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