The ‘phase problem’ in X-ray The ‘phase problem’ in X-ray crystallographycrystallography

What is ‘the problem’?What is ‘the problem’?

How can we overcome ‘the How can we overcome ‘the problem’?problem’?

Fourier TheoryFourier Theory

Diffraction pattern related to objectDiffraction pattern related to objectMathematical operation called Mathematical operation called

FourierFourier TransformTransformCan be inverted to give pattern Can be inverted to give pattern

of electron densityof electron densityRequires amplitude and phaseRequires amplitude and phase

of diffracted waves of diffracted waves

The Phase Problem

The Phase ProblemThe Phase Problem

What we need:What we need:

What we have:What we have:

What we can get:What we can get:

What we miss:What we miss:

Phase and Amplitude of Phase and Amplitude of diffracted wavesdiffracted waves

Number of X-ray Number of X-ray photons in each spotphotons in each spot

Number of Photons Þ Number of Photons Þ Intensity Þ Amplitude2Intensity Þ Amplitude2

Relative phase angles Relative phase angles for different spotsfor different spotsPhase has been

lost

What is the ‘phase problem’?What is the ‘phase problem’?

From diffraction From diffraction experiment; only measure experiment; only measure the intensitiesthe intensities (amplitude(amplitude22))

Phase information is lost!Phase information is lost! ..hence the ‘phase ..hence the ‘phase

problem’problem’ Can we survive without Can we survive without

the lost phase the lost phase information?.....information?.....

Recovering phases…the Patterson Recovering phases…the Patterson functionfunction

Invented by Patterson for small moleculesInvented by Patterson for small molecules Patterson map is calculated with the square of Patterson map is calculated with the square of

structure factor amplitude and a phase of zerostructure factor amplitude and a phase of zero This is an interatomic vector mapThis is an interatomic vector map Each peak corresponds to a vector between Each peak corresponds to a vector between

atoms in the crystalatoms in the crystal Peak intensity is the product of electron Peak intensity is the product of electron

densities of each atomdensities of each atom

Recovering phases experimentally.. Recovering phases experimentally.. Isomorphous replacementIsomorphous replacement (IR) (IR)

Used early 1900’s for small molecules by Groth (1908); Used early 1900’s for small molecules by Groth (1908); Beevers and Lipson (1934)Beevers and Lipson (1934)

Perutz (1956) and Kendrew (1958) used IR on proteinsPerutz (1956) and Kendrew (1958) used IR on proteins Use of heavy atom substitution in a crystal Use of heavy atom substitution in a crystal ““Isomorphous” – same shapeIsomorphous” – same shape ““Replacement” – heavy atom might be replacing light salts Replacement” – heavy atom might be replacing light salts

or solvent moleculesor solvent molecules Why heavy atoms?Why heavy atoms? large atomic numbers; Contribute large atomic numbers; Contribute

disproportionately to the intensitiesdisproportionately to the intensities Principle:Principle: change the crystal (with heavy atoms); perturb change the crystal (with heavy atoms); perturb

the structure factors; conclude phases from how the the structure factors; conclude phases from how the structure factors are perturbedstructure factors are perturbed

Collect two (or more) datasets: ‘native’ and ‘derivative’ Collect two (or more) datasets: ‘native’ and ‘derivative’ datadata

Isomorphous replacement contd…Isomorphous replacement contd…

Calculate the Isormorphous Calculate the Isormorphous difference difference FFiso iso

Use Use FFisoiso in direct or in direct or patterson method to deduce patterson method to deduce position of heavy atomsposition of heavy atoms

Then deduce possible Then deduce possible values for protein phase values for protein phase anglesangles

In Single Isomorphous In Single Isomorphous Replacement (SIR), there Replacement (SIR), there are two possible values of are two possible values of FFP P phases - Phase phases - Phase ambiguityambiguity

Modification of SIR - MIRModification of SIR - MIR

Problems in IRProblems in IR Isomorphism sometimes difficult to achieveIsomorphism sometimes difficult to achieve Must grow more than 1 crystalMust grow more than 1 crystal In some cases, heavy metals distort the crystal lattices so much In some cases, heavy metals distort the crystal lattices so much

that, the crystal nolonger diffractsthat, the crystal nolonger diffracts

Phases experimentally..Anomalous Phases experimentally..Anomalous scatteringscattering

Use heavy atoms which have absorption edges within Use heavy atoms which have absorption edges within the normally used x-ray wavelength – Anomalous the normally used x-ray wavelength – Anomalous scatterersscatterers

They break Friedel’s Law: states that members of a They break Friedel’s Law: states that members of a Friedel pair have equal amplitude and opposite phaseFriedel pair have equal amplitude and opposite phase

Estimate the anomalous differencesEstimate the anomalous differences

Use direct or patterson methods to deduce positions of Use direct or patterson methods to deduce positions of anomalous scatterersanomalous scatterers

- Only one crystal is needed - replace mehionine with Se-- Only one crystal is needed - replace mehionine with Se-met met

Variations of Anomalous scatteringVariations of Anomalous scattering

SAD – Single Anomalous DispersionSAD – Single Anomalous DispersionMAD – (Multiple)- Change the wavelength of MAD – (Multiple)- Change the wavelength of

X-rays, change the degree to which X-rays, change the degree to which anomalous scatterers perturb the dataanomalous scatterers perturb the data

SIRAS – Single Isomorphous Replacement SIRAS – Single Isomorphous Replacement with Anomalous Scattering – Breaking with Anomalous Scattering – Breaking phase ambiguities in SIRphase ambiguities in SIR

MIRAS – Multiple Isomorphous MIRAS – Multiple Isomorphous Replacement with Anomalous ScatteringReplacement with Anomalous Scattering

Improving experimental phasesImproving experimental phases

Experimental phases Experimental phases are never sufficiently are never sufficiently accurate accurate

Density modification Density modification methods used; methods used;

- Solvent flattening, - Solvent flattening, - Histogram matching - Histogram matching and and - Non-crystallographic - Non-crystallographic

symmetry averaging.symmetry averaging.

Recovering phases by guessing Recovering phases by guessing phases – Molecular replacementphases – Molecular replacement

First described by Michel Rossman and David First described by Michel Rossman and David Blow (1962)Blow (1962)

Why the name?Why the name? molecules instead of atoms are molecules instead of atoms are placed in the unit cell. placed in the unit cell.

Sources of search modelSources of search model: :

- Same protein solved in different spacegroup- Same protein solved in different spacegroup

- mutant or complex of known native protein- mutant or complex of known native protein

- homologous protein- homologous protein

- NMR/theoretical models- NMR/theoretical models

- Fragments (domains) of multiple proteins- Fragments (domains) of multiple proteins

Basic principle in MRBasic principle in MR

Orient and position Orient and position search model; concide search model; concide with position of unknown with position of unknown structure in crystalstructure in crystal

In most space groups, 3 In most space groups, 3 rotational & 3 rotational & 3 translational parameters translational parameters need to be determinedneed to be determined

Six-dimensional search: a Six-dimensional search: a big problem! big problem!

Solving orientation problem using Solving orientation problem using patterson functionpatterson function

Break down search into rotation, followed by translation Break down search into rotation, followed by translation search.search.

The two functions use the concept of patterson function.The two functions use the concept of patterson function. Remember:Remember: patterson map; correlation function between patterson map; correlation function between

atoms in the unit cell.atoms in the unit cell. Vectors of two types: self-vectors (intramolecular) and Vectors of two types: self-vectors (intramolecular) and

cross-vectors (intermolecular)cross-vectors (intermolecular) Intramolecular vectors shorter & independent of position Intramolecular vectors shorter & independent of position

– used in rotational searches– used in rotational searches Rotational functions defined by three rotation anglesRotational functions defined by three rotation angles

Self-vectors & cross-vectorsSelf-vectors & cross-vectors

Solving position problem with Solving position problem with patterson functionpatterson function

Also uses patterson correlation function Also uses patterson correlation function Uses cross-vectors (b/w atoms in a molecule & Uses cross-vectors (b/w atoms in a molecule &

atoms in symmetry-related molecule)atoms in symmetry-related molecule) Correlates a model structure and the observed Correlates a model structure and the observed

patterson of the crystalpatterson of the crystal Intermolecular vectors- dependent on both Intermolecular vectors- dependent on both

orientation & positionorientation & position Searches for 3 translational parameters (x,y,z)Searches for 3 translational parameters (x,y,z)

What determines success of MRWhat determines success of MR

The completeness of the search modelThe completeness of the search modelPercentage identityPercentage identityWarning: model-bias problem!!Warning: model-bias problem!!

In difficult cases…In difficult cases… Change the MR search parameters by altering Change the MR search parameters by altering

the program’s default settingsthe program’s default settings Issues with low model identity? Issues with low model identity? - Try mutating model’s sidechains to contain - Try mutating model’s sidechains to contain

those for the unknown protein those for the unknown protein - Try removing highly variable (non-conserved) - Try removing highly variable (non-conserved)

regions from the modelregions from the model - Consider using an ensemble of several - Consider using an ensemble of several

homologues at oncehomologues at once - Consider converting your model to a poly-Ala - Consider converting your model to a poly-Ala

oneone Stuck with no solution? Is the space group right? Stuck with no solution? Is the space group right?

Crystallized the wrong protein?Crystallized the wrong protein? Consider experimental phasing!Consider experimental phasing!

Model building Model building (interpreting e-density map)(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)]

exp(xyz) = Fexp(hkl) exp[ -2 i (hx + ky + lz)]V1

Fourier transform creates e-density exp(xyz):

Now, more common to use computer display, using stereoscopic effects

If resolution and phase determination are good, map can be interpreted easily.

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.

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)

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 well-ordered structure can be seen.

Fitting and RefinementFitting and Refinement Electron density map doesn’t resolve individual atoms – Electron density map doesn’t resolve individual atoms –

Have to fit models to densityHave to fit models to density Graphics programmes: O & XtalView used for fitting/buildingGraphics programmes: O & XtalView used for fitting/building Initial model hence the initial electron density maps have lots Initial model hence the initial electron density maps have lots

of errors – model needs to be adjusted to improve the of errors – model needs to be adjusted to improve the agreement with the measured data - called agreement with the measured data - called RefinementRefinement

Success of atomic model is judged by:-Success of atomic model is judged by:-

1. Crystallographic R-factor (Average error in the calculated 1. Crystallographic R-factor (Average error in the calculated amplitude compared to the observed amplitude. amplitude compared to the observed amplitude.

- A good structure will have an R-factor in the range of 15% - A good structure will have an R-factor in the range of 15% to 25%to 25%

2 Correlation coefficient – Which should go to 1 as the model 2 Correlation coefficient – Which should go to 1 as the model improvesimproves

Two types refinementsTwo types refinements

1. 1. Rigid body refinementRigid body refinement – Refinement of – Refinement of positional and orientational parameters of positional and orientational parameters of the modelthe model

2. 2. Restrained refinementRestrained refinement – Concerned with the – Concerned with the geometry of the amino acids – bond angles, geometry of the amino acids – bond angles, bond lengths and close contactsbond lengths and close contacts

Refining the structureRefining 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

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.

The R-factor compares observed structure amplitudes to those calculated for the current model:

R = Fobs - Fcalc

Fobs

Rfree = test set Fobs - Fcalc

test set Fobs

May also monitor Rfree, based on a random selection of 5% reflections (less statistical bias)

Rfree typically 1.2 x R

R typically <0.15 – 0.2 for well-refined structure

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.

Completed refinement

Validation of structureValidation of structure Need to validate the model to avoid overfittingNeed to validate the model to avoid overfitting Leave out a fraction of the data from use in refinement – Leave out a fraction of the data from use in refinement –

cross-validation datacross-validation data which is free from the effects of which is free from the effects of overfitting: compute R-free; unbiased indication of the overfitting: compute R-free; unbiased indication of the quality of the structurequality of the structure

- R-free should decrease through refinement cycles – Any - R-free should decrease through refinement cycles – Any increase shows over-refiningincrease shows over-refining

Outside refinement:- Outside refinement:- - 1. Ramachandran plot – checks the main-chain torsion - 1. Ramachandran plot – checks the main-chain torsion

angles distributionangles distribution - 2. Distribution of hydrophobic and hydrophilic amino acids- - 2. Distribution of hydrophobic and hydrophilic amino acids-

hydrophobic hidden from solvent, hydrophilic exposed to hydrophobic hidden from solvent, hydrophilic exposed to solventsolvent

-See Protein structure validation suite on: -See Protein structure validation suite on:

http://biotech.ebi.ac.uk:8400/http://biotech.ebi.ac.uk:8400/

Why resolution limits in protein Why resolution limits in protein structures?structures?

- Proteins are fairly flexible; atoms not - Proteins are fairly flexible; atoms not completely stillcompletely still

- Molecules in the crystal are not completely - Molecules in the crystal are not completely in identical conformationsin identical conformations

- Crystal lattices are not completely ordered- Crystal lattices are not completely orderedHence, when looking at finer details by going Hence, when looking at finer details by going

to higher scattering angles, diffraction pattern to higher scattering angles, diffraction pattern starts to cancel out!!starts to cancel out!!

Hence, limited level of fine detailsHence, limited level of fine details

ObjectivesObjectives Describe the nature and growth of protein crystalsDescribe the nature and growth of protein crystals Know the main features of an X-ay diffraction Know the main features of an X-ay diffraction

experimentexperiment Arrive at Bragg’s equation by considering reflection Arrive at Bragg’s equation by considering reflection

from a set of parallel planes and use the equation from a set of parallel planes and use the equation to determine experimental parametersto determine experimental parameters

Describe how a protein structure is builtDescribe how a protein structure is built Give an account of the phase problem and Give an account of the phase problem and

compare different methodscompare different methods Aware of X-ray sourcesAware of X-ray sources Describe the overall strategy of solving a protein Describe the overall strategy of solving a protein

crystal structure and problems to be facedcrystal structure and problems to be faced