1. The ‘phase problem’ in X-ray crystallography What is ‘the problem’? How can we overcome ‘the problem’?

2. Fourier Theory Diffraction pattern related to object Diffraction pattern related to object Mathematical operation called Fourier Transform Mathematical operation called Fourier Transform Can be inverted to give pattern of electron density Can be inverted to give pattern of electron density Requires amplitude and phase of diffracted waves Requires amplitude and phase of diffracted waves The Phase Problem

3. 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 diffracted waves Phase and Amplitude of diffracted waves Number of X-ray photons in each spot Number of X-ray photons in each spot Number of Photons Þ Intensity Þ Amplitude2 Number of Photons Þ Intensity Þ Amplitude2 Relative phase angles for different spots Relative phase angles for different spots Phase has been lost

4. What is the ‘phase problem’? From diffraction experiment; only measure the intensities (amplitude 2 ) From diffraction experiment; only measure the intensities (amplitude 2 ) Phase information is lost! Phase information is lost!..hence the ‘phase problem’..hence the ‘phase problem’ Can we survive without the lost phase information?..... Can we survive without the lost phase information?.....

5. Recovering phases…the Patterson function Invented by Patterson for small molecules Invented by Patterson for small molecules Patterson map is calculated with the square of structure factor amplitude and a phase of zero Patterson map is calculated with the square of structure factor amplitude and a phase of zero This is an interatomic vector map This is an interatomic vector map Each peak corresponds to a vector between atoms in the crystal Each peak corresponds to a vector between atoms in the crystal Peak intensity is the product of electron densities of each atom Peak intensity is the product of electron densities of each atom

6. Recovering phases experimentally.. Isomorphous replacement (IR) Used early 1900’s for small molecules by Groth (1908); Beevers and Lipson (1934) Used early 1900’s for small molecules by Groth (1908); Beevers and Lipson (1934) Perutz (1956) and Kendrew (1958) used IR on proteins Perutz (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 shape “Isomorphous” – same shape “Replacement” – heavy atom might be replacing light salts or solvent molecules “Replacement” – heavy atom might be replacing light salts or solvent molecules Why heavy atoms? large atomic numbers; Contribute disproportionately to the intensities 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 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 Collect two (or more) datasets: ‘native’ and ‘derivative’ data

7. Isomorphous replacement contd… Calculate the Isormorphous difference F iso Calculate the Isormorphous difference F iso Use F iso in direct or patterson method to deduce position of heavy atoms Use F iso in direct or patterson method to deduce position of heavy atoms Then deduce possible values for protein phase angles Then deduce possible values for protein phase angles In Single Isomorphous Replacement (SIR), there are two possible values of F P phases - Phase ambiguity In Single Isomorphous Replacement (SIR), there are two possible values of F P phases - Phase ambiguity

8. Modification of SIR - MIR Problems in IR Isomorphism sometimes difficult to achieve Isomorphism sometimes difficult to achieve Must grow more than 1 crystal Must grow more than 1 crystal In some cases, heavy metals distort the crystal lattices so much that, the crystal nolonger diffracts In some cases, heavy metals distort the crystal lattices so much that, the crystal nolonger diffracts

9. Phases experimentally..Anomalous scattering Use heavy atoms which have absorption edges within the normally used x-ray wavelength – Anomalous scatterers Use heavy atoms which have absorption edges within the normally used x-ray wavelength – Anomalous scatterers They break Friedel’s Law: states that members of a Friedel pair have equal amplitude and opposite phase They break Friedel’s Law: states that members of a Friedel pair have equal amplitude and opposite phase Estimate the anomalous differences Estimate the anomalous differences Use direct or patterson methods to deduce positions of anomalous scatterers Use direct or patterson methods to deduce positions of anomalous scatterers - Only one crystal is needed - replace mehionine with Se- met

10. Variations of Anomalous scattering SAD – Single Anomalous Dispersion SAD – Single Anomalous Dispersion MAD – (Multiple)- Change the wavelength of X-rays, change the degree to which anomalous scatterers perturb the data 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 SIRAS – Single Isomorphous Replacement with Anomalous Scattering – Breaking phase ambiguities in SIR MIRAS – Multiple Isomorphous Replacement with Anomalous Scattering MIRAS – Multiple Isomorphous Replacement with Anomalous Scattering

11. Improving experimental phases Experimental phases are never sufficiently accurate Experimental phases are never sufficiently accurate Density modification methods used; Density modification methods used; - Solvent flattening, - Solvent flattening, - Histogram matching - Histogram matching and and - Non-crystallographic symmetry averaging. - Non-crystallographic symmetry averaging.

12. Recovering phases by guessing phases – Molecular replacement First described by Michel Rossman and David Blow (1962) First described by Michel Rossman and David Blow (1962) Why the name? molecules instead of atoms are placed in the unit cell. Why the name? molecules instead of atoms are placed in the unit cell. Sources of search model: Sources 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

13. Basic principle in MR Orient and position search model; concide with position of unknown structure in crystal 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 In most space groups, 3 rotational & 3 translational parameters need to be determined Six-dimensional search: a big problem! Six-dimensional search: a big problem!

14. Solving orientation problem using patterson function Break down search into rotation, followed by translation search. Break down search into rotation, followed by translation search. The two functions use the concept of patterson function. The two functions use the concept of patterson function. Remember: patterson map; correlation function between atoms in the unit cell. Remember: patterson map; correlation function between atoms in the unit cell. Vectors of two types: self-vectors (intramolecular) and cross-vectors (intermolecular) Vectors of two types: self-vectors (intramolecular) and cross-vectors (intermolecular) Intramolecular vectors shorter & independent of position – used in rotational searches Intramolecular vectors shorter & independent of position – used in rotational searches Rotational functions defined by three rotation angles Rotational functions defined by three rotation angles

15. Self-vectors & cross-vectors

16. Solving position problem with patterson function Also uses patterson correlation function Also uses patterson correlation function Uses cross-vectors (b/w atoms in a molecule & atoms in symmetry-related molecule) 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 Correlates a model structure and the observed patterson of the crystal Intermolecular vectors- dependent on both orientation & position Intermolecular vectors- dependent on both orientation & position Searches for 3 translational parameters (x,y,z) Searches for 3 translational parameters (x,y,z)

17. What determines success of MR The completeness of the search model The completeness of the search model Percentage identity Percentage identity Warning: model-bias problem!! Warning: model-bias problem!!

18. In difficult cases… Change the MR search parameters by altering the program’s default settings Change the MR search parameters by altering the program’s default settings Issues with low model identity? Issues with low model identity? - Try mutating model’s sidechains to contain those for the unknown protein - Try mutating model’s sidechains to contain those for the unknown protein - Try removing highly variable (non-conserved) regions from the model - Try removing highly variable (non-conserved) regions from the model - Consider using an ensemble of several homologues at once - Consider using an ensemble of several homologues at once - Consider converting your model to a poly-Ala one - Consider converting your model to a poly-Ala one Stuck with no solution? Is the space group right? Crystallized the wrong protein? Stuck with no solution? Is the space group right? Crystallized the wrong protein? Consider experimental phasing! Consider experimental phasing!

19. 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 |F obs (hkl)| with best phases,  best (hkl) to give experimentally derived structure factors, F exp (hkl): F exp (hkl) = |F obs (hkl)| exp[ i  best (hkl)]  exp (xyz) =  F exp (hkl) exp[ -2  i (hx + ky + lz)] V 1 Fourier transform creates e-density  exp (xyz):

20. 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)

21. 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.

22. Fitting and Refinement Electron density map doesn’t resolve individual atoms – Have to fit models to density Electron density map doesn’t resolve individual atoms – Have to fit models to density Graphics programmes: O & XtalView used for fitting/building 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 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:- 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% - 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

23. 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

24. 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.

25. 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 =    F obs  -  F calc     F obs  R free =  test set   F obs  -  F calc    test set  F obs   May also monitor R free, based on a random selection of  5% reflections (less statistical bias) R free typically 1.2 x R R typically <0.15 – 0.2 for well-refined structure

26.  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

27. Validation of structure Need to validate the model to avoid overfitting 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 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 - R-free should decrease through refinement cycles – Any increase shows over-refining Outside refinement:- Outside refinement:- - 1. Ramachandran plot – checks the main-chain torsion angles distribution - 1. Ramachandran plot – checks the main-chain torsion angles distribution - 2. Distribution of hydrophobic and hydrophilic amino acids- hydrophobic hidden from solvent, hydrophilic exposed to solvent - 2. Distribution of hydrophobic and hydrophilic amino acids- hydrophobic hidden from solvent, hydrophilic exposed to solvent -See Protein structure validation suite on: http://biotech.ebi.ac.uk:8400/ http://biotech.ebi.ac.uk:8400/

28. Why resolution limits in protein structures? - Proteins are fairly flexible; atoms not completely still - Proteins are fairly flexible; atoms not completely still - Molecules in the crystal are not completely in identical conformations - Molecules in the crystal are not completely in identical conformations - Crystal lattices are not completely ordered - 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, when looking at finer details by going to higher scattering angles, diffraction pattern starts to cancel out!! Hence, limited level of fine details Hence, limited level of fine details

29. Objectives Describe the nature and growth of protein crystals Describe the nature and growth of protein crystals Know the main features of an X-ay diffraction experiment 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 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 Describe how a protein structure is built Give an account of the phase problem and compare different methods Give an account of the phase problem and compare different methods Aware of X-ray sources Aware of X-ray sources Describe the overall strategy of solving a protein crystal structure and problems to be faced Describe the overall strategy of solving a protein crystal structure and problems to be faced