SAXS and Scatter

TRACK B 

SAXS AND SCATTER 

Robert P. Rambo 

Advanced Light Source 

Berkeley CA 

BL 12.3.1 

Thursday, August 1, 13

SAXS 

Small Angle X-ray Scattering 

How does SAXS relate to structure? 

Source 

scattering angle 

Sample 

log I(q) 

∫ 

I molecule 

(q)= p(r)⋅ 

sin(q ⋅ r) 

q ⋅ r 

dr 

q, Å -1 

Set of all distance vectors 

within the particle 

€ 

P(r) ~ Pair Distance Distribution Function 


SAXS 

Small Angle X-ray Scattering 

What can you derive directly from a single curve? 

•Radius-of-gyration, Rg (real and reciprocal space) 

•Porod exponent, PE 

•Volume-of-correlation, Vc 

•Particle volume, V 

•Surface-to-volume ratio, S/V 

•correlation length, lc 

•mass 

•particle shape 


SAXS Invariants 

(structural parameters derived directly from SAXS) 

Q, Porod Invariant 

Directly related to mean square electron density of particle. 

Requires convergence in Kratky plot (q 2 I(q) vs q). 

Vp, Porod Volume 

Requires Q 

lc, correlation length 

Q acts as a normalization constant and corrects for: 

1.concentration 

2.instrumentation parameters 

3.contrast, (Δρ) 2 

Rg, radius-of-gyration 

Does not require Q 

Concentration independent 

Contrast independent (as long as structure does not 

change) 

Essentially normalized to I(0) 


Porod Invariant 

Assessing flexibility 

G. Porod deduced an integral constant contained within a SAXS curve: 

Assumption: 

1.defined Δρ exists between particle and solvent 

2.scatterer has homogenous electron density 

Integration of data transformed as q 2 • I(q) should be constant 

Q = 1 

2π 2 ∫ ∞ 

0 

q 2 · I(q) dq 

Q =2π 2 · (∆ρ) 2 · V 

Q is the direct product of the excess scattering 

electrons of the particle and Vparticle 

Q =2π 2 · c · (∆ρ) 2 · V 

Regardless of beamline, source, or wavelength; 

Data should have the same constant with the same sample at the same concentration. 


Porod Invariant 

Assessing flexibility 

Q = 1 

2π 2 ∫ ∞ 

0 

Kratky Plot 

q 2 · I(q) dq 

• visualization of Q 

• used to interpret samples with flexibility 

A plot of q 2 • I(q) vs q should show a curve that captures a 

defined area, hence the integral. 

Defined area means transformed data converges. 

Asymptotic behavior predicted by Debye 

equation for a Gaussian random coil 

RNA riboswitch (+/- Mg 2+ ) 

flexible-unfolded 

compact-folded 

Defined, captured area 

predicted by Q 

q, Å 

Unfolded particle does not capture a defined area (flexible, unfolded, gaussian chain like) 

Folded particle converges at higher q in Kratky plot (folded, compact particle) 


Defining a new Invariant 

Kratky Plot 

flexible-unfolded 

compact-folded 

q, Å 

plot differently 

q, Å 

Data converges for both 

• compact - folded 

• flexible - unfolded 

7 


The Volume-of-Correlation 

= = 

independent of: 

1. contrast 

2. concentration 

1. substitute for I(q) 

lc is the expected correlation length 

2. collect like terms 

5. collect like terms 

3. integrate by parts 

∞ 

4. substitute P(r) = 4π r 2 γ(r) 

0 

correlation function 


Pair-distance Distribution Function 

Set of all distances measured within the particle. 

∑ 

= ∑ ∫ 

r ij = 

internal distances 

ρ(r) dr 

logically this must be Vparticle 

p(r) = 0 when r > d max 

• defined in real space 

• 

V particle = 

no negative values (R + ) 

• zero except for defined distances 

• expected to be smooth as r → dmax 

peak ⇒〈r〉 = 

∫ 

r · ρ(r) dr 

∫ dmax 

0 

ρ(r) =r 2 γ(r) 

ρ(r) dr 

implies p(r) has units of Å 2 

γ(r): correlation function 

R g 2 = 1 2 · 

∫ 

r2 · ρ(r) dr 

∫ 

ρ(r) dr 

P(r) 

Knowing P(r): 

1. Determine Vparticle 

2. Rg (real space) 

3. Correlation function 

r 


Correlation Function 

Debye and Bueche (1947) 



• displace particle by a small distance, ∆r 









• calculate how well the particle correlates with its previous self 






increase Δr 






increase Δr 




recalculate 



increase Δr 




recalculate 



increase Δr 



γ(r) 

r, Å 

Maximum self-correlation occurs at r = 0 

Correlation decays to 0 at r > dmax 



recalculate 



increase Δr 



ρ(r) =r 2 γ(r) 

Pair-distance Function 

γ(r) 

P(r) 

r, Å 

r, Å 





recalculate 



increase Δr 



ρ(r) =r 2 γ(r) 

Pair-distance Function 

γ(r) 

P(r) 

r, Å 

r, Å 



l c , mean width of γ(r) 

l c = l2 ave 

l ave 


Developed in JAVA 1.6 using: 

• Mac 10.6.8 and greater (RPR) 

• Windows XP and LINUX (Ivan R. ) 

SCÅTTER 

Software for SAXS Analysis 

Free for academic, non-profit, and industry! 

Specific to SAXS and biological materials but can be used with SANS data and non-biological samples. 

Reads in 3 column delimited by /[\s\t]+/ or *.brml (Bruker files) 

• if you have problems, remove header or footer or email me at irodic@lbl.gov! 

Want users to know they have bad data! (minimal automation) 

Available for download at SIBYLS and BioIsis: 

• http://www.bioisis.net/tutorial 

• http://bl1231.als.lbl.gov 

try 1 st ! 


Tutorial links at BIOISIS 


SCÅTTER 


Go to website 

start Scatter 

Load data show basics are automatic 


Detecting Flexibility 

Debye P. Molecular-weight Determination by Light Scattering (1947) J. of Physical and Colloid Chemistry 

Scattering by a Gaussian Coil 

I(q) = 2(e−R2 g·q2 + R 2 g · q 2 − 1) 

(R 2 g · q 2 ) 2 

ASYMPTOTIC CHARACTERISTIC 

lim I(q) · 

q→∞ q2 = 

2 R 2 g 

( 

1 − 1 

q 2 · R 2 g 

) 

q 2 · I(q) ≈ K 

within a limited q range 

where q 2 •Rg 4

Kratky Plot 

Qualitative Assessment of flexibility 

for q•Rg > 1.3, the scattering decays as 1/q 2 

A plot of q 2 •I(q) vs. q 

should approach a constant 

Data must be collected 

to sufficiently high q with 

good S-to-N ratio 


Porod’s Law 

Porod, G. (1951). Kolloid-Z. 124, 83 

Fourth Power law (Porod’s Law) 

I particle (q) =V · 

∫ dmax 

0 

ρ(r) · 

sin(q · r) 

q · r 

dr 

ASSUMING: 

•compact particle 

•discrete en - contrast 

S 

V 

= π · limI(q) 

· q4 

Q 

I(q) =∆ρ 2 V · 1 

l · 8π 

q 4 

I(q) =k · 

1 

q 4 

q 4 · I(q) =constant 

I(q) decays as q -4 scaled by a constant value 

q 4 • I(q) becomes constant at high q 

k proportional to surface area (V/l) 


Power Law Relationship 

log vs. log plot 

if particle is compact, should see a plateau in q 2 • I(q) vs. q 

Porod 


Debye 

constant ≈ q 2 · I(q) 

if particle is compact, should see a plateau in q 4 • I(q) vs. q and q 4 • I(q) vs. q 4 

Defines a power law relationship! 

I(q) = 1 

q P · S′ 

where 2 ≤ P ≤ 4 

lnI(q) =−P · ln(q)+ln(S ′ ) 

Does this work? 

Does a plateau form? 

Can it be quantified? 


Power Law Relationship 

log vs. log plot 

if particle is compact, should see a plateau in q 2 • I(q) vs. q 

Porod 


Debye 

constant ≈ q 2 · I(q) 

if particle is compact, should see a plateau in q 4 • I(q) vs. q and q 4 • I(q) vs. q 4 

Defines a power law relationship! 

I(q) = 1 

q P · S′ 

where 2 ≤ P ≤ 4 

lnI(q) =−P · ln(q)+ln(S ′ ) 

Does this work? 

Does a plateau form? 

Can it be quantified? 


SCÅTTER 


Perform PE Fit: 

•HRP (7 kDa) 

•GI (173 kDa) 

link at BIOISIS 

“Characterizing flexible and intrinsically unstructured biological macromolecules by SAS using the Porod-Debye law” 

Biopolymers 95: 559-571 Rambo, R.P. and Tainer, J.A. 


Single to Many Particle Scattering 

I macromolecules 

(q) = I macromolecule 

(q)⋅c⋅ I e ⋅ N l ⋅ P ⋅ d 

M ⋅ a 2 

substitute constants for k 

c - concentration (gm/cm 3 ) 

I e – 7.9 x 10 -26 (cm 2 ) 

N L - 6.0223 x 10 23 (mol -1 ) 

P – total energy over the irradiated area 

d - sample thickness (cm) 

M – molecular weight (gm • mol -1 ) 

a – distance to detector (cm) 

€ 

€ 



(q)⋅c⋅k 

rearrange 

1 

k ⋅ 1 c ⋅ I macromolecules (q) = I macromolecule (q) 

divide observed intensities by concentration 

extrapolates to single particle 

€ 

I sin gle particle 

(q) = k'⋅ 1 c ⋅ I sample (q) 

limit as q → 0 

I sin gle particle 

(0) = (Δρ) 2 ⋅ V 2 = (Δη e 

) 2 

Allows for the determination of M.W. by either: 

1. Standard curve with proteins of known M.W. 

• assume particles have same packing density 

2. Determination of I(0) on an absolute scale. 

• convert signal into number of electrons 


Scattering Contrast 

Mass Estimation and I(0) 

I particle (0) = (∆ρ) 2 V · 

∫ dmax 

0 

P (r) · 1 dr =(∆ρ) 2 · V 2 

In real experiments, Iparticle(0) has to be scaled by concentration, c 

I particles (0) = c · I particle (0) = c · (∆ρ) 2 · V 2 

Estimated by 

Guinier method 

I particles (0) 

c 

divide both sides by c = [particle] 

=(∆ρ) 2 · V 2 

For a given protein, ratio is a constant at a specific (Δρ) 2 

Mass, kDa 

I particles (0) 

c 

=(∆ρ) 2 · V 2 ∝ Mass 

This relationship can be used to make a standard curve to determine: 

• mass of either protein, RNA or particles of same composition 

• requires accurate knowledge of concentration 

I(0)/c 

20 


Scattering Contrast 

Alternative method for mass determination from I(zero) 

• use a single standard (xylanase) 

• do a dilution series (e.g., 2/3rds) 

• determine slope 

Performed as a 2/3rds dilution series: 

Xylanase in Two Different Buffers 

52 ul sample 

34 ul sample 

17 ul buffer 

34 ul sample 

17 ul buffer 

34 ul sample 

17 ul buffer 

∆I protein (0) 

∆c 

= m protein ∝ Mass 

I(0) 

Mass unkn = m unkn 

m std 

· Mass std 

SAXS data must be collected in same buffer as sample. 

relative concentration 

21 


Vc: A Novel SAS Ratio 

“Accurate Assessment of Mass, Models and Resolution by SAS” Nature, 496, 477-481 Rambo, R.P. and Tainer, J.A. 

Vc 

MD simulation of SAM-1 

• 8 different protein and RNA samples 

• 4 to 7 different concentrations 

67% variance is contained within 2% mean 

Vc sensitive to conformational state like Rg 

CONCENTRATION INDEPENDENCE! 

22 


Direct Mass Determination 

the power law distribution 

9446 PDB entries range from 8 to 400 kDa (protein only) 

• QR scales with mass, linear via power-law distribution. 

• Using actual data, 9% mass error with previously frozen samples. 

• Linear relationship covers a large mass range 20 to 1,000 kDa. 

• Effective for RNA samples 5% error. 

Simulated SAXS 

Use to infer mixtures: 

...expect 26 kDa and get 40 kDa 

• monomer ↔ dimer? 

Protein 

taken from BioIsis.net 

purified via SEC-MALS 

RNA 

purified via SEC-MALS 

23 

Experimental SAXS 


SCÅTTER 


Perform MW 


•preQ1 (25 kDa) 


“Characterizing flexible and intrinsically unstructured biological macromolecules by SAS using the Porod-Debye law” 

Biopolymers 95: 559-571 Rambo, R.P. and Tainer, J.A. 


Distance Distribution Function 

2° Structure Molecular Envelope Distance Distribution P(r) Function 

p(r) 

r, Ǻ 

Goal is to recover 

(decode) signal 

band-limited function (signal) 

I particle (q) =V · 

∫ dmax 

0 

ρ(r) · 

sin(q · r) 

q · r 

dr 

Properties of P(r) 

P(-r) = - P(r) thus P(r) is an “odd” function i.e., f(x) = x 3 , sin(x),... 

Defined on 0 < r < dmax 

How do we calculate P(r) from I(q) data? 


Indirect Fourier Transform 

A measured SAXS curve determines a unique P(r)-distribution. 

A P(r) distribution (from a model) can be used to determine a scattering curve. 

GNOM (Svergun) 

• guess at the distribution 

• uses L2 regularization 

GIFT (Glatter’s method) 

• uses cubic splines 

Expect a smooth curve 

Minimal oscillations 

No negative values 

Iterative process in determining dmax 

Moore’s Method 

• uses Fourier sine series 

• true propagation of I(q) 

uncertainties 

My Method 

• use Shannon theorem 

Implemented in Scatter 

Developing L1 regularization 

26 


Information Content 

Using Shannon Sampling theorem, P. Moore determined the number 

of independent parameters that can be extracted from a single SAXS 

curve. 

N 

I(q) = 8π ⋅ a n 

⋅ 1 q ⋅ π ⋅ n ⋅ d max 

(−1) n +1 N 

⎡⎡ 

⋅ sin(d max 

⋅ q) ⎤⎤ 

⎛⎛ 

∑ ⎢⎢ 

⎥⎥ 

p(r) = 8π ⋅ r ∑ a n 

⋅ sin π ⋅ r ⋅ n ⎞⎞ 

⎜⎜ ⎟⎟ 

⎣⎣ (π ⋅ n) 2 − (d max 

⋅ q) 2 

⎦⎦ 

n=1 ⎝⎝ d max ⎠⎠ 

Moore, P.B. J. Appl. Cryst. (1980). 13, 168-175 

n=1 

How large should N be? 

Represent SAXS data using a Fourier sine series 

€ 

Consider the denominator… 

(π ⋅ n) 2 −(d max 

⋅ q) 2 ⇒ (π ⋅ n) 2 ≠(d max 

⋅ q) 2 

q max d max 

n 

0.32 71 7 

The inequality naturally limits the expansion. 

Notice, increasing d max naturally increases n (same for q max ) 

0.32 

240 

25 

Noisy coding channel theorem 

Recovery of SAXS signal is largely independent of noise 

Δq 

xylanase (dmax: 43) and S-to-N: 0.9 C = 0.068 bits per Å -1 

Δq at SIBYLS ~ 0.006

I/σ 1.06 

I/σ 3.3 

Recover identical P(r) 

distributions (overlay) 

Dependence on how the recovery (decoding) is achieved! 

i.e., GNOM, SCATTER, FOXS, CRYSOL 

28 


SCÅTTER 




Dimensionless Kratky 

scale free analysis 

Receveur-Brechot V, Durand D. How random are intrinsically disordered proteins? A small angle scattering perspective. 

Curr Protein Pept Sci. 2012 Feb;13(1):55-75. 

Durand D, et al. J Struct Biol. 2010 Jan;169(1):45-53. 

Multiply I(q) by (q•Rg) 2 and divide by I(0) 

I(q) is scaled by c and V 



(q)⋅c⋅ k 

€ 

Divide by I(0) 

I particles (0) = c · I particle (0) = c · (∆ρ) 2 · V 2 

I(q) is independent of concentration and 

normalized to V 

q, Å 

Still have units of Å -2 , multiply by Rg 2 

What does it all mean? 

Use Guinier approximation to get some insights... 

30 



Guinier approximation relates scattering to Rg 

Derivation shows all particles that can be 

approximated by Guinier relation should have a 

peak value occurring at: q•Rg = 1.732 

peak value of 3•e -1 = 1.104 

Really about particles that can be 

approximated by the same correlation function 

such as: 

31 



only a button away 

unfolded-flexible 

RAD51-AP1 

1.104 

SAM riboswitch (-Mg 2+ ) 

xylanase 

(21 kDa) 

MBP-RAD51-AP1 

SAM Riboswitch (-SAM) 

SAM Riboswitch (+SAM) 

folded-compact 

√3 

glucose isomerase (173 kDa) 

Flexible, unfolded bounded: 1.104 < peak < 2 (Debye equation Gaussian chain) 

32 



using Volume-of-Correlation 

q 2 ⨯Vc I(q)/I(0) 

glucose isomerase (173 kDa) 

xylanase (21 kDa) 

SAM Riboswitch (+SAM) 

SAM Riboswitch (-SAM) 

SAM riboswitch (-Mg 2+ ) 

similar shapes but not exactly. 

GI is more spherical than XY 

MBP-RAD51-AP1 

RAD51-AP1 

Thursday, August 1, 13 

q 2 ⨯Vc 

Peak is inversely proportional to S-to-V ratio 

Max value is 0.82 (sphere) 

For a fixed molecule, any decrease suggests increase in surface area 

Illustrate differences better than previous (see SAM) 

Fully unfolded particle should have largest S-to-V ratio (low on graph) 

(Presented in our SAXS database paper in 2013) 

33

SCÅTTER 


Perform DK 


•preQ1 (25 kDa) 



References 

“Super-Resolution in SAXS and its application to Structural Systems Biology” 

Annual Review of Biophysics, Volume 42, 415-441 Rambo, R.P. and Tainer, J.A. 

“Accurate Assessment of Mass, Models and Resolution by SAS” 

Nature, 496, 477-481 Rambo, R.P. and Tainer, J.A. 

“Small-Angle Scattering for Structural Biology — expanding the frontier while 

avoiding the pitfalls.” 

Protein Sci. 2010 19(4):642-57. Jacques DA, Trewhella J. 

“Small Angle X-ray Scattering” 

Book circa 1982 Glatter O. and Kratky O. (very technical, freely available online) 

“Structure Analysis by Small-angle X-ray and Neutron Scattering” 

Book circa 1987 Fiegin LA. and Svergun DI. (very technical, freely available online) 

35

SAXS and Scatter

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