Transcript The `phase problem` in X

The ‘phase problem’ in X-ray
crystallography
What is ‘the problem’?
How can we overcome ‘the
problem’?
Fourier Theory
Diffraction pattern related to object
 Mathematical operation called
Fourier Transform
 Can be inverted to give pattern
of electron density
 Requires amplitude and phase
of diffracted waves
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The Phase Problem
The Phase Problem
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What we need:
Phase and Amplitude of
diffracted waves
 What we have:
 Number of X-ray
photons in each spot
 What we can get:  Number of Photons Þ
Intensity Þ Amplitude2
Phase angles
 What we miss:
 Relative phase
has been
lost
for different
spots
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What is the ‘phase problem’?
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From diffraction
experiment; only measure
the intensities
(amplitude2)
Phase information is lost!
..hence the ‘phase
problem’
Can we survive without
the lost phase
information?.....
Recovering phases…the Patterson
function
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Invented by Patterson for small molecules
Patterson map is calculated with the square of
structure factor amplitude and a phase of zero
This is an interatomic vector map
Each peak corresponds to a vector between
atoms in the crystal
Peak intensity is the product of electron
densities of each atom
Recovering phases experimentally..
Isomorphous replacement (IR)
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Used early 1900’s for small molecules by Groth (1908);
Beevers and Lipson (1934)
Perutz (1956) and Kendrew (1958) used IR on proteins
Use of heavy atom substitution in a crystal
“Isomorphous” – same shape
“Replacement” – heavy atom might be replacing light salts
or solvent molecules
Why heavy atoms? large atomic numbers; Contribute
disproportionately to the intensities
Principle: change the crystal (with heavy atoms); perturb
the structure factors; conclude phases from how the
structure factors are perturbed
Collect two (or more) datasets: ‘native’ and ‘derivative’ data
Isomorphous replacement contd…
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Calculate the Isormorphous
difference Fiso
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Use Fiso in direct or
patterson method to deduce
position of heavy atoms
Then deduce possible
values for protein phase
angles
In Single Isomorphous
Replacement (SIR), there
are two possible values of
FP phases - Phase
ambiguity
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Modification of SIR - MIR
Problems in IR
 Isomorphism sometimes difficult to achieve
 Must grow more than 1 crystal
 In some cases, heavy metals distort the crystal lattices so much
that, the crystal nolonger diffracts
Phases experimentally..Anomalous
scattering
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Use heavy atoms which have absorption edges within
the normally used x-ray wavelength – Anomalous
scatterers
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They break Friedel’s Law: states that members of a
Friedel pair have equal amplitude and opposite phase
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Estimate the anomalous differences
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Use direct or patterson methods to deduce positions of
anomalous scatterers
- Only one crystal is needed - replace mehionine with Semet
Variations of Anomalous scattering
SAD – Single Anomalous Dispersion
 MAD – (Multiple)- Change the wavelength of
X-rays, change the degree to which
anomalous scatterers perturb the data
 SIRAS – Single Isomorphous Replacement
with Anomalous Scattering – Breaking
phase ambiguities in SIR
 MIRAS – Multiple Isomorphous
Replacement with Anomalous Scattering
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Improving experimental phases
Experimental phases
are never sufficiently
accurate
 Density modification
methods used;
- Solvent flattening,
- Histogram matching
and
- Non-crystallographic
symmetry averaging.
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Recovering phases by guessing
phases – Molecular replacement
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First described by Michel Rossman and David
Blow (1962)
Why the name? molecules instead of atoms are
placed in the unit cell.
Sources of search model:
- Same protein solved in different spacegroup
- mutant or complex of known native protein
- homologous protein
- NMR/theoretical models
- Fragments (domains) of multiple proteins
Basic principle in MR
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Orient and position
search model; concide
with position of unknown
structure in crystal
In most space groups, 3
rotational & 3
translational parameters
need to be determined
Six-dimensional search: a
big problem!
Solving orientation problem using
patterson function
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Break down search into rotation, followed by translation
search.
The two functions use the concept of patterson function.
Remember: patterson map; correlation function between
atoms in the unit cell.
Vectors of two types: self-vectors (intramolecular) and
cross-vectors (intermolecular)
Intramolecular vectors shorter & independent of position
– used in rotational searches
Rotational functions defined by three rotation angles
Self-vectors & cross-vectors
Solving position problem with
patterson function
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Also uses patterson correlation function
Uses cross-vectors (b/w atoms in a molecule &
atoms in symmetry-related molecule)
Correlates a model structure and the observed
patterson of the crystal
Intermolecular vectors- dependent on both
orientation & position
Searches for 3 translational parameters (x,y,z)
What determines success of MR
The completeness of the search model
 Percentage identity
 Warning: model-bias problem!!
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In difficult cases…
Change the MR search parameters by altering
the program’s default settings
 Issues with low model identity?
- Try mutating model’s sidechains to contain
those for the unknown protein
- Try removing highly variable (non-conserved)
regions from the model
- Consider using an ensemble of several
homologues at once
- Consider converting your model to a poly-Ala
one
 Stuck with no solution? Is the space group right?
Crystallized the wrong protein?
 Consider experimental phasing!
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Model building
(interpreting e-density map)
 Once phases have been determined as accurately as possible,
an e-density map (the Fourier transform of the structure factors with
phases) is prepared and interpreted.
Use observed structure amplitudes |Fobs(hkl)| with best phases, best(hkl) to
give experimentally derived structure factors, Fexp(hkl):
Fexp(hkl) = |Fobs(hkl)| exp[ i best(hkl)]
Fourier transform creates e-density exp(xyz):
exp(xyz) = 1  Fexp(hkl) exp[ -2  i (hx + ky + lz)]
V
Now, more common to use computer display, using stereoscopic effects
If resolution and phase
determination are good,
map can be interpreted
easily.
Extended chain in a 3.7 Å map, using phases
derived from 30-fold redundancy in a virus
structure. Peptide chain was traced for > 400
residues. (Blow 11.6)
Computer programs help
by proposing possible
conformations of main
chain and hence atomic
positions seen in similar
conformations. Side
chains may be put in
later.
e-density at different resolutions:
Quality of map depends on
resolution of data and quality of
phase angles.
Figure shows e-density map of
same structural features at different
resolutions : as resolution is
enhanced, more details of a wellordered structure can be seen.
Fitting and Refinement
Electron density map doesn’t resolve individual atoms –
Have to fit models to density
 Graphics programmes: O & XtalView used for fitting/building
 Initial model hence the initial electron density maps have
lots of errors – model needs to be adjusted to improve the
agreement with the measured data - called Refinement
 Success of atomic model is judged by:1. Crystallographic R-factor (Average error in the calculated
amplitude compared to the observed amplitude.
- A good structure will have an R-factor in the range of 15%
to 25%
2 Correlation coefficient – Which should go to 1 as the model
improves
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Two types refinements
1. Rigid body refinement – Refinement of
positional and orientational parameters of
the model
2. Restrained refinement – Concerned with the
geometry of the amino acids – bond angles,
bond lengths and close contacts
Refining the structure
 Aim is to adjust the structure (built from e-density map) to give best fit
to crystallographic data (intensities of Bragg reflections)
 The refinement parameter gives a measure of the discrepancies
between calculated scattering and observed intensities.
 Idea is to alter the model to give lowest possible refinement parameter,
but
for model to be valid, the number of observations > number of variables.
Also, the model must already be good enough to make the refinement
meaningful.
Principle of refinement: search for a minimum
 The R-factor compares observed structure amplitudes to those
calculated for the current model:
R typically <0.15 – 0.2 for
  Fobs - Fcalc 
R=
well-refined structure
 Fobs 
Either Fourier or Least Squares methods attempt to adjust model to reduce
value of R (fall towards zero as agreement is reached).
Model includes atomic coordinates, site occupancies and thermal motion
parameters.
 May also monitor Rfree, based on a random selection of 5%
reflections (less statistical bias)
Rfree =
test set  Fobs - Fcalc 
test set Fobs 
Rfree typically 1.2 x R
Completed refinement
 When refinement approaches convergence (no further improvement
in R), resolution can be increased.
 Should examine difference map to see if any obvious errors exist.
 Difficult to be sure when to stop refining, this is probably determined by
the quality of your data.
 Result of crystal structure analysis will be a set of coordinates
deposited in PDB.
Validation of structure
Need to validate the model to avoid overfitting
 Leave out a fraction of the data from use in refinement –
cross-validation data which is free from the effects of
overfitting: compute R-free; unbiased indication of the
quality of the structure
- R-free should decrease through refinement cycles – Any
increase shows over-refining
 Outside refinement:- 1. Ramachandran plot – checks the main-chain torsion
angles distribution
- 2. Distribution of hydrophobic and hydrophilic amino acidshydrophobic hidden from solvent, hydrophilic exposed to
solvent
-See Protein structure validation suite on:
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http://biotech.ebi.ac.uk:8400/
Why resolution limits in protein
structures?
- Proteins are fairly flexible; atoms not
completely still
- Molecules in the crystal are not completely
in identical conformations
- Crystal lattices are not completely ordered
 Hence, when looking at finer details by going
to higher scattering angles, diffraction pattern
starts to cancel out!!
 Hence, limited level of fine details
Objectives
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Describe the nature and growth of protein crystals
Know the main features of an X-ay diffraction
experiment
Arrive at Bragg’s equation by considering reflection
from a set of parallel planes and use the equation
to determine experimental parameters
Describe how a protein structure is built
Give an account of the phase problem and
compare different methods
Aware of X-ray sources
Describe the overall strategy of solving a protein
crystal structure and problems to be faced