1
Structural Biology
Methods
Fall 2024
Lecture #4
Phase problem
[0;0;0]
x
y
z
Multiple/Single Isomorphous Replacement (MIR/SIR)
• source of phases – intensity differences between data from native and
derivative (heavy atom containing) crystals
• Positions of heavy atoms identified from isomorphous difference
Patterson maps
Solving the phase problem by:
Multiple/Single-wavelength anomalous diffraction
(MAD/SAD)
• source of phases – intensity differences between structure factors due to the
presence of atom that specifically interacts with X-rays of a given wavelength
• Positions of heavy atoms identified from anomalous difference Patterson
maps
Solving the phase problem 3
Patterson function
Phase problem
Patterson map of a macromolecule is a mess!
Isomorphous replacement method
Isomorphous replacement method
Determination of heavy atom positions from centric
reflections (centro-symmetric projections)
Scale!
Determination of heavy atom positions from
acentric reflections
Determination of protein phase angles
The “lack of closure" error
Ideal:
Real life:
Phase probability
Anomalous scattering
The anomalous Patterson map
Molecular replacement
1. source of initial phases is a model
2. the model is oriented and positioned to obtain the best agreement with
the x-ray data
3. phases are calculated from the model
4. The calculated phases are combined with the experimental data
Solving the phase problem by:
Molecular Replacement was invented by
Michael Rossmann
28
Molecular Replacement
Known crystal structure New crystal structure
Given: • Crystal structure of a homologue
• New X-ray data
Determine: • The new crystal structure
29
MR Technique
Method: • 6-dimensional global optimisation
– one 6-d search for each molecule in the AU
>> split further to orientation + translation searches = 3 + 3?
Required: • Scoring
– the match between the data and (incomplete) model
– ideally: the highest score = the correct model
Search model
Known crystal structure New crystal structure
First, consider the model Patterson
We put the model in a large P1 box and calculate the Patterson
from the structure factors of the model in the P1 box.
ROTATION?
model in large P1 box
Patterson of
model in large
P1 box
“search” model
Clusters
separated by
P1 cell
dimensions
ROTATION FUNCTION
ROTATION FUNCTION
The Patterson of our unknown structure contains self-vectors and cross-vectors, but
because the cell was large, the self-vectors and cross vectors are well separated from
one another.
self vector
cross
vector
Just as we generated the Patterson for our model in the first orientation, we can
generate the Patterson for the model in any orientation in any sized box.
model in same large P1 box
in different orientation Patterson of
model in large P1
box
in different
orientation
ROTATION FUNCTION
When the models are in different orientations the Pattersons will not match one
another.
=X
ROTATION FUNCTION
However, when the second model is in the same orientation parts of the Pattersons
will match one another, and we can “solve” the rotation function for the model.
=
ROTATION FUNCTION
If the model were in a different sized box, the Patterson of the intramolecular
vectors, which are located in a sphere centred on the origin, can be overlaid. We can
cut out the peaks corresponding to the inter-molecular vectors from each Patterson
and just compare the central parts of the Pattersons.
``
``
ROTATION FUNCTION
=X
=
Now, the Pattersons of the intramolecular
vectors will match when
the model is in the correct
orientation.
ROTATION FUNCTION
Translation function
Observed
amplitudes
Fourier amplitudes
and phases
Real space cat
Fourier
transform
Circular rainbow scale of
phases
Linear intensity scale
of amplitude size
Fourier cat
Observed
amplitudes Phases
unknown!
Unknown structure,
unknown orientation
Known structure
Fourier catCat
Fourier
transform
Diffraction
experiment
Manx cat
Wrong orientation!
Calculated amplitudes
and phases
FT of Manx cat
Observed
amplitudes Phases
unknown!
Known structure
Fourier cat
Fourier transform,
try different orientations
Manx cat
Wrong orientation!
Calculated amplitudes
and phases
FT of Manx cat
Observed amplitudes (tailed cat),
calculated phases (Manx cat)
Even the tail becomes visible!
Inverted
Fourier
transform
Duck amplitudes
+ cat phases
Duck Fourier transform
of duck
Looks like a cat!!
Model Bias
Fourier
transform
Inverted
Fourier
transform
Model building & resolution
Phase improvement
When to use:
Refinement of the Model Structure
Parameter to measurement ratio – x, y, z, B (anisotropic B-> 8 parameters)
- other data than |Fobs|: stereochemical data (bond lengths and angles)
solvent flattening
NCS
Constrains x Restrains
R-factor, Rfreefactor
R-factor Rfree factor
Ramachandran plot
Ψ
φ
φ
Ψ
ω
The Ramachandran Plot
Geometry and stereochemistry
Bond lengths Dihedral angles
Quality of diffraction data