1. Macromolecular Small-Angle X-ray Scattering

2. What is SAXS? Small Angle X-ray Scattering
Scattering proportional to l/Molecular size
Typical x-ray wavelengths ~ 0.1 nm
Typical molecular dimensions nm
Scattering angles are small
0-2o historically.
Now 0-15o range is of increasing experimental interest

3. Why SAXS ?
Atomic level structures from crystallography or NMR “gold standard” for structural inferences
Crystallography, by definition, studies static structures
Most things crystallize only under rather specific, artificial conditions
Kinetics of molecular interactions frequently of interest
SAXS can provide useful, although limited, information on relatively fast time scales

4. Why not visible light? Diffraction effects limit to ~ /50 (5)
i.e. 14 nm visible light vs 1- 2nm RG for small proteins
Water absorbs many wavelengths of light strongly
0.2 <  < 160 nm experimentally inaccessible

5. What is SAXS Used for? Estimating sizes of particulates
Interactions in fluids
Sizes of micelles etc in emulsions
Size distributions of subcomponents in materials
Structure and dynamics of biological macromolecules

6. SAXS and Biological Macro-molecules
How well does the crystal structure represents the native structure in solution?
Can we get even some structural from large proportion of macro-molecules that do not crystallize?
How can we test hypotheses concerning large scale structural changes on ligand binding etc. in solution
SAXS can frequently provide enough information for such studies
May even be possible to deduce protein fold solely from SAXS data

7. Scattering from Molecules
Molecules are much larger than the wavelength (~0.1 nm) used => scattered photons will differ in phase from different parts of molecule
Observed intensity spherically averaged due to molecular tumbling
destructive interference e-
Constructive interference

8. Intensity in SAXS Experiments:
Sum over all scatterers (electrons) in molecule to get structure factor (in units of scattering 1 electron)
F(q) = i e i q • ri
Intensity is square (complex conjugate) of structure factor
I(q) = F F* = ji e iq • ri,j
Isotropic, so spherical average ( is rotation angle relative to q)
I(q) =  ji e iq • ri,j sin d
Debye Eq.
<I>(q) = ij sin q ri,j/ q ri,j
where q = 4sin q/l

9. In Scattering Experiments, Particles are Randomly Oriented
Intensity is spherically averaged
Phase information lost
Low information content fundamental difficulty with SAXS
Only a few, but frequently very useful, structural parameters can be unambiguously obtained.

10. Structural Parameters Obtainable from SAXS
Molecular weight*
Molecular volume*
Radius of gyration (Rg)
Distance distribution function p( r )
Various derived parameters such as longest cord from p ( r )
* requires absolute (or calibrated) intensity information

11. Experimental Geometry
Collimated
X-ray beam
200 cm
Sample
in 1 mm capillary
Backstop
“long camera ~1o
30 cm
Detector
short camera ~ 15o

12. The data:
Shadow of lead beam stop
2-D data needs to be radially integrated to produce 1-D plots of intensity vs q

13. Scattering Curves From Cytochrome C
Red line = sample +buffer
Blue = buffer only
Black = difference I
ln I
q nm-1

14. What does this look like for a typical protein ?
Since a Fourier transform, inverse relationship:
Large features at small q
Small features at large q
Globular size
Domain folds
2o structure

15. What’s Rg? Analogous to moment of inertia in mechanics
Rg2 = p(r)  r2  dV
p(r)  dV

16. Rg for representative shapes
Sphere
Rg2 = 3/5r2
Hollow sphere (r1 and r2 inner and outer radii)
Rg2 = 3/5 (r25-r15)/(r23-r13)
Ellipsoid (semi-axis a, b,c)
Rg2 = (a2+b2+c2)/5

17. Estimating Molecular Size from SAXS Data
I(q)/I0 = (1/n2)ij sin q ri,j/ qri,j
I(q)/I0= 1-q2RG2/3+…..
Guinier approximation:
e-q2Rg2/3 = 1 – q2Rg2/3 + (q2Rg2 /3 )2/2! … so
ln I/I0 = -q2Rg2/3 for small enough q

18. Guinier Plot Plot ln I vs. q2 Inner part will be a straight line
Slope proportional to Rg2
Only valid near q = 0 (i.e. where third term is insignificant)
For spherical objects, Guinier approximation holds even in the third term… so the Guinier region is larger for more globular proteins
Usual limit: Rg qmax <1.3

19. Guinier Fits Pg Rg 30.6 49.1 37.1 PBS +EACA +Benzamidine
Plasminogen data courtesy N. Menhart IIT

20. Need for Series of Concentrations
SAXS intensity equations valid only at infinite dilution
Excess density of protein over H2O very low
Need a non-negligible concentration ( > 1 mg/ml) to get enough signal.
In practice use a concentration series from ~ mg/ml and extrapolate to zero by various means
Only affects low angle regime
Can use much higher concentrations for high angle region (where scattering weak anyway)

21. Effect of Concentration

22. Correcting for Concentration

23. Shape information
SAXS patterns have relatively low information content
Sources of information loss:
Spherical averaging
X-ray phase loss, so can’t invert Fourier transform
In general cannot recover full shape, but can unambiguously compute distribution of distance s within molecule: i.e. p(r) function

24. p(r) Distribution of distances of atoms from centroid
Autocorrelation function of the electron density
1-D: Only distance, not direction
No phase information
Can be determined unambiguously from SAXS pattern if collected over wide enough range
20:1 ratio qmin :qmax usually ok

25. P( r ) & Intensity related by Fourier transform pair*
* This is a fourier sine transform because of symmetry (see Glatter & Kratky)

26. Relationship of shape to p(r)
Fourier transform pair
p(r)  I(q)
Can unambiguously calculate p( r ) from a given shape but converse not true
shape

27. Inversion intensity equation not trivial
Need to worry about termination effects, experimental noise and various smearing effects
Inversion of intensity equation requires use of various “regularization approaches”
One popular approach implemented in program GNOM (Svergun et al. J. Appl. Cryst. 25:495)

28. Example of p(r ) Analysis

29. Troponin C structure
Does p(r) make sense?

30. Scattering Pattern from Troponin C
q nm -1

31. Troponin C: Bimodal Distribution

32. Hypothesis Testing with SAXS
p (r ) gives an alternative measure of Rg and also “longest cord”
Predict Rg and p( r ) from native crystal structure (tools exist for pdb data) and from computer generated hypothetical structures under conditions of interest
Are the hypothesized structures consistent with SAXS data?

33. SAXS Data Alone Cannot Yield an Unambiguous Structure
One can combine Rg and P( r ) information with:
Simulations based on other knowledge (i.e. partial structures by NMR or X-ray)
Rigid body refinement Or
Whole pattern simulations using various physical criteria:
Positive e density,
finite extent,
Connectivity
chemically meaningful density distributions

34. Reconstruction of Molecular Envelopes
Very active area of research
3 main approaches:
Spherical harmonic-based algorithms (Svergun, & Stuhrmann,1991, Acta Crystallogr. A47, 736), genetic algorithms (Chacon et al, 1998, Biophys. J. 74, 2760), simulated annealing (Svergun,1999Biophys. J. 76, 2879), and “give ‘n take” algorithms (Walter et al, 2000, J. Appl. Cryst 33, 350).
Latter three make use of “Dummy atom approach” using the Debye formula.

35. Configuration Changes in Plasminogen
EACA Bz

36. Pg Complete Scattering curves-1

37. Shape Reconstruction using SAXS3D *:
+EACA
+BNZ 2Å 2Å
* D. Walther et. al., UCSF

38. Relationship of scattered intensity distribution to structure
Globular size
Domain folds
2o structure

39. GASBOR Reconstruction
NtrC in BeF
P(r)
GASBOR Reconstruction

40. Activated NtrC
De Carlo et al 2006, Genes and Development, in press

41. NtrC in various nucleotide states
Courtesy B.T. Nixon Penn State

42. Technical Requirements for SAXS
Monodispersed sample (usually)
Very stable, very well collimated beam
Very mechanically stable apparatus
Methods to assess and control radiation damage and radiation induced aggregation (flow techniques)
Ability to accurately measure and correct for variations in incident and transmitted beam intensity
High dynamic range, high sensitivity and low noise detector

43. Detectors For SAXS
1-D or 2 D position sensitive gas proportional counters
Pros: High dynamic range, zero read noise
Cons: limited count rate capability typically cps, 1-D detectors very inefficient high q range
2D CCD detectors
Pros: integrating detectors - no intrinsic count rate limit, 2-D so can efficiently collect high q data
Cons: Significant read noise, finite dynamic range
Most commercial detectors designed for crystallography too high read noise
BioCAT has special purpose, high sensitivity, low noise CCD detectors
Now commercially available from Aviex

44. Why Do You Need a Third Generation Source for SAXS?
Time resolved protein folding studies using SAXS
=> The “Protein Folding Problem”
High throughout molecular envelope determinations using SAXS
=> “Structural genomics”

45. LC-SAXS PROBLEM: Combine size exclusion chromatography with SAXS
Non linear Guinier plots
Combine size exclusion chromatography with SAXS
remove aggregates
reversibly associating proteins
facile extrapolation to low concentrations
Implementation
Superose-6 1 x 10 cm ; FPLC ; ml/min; 2-10mg bolus injections

46. Resolution Test Control mixture: Blue dextran BSA Cyto c MW > 1 MDa
65kDa Rg 3.0 nm
Dimer and higher oligomers
Cyto c
12 kDa Rg 1.3 nm BD
BSA Cc

47. Plasminogen Sticky protein… curved Guinier plots
conformational change…DRg 3040 A
Need accurate, reproducible Rgs to see these modest changes
static LC-SAXS

48. Plasminogen
Reliable data down to 0.3 mg/ml
Rg0 = 32.1 nm

49. Time resolved studies

50. Time-resolved Stopped-flow Experiment
Jacob et al.,2004 J Mol Biol338:369-82

51. Turbulent Continuous-Flow Mixer Design
T-shape mixer has a design adapted from Akiyama et al., The mixer dimensions are 200 x 350 x 2000 μm3 .
Stainless steel
plates
Kapton™ or Mylar
windows
Protein
Buffer
Waste
X-ray beam
200 μm
Capillary mixer by P.Regenfuss et al, Rev.Sci.Instrum. (1985) 56, 283;
Needle mixer by C.K.Chan et al. Proc. Natl. Acad. Sci. USA (1997), 94, 1779;
Mixer by L.Pollack et al. Proc. Natl. Acad. Sci. USA (1999) 96, 10115, with diffusion-based mixing.
T-shape turbulent mixer by S.Akiyama et al. Proc. Natl. Acad. Sci. USA (2002), 99, 1329.
Arrow-shape mixer by O.Bilsel et al. 2005, Rev.Sci.Instrum. 76,
Courtesy O. Bisel U. Mass

52. Cytochrome c Folding Probed by TR-SAXS
20 mg/ml solution of cytochrome c unfolded in GdnHCl at 4.5 M concentration was injected into the mixer at flow rate 2 ml/min and mixed with Tris buffer flowing at 18 ml/min. This provided 10-fold dilution and final concentration 2 mg/ml of cytochrome c and 0.45 M of GndHCl. Other parameters: pH 7, 0.2 M imidozole.
Guinear plots
Radius of gyration vs time
Rg exponential decay at t>150 μs is consistent with previous GdnHCl-jump CF-FL and pH-jump CF-SAXS results (M.C.Shasty et al. 1998, S.Akiyama et al. 2000).
60 µs time resolution is achieved. The ambiguity of results at t~<150 µs might be attributed to incomplete mixing.
ACA meeting, 2005

53. TR-SAXS results on Cytochrome c Folding
Kratky plots
Distance distribution functions
Collapsed state is globular and significantly more compact (~40%) than the denatured ensemble.
Channel scattering and noise are considerably low allowing to reliably measure SAXS signal at < q > A-1 at protein concentration of 2 mg/ml.

54. For further reading…..
A Guinier “X-ray Diffraction in Crsytals, Imperfect Crystals and Amorphous Bodies” Freeman, 1963
C. Cantor and P. Schimmel “Biophysical Chemistry part II: Techniques for the study of Biological Strcutre and Function” Freeman, 1980
O. Glatter and O. Kratky “Small-angle X-ray Scattering” Academic Press 1982
See Dmitri Svergun’s web site at