Scaling, Phasing, Anomalous,
Density modification, Model
building, and Refinement
What we’ve learned so far
•Electrons scatter Xrays.
•Scattering is a Fourier transformation.
•Inverting the Fourier transform gives the image of the electron
density.
•Waves have amplitude and phase. And we can’t measure phase.
•Inverting the Fourier transform without the phases gives the Patterson
map, which is the map of all inter-atom vectors.
•Space groups are groups of symmetry operations in 3D.
•The Patterson plus symmetry gives us heavy atoms positions.
•Heavy atom positions plus amplitudes gives us phases.
What we can do.
•Sum waves.
•Calculate the phase given atom position and scattering vector.
•Index spots on an X-ray photograph.
•Draw Bragg planes.
•Invert the Fourier transform using Bragg planes.
•Calculate cell dimensions from an X-ray photograph.
•Describe the symmetry of a crystal or periodic pattern.
•Convert a simple Patterson to heavy atom positions.
•Calculate heavy atom vectors given heavy atom positions.
•Solve for phases given amplitudes and heavy atom vectors.
From data to model
Collect native data: Fp
Collect heavy atom data: Fph
Estimate phases
no
Calculate r
Is the map traceable?
Trace the map
Refine
yes
density
modification?
From data to phases
native data: Fp
heavy atom data: Fph
Calculate difference Patterson
Find heavy atom peaks on Harker sections
Solve for heavy atom positions using symmetry
Calculate heavy atom vectors
Estimate phases
From data to Patterson map
native data: Fp
heavy atom data: Fph
Find the best scale factor, k
Calculate Fdiff = k*|Fph| – |Fp|
Calculate difference Patterson
From crystal to data
Indexed film
Internal scaling
Intensity, Ip(hkl) = F2
I is relative
Bigger crystal, higher I
Better crystal, higher I
Longer exposure, higher I
More intense Xrays, higher I
native data: Fp
Because there is no absolute scale:
Fp and Fph are on different scales
What happens to the phase
calculation if the scaling is off?
Radii are Fp and
k*Fph
Scaling two datasets
hkl
1
2
3
4
0
0
0
0
...
Fp
s
0 3233.
0 2028.
0 2179
0 ....
hkl
100.
98.
88.
1
2
3
4
0
0
0
0
Fph
0 1122.
0 1014.
0 1081.
0 ....
s
50.
49.
44.
...
1st approximation: The intrinsic average amplitude of scattering is
constant for different crystals. A simple scale factor k corrects for crystal
differences:
<Fp> = <Fph>*k, therefore: k = <Fp>/<Fph>
Basic assumption
…when scaling two crystals.
The total number of electrons in the unit cell
is the same for each (isomorphous) crystal.
Note: isomorphous means “same space group, same cell
dimensions.”
Better scaling: Wilson B-factor
Low-resolution features
(ignored in slope calc)
water peak (ignored in slope calc)
Region of linear
dependence of
amplitude with
resolution. Slope =W=
“Wilson B-factor”
<I>
=<F2>
Averaged in
resolution
bins
Scaled separately
X-axis=Two sine-theta over lambda = 1/d
d = 20Å 5Å
3Å
2Å
Two sets of F’s might have different overall B-factors, because the
crystals may have different degrees of mosaicity
So Wilson scaling is better than simple scaling.
<Fp> = <Fph>*k,
k=W(1/d) + C
How good is scaling?
After solving the structure, we can go back and see how good
the scaling was. Typically, error in scaling < 10%. In best cases
< 2%.
Scaling error is worse if:
(1) crystals are non-isomorphous
(2) too many heavy atoms present (“basic assumption” is
wrong).
Heavy atom difference Fourier
Fph = Fp + Fh (vector addition)
•The amplitude of Fh is only approximately Fph-Fp.
•The true difference |Fph - Fp| depends on the phase of Fh
relative to Fp
Centrosymmetric reflections
•If the crystal has centrosymmetric symmetry, all reflections are
centrosymmetric. Phase = 0° or 180°
•If the crystal has 2-fold, 4-fold or 6-fold rotational symmetry,
then the reflections in the 0-plane are centrosymmetric.
(Because the projection of the density is centrosymmetric)
For centrosymmetric reflections:
|Fph| = |Fp| ± |Fh|
This means the amplitude |Fh| is exact*
for centrosymmetric reflections.
*assuming perfect scaling.
0-plane
R
R
Draw any set of
Bragg planes parallel
to the 2-fold. The
projected density is
centrosymmetric.
Therefore, phase is 0 or 180°.
Initial phases
The most probable phase is
not necessarily the “best” for
computing the first e-density
map.
Shaded regions are possible Fp and
Fph solutions.
weighted
average,
best phase
Figure of merit
Figure of merit “m” is a measure of how good the phases are.
C is the “center of mass” of a
ring of phase probabilities (that’s
the “mass”). The radius of the
ring is 1. So m=1 only if the
probabilities are sharply
distributed. If they are distributed
widely, m is small.
Fbest(hkl) = F(hkl)*m*e-iabest
In class exercise: phase error
FP=5.00
s=0.5
FPH1=5.50
s=0.8 FH1=2.23
aH1=-63.4°
FPH2=4.50
s=0.9 FH2=0.50
aH2=-164°
(1) Draw three circles separated by vectors FH1 and FH2.
(2) Draw circular “error bars” of width 2s.
(3) Draw circle plot of Fp phase probabilities.
(4) Estimate the centroid c of probabilit.
(5) What is the Figure of Merit, m?
Anomalous dispersion
Inner electrons
scatter with a time
delay. This is a
phase shift that is
always counterclockwise relative
to the phase of the
free electrons.
bound electrons
Heavy atom
free electrons
Anomalous dispersion
Anomalous dispersion
SIR = single isomorphous replacement
Advantages: only one derivative crystal
is needed. (fewer scaling problems)
Anomalous dispersion has a greater effect at
higher resolution. (because the inner
electrons are more like a point source)
Is the initial map good enough?
(1) The map is calculated using
abest.
(2) The map is contoured and
displayed using {InsightII,
MIDAS, XtalView, FRODO,
O, ...}
(3) A “trace” is attempted.
Model building
e- density cages (1 s contours) displayed using InsightII
Information used to build the first
model:
Sequence and Stereochemistry
...plus assorted disulfide and ligand information.
Models are built initially by identifying characteristic sidechains (by their
shape) then tracing forward and backward along the backbone density
until all amino acids are in place.
Alpha-carbons can be placed by hand, and numbered, then an automated
program will add the other atoms (MaxSprout).
Class exercise:
Tracing an electron density map
sequence: AGDLLEHEIFGMPPAGGA
Can you locate the density above in the sequence?
R-factor: How good is the model?
Calculate Fcalc’s based on the model.
Compute R-factor
F
obs
R
h
h  Fcalc h
F
h
obs
h
Depending on the space group, an R-factor of ~55% would be
attainable by scaled random data.
The R-factor must be < ~50%.
Note: It is possible to get a high R-factor for a correct model.
What kind of mistake would do this?
What can you do if the phases are
not good enough?
1. Collect more heavy atom derivative data
2. Try density modification techniques.
initial
phases
Fo’s and
(new) phases
Density modification :
Map
Modified
map
Fc’s and
new phases
Density modification techniques
Solvent Flattening: Make the water part of the map flat.
(1) Draw envelope
around protein part
(2) Set solvent r to <r>
and back transform.
Solvent flattening
Requires that the protein part can be distinguished from
the solvent part. BC Wang’s method: Smooth the map
using a 10Å Guassian. Then take the top X% of the
map, where X is calculated from the crystal density.
Skeletonization
(1) Calculate map.
(2) Skeletonize it (draw ridge lines)
(3) Prune skeleton so that it is “protein-like”
(4) Back transform the skeleton to get new phases.
Protein-like means: (a) no cycles, (b) no islands
Non-crystallographic symmetry
If there are two molecules in the ASU, there is a matrix and
vector that rotate one to the other: Mr1 + v = r2
(1) Using Patterson Correlation Function, find M and v.
(2) Calculate initial map.
(3) Set r(r1) and r(r2) to (r(r1) + r(r2) )/2
(4) Back transform to get new phases.
R
R
What does a good map look like?
plexiglass
stack
brass parts
model
Before computers, maps were contoured on
stacked pieces of plexiglass. A “Richards
box” was used to build the model.
halfsilvered
mirror
Low-resolution
At 4-6Å resolution, alpha helices look like
sausages.
Medium resolution
~3Å data is good enough to se the backbone with
space inbetween.
The program BONES traces the density automatically,
if the phases are good.
BONES models need to be manually connected and sidechains attached.
MaxSPROUT converts a fully connected trace to an all-atom model.
Errors in the
phases make
some
connections
ambiguous.
Contouring at two density cutoffs sometimes helps
Holes in rings are a good thing
Seeing a hole in a tyrosine or phenylalanine ring is universally
accepted as proof of good phases. You need at least 2Å data.
Can you see in stereo?
Try this at home. In 3D, the density is much easier
to trace.
New rendering programs
“CONSCRIPT: A program for generating electron density
isosurfaces for presentation in protein crystallography.” M. C.
Lawrence, P. D. Bourke
Great map: holes in rings
Superior map: Atomicity
Rarely is the data this good. 2 holes in Trp. All atoms separated.
Only small molecule structures
are this good
Atoms are separated down to several contours. Proteins
are never this well-ordered. But this is what the density
really looks like.
Refinement
•The gradient* of the R-factor with respect to each atomic
position may be calculated.
•Each atom is moved down-hill along the gradient.
•“Restraints” may be imposed.
*
dR factor
dvi
What is a restraint?
A restraint is a function of the coordinates that is lowest when the coordinates
are “ideal”, and which increases as the coordinates become less ideal..
Stereochemical restraints
also...
planar
groups
B’s
bond lengths
bond angles
torsion angles
Calculated phases, observed
amplitudes = hybrid F's
•Fc’s are calculated from the atomic coordinates
•A new electron density map calculated from the Fc's would only
reproduce the model. (of course!)
•Instead we use the observed amplitudes |Fo|, and the model phases, ac.
Hybrid back transform:
r(r )   Fobs h e
  
 
i 2  h •r a cal c h
h
.
Hybrid maps show places where the current model is wrong and needs to be changed.
Difference map: Fo-Fc amplitudes
The Fo “native” map r(Fo) differs from the Fc map r(Fc) in
places where the model is wrong. So we take the difference. In
the difference map:
Missing atoms?
r(Fo-Fc) > 0.0
Wrongly placed atoms?
r(Fo-Fc) < 0.0
Correctly modeled atoms?
r(Fo-Fc) = 0.0
Q: Subtracting densities (real space) is the same as subtracting
amplitudes (reciprocal space) and transforming. T or F?
2Fo-Fc amplitudes
The Fo map plus the difference map is
Fo where the differences are zero (the atoms are correct)
Less than Fo where the model has wrong atoms.
Greater than Fo where the model is missing atoms.
Fo + (Fo-Fc) = 2Fo-Fc
The “free R-factor”: crossvalidation
The free R-factor is the test set residual, calculated the same as
the R-factor, but on the “test set”.
Free R-factor asks “how well does your model predict the data
it hasn’t seen?”
F
obs
R free  hT
h  k Fcalc h 
F
obs
h
hT
Note: the only difference is which hkl are used to calculate.
Why cross-validate?
If you have three points, you can fit them to a quadratic
equation (3 parameters) with zero residual, but is it right?
Observed data
calculated
R-factor = 0.000!!
Fitting and overfitting
Fit is correct if additional data, not used in fitting the
curve, fall on the curve.
Low residual in the “test set” justifies the fit.
residual≠0
cross-validation
=Measuring the residual on data (the “test set”) that were not
used to create the model.
The residual on test data is likely to be small if
data
parameters
is large.
a line has 2 parameters
parameters versus data
Example from Drenth, Ch 13:
Papain crystal structure has 25,000 reflections.
Papain has 2000 non-H atoms
times 4 parameters each (x, y, z, B)
equals 8000 parameters
data/parameters = 25,000/8000 ≈ 3 <-- this is too small!
restraints are “data”
Bond lengths, angles, etc. are “measurements”
that must be fit by the model. The true
“residual” should include deviations from ideal
bond lengths, angles, etc.
In practice, residual in restraints (e.g.
deviations from ideal bond lengths, angles) is
very low. This means that restraints are
essentially “constraints”.
van der Waals
planar groups
bond lengths
bond angles
torsion angles
constraints reduce the number of
parameters
Bond lengths, angles, and planar groups may
be fixed to their ideal values during refinement
(“Torsion angle refinement”).
bond lengths
Using constraints, Ser has 3 parameters, Phe 4,
and Arg 6.
bond angles
There are an average 3.5 torsion angles per
residue.
Papain has ~700 torsion angle parameters.
\ data/parameter=25,000/700≈35
planar groups
radius of convergence
total
residual
parameter space
...=How far away from the truth can it be, and still find the truth?
radius of convergence depends on data & method.
More data = fewer false (local) minima
Better method = one that can overcome local minima
Molecular dynamics w/ Xray
refinement
MD samples conformational space while maintaining good
geometry (low residual in restraints).
E = (residual of restraints) + (R-factor)
dE/dxi is calculated for each atom i, then we move i downhill.
Random vectors added, proportional to temperature T.
The simulated annealing MD method:
(1) start the simulation “hot”
(2) “cool” slowly, trapping structure in lowest minimum.
“X-plor” Axel Brünger et al
Phase bias, and how to fix it.
The model biases the phases.
The effect of phase bias is local to the errors.
To correct a part of the model, we must first remove
that part .
An “OMIT MAP” is calculated. The phases for an
omit map are derived from a partial model, where
some small part has been omitted.
Omit maps
This residue has
been removed
before calculating
Fc.
2Fo-Fc density =
Fo + (Fo - Fc) =
The native map
plus the difference
map.
Two inhibitor
peptides bound to
thrombin. The
inhibitors were
omited from the
Fc calculation.
(stereo images)
FÉTHIÈRE et al, Protein Science (1996), 5: 1174- 1183.
The final model
Mond ay, Janu ary 2 9,2 001
Stru cture Explor er - 1 AAJ
Pag e:1
Structure Explorer - 1AAJ
Summary Information
Summary Information
View Structure
Download/Disp lay File
Structural Neighbors
Geometry
Other Sources
Title:
Compound:
Authors:
Exp. Method:
Classification:
Source:
Primary Citation:
not available
Amicyanin (Apo Form)
R. C. E. Durle y, L. Che n, L. W. Lim, F. S . Math ews
X-ray Diffraction
Ele ctron Tran sport
Paracoccus Denitrifican s
Durley, R., Chen , L., Lim, L. W., Mathe ws, F. S ., Davidson, V. L.: Crystal structure
an alysis of amicyanin and apoamicyan in from Paracoccus denitrifican s at 2.0 A and 1.8 A
resolu tion.Protein Sci 2 pp. 739 (1993)
[ Me dline]
Sequence Details
Deposition Date: 09-Apr-1992
Explore
SearchLite SearchFields
Resolution [Å]: 1.80
Space Group: P 21
Unit Cell:
dim [Å]:
a
angles [¯]: alpha
Polymer Chains: 1AAJ
Atoms: 905
HET groups: HOH
Release Date: 31-Oct-1993
R-Value: 0.155
28.95
90.00
b
beta
56.54
96.38
c
gam ma
27.55
90.00
Residues: 105
© RCSB
Other data commonly reported: total unique reflections, completeness, free R-factor