Outline Model fitting and least-squares methods Available form - - PDF document

1

Jan Skov Pedersen, Department of Chemistry and iNANO Center University of Aarhus Denmark

Form and Structure Factors: Modeling and Interactions

SAXS lab

2

Outline

• Concentration effects and structure factors

Zimm approach Spherical 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

Literature

Jan 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 Pedersen Monte 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 Pedersen Modelling 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 Klein Interacting 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

[ ]

N i q I q I q

i i i

,... 3 , 2 , 1 ) ( ), ( , = σ ) (

i

q I

[ ]

) (

i

q I σ

i

q calibrated calibrated, i.e. on absolute scale

• noisy, (smeared), truncated

Statistical standard errors: Calculated from counting statistics 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 to i.e. statistical agreement between model and data

9

Cross section

d(q)/d : 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 particles d(q) d = I(q) = n ∆2V 2P(q)S(q) n is the number density of particles, ∆ is the excess scattering length density, given by electron density differences V is the volume of the particles, P(q) is the particle form factor, P(q=0)=1 S(q) is the particle structure factor, S(q=∞)=1

• V ∝ M
• n = c/M
• ∆ can be calculated from partial specific density, composition

= c M ∆m

2P(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 Lens

• 10. Cube with terraces:
• 11. Cylinder:
• 12. Cylinder with elliptical cross section:
• 13. Cylinder with hemi-spherical end-caps:

13x Cylinder with ‘half lens’ end caps

• 14. 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 III

• 40. 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

[ ]

( )

( )

3 3 3 2 2 2

) ( )] cos( ) [sin( 3 3 4 cos sin 4 sin cos 4 sin cos 4 ' ' ) sin( 4 ) sin( 4 ) sin( ) ( 4 ) ( qR qR qR qR R qR qR qR q q qr q qR R q q qr q qR R q dx fg fg dx g f rdr qr q dr r qr qr dr r qr qr r q A

R R R

− = − =

• +

− =

• +

− = − = = = = =

• ∞

π π π π π π ρ π

(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 spheres Smeared fit Ideal fit

Data from Wiggnal et al.

2 3 2 2

) ( )] cos( ) [sin( 3 ) (

• −

∆ = Ω qR qR qR qR V q d d ρ σ

Instrumental smearing is routinely included in SANS data analysis

17

And:

Ellipsoid

18

Glatter

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 + background R = 15.48 Å ε = 1.61 (prolate)

χ2=2.4 (χ=1.55)

20

where Vout= 4πRout

3/3 and Vin= 4πRin 3/3.

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

Core-shell particles:

[ ]

) ( ) ( ) ( ) (

in in core shell

• ut
• ut

shell core shell core

qR V qR V q A Φ ∆ − ∆ − Φ ∆ ∆ =

−

ρ ρ ρ ρ

[ ]

3

cos sin 3 ) ( x x x x x − = Φ

= − +

21

Glatter

P(q): Core-shell

22

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

SDS micelle

20 Å Hydrocarbon core Headgroup/counterions

23

q [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/cm Rcore = 13.5 ± 2.6 Å ε = 1.9 ± 0.10 dhead = 7.1 ± 4.4 Å ρhead/ρcore = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm

SDS micelles:

prolate ellispoid with shell of constant thickness

24

Molecular constraints:

O H e O H tail e tail e tail

V Z V Z

2 2

− = ∆ρ

tail agg core

V N V =

O H e O H O H head O H e head e head

V Z nV V nZ Z

2 2 2 2

− + + = ∆ρ

n water molecules in headgroup shell

) (

2O H head agg shell

nV V N V + =

surfactant

M N c n

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( ) (

j i ij j j

s r q i b r q i b q A

• +

⋅ − = ⋅ − =

• s

r r

• +

= '

center of sphere

Debye, 1915

self-term interference term phase factor

)) ' ' ( exp( ) (

l k j i kl ij

s r s r q i b b q I

• −

− + ⋅ − = )) ' ' ( exp( ) ' exp( ) ' exp(

k j k kl i ij

s s q i r q i b r q i b

• −

⋅ − ⋅ ⋅ − =

jk jk k k k k j j j j j j j

qd qd qR V qR V qR P V sin ) ( ) ( ) (

2 2

Φ ∆ Φ ∆ + ∆ =

• ≠

ρ ρ ρ

jk jk k k k j j j

qd qd qR V qR V sin ) ( ) (

• Φ

∆ Φ ∆ = ρ ρ

30

• Generate points in

space by Monte Carlo simulations

• Select subsets by

geometric constraints

• Caclulate histograms

p(r) functions

• (Include polydispersity)
• Fourier transform

MC 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

• blate ellipsoids

R=20.3±0.3 Å ɛ ɛ ɛ ɛ=0.663±0.005

32

Polymer chains in solution

Gigantic ensemble of 3D random flights

• all with different configurations

33

Gaussian polymer chain

Look at two points: Contour separation: L Spatial separation: r Contribution to scattering:

r

• L

qr qr q I ) sin( ) (

2

=

• >

< − ∝

2 2

2 3 exp ) ( r r r D

For an ensemble of polymers, points with L has <r2>=Lb and r has a Gaussian distribution: ’Density of points’: (Lo- L)

Lo L L Add scattering from all pair of points

[ ]

6 / exp

2Lb

q −

34

Gaussian chains: The calculation

[ ]

2 2 2 2 2 1 2 2 1 2

] 1 ) [exp( 2 6 / exp ) ( 1 ) sin( ) , ( ) ( 1 ) sin( |) | , ( 1 ) ( q R x x x x Lb q L L dL L r qr qr L r D L L dL dr L r qr qr L L r D dL dL dr L q P

g L

• L
• L

L

• =

+ − − = − − = − = − =

• ∞

∞

How does this function look?

Rg

2 = 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 g

R q / 1 ≈

2 1 − −

= q q

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+ background Rg = 21.3 Å

χ2=1.8

37

Self avoidence

No excluded volume No excluded volume => expansion

2 2 2

] 1 ) [exp( 2 ) ( x x x q R x P

g

+ − − = =

no analytical solution!

38

Monte Carlo simulation approach

(1) Choose model (5) Fit experimental data using numerical expressions for 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

2 1

= =

− −

ν

ν

q q

588 .

70 . 1 / 1

= =

− −

ν

ν

q q

Excluded volume chains: Gaussian chains

.) . ( / 1 vol excl R q

g

≈

) ( / 1 Gaussian R q

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 persistence length with ionic strength works also for polyelectrolytes like unfolded proteins

41

Models II

Polydispersity: Spherical particles as e.g. vesicles No interaction effects: Size distribution D(R)

• ∆

= Ω

i i i i

q P f M c d q d ) ( ) (

2

ρ σ

Oligomeric mixture: Discrete particles

fi = mass fraction

=

i i

f 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-crystallin eye lens protein

44

p(r) by Indirect Fourier Transformation (IFT)

At high concentration, the neighborhood is different from the average 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 ~ concentration P’(q) = P(q) - υ P(q)2 = P(q)[1 - υ P(q)]

) ( 1 ) ( q P q P 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 ) ( ) ( q P q P K q I υ + =

• +

= + =

− − −

υ υ ) ( 1 ) ( ) ( 1 ) (

1 1 1

q P K q P q P K q I 3 / 1 1 ) (

2 2 g

R q q P + ≈

[ ]

υ + + =

− −

3 / 1 ) (

2 2 1 1 g

R q K q I

With

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 Wunderlich PS in toluene

q = 0 P(q) c = 0

48

My suggestion:

) 3 / exp( 1 ) (

2 2 2 1

a q ca P c q I

i i i

− + =

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 principles governing aggregation and crystallization Why is the eye lens transparent despite a protein concentration

• f 30-40% ?

50

Structure factor

) ( ) ( sin ) ( ) (

2 2 2 2

q S q P V qr qr q P V q I

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 j

qr qr r p q P V sin ) ( ) (

2 2

• ∆

= ρ

51

Correlation function g(r)

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

52

g(r) and S(q)

q

dr r qr qr r g n q S

2

) sin( ) 1 ) ( ( 4 1 ) (

• −

+ = π

53

GIFT

Glatter: Generalized Indirect Fourier Transformation (GIFT)

• =

dr qr qr r p q I ) sin( ) ( 4 ) ( π

• =

dr qr qr r p c Z R R q S q I

salt eff

) sin( ) ( 4 ) , ), ( , , , ( ) ( π σ η

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

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 trimers monomers Home-written software

55

Osmometry (Second virial coeff A2)

c Π/c Ideal gas A2 = 0 repulsion A2 > 0 attraction A2 < 0

56

S(q), virial expansion and Zimm

... 3 2 1 1 ) (

3 2 2 1

+ + + =

• ∂

Π ∂ = =

−

MA c cMA c RT q S

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 sphere Debye-Hückel screened Coulomb potential

59

Depletion interactions

π

Asakura & Oosawa, 58

π

60

Theory for colloidal stability

Debye-Hückel screened 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) !!!

) ( 1 1 q nc −

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-crystallin solution pH 6.8, 150 mM ionic strength.

65

Anisotropy

Small: decoupling approximation (Kotlarchyk and Chen,1984) : Measured structure factor:

[ ]

) ( 1 ) ( ) ( 1 ) ( ) ( ) (

2

q S q S q q P n q q Smeas ≠ − + = ∆ Ω ∂ ∂ ≡ β ρ σ

!!!!!

66

Large anisotropy

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

) ( 1 ) ( ) (

2 2

q P q P V n q d d ν ρ σ + ∆ = Ω

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

) ( ) ( 1 ) ( ) (

2 2

q P q nc q P V n q d d − ∆ = Ω ρ σ

c(q) direct correlation function related to FT of V(r) Polymers, cylinders…

67

Empirical PRISM expression

Arleth, Bergström and Pedersen ) ( ) ( 1 ) ( ) (

2 2

q P q nc q P V n q d d − ∆ = Ω ρ 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 approximation 3) 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 approximation 5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation 6) 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 factors

Zimm approach Spherical 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

Literature

Jan 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 Pedersen Monte 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 Pedersen Modelling 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 Klein Interacting Colloidal Suspensions in Neutrons, X-Rays and Light

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