Bio-SAXS
Absolute Units, Mass Retrieval,
Globularity, Distribution Function
Javier Pérez
Beamline SWING, Synchrotron SOLEIL, Saint-Aubin, France
BioSAXS Workshop, Honolulu, July 2013
Mass retrieval from Guinier analysis
  Q 2 Rg 2 

I (Q)  I (0) exp 
3


Prof. André Guinier
1911-2000
Orsay, France
Absolute Unit : cm-1
Classical electron radius


c  M  r02
2
I (0) 
 v p  prot  buf 
NA
Rg
2


  
  r    d r
V
r 2  prot r  buf d r
V
prot
buf
Mass concentration Electronic density contrast
Protein specific volume
I(0) gives an independent estimation
of the molar mass of the protein
(only if the mass concentration, c,
is precisely known …)
Rg depends on the volume
AND on the shape of the particle
1
3
For globular proteins : Rg (Å) ≈ 6.25 ∗ 𝑀 , 𝑀 𝑖𝑛 𝑘𝐷𝑎
For unfolded proteins : Rg (Å) ≈ 8. 05 ∗ 𝑀0.522
Bernado et al. (2009), Biophys. J., 97 (10), 2839-2845.
Typically :
M (kDa) = 1500 * I0 (cm-1) / C (mg/ml)
BioSAXS Workshop, Honolulu, July 2013
Calibration of the set-up using water scattering
SWING Liquid scattering (theory): I(Q)  constant at small Q  r0
IH2O,theory = 0.0163 cm-1
Molecular density
2
Z 2  A  kTT
2
Isothermic compressibility
Water is used as primary reference
to get the absolute intensity scale
•Capillary diameter =1.6 mm
•Average of 2 frames of 2s
•Empty capillary subtracted
•Normalized by solid angle
•Normalized by transmitted intensity
Example:
IH2O,exp
= 0.042 Exp. Units
IH2O,exp
= Kexp* IH2O,theory
 Here : Kexp=2.56 Exp.Units / cm-1
For any sample in that capillary : Itheory(cm-1) = Iexp / Kexp = Iexp / 2.56
BioSAXS Workshop, Honolulu, July 2013
Example of Mass retrieval from Guinier analysis
Hen egg-white lysozyme
M=14.3 kDa
•C =5.6 g/l
•Average of 8 frames of 2s
Ln I
•Buffer subtracted
•Normalized by solid angle
•Normalized by transmitted intensity
Rg 2 2
LnI (Q)  LnI (0) 
Q
3
Rg = 15.1 ± 0.03 Å
Iexp(0) = = 0.0543 cm-1
From I(0) provided the set-up was calibrated to give
I(Q) in absolute units (cm-1).
Mexp(kDa) = Iexp (0) *1500 / c,
 Mexp = 14.6 kDa
Q2
From Rg, supposing the protein is globular:
MRg(kDa) = (Rg / 6.3)3
 MRg = 13.8 kDa
BioSAXS Workshop, Honolulu, July 2013
Kratky Plot
SAXS provides a sensitive means to evaluate the degree of compactness of
a protein:
o To determine whether a protein is globular, extended or unfolded
Prof. Otto Kratky
1902-1995
Graz, Austria
o To monitor the folding or unfolding transition of a protein
This is most conveniently represented using the so-called Kratky plot:
Q2 I(Q) versus Q
Folded particle : bell-shaped curve (asymptotic behaviour I(Q)~Q-4 )
Random polymer chain : plateau at large q-values (asymptotic behaviour in I(Q)~ Q-2 )
Extended polymer chain : increase at large q-values (asymptotic behaviour in I(Q)~ Q-1.x )
BioSAXS Workshop, Honolulu, July 2013
Kratky Plots of folded proteins
0.0025
G-Actin
ASNP
ASDG
CDA2
BCDA3
0.0015
2
Q I(Q) / I(0)
0.002
0.001
0.0005
0
0
0.1
0.2
0.3
0.4
0.5
Q
Folded proteins display a bell shape. Can we go further?
BioSAXS Workshop, Honolulu, July 2013
Dimensionless Kratky Plots of folded proteins
Introduced for biology in Durand et al. (2010), J. Struct. Biol. 169, 45-53.
The relation MRg(kDa) ≈ (Rg / 6.3)3 only works
for the globular structures, not the elongated
For globular structures, DLKPs
fold into the same maximum
1.6
G-Actin
Rg=23.2 Angs, Mass=41.7 kDa
ASNP
Rg=26.0 Angs, Mass=71.4 kDa
ASDG
Rg=35.6 Angs, Mass=146.6 kDa
CDA2
Rg=39.1 Angs, Mass=98.9 kDa
BCDA3
Rg=51.7 Angs, Mass=144.4 kDa
1.4
2
(QRg) I(Q) / I(0)
1.2
1.1
1
0.8
0.6
0.4
0.2
0
0
2
1.75
4
6
8
10
QRg
The maximum value on the dimensionless bell shape tells if the protein is globular.
BioSAXS Workshop, Honolulu, July 2013
Dimensionless Kratky Plots of (partially) unfolded proteins
Receveur-Bréchot V. and Durand D (2012), Curr. Protein Pept. Sci., 13:55-75.
unfolded
3.5
PolX
p47
p67
XPC
IB5
2.5
2
g
2
(qR ) I(q)/I(0)
3
1.5
1.1
1
0.5
0
0
1.75 2
4
qR
6
8
10
globular
g
The bell shape vanishes as folded domains disappear and
flexibility increases.
The curve increases at large Q as the structure
extends.
BioSAXS Workshop, Honolulu, July 2013
Kratky Plot : NCS heat unfolding
!
In practice, thin Gaussian
chains do not exist.
In spite of the plateau at T=76°C,
NCS is not a Gaussian chain when
unfolded, but a thick chain with
persistence length
Pérez et al., J. Mol. Biol.(2001), 308, 721-743
BioSAXS Workshop, Honolulu, July 2013
Cytochrome c folding kinetics
44 ms after mixing
160 µs after mixing
S. Akiyama et al. (2002), PNAS, 99, 1329-1334.
ApoMb : T. Uzawa et al. (2004), PNAS, 101, 1171-1176
BioSAXS Workshop, Honolulu, July 2013
Porod Invariant and Porod Volume
Porod law:
Homogeneous object  I(Q) ~ Q-4 at high Q values

Q   I (Q)  Q dQ  2   1    r  
2
2
2
e
2

0
Porod invariant
Number of proteins
N  Vobj
V
Protein volume
Solution volume
Protein volumic concentration
• calculated from experimental data in absolute units
• does not depend on shape, only on contrast
Since
Then
I0  re φVobj 
2
Vobj 
2
2  I (0)
2
Q
• valid for diluted systems
• does not require absolute units
I(0) / Q therefore gives an independent
estimation of the volume of the protein
But :
• Requires Porod law is fulfilled
• Not valid for unfolded proteins
BioSAXS Workshop, Honolulu, July 2013
Porod Invariant

Q   I (Q)  Q dQ
2
Porod invariant
0
Kratky representation : I(Q)·Q2 vs Q
I(Q)·Q2
The Porod Invariant is the
integral of this curve
Program Primus
Atsas suite of programs
www.embl-hamburg.de/biosaxs/software.html
Q
BioSAXS Workshop, Honolulu, July 2013
Molecular Weight estimation based on Porod invariant
http://www.ifsc.usp.br/~saxs/saxsmow.html
• does not require knowledge of concentration
• relies on Porod Volume theory + structural database
• does not work for proteins with unfolded domains
Recent methods for MW estimation based on similar though different grounds were developed  Track B.
Rambo R. And Tainer J. (2013), Nature, 496, 477-481.
BioSAXS Workshop, Honolulu, July 2013
Distance Distribution Function p(r)
The distance distribution function p(r) is
proportional to the average number of
atoms at a given distance, r, from any
given atom within the macromolecule.
P(R)
Cylindre
Sphere
solide
Disque
Domaines
Protein
Dmax
R
The pair distribution function characterises the shape of the particle in real space
BioSAXS Workshop, Honolulu, July 2013
Relation between p(r) and I(Q)
sin Qr 
IQ  4 re φ   obj (r )r
dr
Qr
Vobj
Intensity is the Fourier Transform
of self-correlation function γobj(r):
It can be shown that :
Then :
2
p(r )   obj (r )r 2
IQ  4 re φ  p(r )
D
2
0
And :
2
r2
p(r) 
2
2 2φ re


0
sin Qr 
dr
Qr
Fourier Transform for
isotropic samples
sin Qr 
Q I (Q)
dQ
Qr
2
p(r) could be directly derived from I(Q). Both curves contain the same information.
However, direct calculation of p(r) from I(Q) is made difficult and risky
by [Qmin,Qmax] truncation and data noise effects.
BioSAXS Workshop, Honolulu, July 2013
Back-calculation of the Distance Distribution Function
Glatter, O. J. Appl. Cryst. (1977) 10, 415-421.
Main hypothesis : the particle has a « finite » size, characterised by Dmax.
Prof. Otto Glatter
Guinier Prize 2012
Graz, Austria
• Dmax is proposed by the user
• p(r) is expressed over [0, DMax] by a linear combination of orthogonal functions
M
ptheoret (r )   cn n (r )
1
• I(Q) is calculated by Fourier Transform of ptheoret(r)
I (Q)  4 re 
2
Dmax

0
sin(Q  r )
ptheoret (r )
dr
Qr
Svergun (1988) : program "GNOM"
Dr. Dmitri Svergun
M ~ 30 - 100  ill-posed LSQ  regularisation method
Hamburg, Germany
+ "Perceptual criteria" : smoothness, stability, absence of systematic deviations
• Each criterium has a predefined weight
• The solution is given a score calculated by comparison with « ideal values »
BioSAXS Workshop, Honolulu, July 2013
Distance Distribution Function
Experimental examples
Heat denaturation of Neocarzinostatin
GBP1
Pérez et al., J. Mol. Biol. (2001) 308, 721-743
BioSAXS Workshop, Honolulu, July 2013
Distance Distribution Function
Experimental examples
Bimodal distribution
Topoisomerase VI
0.0008
0.0007
70 Å
P(r) / I(0)
0.0006
0.0005
0.0004
0.0003
0.0002
0.0001
0
0
50
100
150
200
250
r (Å)
M. Graille et al., Structure (2008), 16, 360-370.
BioSAXS Workshop, Honolulu, July 2013
Distance Distribution Function
Scattering curves obtained on different complexes Spire-Actin and Actin alone
Complexes
Radius of gyration
Maximum diameter
75.5 Å
285 Å
55.5 Å
210 Å
38.9 Å
130 Å
25 Å
75 Å
23.1 Å
70 Å
Histogram of intramolecular distances and ab initio molecular enveloppes determined using DAMMIF
P(R)
KindABCD-A4
P(R)
Dmax = 285
BCD-A3
P(R)
Dmax = 210
r in Å
• Organization of actin oligomers
CD-A2
P(R)
Dmax = 130
r in Å
D1-A1
Dmax = 75
r in Å
r in Å
BioSAXS Workshop, Honolulu, July 2013
Distance Distribution Function
The radius of gyration and the intensity at the origin can be derived from p(r)
using the following expressions :
R
2
g


Dmax
0
2
r 2 p (r )dr
Dmax
0
p (r )dr
and
I0   4 re φ  p (r )dr
2
D
0
This alternative estimate of Rg makes use of the whole scattering curve, and is less
sensitive to interactions or to the presence of a small fraction of oligomers.
Comparison of estimates from Guinier
analysis and from P(r) is a useful cross-check.
BioSAXS Workshop, Honolulu, July 2013
To what extent does SAXS give information of flexible proteins ?
The example of IB5
• Salivary, proline-rich protein, 70 residues (pink)
First analyzed as a thick worm-like polymer chain
• Intrinsically Disordered Protein (IDP)
L = 190 Å
b = 30 Å
Rc = 2.7 Å
Rg = 30 Å
Rg larger than usual
IDP :
Rg = 2.54(n)0.522 Å
 Rg = 23 Å
H. Bose et al., Biophys. J. (2010), 99,656-665
BioSAXS Workshop, Honolulu, July 2013
To what extent does SAXS give information of flexible proteins ?
The example of IB5
• Salivary, proline-rich protein, 70 residues (pink)
• Intrinsically Disordered Protein (IDP)
Data-compatible average structure
Model dependent structure distribution
Which is the best way to present the results is an open question
BioSAXS Workshop, Honolulu, July 2013
Comments
 Analysis and modeling require a monodisperse and
ideal solution, which has to be checked
independently.
 SAXS is at his best when it is used to distinguish
between several preconceived hypotheses.
BioSAXS Workshop, Honolulu, July 2013
BioSAXS Workshop, Honolulu, July 2013