diffracting direction or a
combination of both.
“B”s can be positive or negative.
Anisotropic Scale Factor
Anisotropic tensor
[ ]
b11 b12 b13
b12 b22 b23
b13 b23 b33
The anisotropic scale factor components are obtained
from a least-squares fit of the elements of an
anisotropic tensor to Fobs.
S(|Fobs|-k|Fcalc|)2 → min
k=e-
The value of k changes in the form of concentric elliptical shells
from the center of the reciprocal lattice. The parameters b11
b22 and b33 correspond to the principal axes of the ellipse.
Anisotropic scaling is increasingly employed in
crystallography.
–
Molecular replacement
•
c*
–
b*
•
Phaser
(MR_ANISO keyword)
Refinement
•
•
a*
(b11a*2h2+2b12a*b*hk+2b12a*c*hl+b22b*2k2+2b23b*c*kl+b33c*2l2)
Refmac
CNS
Anisotropic scaling dramatically improves R-factors
(see The Effect of Overall Anisotropic Scaling in
Macromolecular Refinement. Murshudov, Davies, Isupov,
Krzywda and Dodson CCP4 Newsletter on Protein Crystallography
Number 35. July 1998)
–
,
But, a shortcoming in its formulation was newly
revealed by the severe degree of anisotropy in our
data set… and refinement was stalled.
Crystal structure of a PE-PPE protein
complex from M. tuberculosis.
• PE and PPE are 2 families
named for the conserved
proline (P) and glutamate
residues (E) near the Ntermini.
• Large families
– 100 PE members
– 60 PPE members
• Precise function not known
– Associated with cell wall
– Linked to virulence
– Immune evasion by antigenic
variation?
• Prevalent in M.tb. and absent
in humans
– Drug target
Domain organization of
the PE and PPE proteins
as reported in Nature
393:537-44. (1998)
PE-PPE project
• Michael Strong
– Characterization of the complex
- 28 different individual proteins tested –
insoluble.
- A complex of Rv2430c and Rv2431c guided
by bioinformatics
– Purification
– Crystallization and Structure Solution
PE-PPE Crystal parameters
• Crystals are plates
• Selenomethionine
derivative for MAD
• Long, rod shaped unit cell
a=40.8
b=46.7
c=283.1
Rebecca Page Screen
14% iso-Propanol
0.07M Sodium Acetate trihydrate pH 4.6
0.14M Calcium Chloride dehydrate
30% Glycerol anhydrous
Two complexes/asu
•Space group P2221
–Fairly rare in PDB (0.03%)
•Solvent content 42%
PE-PPE crystals diffract anisotropically
mean |F| vs.
resolution

ALS beamline 8.2.2
c -strong
a- medium
b-weakest
a*
b*
c*
Data used for phasing and
refinement
Data Collection
Se used for
refinement
Se (peak)
Se
(inflection)
Se (remote)
Wavelength (Å)
1.0000
0.9796
0.9794
0.9719
Resolution limit (Å)
2.2
2.4
2.4
2.4
Rsym(%) (last shell)
16.8 (34.2)
10.8(35.6)
9.8(43.8)
10.5(42.6)
I/s (last shell)
2.0
2.0
1.8
1.5
Total observations
233,050
120,481
58,582
112,996
Unique reflections
27,342
18,561
17,315
18,154
Completeness (%)
92.7 (78.6)
80.8(46.3)
74.7(35.5)
78.3(41.9)
• Using standard spherical bins of resolution
R-sym
Data statistics for best data set
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
R-linear
R-squared
4.7 3.8 3.3 3.0 2.8 2.6 2.5 2.4 2.3 2.2
16
14
I/sigma
12
I/s
10
In highest resolution shell
8
a*
|F|/s 6.0
6
4
2
0
4.7 3.8 3.3 3.0 2.8 2.6 2.5 2.4 2.3 2.2
Resolution (Å)
b*
1.8
c*
15.0
Phasing Statistics
Phase Determinationb
Se (peak)
Se
(inflection)
Se (remote)
Rcullisc (%, 20-2.4 Å, acentric/centric, isomorphous)
0.97/0.95
-
0.92/0.85
Rcullisd (%, 20-2.4 Å, anomalous)
0.76
0.94
0.85
Phasing powere(20-2.6 Å, acentric/centric)
0.50/0.35
-
0.72/0.55
Number of sites
Mean overall figure of merit (before/after DM)
• Just adequate
11
0.39/0.60
2.4 Å experimental electron density map
• Connectivity good enough to
see the helical fold.
• Side chain density is weak or
non existent.
• Use Se sites as reliable markers
for sequence registration
• Go forward with
refinement…maps should
improve.
PE protein
PPE protein
PPE motif
Refinement yields only marginal
improvement in electron density map
•
•
Side chain density is still missing
Refinement stuck
–
–
–
–
•
Check for twinning
–
•
Refinement in P21 or P1 yielded no
improvement in R factors
Use TLS
–
•
Twinning not indicated
Check for pseudosymmetry
–
•
Rwork=38.5%
Rfree=43.4%
No apparent way to improve the
coordinates/R-factors.
No new features apparent in
electron density map.
Unstable, R-factor shot up.
Use 3.0 Å cutoff
–
R-factors improved, but map does
not improve.
Experimental
2.4 Å
2Fo-Fc
2.2 Å
Looking to the literature for help
Science, vol 300, pp. 1256-1262
•
Lodowski et al. Supplemental
methods,
–
•
Zhang et al.
–
–
–
•
“Because the diffraction pattern
exhibited severe anisotropy, a 3-D
ellipsoid was defined and
merging R-factors and I/s were
calculated in ellipsoidal shells.
Diffraction data were then
limited to the outermost shell
that still contained significant
data…”
“Data observed to 2.5A resolution
in the c* direction, but to only 3.3 A
in the plane perpendicular to c*.”
An ellipsoid of diffraction data,
rather than the usual sphere, was
used for scaling and refinement.
Refers to Lodowski et al. for
method.
Let’s do the same
Acta D, vol 60, pp. 1512-1518
Solution proposed by literature
Equation of an ellipsoid
1=x2/a2 + y2/b2 + z2/c2
Where a, b, and c are the vertices of
the ellipse.
Set the following:
a= 1/resolution limit along a*=1/2.2Å
b=1/resolution limit along b*=1/3.2Å
c=1/resolution limit along c*=1/2.2Å
Resolution limits determined by the
point were mean |F|/s drops below 2
for the given axis. See truncate
output.
To test whether a given reflection falls
within the ellipsoid, calculate:
x=component of d* along a*
y=component of d* along b*
z=component of d* along c*
Plug a,b,c,x,y,z into equation above.
Where the sum>1, discard reflection.
Reflections before truncation 27,293
Reflections after truncation 20,053
Elliptical truncation produced a sharp drop
in R-factors but no improvement in map.
•
Elliptical truncation yielded a
–
–
•
Details:
–
–
•
•
TLS refinement is now stable, so it
also contributes to improvement in
R-factors.
Most of the drop is in the high
resolution shells 3.0-2.2Å, where
much of the poorly measured data
was discarded.
2Fo-Fc maps are still not improved.
–
–
•
Rwork= 38.5% →32.5%
Rfree = 43.4% →36.2%
Side chain density is still blobby as if
only 3.5A resolution.
No new features. Can’t improve
model! Panic!!
Clue: Average B of model
coordinates =75 Å2. An effect
artificially produced by the
anisotropic scale factor.
Big drop
R-work
•
6.0% drop in Rwork
7.2% drop in free Rfree
Before truncation
After truncation
Adverse Side Effect of Anisotropic Scaling
•
•
•
•
The effect of anisotropic
scaling was observed by
plotting the scaled |Fobs| as
a pseudo precession
photograph; appearance
was compared before and
after scaling.
The adverse side effect of
anisotropic scaling is to
diminish the amplitude of
well measured, high
resolution reflections in the
a*c* plane.
These reflections contribute
almost nothing to the
electron density because
anisotropic scaling
diminished their
amplitudes.
The diminished contribution
of these high resolution
|Fobs| to the Fourier
synthesis results in a map
that appears to be low
resolution.
Fobs
Fobs
b*
after scaling
b*
c*
c*
Isotropic, but high resolution
|F|obs near c* are diminished
Why anisotropic scaling might
diminish high resolution |Fobs|
•
•
•
Imagine an ellipsoidal shell (Red ellipsoid)
encapsulating all reflections in the data set
were |Fobs|>2.
The goal of anisotropic scaling is to
transform the ellipsoid into a sphere (Blue
sphere) by scaling |Fobs| by the “appropriate
amounts” in the three principal directions.
The “appropriate amounts” may be derived
from 3 different approaches:
–
–
–
•
•
•
decrease |Fobs| in the strong diffracting
directions (SBij ≥0)
Increase |Fobs| in the weak diffracting
directions (SBij ≤0)
A combination of both of the above (SBij =0).
SBij ≥0
B11 (Å2): +
B22 (Å2): 0
B33 (Å2): +++
SBij =0
++
The choice of approach appears arbitrary;
the results differ only by an isotropic B-factor
(i.e. the radii of the blue spheres).
REFMAC encodes the last option,
constraining the amplitude gains in the weak
diffracting direction to be equal to the
amplitude decreased in the strong diffracting
direction.
Mathematically, this is equivalent to
constraining the sum of the principle
components of the anisotropic scale factor
to be zero.
–
i.e. B11+B22+B33=0.0
REFMAC
SBij ≤0
--0
Constraining SBij≤0 improved map, model
building resumed.
•
•
•
•
It seemed important to maintain
the contribution of the well
measured, high resolution
reflections in the a*c* plane so
that they may contribute to the
electron density map and reveal
new details.
Of the three approaches, this
effect can be best achieved by the
constraint SBij≤0.
In practice, the SBij≤0 constraint
was achieved by first applying the
REFMAC derived anisotropic
scale factor to |Fobs|, followed by a
negative isotropic B-factor (-10Å2).
The anisotropically scaled |Fobs|
was used as input for REFMAC
refinement.
SBij ≤0
SBij ≥0
SBij =0
B11 (Å2): +
B22 (Å2): 0
B33 (Å2): +++
++
x x
REFMAC
--0
2Fo-Fc maps showed a marked
improvement.
• 2Fo-Fc maps
began to reveal
carbonyl bumps,
side chain
density, and the
presence of 72
waters, where
previously we
could see none.
2Fo-Fc using Automatic
Anisotropic
Scaling
2Fo-Fc using Improved
Anisotropic
Scaling
Elliptical truncation produced a sharp drop
in R-factors but no improvement in map.
• Further model
building yielded a
Before truncation
– 7.7% drop in Rwork
– 4.9% drop in free Rfree
– Rwork= 38.5% →32.5% →24.8%
– Rfree = 43.4% →36.2% →31.3%
• R-factor dropped in
both high and low
resolution shells
R-work
• Details:
After truncation
After negative B-factor
correction and additional
refinement
R-factor improved throughout resolution range
Refinement statistics
• final
Model Refinement
Native
Rworkf( 20 Å-2.1 Å)
24.7
Rfreeg( 20 Å –2.1 Å)
31.2
PDB ID code
2G38
Number of residues
(protein/water)
499/73
Average B (main
chain/side chain)
20.7/22.3
Rmsd bonds (Å)
0.007
Rmsd angles (°)
1.4
B-values (Å2 bonded) 1.6
Origins of diffraction anisotropy
resemble those in myohemorythrin
Strong diffraction
Poor diffraction
Anisotropic scaling of other
proteins
Protein
Resolution in Resolution
strongest
in weakest
direction
direction
Rwork
Before & After
Correction
Rfree
Before & After
Correction
PE-PPE
2.2
3.2
38.5->24.7 ↓13.8%
43.4->31.2 ↓12.2%
Actin dimer
2.7
3.8
29.1->24.2 ↓4.9%
30.8->28.6 ↓2.2%
Tim8-13
2.6
3.3
30.8->26.5 ↓4.3%
35.3->30.6 ↓4.7%
•
•
•
The technique of applying anisotropic scaling with SBij≤0 has helped in the
refinement of structures of Actin dimer, and Tim8/13 complex.
The improvement in R-factors and electron density maps have been
more modest in these cases, as the anisotropy is less severe.
The technique appears to be most helpful when the best and worst
diffracting directions extend between 2.5 to 3.0, where water
molecules are discernable.
Procedures
•
Judge whether anisotropy is a problem.
–
–
•
If anisotropy is significant, determine the resolution limits along the three
principle cell directions.
–
•
For example B11= +6, B22= -14, B33= +9
use cad from CCP4
Apply a negative isotropic scale factor to the newly isotropic Fobs to restore the
magnitude of those reflections weakened by the previous step.
–
–
•
•
•
Perform a cycle of refinement with Refmac or CNS.
Note the anisotropic scale parameters (B11,B22,B33,etc.) listed in the PDB header
For example B11= -6, B22= +14, B33= -9
Apply the negated scale factors to Fobs to create an isotropic data set.
–
–
•
I’ll make my truncation program available from http:www.doe-mbi.ucla.edu/~sawaya.
Calculate the anisotropic scale parameters (for Fcalc).
–
–
–
•
Note where mean F/s drops below 2 along the three principle directions.
Truncate data using ellipsoidal limits.
–
•
Look at the anisotropy graph from truncate (loggraph truncate.log)
Does mean F/s drop with different slopes along the 3 principle directions?
Negate the most positive component from the previous step (e.g. +9 → -9).
Use cad again.
Use this scaled Fobs for refinement.
-all these steps are performed by the diffraction anisotropy server
http://www.doe-mbi.ucla.edu/~sawaya/anisoscale/
Acknowledgements
•
•
•
•
•
Michael Strong
Shuishu Wang
Duilio Cascio
Alex Lisker
David Eisenberg