Macromolecular Small-Angle Scattering with Synchrotron

Radiation

Tom Irving

BioCAT, Dept. BCPS and CSRRI

Illinois Institute of Technology

Scope of Lecture

• Why do SAXS?

• Physical Principles

• Experimental methods

• Data interpretation

• Advantages of Third Generation Synchrotrons for SAXS

• References for learning more

What is SAXS?

• Small Angle X-ray Scattering• Scattering proportional to /Molecular size• Typical x-ray wavelengths ~ 0.1 nm• Typical molecular dimensions 1 -100 nm• Scattering angles are small• 0-2o historically. • Now 0-15o range is of increasing experimental interest

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

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

SAXS and Biological Macro-molecules

• How well does the crystal structure represents the native structure in solution?

• Can we get even some structural information 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

Scattering from MoleculesMolecules are much larger than the wavelength (~0.1 nm) used => scattered photons will differ in phase from different parts of moleculeObserved intensity spherically averaged due to molecular tumbling

e-

e-

e-

e-

e-

e-

Constructive interference

destructive interference

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* = ji e iq • ri,j • Isotropic, so spherical average ( is rotation angle relative to q)

I(q) = ji e I q • ri,j sin d

• Debye Eq.

<I>(q) = ij sin q ri,j/ q ri,j

where q = 4sin /

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.

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 intensity information

Experimental Geometry

200 cm

30 cm

“long camera ~1o

short camera ~ 15o

Detector

Samplein 1 mm capillary

Collimated X-ray beam

Backstop

The data:Shadow of lead beam stop

2-D data needs to be radially integrated to produce 1-D plots of intensity vs q

Scattering Curves From Cytochrome C

q nm-1

ln I

Red line = sample +buffer

Blue = buffer only

Black = difference

I

What does this look like for a typical protein ?

0 2 4 6 8 10 12 14 16 18

100

1000

10000

I

q (nm-1)

Since a Fourier transform, inverse relationship:

Large features at small q

Small features at large q

Globular size

2o structure

Domain folds

What’s Rg?

• Analogous to moment of inertia in mechanics

• Rg2 = p(r) r2 dV

p(r) dV

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

Estimating Molecular Size from SAXS Data

<I>() = ij sin q ri,j/ qri,j

Taylor series expansion

= 1 - (qrij )2/6 + (qrij )4/120 ….Guinier approximation:

e-q2Rg2/3 = 1 – q2Rg2/3 + (q2Rg

2 /3 )2/2! …

Equate first two terms

1 - (qrij )2/6 = 1 – q2Rg2 3

Or

ln I/I0 = q2Rg2/3

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, Gunier approximation holds even in the third term… so the Guinier region is larger for more globular proteins

– Usual limit: Rg qmax <1.3

Configuration Changes in Plasminogen

EACA

Bz

Pg Rg

PBS 30.6

+EACA 49.1

+Benzamidine 37.1

Guinier Fits

1.00E+02

1.00E+03

1.00E+04

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

7.00E-02

8.00E-02

9.00E-02

1.00E-01

blank

eaca

Benz

Plasminogen data courtesy N. Menhart IIT

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 ~ 3 - 30

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)

Effect of Concentration

Correcting for Concentration

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

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

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e-

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Relation of p( r ) to Intensity

I(q) = 4 0D p( r )sinqr dr

Relationship of shape to p(r)

• Fourier transform pairp(r) I(q)

shapeCan unambiguously calculate p( r ) from a given shape but converse not true

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)

Example of p(r ) Analysis

Troponin C structure• Does p(r) make sense?

Scattering Pattern from Troponin C

q nm -1

I

Troponin C: Bimodal Distribution

0 20 40 60 80

spurious water peak @ 3 A

41 A15 A

r (A)

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?

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)Or Whole pattern simulations using various physical

criteria:– Positive e density, – finite extent, – Connectivity– chemically meaningful density distributions

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.

Configuration Changes in Plasminogen

EACA

Bz

Pg Rg

PBS 30.6

+EACA 49.1

+Benzamidine 37.1

Guinier Fits

1.00E+02

1.00E+03

1.00E+04

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

7.00E-02

8.00E-02

9.00E-02

1.00E-01

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eaca

Benz

Plasminogen data courtesy N. Menhart IIT

Pg Complete Scattering curves

0 2 4 6 8 10 12 14 16 18 20

100

1000

10000

I

q (nm)

unliganded Bz EACA

-1

+EACA +BNZ+BNZ

2Å

2Å

Shape Reconstruction using SAXS3D *:

* D. Walther et. al., UCSF

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

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 105 - 106

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

SAXS at Third Generation Synchrotron Sources

The Advanced Photon Source

The APS is Optimized for Producing Undulator Radiation

Why is APS Undulator Radiation Good for Biological Studies?

• Wide energy range available for spectroscopy

• High flux for time resolved applications

• Very low beam divergence for high quality diffraction/scattering patterns

• Can focus to very small beams to examine small samples or regions within samples

What is BioCAT?

• A NIH-supported research center for the study of partially ordered and disordered biological materials

• Supported techniques are X-ray Spectroscopy (XAS and high resolution), powder diffraction, fiber diffraction, and SAXS

• Comprises an undulator based beamline, (18-ID) associated laboratory and computational facilities.

• Available to all scientists on basis of peer-reviewed beamtime proposals

BioCAT

A NIH Supported Research Center

The BioCAT Sector at the APS

SAXS Instrument on the BioCAT 18ID - Undulator Beamline

0 m52.6 m 56 m63 m68 m

Collimator Slits

Monochromator:

Source size(FWHM) anddivergence:597 x 28µm16 x 3µrad

Working beam size

145 x 40 µm0.19 x 0.16 rad

and divergence:

µ

Mirror,verticallyfocusing

Beam Stop

CCD

Sampleflowcell

Undulator18ID

Si (111) or (400),horizontallyfocusing

Slow and FastShutters

Al-filters

Guard Slits

Scatteringchamber250 - 5000 mm

Si (111) or (400),flat

BeamMonitor

BioCAT PERFORMANCE FOR SAXS

• 3 m camera can access a range of q from ~0.04 to 1.3 nm-1 • 0.3 m camera accesses range of q from ~0.8 to 20.0 nm-1 • 55 x 88 mm high sensitivity CCD detector can detect

single photons• Useful SAXS patterns can be collected from 5 mg./ml

cytochrome c in 300 ms => can do time resolved experiments on ms time scales or less

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”

Radius of gyration (Rg) obtained fromGuinier analysis as a function ofdenaturant concentration. Black squaresdenote equilibrium data and red circlesindicate values obtained ~1 msec afterinitiation of refolding at different GdmClconcentrations.

Time-resolved Stopped-flow ExperimentTime-resolve Stopped Flow Experiment

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 http://www.embl-hamburg.de/Externalinfo/Research/Sax