Post on 03-Apr-2020
1
A brief introduction to SAXS
M.H.J. Koch
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SAXS
02
20
02
22
0
12
2cos12 I
r
rI
r
)θ(r)θ(Ie =
+=
classical electron radius = 2.82 10-15 m
For SAXS this factor =1
sin2θ = 2θ cos 2θ = 1
I0=Iinexp(-µt)
r>>t
∆λ/λ~0
For a single electron:very small!!
3
Electrons in an electromagnetic fieldare accelerated and therefore emit radiation: they scatter. The spatial distribution of the scattered intensity depends on the geometry of theexperiment. For unpolarized incident radiation the spatial distributionon the equator is:
2Ө
(1) I)ω(ω
ωr1
2)θ2(cos1
r)I(2 02220
4
2
220
−⋅⋅
+⋅=θ
where 2θ is the angle between the incident and the scattered beam.
corresponds to the
natural frequency (ν0=ω0/2π) of the oscillator and ω to the frequency of the incident radiation.
Obs. m
k=0ω
4
) ωω (ω
220
2
−
in the previous equation is the one describing the frequencydependence. For the AMPLITUDE (E with I= E·E*)
The natural frequency of the oscillator (ω0) corresponds to the binding strength of electrons in atoms and lies somewhere in the UV to X-ray region. If the incident radiation is visible light (λ ≈ 500 nm),ω << ω0 and the factor reduces to:
20
2
ωω
The amplitude of the scattered radiation at r is proportional to ω2 andin phase with the incident radiation. This is
Rayleigh scattering.
The scattered intensity is proportional to ω4 hence the blue sky.
The most interesting factor
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X-RaysIf the incident radiation is X-rays (λ ≈ 0.1 nm), ω0 << ωand the factor
) ωω (ω
220
2
−
The scattering amplitude is independent of the frequency and its phase is shifted by 180 degrees relative to the incident radiation. This is
Thomson scattering
= -1
As it is independent of the frequency of the incident radiation, the worldof X-rays is colorless with shades of gray (i.e. contrast) only and Eq.1 above simplifies to:
radiuselectron classical10817.2mce
r 152
2
0 m−==
02
220 I
12
)2(cos1r)I(2
r⋅
+⋅=
θθ
6
Energy out /energy in
0220e I
r1
r)(0I =For unpolarized X-rays at 2Ө = 0:
I0
1m2
r
Ie(2θ) r2dΩ
Detector
sample
22
0
e brI
) (2θIdΩdσ
=⋅=
b: scattering length =
r0 = 2.810-15 m for the electron
Energy scattered/unit solid angle/unit time
Energy incident/unit area/unit time
The differential scattering cross-section
has the dimension of an area and represents
For one electron: the amplitude of scattering |Ie(0)|1/2= 2.810-15|I0|1/2 and as the scattering amplitude ≡ f =
amplitude scattered by an objectamplitude scattered by an electron in identical conditions
fe =1
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Particle size (D) and wavelength of the radiation
When λ >> D all N electrons in the particle are accelerated in phasethe scattering amplitude is N times that of one electron.
When λ < D the electrons in the particle are no longer moving in phaseand one has to take the phase shift of the waves into account.
ObserverObserver
λ >> D Light scattering λ < D X-ray scattering
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Waves and Interference
Interferences lead to fringe patterns. This is
illustrated here with water waves.
When “solving” a structure the problem is to
go from the fringe pattern – in the case of
X-ray diffraction from the intensities of the
fringes – to the distribution of sources i.e.
of scatterers.
Similar effects are observed with optical
transforms obtained by shining coherent visible
(laser) light through small apertures (see e.g.
Cantor and Schimmel , Biophysical Chemistry,
Part II, Ch. 13).
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Interference and coherent scattering
r
2Θ
r.S
r.S0
λ)( 0SS
s−
=
λ2sinθ
=s
S/λ
S0/λ
The total amplitude from two centers (one at the origin and one at r) is thus:
)rs i exp(2πff)sr i exp(2πfF(s) 2eei
2
1ie ⋅+==∑
=
0SrSr ⋅−⋅Path difference =
In Thomson (coherent scattering) the scattered wave is 180° out of phasewith the incident wave.
Phase difference = )SrS(r 0⋅−⋅λ
π2
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The sum of amplitudes for N electrons:
F(s) is the Fourier transform of the distribution of electrons
) i exp(2πf)F(N
1ie irss ⋅=∑
=
The average of the exponential factor over all orientations of r relative to s
for randomly oriented particles (e.g. in solution) is:
sr
sr
π
π
2)2(sin
) i π2(exp =>⋅< irs
As this is a real number there is no phase problem but one has lost most of the structural information.
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Scattering factor
For λ = 0.15 nm f’ = 0.3641f” = 1.2855
drsrπ2
)srπsin(2rρ(r)π4f(s)F(s) 2
0∫∞
=≡
For an atom with a continuous radial electron density ρ(r):
1sin
lim0
=
→xx
x
and since
f(0) = Z
( )( ) ( )( )sRsRsR
sRsf sphere πππ
π2cos22sin
2
3)( 3 −=
For modeling purposes one oftenuses larger spherical subunits (beads,dummy residues, etc) for which:
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Anomalous scattering
The scattering factor fe must be modified to take anomalous scattering into account
fe = 1 + fe’ + ife”(f” is always π/2 ahead of the phase of the real part)
Note that for most practical purposes, f’ and f” are independent ofs = 2sinθ/λ
KronigKramers'dωω'ω
)'(ω"fω'π
2)(ω'f
weightatomicA number, sAvogadro' :N
t,coefficien absorption tric)(photoeleclinear :µ )ω(µcNr4πωA
0)ω,("f
22
A
A0
−−
=
=
∫
Near an absorption edge, the dissipative effects due to the rearrangement of the electrons can no longer be neglected.
13
Selenium f‘ and f‘‘
SeK
)ω(µcNr4πωA
0)ω,("fA0
=
'dωω'ω
)'(ω"fω'π
2)(ω'f
22∫ −=
Note the sign and absolute value of the corrections.
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Scattering from N spherical atoms:
F(s) is the Fourier transform of the distribution of the spherical atoms.Crystallographers call this the structure factor. Note that in SAXS the structure factor refers to structure of the solution. The intensity is, of course:
) i (s)exp(2πf)F(N
1ii irss ⋅=∑
=
ij
ijj rsπ2
)rsπ2(sin)-( i π2(exp =>⋅< rrs i
))-( i (s)exp(2π(s)ff)I(1i
j1j
ji∑∑= =
⋅=N N
rrss i
For random orientation
ij
ijN
1i
N
1jji sr2π
)srsin(2π)(s(s)ffI(s) ∑∑
= =
=
and
Debye (1915)
Crystal structure -atomic coordinates
Solution –Distance distribution only!
This is a real number!
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Short distances >> low frequencies dominate athigh anglesLarge distances >> high frequencies contribute only at low angles.
In isotropic systems, each distance d = rij contributes a sinx/x –like term to the intensity. A scattering pattern is a continuous function of s.
SAXS:ij
ijN
1i
N
1jji sr2π
)srsin(2π)(s(s)ffI(s) ∑∑
= =
=
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The wider a function in real space the narrower its transform in reciprocal space
1) The Fourier transform of the Dirac delta function
is the 1(x) function (i.e. the function which has a constant value of 1 over the interval [-∞,∞].2) Obviously, the Fourier transform of 1(x) is δ(x).
∫∞
∞−=∞= 1δ(x)dx δ(0)
3) The Fourier transform of a Gaussian is also a Gaussian
)/exp()())(exp( 222ak
akFaxFT π
π−==−
Note the relationship between the widths. If the Gaussian has a widthσR=(1/2a)½, its transform has a width σF=(a/2π2)½ and σRσF=1/2π.
The δ-function is an infinitely narrow Gaussian.
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Inte
nsi
ty
10.0
1.0
0.10.02 0.04 0.06Q (Å-1)
0.1 s ID2 (ESRF)
Kinetics of the Ca2+-dependent swelling transition of Tomato Bushy Stunt Virus
Perez et al.
Larger objects scatter at lower angles!
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In an ideal solutionThe solute particles are randomly oriented and their positionsare uncorrelated in space and time. Consequently their scattering inisotropic and incoherent. The total scattering intensity is the sum ofthe coherent scattering intensity of all molecules. It is a function of the scattering angle or modulus of the scattering vector only: I(s).
Usually one plots log(I(s)) vs s, because the intensity falls off rapidly due to the interferences.
i.e. for atoms one can neglect the s-dependence of fi
If one uses a continuous density distribution ρ(r) this becomes
ij
ijN
1i
N
1jji1 sr2π
)srsin(2πff(s)i ∑∑
= =
=
2112
1222111 2
)2sin()()()( rrrr dd
sr
srsi
V π
πρρ∫ ∫=
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Interactions of X-rays with matter
Sample
fluorescence
Incident beamTransmitted beam
Coherent scattering
I0
I=I0 exp(-µt)
Incoherent scattering
Structural information at the atomic/molecular level is in: coherent scattering
and to a limited extent in absorption/fluorescence near edges (EXAFS, XANES)
At lower resolution transmission/phase contrast imaging is also useful
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Choice of wavelength
Absorption!!Incoherent!!
Optimal thickness of the sample: topt =1/µ ≈ 1 mm for H2O @ 1.5Å
For oxygen
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Background
solution
buffer
with air gap
vacuum
Background arises from the incoherent(Compton) scattering from the sample and from coherent and incoherent scattering due to air gaps, windows, solvent ….
To obtain the coherent scattering of the solute normalize the intensities ofsolution and buffer to transmitted beam and subtract :
I(s)= [I(s)/I0T]solution – [I(s)/I0T]buffer
Divide by c to normalize for concentration
Lysozyme : 14 289 Da MW5 mg/ml solution in Acetatebuffer pH 4.5
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CONTRAST: < ρparticle(r) > - ρbuffer
Homogeneous solvent
Particle:
Solvent:
Only fluctuations in electron density contribute to the scattering:
rrsrs )diexp(2π)(ρ)(FV
pp ⋅= ∫
rrss )diexp(2πρ)(FV
bb ⋅= ∫ Ideally, a Dirac δ(0) and constantin practice not
Iobs(s) = Ip(s) –Ib(s)
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Isolution(s) Isolvent (s) Iparticle(s)
♦ To obtain scattering from the particles, solvent scattering must be subtracted to yield the excess scattering
ρp(r) = ρsolution(r) - ρb
where ρb is the scattering density of the solvent
Solvent scattering and contrast
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Scattering arises from fluctuations
In a perfectly homogeneous body the contribution of each smallscattering volume element to the scattering amplitude would always becancelled out by that of another one which is out of phase. This is why perfect crystals do not scatter visible light.Solutions of macromolecules or colloidal suspensions strongly scattervisible light because of fluctuations in concentration. The fluctuations are also what links scattering to diffusion (e.g. DLS) andthermodynamics (compressibility).
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EXCESS SCATTERING DENSITY
(Stuhrmann and Kirste, 1965)
)(ρρ-)(ρ)(ρρ) ρ( cbscx rrrr +=
shape
internalstructure
equivalent volume of solvent
)(ρ)(ρ )ρ-ρ( )ρ( scbx rrr += )(ρ)(ρ ρ sc rr +=
contrast
(s)I(s)Iρ(s)IρI(s) scsc2 ++=The corresponding intensity is:
For a homogeneous particle: Is(s) =Ics(s) =0
ρc (r) has a value of 1 inside the particle and 0 outside and thusrepresents the shape. ρs(r), the internal structure, represents the fluctuations around the average electron density of the particle ρx.
26
A communication problem
(s)I(s)Iρ(s)IρI(s) scsc2 ++=
For a homogeneous particle (i.e. a shape): Is(s) =Ics(s) =0
Is(0) =Ics(0)=0 always!
These two terms are the mostimportant in SAXS
This is essentially whatcrystallographers talk about
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Scattered intensity
)F()F()(F)F()(i *1 sssss −⋅=⋅=
)()()]([
)()]([* ssr
sr
FFFT
FFT
=−=−
=
ρ
ρρ(r) is the excess electron density
Hence )]()([)]([)]([)( *1 rrrrs −=−⋅= ρρρρ FTFTFTi
γ(r): Autocorrelation function of the excess electron density and:
γ (r)
∫ ⋅==
rV
rdViFTsi )2exp()())(()(1 rsrr πγγ
drrs
rsrpsi ∫
∞
=0
1 2)2sin(
)(4)(π
ππ )()( 2
rrrp γ=
After spherical averaging:
with( ) ( )rγ γ= r
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Autocorrelation: Shift-Multiply-Integrate
∫ +==−=
u
uuurrrrrrV
dV)()()()()()()( * ρρρρρργ o
-r
u
Dmax
2Dmax
r
ρ(u)
γ(r) p(r)
r
× r2Dmax
0
29
2 2 20( ) ( ) ( )p r r r r r Vγ γ ρ= =
rij ji
r
p(r)
Dmax
For a homogeneous body ρ(r) = ρ, γ(0) = ρ2V
γ(r) : probability of finding a point at r from a given point (i).Number of volume elements i ∝ V;Number of volume elements j ∝ 4πr2.Number of pairs (i,j) separated by the distance r ∝ 4πr2Vγ0(r)=(4π/ρ2) p(r).
0( ) ( ) (0)r rγ γ γ= Characteristic function
Distance distribution function
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I(s) and p(r)dr
sr
srrpsI
D
π
ππ
2)2sin(
)(4)(max
0∫=
drsr
srsIrrp ∫
∞
=0
2
2)2sin(
)(2)(π
π
For a homogeneous particle p(r) represents the histogram of distancesbetween pairs of points within the particle.
31
At low angles
( ) ( )L−+−=
1202
62
12
)2sin( 42srsr
sr
sr ππ
π
π
ij
ijN
1i
N
1jji1 sr2π
)srsin(2πff(s)i ∑∑
= =
=
At very low angles one can use the approximation: exp(x) = 1+ x + x2/2 +… which yields the Guinier formula
)sRπ34
(1Vρ(s)i 22g
2221 K+−=
massMolecular ~Vρ)0( )sRπ34
exp( I(0))( 2222g
2 =−= IwithsI
Since2
∑
i
if
Rg is the mean squared distance to the centre of scattering massweighted by the excess electron density.
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Valid for a sphere for
0 < 2ππππRgs<1.2
The plot of ln[I(s)] vs s2
[ ] [ ]24
3π
≅ − 2 2
gln I(s) ln I(0) R s
yields two parameters :
y-intercept: I(0)Rg: from the slope
Guinier plot for ideal solutions
33
Forward scattering I(0) and molar mass (M)
2
´
´
´)()()0(excessi
ir
V V
r fdVdVrrI
r r
== ∑∫ ∫ ρρ
2
02
0 )()()0(
−=−= ρρν ρ
AN
MmmI
VN
NMc
A
=
[ ]20 )()0( ρρν ρ −=
AN
cMVI
concentration (w/v)
Molar mass
Avogadro’s number
Number of electrons contrastPartial specific volume
∫=max
0
)(4)0(D
drrpI πor
number ofmolecules
Tip: Use a sample of similar contrast e.g.a known protein ( BSA) as standard to calibrate the measurements:MMsample= MMstand.I(0)/c)sample/(I(0)/c)stand.
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Radius of gyration
2121
21
2
21212
)()(2
)()(
rrrr
rrrrrr
dd
ddRg
∫ ∫∫ ∫ −
=ρρ
ρρ
If one places the origin of the coordinates at r0,,the centre of scatteringmass of the particles, this yields the expression for the radius of gyrationwhich is obtained from a Guinier plot, Or even better, calculated fromthe whole experimental scattering pattern:
∫
∫=
V
Vg
d
dr
Rrr
rr
)(
)( 2
2
ρ
ρ
∫
∫=
max
max
0
0
2
2
)(2
)(
D
D
g
drrp
drrpr
R
Rg is the second moment (standard deviation) of the electron density distribution.
35
Guinier’s formula and the contrast
1sRπ34
for sRπ34
exp Vρ )sRπ34
(1VρI(s) 2222222222222 ≤
−=+−= K
where R is the radius of gyration and I(0) the forward scattering is: 22Vρ
If the contrast changes so do I(0) and R:
2
2c
2
ρ
β
ρ
αRR −+=
second moment ofinternal structure
displacement of centre of scattering mass with contrast
α = 0α = 0α = 0α = 0 α > 0α > 0α > 0α > 0 β=0β=0β=0β=0 β>0β>0β>0β>0αααα < 0
radius of gyration at infinite contrast
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Contrast: X-rays
Substance
X-
rays
Proteins 2.5
Nucleic acids 6.7
Fatty acids -1.1
Carbohydrates 4.5
Average contrast (X 1010 cm-2) of biological macromolecular assemblies in water.
One can change the contrast e.g. by adding salts like CsBr but this has disadvantages like increasing absorption,fluorescence and changing ionic strength.
The alternative is to use anomalous scattering (see e.g Stuhrmann HB.Acta Crystallogr A. 2008, 64:181-91)but beware of radiation damage.
TIP: If you need to change the contrastUSE NEUTRONS!
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Typical values of radii of gyrationMw Rg(nm)
Ribonuclease 12700 1.48Lysozyme 14800 1.45B-lactoglobulin 36700 2.17Bovine serum albumin 68000 2.95
Myosin 493000 46.8Brome mosaic virus 4.6 106 13.4Tobacco mosaic virus 3.9 107 92.4
For a sphere of radius R: 22
53
RRg =
38
For an infinite rod
)(1)()(),0,0( zyxf δδ=∞ )()(1)(1)0,,( ZYXF δ=∞∞
or a long fiber with its axis along z, the transform is limited to theX,Y plane (i.e. the cross-section)
z
xy
Z
X
YFT
Remember δ(x)!
In such a case the radius of gyration of the cross-section and themass/unit length can be derived using a representation analogousto the Guinier plot with a plot of sI(s) vs s2 to obtain
)2exp()( 222sR
L
mssI cπ−= For a circular cross-section
Rc=R/√2
39
For a flat object like a membrane
)()(1)(1)0,,( zyxf δ=∞∞ )(1)()(),0,0( ZYXF δδ=∞
With a small thickness (T) along z, the transform is limited to theZ-axis (i.e. the thickness)
z
xy
Z
X
YFT
Remember δ(x)!
In such a case the radius of gyration of the thickness and themass/unit area are obtained from a plot of s2I(s) vs s2 to obtain
)4exp()( 2222sR
A
msIs Tπ−=
12/TRT =with
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Shape scattering
At larger s-values the scattering ofa particle with ρs ≠ 0 oscillates around a straight line given byPOROD’s law:
s4I(s)= Bs4 +A
Subtract a constant equal to theslope of the Porod plot toobtain an approximation to the
SHAPE scattering
41
Mixtures
)0(/)0( )()(N
1i
22
1
IRInRsInsI iii
N
i
ii ∑∑==
==
ni: number concentration of the ith species, with forward scatteringIi(0) and radius of gyration Ri.
These formula illustrate that a small quantity of high molecularmass or hight contrast particles will have a large influence onthe scattering curve.
For modeling it is thus indispensable to make sure that the solutions do not contain aggregates!
42
Proteins at low resolutionHydration shell
Crowding max. conc. 300-500mg/ml
IONS:Kosmotropes e.g. Na+
Chaotropes e.g. K+
OSMOLYTESe.g. free amino acidspolyhydroxy alcoholsmethylated ammoniumand sulfonium compoundsurea.
solvent
Interactions/ stability/activitymodulated by
FOLDING
Coupled equilibriaNon-contact interactions
43
In the case of an unfolded protein : models developed for polymers
Gaussian chain : linear association of N monomers of length l with no persistence length (no rigidity due to short range interactions between monomers) and no excluded volume (i.e. no long-range interactions).
Debye formula : where
I(s) depends on a single parameter, Rg .
Valid over a restricted s-range in the case of interacting monomers
Limit at large s :
2
( ) 2( 1 )
(0)xI s
x eI x
−= − + ( )2
2 gx R sπ=
22
2 2
1 1 (2 )lim [ ( )]
2g
s
g
R ss I s
R
π
π→∞
−=
I(s) varies like s-2 instead of s -4 for a globular particle (Porod law).
Scattering by an extended chain
44
arrows : angular range
used for Rg determination
Pérez et al., J. Mol. Biol.(2001)
308, 721-743
Debye law for extended chains: NCS unfolding
2
( ) 2( 1 )
(0)xI s
x eI x
−= − +
( )2
2 gx R sπ=
Neocarzinostatine : small (113 residue long) all-β protein.
45
Is sensitive to the degree of compactness of a protein.
Globular particle : bell-shaped curve
Gaussian chain : plateau at large s-values but a plateau does not imply a Gaussian chain!
Kratky plot: s2I(s) vs s
Pérez et al., J. Mol. Biol.(2001), 308, 721-743
In thermal unfolding of NCS the Kratky plot has a plateau althoughunfolded NCS is not a Gaussian chain when unfolded.A thick persistence chain is a bettermodel in this case.
46
Protein folding: cytochrome c
Akiyama, S. et al. (2002) Proc. Natl. Acad. Sci. USA, 99, 1329-34
47
Protein folding: cytochrome c
Akiyama, S. et al. (2002) Proc. Natl. Acad. Sci. USA 99, 1329-34
?InitialEnsemble ?
expect >900 Å2
for randomchain !
48S. Akiyama et al. (2002), PNAS, (2002), 99, 1329-1334.
160 µs after mixing
44 ms after mixing
cytochrome c folding
49
Labeling
D L1 L2D X X XL1 X X XL2 X X X
D L1 L2D X 0 0 L1 0 0 0L2 0 0 0
D L1 L2D X X 0L1 X X 0L2 0 0 0
D L1 L2D X 0 XL1 0 0 0 L2 X 0 X
+ +-
L1 D L2
L1-D or L2-D
L1-D-L2
D InterferenceFT
Distance betweenL1 and L2
Matthew-Fenn R.S. et al. (2008)Science 232, 446-9
50
35
30252015
10bp
Note the difference in position and width(variance) of the distribution of end to end distances as a function of the number ofbase pairs between labels. In absence of applied force DNA is at least1000 softer than in single molecule stretching experiments. Stretching is cooperative over more than two turns of thedouble helix. DNA is not an elastic rod.
Interference
FT
51
What SAXS/SANS have to offer
SAXSSANS
Electron microscopy
NMR
Modelling
Light scattering
Molecular Dynamics
ultracentrifugation
viscosity
Electric or flowdichroism
chromatography
Perturbation methodsLabeling methods