1

Jan Skov Pedersen,

Department of Chemistry and iNANO CenterUniversity of Aarhus

Denmark

Form and Structure Factors:

Modeling and Interactions

SAXS lab

2

Outline

• Concentration effects and structure factorsZimm approachSpherical particles Elongated particles (approximations)Polymers

• Model fitting and least-squares methods• Available form factors

ex: sphere, ellipsoid, cylinder, spherical subunits…ex: polymer chain

• Monte Carlo integration for form factors of complex structures

• Monte Carlo simulations for form factors of polymer models

3

Motivation

- not to replace shape reconstruction and crystal-structure based modeling – we use the methods extensively

- describe and correct for concentration effects

- alternative approaches to reduce the number of degrees of freedom in SAS data structural analysis

- provide polymer-theory based modeling of flexible chains

(might make you aware of the limited information content of your data !!!)

4

LiteratureJan Skov Pedersen, Analysis of Small-Angle Scattering Data from Colloids and

Polymer Solutions: Modeling and Least-squares Fitting

(1997). Adv. Colloid Interface Sci., 70, 171-210.

Jan Skov PedersenMonte Carlo Simulation Techniques Applied in the Analysis of Small-Angle

Scattering Data from Colloids and Polymer Systems

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 381

Jan Skov PedersenModelling of Small-Angle Scattering Data from Colloids and Polymer Systems

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 391

Rudolf KleinInteracting Colloidal Suspensions

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 351

5

Form factors and structure factors

Warning 1:Scattering theory – lots of equations!= mathematics, Fourier transformations

Warning 2:Structure factors:Particle interactions = statistical mechanics

Not all details given - but hope to give you an impression!

6

I will outline some calculations to show that it is not black magic !

7

Input data: Azimuthally averaged data

[ ] NiqIqIq iii ,...3,2,1)(),(, =σ

)( iqI

[ ])( iqIσ

iq calibrated

calibrated, i.e. on absolute scale - noisy, (smeared), truncated

Statistical standard errors: Calculated from countingstatistics by error propagation- do not contain information on systematic error !!!!

8

Least-squared methods

Measured data:

Model:

Chi-square:

Reduced Chi-squared: = goodness of fit (GoF)

Note that for corresponds toi.e. statistical agreement between model and data

9

Cross sectiondσ(q)/dΩ : number of scattered neutrons or photons per unit time,

relative to the incident flux of neutron or photons, per unit solid angle at q per unit volume of the sample.

For system of monodisperse particlesdσ(q)dΩ = I(q) = n ∆ρ2V 2P(q)S(q)

n is the number density of particles, ∆ρ is the excess scattering length density,

given by electron density differencesV is the volume of the particles, P(q) is the particle form factor, P(q=0)=1S(q) is the particle structure factor, S(q=∞)=1

• V ∝ M

• n = c/M

• ∆ρ can be calculated from partial specific density, composition

= c M ∆ρm2P(q)S(q)

10

Form factors of geometrical objects

11

Form factors I

1. Homogeneous sphere 2. Spherical shell: 3. Spherical concentric shells: 4. Particles consisting of spherical subunits: 5. Ellipsoid of revolution: 6. Tri-axial ellipsoid: 7. Cube and rectangular parallelepipedons: 8. Truncated octahedra: 9. Faceted Sphere: 9x Lens10. Cube with terraces: 11. Cylinder: 12. Cylinder with elliptical cross section: 13. Cylinder with hemi-spherical end-caps: 13x Cylinder with ‘half lens’ end caps14. Toroid: 15. Infinitely thin rod: 16. Infinitely thin circular disk: 17. Fractal aggregates:

Homogenous rigid particles

12

Form factors II

18. Flexible polymers with Gaussian statistics: 19. Polydisperse flexible polymers with Gaussian statistics: 20. Flexible ring polymers with Gaussian statistics: 21. Flexible self-avoiding polymers: 22. Polydisperse flexible self-avoiding polymers: 23. Semi-flexible polymers without self-avoidance:24. Semi-flexible polymers with self-avoidance: 24x Polyelectrolyte Semi-flexible polymers with self-avoidance: 25. Star polymer with Gaussian statistics: 26. Polydisperse star polymer with Gaussian statistics: 27. Regular star-burst polymer (dendrimer) with Gaussian statistics: 28. Polycondensates of Af monomers: 29. Polycondensates of ABf monomers: 30. Polycondensates of ABC monomers: 31. Regular comb polymer with Gaussian statistics: 32. Arbitrarily branched polymers with Gaussian statistics: 33. Arbitrarily branched semi-flexible polymers: 34. Arbitrarily branched self-avoiding polymers: 35. Sphere with Gaussian chains attached: 36. Ellipsoid with Gaussian chains attached: 37. Cylinder with Gaussian chains attached: 38. Polydisperse thin cylinder with polydisperse Gaussian chains attached to the ends: 39. Sphere with corona of semi-flexible interacting self-avoiding chains of a corona chain.

’Polymer models’

(Block copolymer micelle)

13

Form factors III40. Very anisotropic particles with local planar geometry: Cross section:(a) Homogeneous cross section (b) Two infinitely thin planes(c) A layered centro symmetric cross-section (d) Gaussian chains attached to the surface

Overall shape:(a) Infinitely thin spherical shell(b) Elliptical shell (c) Cylindrical shell(d) Infinitely thin disk

41. Very anisotropic particles with local cylindrical geometry: Cross section:(a) Homogeneous circular cross-section (b) Concentric circular shells (c) Elliptical Homogeneous cross section. (d) Elliptical concentric shells (e) Gaussian chains attached to the surface

Overall shape:(a) Infinitely thin rod(b) Semi-flexible polymer chain with or without excluded volume

P(q) = Pcross-section(q) Plarge(q)

14

From factor of a solid sphere

R

r

ρ(r)1

0

[ ]( )

( )

33

3

2

0

0

2

0

2

0

)(

)]cos()[sin(334

cossin4

sincos4

sincos4

''

)sin(4

)sin(4

)sin()(4)(

qR

qRqRqRR

qRqRqRq

q

qr

q

qRR

q

q

qr

q

qRR

q

dxfgfgdxgf

rdrqrq

drrqr

qrdrr

qr

qrrqA

R

R

R

−=

−=

+−=

+−=

−=

==

==

∫ ∫

∫

∫∫∞

π

π

π

π

π

πρπ

(partial integration)…

spherical Bessel function

15

q-4

1/R

Form factor of sphere

C. Glinka

Porod

P(q) = A(q)2/V2

16

Measured data from solid sphere (SANS)

q [Å-1]

0.000 0.005 0.010 0.015 0.020 0.025 0.030

I(q)

10-1

100

101

102

103

104

105

SANS from Latex spheresSmeared fitIdeal fit

Data from Wiggnal et al.

2

322

)(

)]cos()[sin(3)(

−∆=

Ω qR

qRqRqRVq

d

dρ

σ

Instrumental smearing is routinely includedin SANS dataanalysis

17

And:

Ellipsoid

18Glatter

P(q): Ellipsoid of revolution

19

Lysosyme

Lysozyme 7 mg/mL

q [Å-1]

0.0 0.1 0.2 0.3 0.4

I(q)

[cm

-1]

10-3

10-2

10-1

100

Ellipsoid of revolution + backgroundR = 15.48 Åε = 1.61 (prolate) χ2=2.4 (χ=1.55)

20

where Vout= 4πRout

3/3 and Vin= 4πRin3/3.

∆ρcore is the excess scattering length density of the core, ∆ρshell is the excess scattering length density of the shell and:

Core-shell particles:

[ ])()()()( inincoreshelloutoutshellcoreshellcore qRVqRVqA Φ∆−∆−Φ∆∆=− ρρρρ

[ ]3

cossin3)(

x

xxxx

−=Φ

= − +

21

Glatter

P(q): Core-shell

22www.psc.edu/.../charmm/tutorial/ mackerell/membrane.htmlmrsec.wisc.edu/edetc/cineplex/ micelle.html

SDS micelle

20 Å

Hydrocarbon core

Headgroup/counterions

23q [Å-1]

0.0 0.1 0.2 0.3 0.4

I(q)

[cm

-1]

0.001

0.01

0.1

χ2 = 2.3

I(0) = 0.0323 ± 0.0005 1/cmRcore = 13.5 ± 2.6 Åε = 1.9 ± 0.10 dhead = 7.1 ± 4.4 Åρhead/ρcore = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm

SDS micelles:prolate ellispoid with shell of constant thickness

24

Molecular constraints:

OH

e

OH

tail

e

taile

tailV

Z

V

Z

2

2−=∆ρtailaggcore VNV =

OH

e

OH

OHhead

OH

e

heade

headV

Z

nVV

nZZ

2

2

2

2 −+

+=∆ρ

n water molecules in headgroup shell

)( 2OHheadaggshell nVVNV +=

surfactantMN

cn

agg

micelles =

25

Cylinder

26

E. Gilbert

P(q) cylinder

27

Glucagon Fibrils

Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher,

Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387, 147–161

R=29Å

R=16 Å

28

Primus

29

Collection of particles with spherical symmetry

))'(exp()exp()( jiijjj srqibrqibqArrrrrr

+⋅−=⋅−= ∑∑

srrrr

+= '

center of sphere

Debye, 1915

self-term interference term phase factor

))''(exp()( lkjiklij srsrqibbqIrrrrr

−−+⋅−=∑

))''(exp()'exp()'exp( kjkkliij ssqirqibrqibrrrrrrr

−⋅−⋅⋅−=∑

jk

jk

kkk

kj

jjjjjjqd

qdqRVqRVqRPV

sin)()()(22 Φ∆Φ∆+∆= ∑∑

≠

ρρρ

jk

jk

kkkjjjqd

qdqRVqRV

sin)()(∑ Φ∆Φ∆= ρρ

30

•Generate points in space by Monte Carlo simulations

•Select subsets by geometric constraints

•Caclulate histogramsp(r) functions

•(Include polydispersity)

•Fourier transformMC points

Sphere

x(i)2+y(i)2+z(i)2 ≤ R2

Monte Carlo integration

in calculation of

form factors for

complex structures

31

Pure SDS micelles in buffer:

C > CMC (∼ 5 mM)

C ≤ CMC

Nagg= 66±1 oblate ellipsoidsR=20.3±0.3 Åɛɛɛɛ=0.663±0.005

32

Polymer chains in solutionGigantic ensemble of 3D random flights

- all with different configurations

33

Gaussian polymer chain

Look at two points: Contour separation: LSpatial separation: r

Contribution to scattering:rr

L qr

qrqI

)sin()(2 =

><−∝ 2

2

23

exp)(r

rrD

For an ensemble of polymers, points withL has <r2>=Lb

and r has a Gaussian distribution:

’Density of points’: (Lo- L) Lo

LL

Add scattering from all pair of points

[ ]6/exp 2Lbq−

34

Gaussian chains: The calculation

[ ]

222

0

2

2

00

0

122

0

1

02

]1)[exp(2

6/exp)(1

)sin(),()(

1

)sin(|)|,(

1)(

qRxx

xx

LbqLLdLL

rqr

qrLrDLLdLdr

L

rqr

qrLLrDdLdLdr

LqP

g

L

o

o

L

o

o

LL

o

o

o

oo

=+−−

=

−−=

−=

−=

∫

∫∫

∫∫∫∞

∞

How does this function look?

Rg2 = Lb/6

35

Polymer scattering

Form factor of Gaussian chain

qRg

0.1 1 10 100

P(q

)

0.0001

0.001

0.01

0.1

1

10

gRq /1≈

21

−− = qq ν

36

Lysozyme in Urea 10 mg/mL

q [Å-1]

0.01 0.1 1

I(q)

[cm

-1]

10-3

10-2

10-1

Lysozyme in urea

Gaussian chain+ backgroundRg = 21.3 Å

χ2=1.8

37

Self avoidence

No excluded volume No excluded volume=> expansion

222 ]1)[exp(2)(

x

xxqRxP g

+−−== no analytical solution!

38

Monte Carlo simulation approach

(1) Choose model

(5) Fit experimental data using numerical expressionsfor P(q)

(2) Vary parameters in a broad range Generate configs., sample P(q)

(3) Analyze P(q) using physical insight

(4) Parameterize P(q) using physical insight

Pedersen and Schurtenberger 1996

P(q,L,b)

L = contour length

b = Kuhn (‘step’) length

l

39

Expansion = 1.5

Form factor polymer chains

qRg (Gaussian)

0.1 1 10 100

P(q

)

0.0001

0.001

0.01

0.1

1

10

2/1

21

=

= −−

ν

ν qq

588.0

70.1/1

=

= −−

ν

νqq

Excluded volume chains:

Gaussian chains

.).(/1 volexclRq g≈

)(/1 GaussianRq g≈

q-1 local stiffness

40

C16E6 micelles with ‘C16-’SDS

Sommer C, Pedersen JS, Egelhaaf SU, Cannavaciuolo L, Kohlbrecher J, Schurtenberger P.

Variation of persistencelength with ionic strength

works also for polyelectrolyteslike unfolded proteins

41

Models IIPolydispersity: Spherical particles as e.g. vesicles

No interaction effects: Size distribution D(R)

∑∆=Ω i

iii qPfMcd

qd)(

)( 2ρσ

Oligomeric mixture: Discrete particles

fi = mass fraction ∑ =i

if 1

Application to insulin:

Pedersen, Hansen, Bauer (1994). European Biophysics Journal 23, 379-389.

(Erratum). ibid 23, 227-229.

Used in PRIMUS

‘Oligomers’

42

Concentration effects

43

Concentration effects in protein solutions

= q/2π

α-crystallineye lens protein

44

p(r) by Indirect Fourier Transformation (IFT)

At high concentration, the neighborhood is different from theaverage further away!

(1) Simple approach: Exclude low q data.(2) Glatter: Use Generalized Indirect Fourier Transformation (GIFT)

I. Pilz, 1982

- as you have done by GNOM !

45

Low concentraions

= Random-Phase Approximation (RPA)

P’(q) = -

Zimm 1948 – originally for light scattering

Subtract overlapping configuration

υ ~ concentrationP’(q) = P(q) - υ P(q)2

= P(q)[1 - υ P(q)]

)(1)(qP

qP

υ+=

…= P(q)[1 - υ P(q )+ υ 2P(q)2 - υ 3P(q)3…..]

Higher order terms:P’(q) = P(q)[1 - υ P(q)1 - υ P(q)]

= P(q)[1 - υ P(q)+ υ 2P(q)2]

46

Zimm approach

)(1)(

)(qP

qPKqI

υ+=

+=

+= −−− υ

υ

)(1

)()(1

)( 111

qPK

qP

qPKqI

3/1

1)(

22gRq

qP+

≈ [ ]υ++= −− 3/1)( 2211gRqKqIWith

Plot I(q)-1 versus q2 + c and extrapolate to q=0 and c=0 !

47

Zimm plot

I(q)-1

q2 + c

Kirste and WunderlichPS in toluene

q = 0

P(q)

c = 0

48

My suggestion:

)3/exp(1)(

22

21 aqca

P

c

qI

i

ii

−+=

With a1, a2, and Pi fit parameters

( - which includes also information from what follows)

• Minimum 3 concentrations for same system.• Fit data simultaneously all data sets

49

But now we look at the information content related to

these effects…

Understand the fundamental processes and principlesgoverning aggregation and crystallization

Why is the eye lens transparent despite a protein concentrationof 30-40% ?

50

Structure factor

)()(

sin)()(

22

22

qSqPV

qr

qrqPVqI

jk

jk

ρ

ρ

∆=

∆= ∑

Spherical monodisperse particles

S(q) is related to the probability distribution function of inter-particles distances, i.e. the pair correlation function g(r)

j

j

jqr

qrrpqPV

sin)()(22 ∑∆= ρ

51

Correlation function g(r)

g(r) =Average of the normalized density of atoms in a shell [r ; r+ dr]from the center ofa particle

52

g(r) and S(q)

q

drrqr

qrrgnqS 2)sin(

)1)((41)( ∫ −+= π

53

GIFTGlatter: Generalized Indirect Fourier Transformation (GIFT)

∫= drqr

qrrpqI

)sin()(4)( π

∫= drqr

qrrpcZRRqSqI salteff

)sin()(4),),(,,,()( πση

With concentration effects

Optimized by constrained non-linear least-squares method

- works well for globular models and provides p(r)

54

A SAXS study of the small hormone glucagon: equilibrium aggregation and fibrillation

29 residue hormone, with a net charge of +5 at pH~2-3

0 10 20 30 40 50 60 70

p(r)

[arb

. u.]

r [Å]0,01 0,1

10.7 mg/ml

6.4 mg/ml

5.1 mg/ml

2.4 mg/ml

I(q)

[arb

. u.]

q [Å-1]

1.0 mg/ml

Hexamers

trimersmonomers

Home-written software

55

Osmometry (Second virial coeff A2)

c

Π/c

Ideal gas A2 = 0repulsio

n A 2> 0

attraction A2 < 0

56

S(q), virial expansion and Zimm

...3211

)0(3

22

1

+++=

∂

Π∂==

−

MAccMAcRTqS

In Zimm approach ν = 2cMA2

From statistical mechanics…:

57

A2 in lysozyme solutions

O. D. Velev, E. W. Kaler, and A. M. Lenhoff

A2

Isoelectric point

58

Colloidal interactions• Excluded volume ‘repulsive’ interactions (‘hard-sphere’)• Short range attractive van der Waals interaction (‘stickiness’) • Short range attractive hydrophobic interactions

(solvent mediated ‘stickiness’) •Electrostatic repulsive interaction (or attractive for patchy charge distribution!)

(effective Debye-Hückel potential)• Attractive depletion interactions (co-solute (polymer) mediated )

hard sphere sticky hard sphereDebye-Hückelscreened Coulomb potential

59

Depletion interactions

π

Asakura & Oosawa, 58

π

60

Theory for colloidal stability

Debye-Hückelscreened Coulomb potential+ attractive interaction

DLVO theory: (Derjaguin-Landau-Vewey-Overbeek)

61

Integral equation theory

Relate g(r) [or S(q)] to V(r)

At low concentration g(r) = exp(− V (r) / kBT)Boltzmann approximation

c(r) = direct correlation function

Make expansion around uniform state [Ornstein-Zernike eq.]

g(r) = 1 – n c(r) –[3 particle] – [4 particle] - …= 1 – n c(r) – n2 c(r) * c(r) – n3 c(r) * c(r) * c(r) − …

[* = convolution]

≈1 − V (r) / kBT

(weak interactions)

S(q) = 1− n c(q) – n2 c(q)2 – n3 c(q)3 − …. =

⇒ but we still need to relate c(r) to V(r) !!!

)(11

qnc−

62

Closure relations

Systematic density expansion….

Mean-spherical approximation MSA: c(r) = − V(r) / kBT

(analytical solution for screened Coulomb potential- but not accurate for low densities)

Percus-Yevick approximation PY: c(r) = g(r) [exp(V (r) / kBT)−1](analytical solution for hard-sphere potential + sticky HS)

Hypernetted chain approximation HNC: c(r) = − V(r) / kBT +g(r) −1 – ln (g(r) )

(Only numerical solution - but quite accurate for Coulomb potential)

Rogers and Young closure RY:Combines PY and HNC in a self-consistent way(Only numerical solution

- but very accurate for Coulomb potential)

63

α-crystallin S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

64

α-crystallin + PEG 8000 S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

Depletion interactions:

40 mg/ml α-crystallinsolution pH 6.8, 150 mM ionic strength.

65

Anisotropy

Small: decoupling approximation (Kotlarchyk and Chen,1984) :

Measured structure factor:

[ ] )(1)()(1)(

)(

)( 2 qSqSqqPn

q

qSmeas ≠−+=∆

Ω∂

∂

≡ βρ

σ

!!!!!

66

Large anisotropy

Anisotropy, large: Random-Phase Approximation (RPA):

)(1)(

)( 22

qP

qPVnq

d

d

νρ

σ

+∆=

Ω υ ~ concentration

Anisotropy, large: Polymer Reference Interaction Site Model (PRISM)Integral equation theory – equivalent site approximation

)()(1)(

)( 22

qPqnc

qPVnq

d

d

−∆=

Ωρ

σ c(q) direct correlation functionrelated to FT of V(r)

Polymers, cylinders…

67

Empirical PRISM expression

Arleth, Bergström and Pedersen

)()(1)(

)( 22

qPqnc

qPVnq

d

d

−∆=

Ωρ

σ

c(q) = rod formfactor- empirical from MC simulation

SDS micelle in 1 M NaBr

68

Overview: Available Structure factors

1) Hard-sphere potential: Percus-Yevick approximation 2) Sticky hard-sphere potential: Percus-Yevick approximation3) Screened Coulomb potential:

Mean-Spherical Approximation (MSA).Rescaled MSA (RMSA). Thermodynamically self-consistent approaches (Rogers and Young closure)

4) Hard-sphere potential, polydisperse system: Percus-Yevick approximation5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation6) Screened Coulomb potential, polydisperse system: MSA, RMSA, 7) Cylinders, RPA

8) Cylinders, `PRISM':

9) Solutions of flexible polymers, RPA:

10) Solutions of semi-flexible polymers, `PRISM':

11) Solutions of polyelectrolyte chains ’PRISM’:

69

Summary

• Concentration effects and structure factorsZimm approachSpherical particles Elongated particles (approximations)Polymers

• Model fitting and least-squares methods• Available form factors

ex: sphere, ellipsoid, cylinder, spherical subunits…ex: polymer chain

• Monte Carlo integration for form factors of complex structures

• Monte Carlo simulations for •form factors of polymer models

70

LiteratureJan Skov Pedersen, Analysis of Small-Angle Scattering Data from Colloids and

Polymer Solutions: Modeling and Least-squares Fitting

(1997). Adv. Colloid Interface Sci., 70, 171-210.

Jan Skov PedersenMonte Carlo Simulation Techniques Applied in the Analysis of Small-Angle

Scattering Data from Colloids and Polymer Systems

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 381

Jan Skov PedersenModelling of Small-Angle Scattering Data from Colloids and Polymer Systems

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 391

Rudolf KleinInteracting Colloidal Suspensions

in Neutrons, X-Rays and Light

P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 351