X-ray Crystallography

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1 2009/11/25 [ 1 ] X-ray Crystallography Andrew Torda, wintersemester 2009 / 2010 X-ray numerically most important more than 4/5 structures Goal a set of x, y, z coordinates different properties to NMR History 1896 X-rays from Wilhelm von Röntgen 1913 Bragg first small molecule 1950's or early 60's first proteins (myoglobin)

2 Andrew Torda 2009/11/25 [ 2 ] Where to learn best book "Crystallography made crystal clear", Rhodes, G, Academic press lectures Prof Betzel? Hamburg full of crystallographers

3 Andrew Torda 2009/11/25 [ 3 ] Steps From protein crystal crystal in front of X-ray source diffraction pattern (structure factors) structure factors coordinates fourier tranform + phasing coordinates better coordinates

4 2009/11/25 [ 4 ] Proteins and crystals Proteins can form crystals like table salt or sugar just much more difficult a, b, c define the unit cell may not be perpendicular may contain several copies usually - protein + water + salt sometimes ligand b a

5 2009/11/25 [ 5 ] Making crystals do you normally see protein crystals? what scale? << mm usually rather difficult concentrated, rather pure protein put in a drop, concentrate more by diffusion / evaporation repeat many times with different conditions buffer, solvent, ph, temperature partially automated (expensive) rather empirical

6 photo from Manfred Weiss Andrew Torda 2009/11/25 [ 6 ] 1mm

7 diagram from www-structmed.cimr.cam.ac.uk/course/overview/overview.html Andrew Torda 2009/11/25 [ 7 ] The measurement x-ray source detector CCD detector (/ film) from this data to a set of coordinates

8 Andrew Torda 2009/11/25 [ 8 ] Some necessary terms ρ is electron density ρ x,y,z where the atoms are F h,k,l 's are what comes from x-rays + proteins they have a phase one records an amplitude without phase h, k, l are indexes in the recorded data

9 2009/11/25 [ 9 ] Proteins and X-rays light wavelength x 10-9 m (about 4000 bonds) x-rays have wavelengths near 1 Å (10-10 m) no such thing as X-ray lens cannot focus, cannot record an image x-rays are diffracted by electron clouds H atoms hardly any electrons (almost invisible) C, N, O (+ P, S,..) do diffract heavy atoms less biological diffract most (Hg, Se)

10 Andrew Torda 2009/11/25 [ 10 ] From protein to diffraction Start with reverse problem shine light on two holes important for pattern r AB λ (wavelength)

11 Andrew Torda 2009/11/25 [ 11 ] Diffraction by a grid pattern on this side is regular r AB, λ (wavelength)

12 Andrew Torda 2009/11/25 [ 12 ] One and more waves What does one wave look like? 2π y ( x ) = A cos x + α λ For some amplitude A, phase shift α often neglect wavelength λ wavelengths are the same If one adds some waves together

13 Andrew Torda 2009/11/25 [ 13 ] amplitude Adding waves (same frequency) same frequency diff amplitudes x amplitude sum one wave same frequency different amplitude x amplitude of this wave? will depend on phases

14 Andrew Torda 2009/11/25 [ 14 ] What one records lots of scatterering atoms / electron clouds lots of waves b amplitude x each one of these waves is a sum of others has a contribution from all atoms a

15 Andrew Torda 2009/11/25 [ 15 ] From protein to diffraction If we know the protein coordinates x-ray source detector One should be able to calculate the pattern scattering from one site one dimension 2π y ( x ) = A cos x + α λ

16 Andrew Torda 2009/11/25 [ 16 ] earlier formalise the relation 2π y ( x ) = A cos x + α λ here wavelength (λ) is the same 2πihx change nomenclature f = e phase information is hiding in 2πhx 2 i( hx+ ky+ lz) go to three dimensions f = e π sum over lots of waves n = scatterers F hkl e j= 1 2πi ( hx + ky + lz ) j j j phase information is hiding in hkl

17 Andrew Torda 2009/11/25 [ 17 ] From protein to diffraction x-ray source detector F is the structure factor n indexed by hkl F hkl = e j= 1 summation runs over all scatterers electron clouds periodic functions each scatterer contributes to all reflections 2πi ( hx + ky + lz ) j j j

18 Andrew Torda 2009/11/25 [ 18 ] before What is recorded no phases F hkl = n j= 1 e 2πi ( hx + ky + lz ) j j j make the volume explicit put in the electron density ρ Fhkl F has phase not recorded really F 2 is recorded (no phase) = V ρ x, y, z e 2πi ( hx+ ky+ lz ) dv we want ρ x,y,z if we know the phases can reverse Fourier transform ρ x, y, z = V 1 h k l F h, k, l e 2πi ( hx+ ky+ lz )

19 Andrew Torda 2009/11/25 [ 19 ] From scattering to coordinates from coordinates to data (no phases) F hkl = V ρ x, y, z e 2πi ( hx+ ky+ lz ) dv from data to coordinates if you knew the phases of F h,k,l 's each reflection contains contribution from all atoms each atom Stop here make points clear ρ x, y, z = V 1 h k l F h, k, l e 2πi ( hx+ ky+ lz )

20 Andrew Torda 2009/11/25 [ 20 ] Summary so far Given the coordinates for atoms in a crystal can calculate the expected reflections (used later) want the electron density ρ x,y,z (atomic coordinates) fourier transform of reflections with phases from somewhere else each reflection depends on all atoms (ρ x,y,z ) electron density ρ x,y,z depends on all observed reflections

21 Andrew Torda 2009/11/25 [ 21 ] Summary so far most things are additive add more scatterers, add more components to F F hkl is somewhat abstract an amplitude with a phase the phase is not measured if one does know the phase one can get F hkl

22 Andrew Torda 2009/11/25 [ 22 ] How to get phases two methods dominate MR (molecular replacement) about 80 % of structures MIR (multiple isomorphous replacement) MIR

23 Andrew Torda 2009/11/25 [ 23 ] MIR structure with two or three atoms (not 1000's) one could record data get the phases by brute force (direct) what if one has some huge atoms in a protein? they would affect all the reflections one could still get their phases directly heavy atoms

24 Andrew Torda 2009/11/25 [ 24 ] MIR Record data from protein (F P ) Record data from protein + heavy atoms (F PH ) get difference (F PH -F P ) call this F H use this for coordinates and phases of heavy atoms call these phases α H go back to (F PH ) get phases of original data you know the positions of heavy atoms For each original reflection F P = F PH F H now have phased reflections for protein back Fourier transform to get ρ

25 Andrew Torda 2009/11/25 [ 25 ] MIR some details and formalism skipped really need more than one heavy atom type ambiguities chemistry what are heavy atoms? something with lots of electrons should bind at specific positions to protein

26 Andrew Torda 2009/11/25 [ 26 ] MIR examples Selenomethionine replaces methionine very little change to protein about 5000 examples in PDB very specific positions in protein special molecular biology techniques used put in a solution with heavy metal salt fast, cheap put in solution with large anion Br, I put under pressure with noble gas Xe, Kr

27 Andrew Torda 2009/11/25 [ 27 ] MIR problems chemical problems not always easy fundamental problems protein, protein + heavy atoms not identical maths assumes that F PH = F P + F H initial phases should be good, but not perfect refinement necessary (later)

28 Andrew Torda 2009/11/25 [ 28 ] phasing by MR / molecular replacement Problem you have crystals recorded reflections want phases to calculate ρ x,y,z what one needs for MR structure of similar molecule

29 Andrew Torda 2009/11/25 [ 29 ] MR (simplified) if we know related structure can calculate F with phases can calculate amplitudes (observed data) calculated calculated

30 Andrew Torda 2009/11/25 [ 30 ] MR use a simplified function to get rotations and translations call these correction factors we have F model F protein = F model corrections problems model is not identical to protein of interest 90 % sequence identity, similar length no problem 10 % sequence identity, different sizes will not work the initial phases may be roughly right but not perfect remember this was the case with MIR - refinement

31 Andrew Torda 2009/11/25 [ 31 ] more phasing / refinement whatever method we have initial phases calculate F hkl inverse fourier transform yields electron density not finished density is not very accurate (poor phases) try to adjust phases so as to get sharpest electron density

32 Andrew Torda 2009/11/25 [ 32 ] more refinement one has electron density ρ the atoms have to be fit into it electron density is not perfect where the atoms go is not simple atoms are mobile high mobility B factor higher we can calculate electron density from coordinates variables (at least) x, y, z, B

33 Andrew Torda 2009/11/25 [ 33 ] more refinement model / coordinates calculate density ρ adjust coordinates of atoms maybe adjust B factors calculate F hkl gives F calc to adjust, compare F calc and F obs calculated from model, observed measured

34 Andrew Torda 2009/11/25 [ 34 ] Refinement from my model I can calculate F from F I can calculate the amplitude I would measure call this F calc from my measurements I have F obs if model and phases are perfect each F calc = F obs refinement adjust coordinates of atoms in model adjust phases minimise F calc F obs I have many F 's my function summed over all reflections

35 Andrew Torda 2009/11/25 [ 35 ] R factor what I care about is j h, k, l calc F j F obs j normalise and use standard definition R = 100 j h, k, l F calc j j h, k, l F obs j F obs j slightly change recipe for refinement

36 Andrew Torda 2009/11/25 [ 36 ] model / coordinates calculate density ρ adjust coordinates of atoms maybe adjust B factors calculate F hkl gives F calc calculate R factor

37 Andrew Torda 2009/11/25 [ 37 ] R factor most structures are % F j h, k, l R = 100 is this enough? j what if there is over fitting? how to test? common form of cross validation take maybe 10 % of data and remove refine on 90 % of data calculate R based on the 10 % not used called R free calc j h, k, l F obs j F obs j

38 Andrew Torda 2009/11/25 [ 38 ] Resolution nλ = = length ABC 2d sinθ nλ d = 2sinθ A 2θ B C d d is the smallest distance one can resolve wavelength λ - smaller better angle θ can one adjust θ? not really...

39 Andrew Torda 2009/11/25 [ 39 ] diffraction angle / resolution high resolution few Å x-ray source low resolution different picture

40 Andrew Torda 2009/11/25 [ 40 ] diffraction and resolution the higher resolution spots are harder to see

41 Andrew Torda 2009/11/25 [ 41 ] Different resolution - effects Movie time - ucxray.berkeley.edu/~jamesh/movies/ At <1.0 Å see atoms at 5 Å tryptophans disappear typical for big and difficult structures

42 Andrew Torda 2009/11/25 [ 42 ] diffraction and resolution typical resolution in PDB 1.5 to 2.5 Å resolution could you just record longer? molecule would be toasted middle would be black how far can you go? limits for each protein

43 Andrew Torda 2009/11/25 [ 43 ] Atomic motion what if the atoms move by 2 Å? can never really define the coordinates of the atom typical of loops and termini

44 Andrew Torda 2009/11/25 [ 44 ] Static disorder perfect crystal imperfect crystal In each copy of the molecule atoms are in different place coordinates of atoms are not defined to 1, 2,.. Å you cannot collect high resolution data

45 Andrew Torda 2009/11/25 [ 45 ] disorder static versus dynamic not easy to distinguish model for atomic motion? gaussian real meaning of B factor width of Gaussian density = ( x µ ) σ 2 2πσ e density space

46 Andrew Torda 2009/11/25 [ 46 ] Back to PDB files x, y, z coordinates B factors sometimes for each atom for each residue at top of file R factor, R free optional information about conditions, phasing amount of data (1000's of reflections) software used for refinement still some errors.. example

47 2009/11/25 [ 47 ] how difficult is fitting density? errors can be made backwards wrong sequence cannot tell O from N

48 Andrew Torda 2009/11/25 [ 48 ] Summarise and compare with NMR overall procedure crystallise collect data phase refine (x, y, z, B) check R

49 Andrew Torda 2009/11/25 [ 49 ] NMR and X-ray NMR X-ray resolution model how different are the solutions formally a Gaussian resolution not well defined 1.5 to 2.5 Å size many examples >10 3 residues few examples > 200 residues

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