Small Angle Neutron Scatteringcins.ca/docs/ss2011/lectures/Kucerka.pdf · Small Angle Scattering...
Transcript of Small Angle Neutron Scatteringcins.ca/docs/ss2011/lectures/Kucerka.pdf · Small Angle Scattering...
Small Angle Neutron ScatteringNorbert Kučerka
NRC – Canadian Neutron Beam Centre at Chalk River
CINS/CNBC Summer School 2011
Outline
• Common Features of SAS• Advantage of SANS• Scattering Principles• SANS Data
• Repeating Lamellae• Model-Independent Analysis• Model-Based Analysis• Scattering Form Factors• Joint Refinement of SANS and SAXS
• Summary
Common Features ofSmall Angle Scattering
(SAS)
- radiation (e.g. neutrons, X-rays, light) is elastically scattered by a sample and the resulting scattering pattern is analysed to provide information about the size, shape and location of some components of the sample
- Q range is usually between 0.003 and 0.3 Å-1 – providing a wide range of length scales (~10 to 1000 Å)
- powerful tool for investigating the average structure of the entire ensemble of particles in solution (biologically relevant environment)
- orientation of the studied structures is either isotropic or poorly ordered
- type of the sample, sample environment and the information that can ultimately be obtained depend on the nature of the radiation, however different radiations often provide complementary results
NeutronSmall Angle Scattering
Scattered intensity is proportional to
“the square of the difference between scattering length density of studied material and medium”
0 20 40 60 80 100-2.0
0.0
2.0
4.0
6.0
8.0-CD2-
-CH2-
phospholipid
proteinwater solution
RNADNA
deuterated RNA
deuterated protein
D2O / (D2O + H2O) [%]
Neu
tron
Scat
terin
g Le
ngth
Den
sity
[10
-6xÅ
-2]
low contrast
2 ways to increase contrast
Neutron contrast variation can be done without changing the chemical properties of the system, because the neutron scattering lengths of isotopes can be very different.
Two ways to get small scattering vectors:
1.) large wavelengths λ
2.) small angles
Scattering PrinciplesSANS Geometry
Scattering vector, q
ki
ks
θ q
ki
2θsinλ
4πq|| ==q
q = ks – ki
θ R (detector cell)D (sample-to-detector)
2DR
2θsin ~
Scattering Principles2D SANS Instrument
For λ = 8 Å and θmin~ 0.25o, qmin ~ 0.003 Å-1.Max. attainable length scale = 2π/qmin ~ 2000 Å.
VelocitySelector
NeutronGuide
2-DDetector
Sampletable
circularly averaged into 1-D
Neutron Beam from Reactor
q = 4πλ
sin θ2
Scattering PrinciplesSANS modified TAS
For λ = 5.2 Å and θmin~ 0.28o, qmin ~ 0.006 Å-1.Max. attainable length scale = 2π/qmin ~ 1000 Å.
near-source end
near-sample end
M.-P. Nieh, Z. Yamani, N. Kučerka, J. Katsaras, D. Burgess and H. Breton, RSI 79 (2008) 095102
θΩο
Scattering PrinciplesSANS Intensity
measured intensity at q (or θ)
Im(q) = IF · Ωo · ε · T ·
Differentialcross-section
Flux Solidangle Detector
efficiency Sampletransmission
dσdΩ( )v ·A ·t
Beam areaon the sample
Path lengthDifferentialcross-section
number of neutrons scattered per second into a solid angle dΩ with the final energy between E and E+dE
neutron flux of the incident beam • dΩdE
d2σ(Ω,Ε)dΩdE =
For (quasi-)Elastic Scattering we assume no energy transfer
∫∞
0
dσdΩ = d2σ
dΩdE dEnumber of neutrons scattered per second into dΩ
neutron flux of the incident beam • dΩ=
[time-1area-1]
[time-1]
[area]
SANS Datafactorization
( )2
i
rqii
i j
rrqiji
iji ebebbdΩdσ ∑∑∑ ⋅−⋅ ==
Sρ)rρ()rΔρ( −=
ρp
ρoVp
V
N particlesR
O
x
( )2N
k i
xRqiiV
ikebV1)
dΩdσ( ∑∑ +⋅=
;dVe)xΔρ(eV1)
dΩdσ(
2N
kP
xqiRqiV
k∑ ∫ ⋅⋅=
)q)P(qS(VN)
dΩdσ( V
=
Inter-particle structure factor
square of amplitude of (intra-)particle form factor
∫ ⋅== 2P
xqi2 |dV)exΔρ(||)qF(||)qP(|
orientationalaverage
1...Nk kd;R k ==
SANS Datarepeating lamellae
2N
kP
xqiRqiV dVe)xΔρ(e
V1)
dΩdσ( k∑ ∫ ⋅⋅=
real space
-0.1 0.0 0.1 0.2 0.3 0.4 0.5
F1(q)
q [Å-1]
0 50 100 150
Dρ2(r)
r [Å] 0.0 0.1 0.2 0.3 0.4 0.5
2π/DF2(q)
q [Å-1]
spacing D spacing 2π/D
)()( 21 rfrf ∗
( ) ( ) ( ))()()()( 2121 rfFTrfFTrfrfFT =∗
-20 0 20 40 60 80 100 120 140 160
ρ(r)
r [Å]
)()( 21 qFqF
-0.1 0.0 0.1 0.2 0.3 0.4 0.5
F(q)
q [Å-1]
inverse space
?
D
-30 -20 -10 0 10 20 300.0
0.2
0.4
0.6
0.8
1.0water
bilayerD
istrib
utio
n P
robabili
ty
Distance form Bilayer Centre [Å]
SAND Datarepeating lamellae
0 2 4 6 8 10 12 14 16 180.01
0.1
1
10
100
1000
Inte
nsity
100% D2O 50% D2O 10% D2O 0% D2O
2theta
-30 -20 -10 0 10 20 30-1
0
1
2
3
4
5
6
7
100% D2O 50% D2O 10% D2O 0% D2O
Neutr
on S
catte
ring L
ength
Den
sity
[10-6 Å
-2 ]
Distance form Bilayer Centre [Å]
∑=
+=
1hh0 D
2ππhcosFD2F
D1(z)Δρ
EXAMPLE:Water penetration in lipid bilayersdetermined from contrast variation(more details presented in N5 spectrometer demonstration)
Water Distribution
)()()( 11 zPzPz BBWW ρρρ +=
The difference profiles of contrastvaried SDPs provide distributionsof bilayer/water probability.
)()()( 22 zPzPz BBWW ρρρ +=
)()()()( 2121 zPzz WWW ρρρρ −=−
SAND Datarepeating lamellae
EXAMPLE:The orientation of cholesterol in PUFA bilayers determined from D-labeling
)()()()( zPzPzPz LLDWWBBD ρρρρ ++=
)()()()( zPzPzPz LLHWWBBH ρρρρ ++=
)()()()( zPzz LLHLDHD ρρρρ −=−
Cholesterol in PUFAsCholesterol can be made to change itsorientation in model membranes bychanging the ratio of PUFA tosaturated chain lipids.
N. Kučerka, D. Marquardt, T.A. Harroun, M.-P. Nieh, S.R. Wassall and J. Katsaras, JACS 131 (2009)
SANS DataModel-Independent Analysis
2rqi
V dVe)rΔρ(VN)
dΩdσ( ∫ ⋅=
22
P
2
V ...2
)rq(-1ΔρVVNdV)rqcos(Δρ
VN)
dΩdσ( >+
⋅<>−⋅= ∫
Assume a cento-symmetric particle with homogeneous ρ
;...3
Rq1VΔρVN)
dΩdσ(
2G22
P2
V
+−=
(dilute system: S(q)=1)
Interpretation of Slopes
Decay of the scattering function depends on the system’s overall structure (dimensionality)
cylinder
disk
;q~)dΩdσ( m-
V
m=1 - cylindersm=2 - disks, lamellaem=3 - spheresm=4 - sharp interfaces
SANS DataModel-Independent Analysis
Modified Guinier Plots
...3
Rq)log(Ilog(I(q))
2sphereG,2
0 +−=
...2
RqAI(q))log(q
2cylinderG,2 +−=⋅
...RqBI(q))log(q 2lamellaG,
22 +−=⋅
Radius of Gyration corresponds to the “scattering size” of object
(substitute “scattering amplitude” for “mass”)
SANS DataModel-Based Analysis
Select a model forpossible structure of the aggregates
Fit the experimental data using the selected model
(Fix the values of any“known” physical parameters - as
many as possible)
Is it a good fit
? No
Change the model
Yes
A PossibleStructure
SANS DataModel-Based Analysis
PathOblate shells: a=180 Å, b=62 Å
Spherical shells: R=133 Å, p= 0.15
Bilayer disks: R=156 Å, L= 45 Å
EXAMPLE:Bicelles forming mixture (DMPC, DHPC, and DMPG in D2O at the total lipid concentration of 0.1 wt.%)
SANS DataScattering Form Factors
- Spheres -
Since spheres are isotropic, there is no need to do orientational average∫∫∫ ρ( r )·e-iq · r drP(q) =
1
Vsphere vsphere
2
2
= ∫ ∫ ∫ e-iqr·cosθ r2 sinθ dϕ dθ dr1
Vsphere2
2
r= 0
r= R
θ= 0
θ = π
ϕ= 0
ϕ= 2π
= [sin(qR) - qR·cos(qR)]29
(qR)6
R= 100 Å∆ρ = 1x10-6 Å-2
φ= 0.1
dσdΩ( )v = (ρsphere – ρo) · Vsphere· P( q )
NV
22
=φsphereVsphere∆ρ2 [sin(qR) - qR·cos(qR)]29
(qR)6
rθ
ϕ
SANS DataScattering Form Factors
Common form factors of particulate systems
Cylinders(radius: Rlength: L)
Spherical shells(outer radius: R1inner radius: R2)
Morphologies P(q)
Triaxial ellipsoids(semiaxes: a,b,c)
RemarksSpheres
(radius :R) [sin(qR) - qR·cos(qR)]2 =Asph(qR)9
(qR)62
(R13 – R2
3)2
[R13·Asph(qR1)– R2
3·Asph(qR2)]2
∫ ∫ Asph[q a2 cos2(πx/2) + b2sin2(πx/2)(1-y2)1 + c2y2 ] dx dy0 0
1 1 2 • Integration of x and y are for orientational average.
• J1(x) is the first kind Bessel function of order 1
∫0
1 J12[qR 1-x2 ]
4[qR 1-x2 ]2
dxsin2(qLx/2)
(qLx/2)2
Disk (radius: Rinfinitely thin)
Rod (length: Linfinitely thin)
By setting L = 0 2 - J1(2qR)/qR
q2R2
By setting R = 0 qL ∫0
qL2 sin(t)
tdt -
sin2(qL/2)
(qL/2)2
“Structure Analysis by Small Angle X-Ray and Neutron Scattering” L. A. Feigen and D. I. Svergun
SANS DataSimple Models
ρC
rhoHrhoCrhoH
MD simulation model
Water
HEAD
Water
HEADrhoC
MD simulation model
CH2 Water
HEADCH2Water
HEADrhoC
MD simulation model
0.01 0.110-5
10-3
10-1
101
103
I(q) [
cm-1]
q [Å-1]
vesicle size andsize distribution
bilayer thickness
bilayer innerstructure
N. Kučerka, J.F. Nagle, S.E. Feller and P. Balgavý, Phys Rev E 69, 051903 (2004)
rho
MD simulation model
•The neutron scattering length density profiles of fluid bilayers in solution are inherently quite featureless (compared to X-ray scattering profiles)•Nevertheless, the mid-q region provides high quality information, reflecting the large scattering contrast between the lipid bilayer (a lot of H) and solvent (D2O)
SANS DataAdvanced Models
X
DLPC, DMPC: N. Kučerka, Y. Liu, N. Chu, H. I. Petrache, S. Tristram-Nagle, and J.F. Nagle, Biophys. J (2005)DOPC, POPC, DEPC: N. Kučerka, S. Tristram-Nagle, and J.F. Nagle, J Mem Biol (2005)DPPC: N. Kučerka, S. Tristram-Nagle, and J.F. Nagle, Biophys J Lett (2006)
D'B2DC
Total
MethylCG
P
CH2
Water + Choline
-30 -20 -10 0 10 20 300.0
0.1
0.2
0.3
0.4 e
lect
ron
dens
ity [
e/Å3 ]
z [Å]
A combined global analysis approach takes advantage of the complementarity of ULVs and oriented samples, enhancing the spatial resolution of the bilayer structure.
SANS and SAXS Datain Joint Refinement
0.0
0.1
0.2 PCN_d4
PCNCholCD3
CGglycerol
carbonylphosphatecholine_d9
choline_d13
CH
NSLD
[10
-5 Å
-2]
-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 300.0
0.1
0.2
CholCH3
CG PCN
CholCH3
carbonyl
glycerol
phosphate
choline
choline CH3 CH
CH2CH3
ED [
e/Å3 ]
z [Å] -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
CH
Cwater
CH3
CH2
CholCH3PCN
CG
Volu
me
Prob
abilit
y
z [Å]
0.0 0.2 0.4 0.6 0.8
0.0
0.5
1.0
1.5
2.0
|F(q
)| [e
/Å2 ]
q [Å-1]
-5 0 5 10 15 20 25 300.20
0.25
0.30
0.35
0.40
0.45 ADHH2
ED [e
/Å3 ]
z [Å]
0.0 0.1 0.2 0.3
0.0
0.5
1.0
1.5
2.0
2.5
DOPC 50% D2O
DOPC 100% D2O
|F(q
)| [1
0-4 Å
-1]
q [Å-1]
-5 0 5 10 15 20 25 30
0.0
0.2
0.4
0.6 B
NSLD
[10-5
Å-2]
z [Å]
•Each of the component groups has nearly the same functional form for all of the different contrast conditions•Volume distributions satisfy a spatial conservation principle
N.Kučerka, J.F.Nagle, J.N.Sachs, S.E.Feller, J.Pencer A.J.Jackson, and J.Katsaras, Biophys. J (2008)
The SDP model was fit (with only one set of parameters) simultaneously to the set of scattering data obtained at different contrast conditions (X-rays and neutron contrast variation).
Concluding Remarks• SANS is a valuable tool for structural biophysics allowing the in
situ measurements of systems between 10 to 1000 Å
• The scattering function is proportional to the product of form factor (intraparticle scattering function) and structure factor (interparticle interactions)
• Overall morphology can be obtained through model independent approach, while model-based analysis provides further details on various structural parameters
• Contrast variation and specific labeling greatly enhances resolution of determined structures