Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and Inano Center University of Aarhus Denmark SAXS Lab

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Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and iNANO Center University of Aarhus Denmark SAXS lab

Outline • 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 • Concentration effects and structure factors Zimm approach Spherical particles Elongated particles (approximations) Polymers 2 Motivation

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

- alternative approaches to reduce the number of degrees of freedom in SAS data structural analysis (might make you aware of the limited information content of your data !!!) - provide polymer-theory based modeling of flexible chains

- describe and correct for concentration effects

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 4 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!

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

6 Input data: Azimuthally averaged data

qi , I(qi ), σ [I(qi )] i = 3,2,1 ,... N

qi calibrated

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

σ [I(qi )] Statistical standard errors: Calculated from counting statistics by error propagation - do not contain information on systematic error !!!!

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

8 Cross section

dσ(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 particles dσ(q) 2 2 2 dΩ = I(q) = n ∆ρ V P(q)S(q) = c M ∆ρm P(q)S(q)

n is the number density of particles, ∆ρ 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 • ∆ρ can be calculated from partial specific density, composition 9

Form factors of geometrical objects

10 Form factors I Homogenous rigid particles

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:

Form factors II ’Polymer models’

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 AB f 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: (Block copolymer micelle) 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. 12 Form factors III P(q) = Pcross-section (q) Plarge (q) 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

From factor of a solid sphere ρ(r) 1

0 R ∞ sin( qr ) R sin( qr ) A(q) = 4π ρ(r) r 2 dr = 4π r 2 dr ∫ qr ∫ qr 0 0 r 4π R = ∫sin( qr ) rdr = q 0 ()∫f ' g dx = []fg − ∫ fg 'dx (partial integration)…  R  4π  R cos qR sin qr   = − +   q  q q     0    4π  R cos qR sin qr  = − + 2  q  q q  4π = ()sin qR − qR cos qR q 3 4 3[sin( qR ) − qR cos( qR )] = π R 3 spherical Bessel function 14 3 (qR )3 Form factor of sphere P(q) = A(q)2/V2

1/R

q-4

Porod

15 C. Glinka

Measured data from solid sphere (SANS)

2 dσ 3[sin( qR ) − qR cos( qR )]  (q) = ∆ρ 2V 2   dΩ  (qR )3 

10 5

SANS from Latex spheres 10 4 Smeared fit Ideal fit 10 3 Instrumental smearing is 10 2 I(q) routinely included

10 1 in SANS data analysis

10 0

10 -1 0.000 0.005 0.010 0.015 0.020 0.025 0.030

-1 q [Å ] 16 Data from Wiggnal et al. Ellipsoid

And:

P(q): Ellipsoid of revolution

Glatter 18 Lysosyme Lysozyme 7 mg/mL

10 0 Ellipsoid of revolution + background R = 15.48 Å 2 ε = 1.61 (prolate) χ =2.4 ( χ=1.55) 10 -1 ] -1 I(q) [cm I(q) 10 -2

10 -3

0.0 0.1 0.2 0.3 0.4 q [Å -1 ] 19

Core-shell particles:

= − +

Acore −shell (q) = ∆ρcore [∆ρshell Vout Φ(qR out ) − (∆ρshell − ∆ρcore )Vin Φ(qR in )] where 3 3 Vout = 4 πRout /3 and V in = 4 πRin /3.

∆ρcore is the excess scattering length density of the core,

∆ρshell is the excess scattering length density of the shell and: 3[sin x − x cos x] Φ(x) = x3

20 P(q): Core-shell

Glatter 21

SDS micelle Hydrocarbon core

Headgroup/counterions

20 Å

www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html mrsec.wisc.edu/edetc/cineplex/ micelle.html 22 SDS micelles: prolate ellispoid with shell of constant thickness 0.1

χ2 = 2.3

I(0) = 0.0323 ± 0.0005 1/cm

Rcore = 13.5 ± 2.6 Å ε = 1.9 ± 0.10

] d = 7.1 ± 4.4 Å 0.01 head -1 ρhead /ρcore = − 1.7 ± 1.5 backgr = 0.00045 ± 0.00010 1/cm I(q) [cm I(q)

0.001

0.0 0.1 0.2 0.3 0.4 23 q [Å -1 ]

Molecular constraints:

Z e Z e ∆ρ e = tail − H 2O tail Vcore = N agg Vtail Vtail VH 2O

e e n water molecules in headgroup shell e Z head + nZ H 2O Z H 2O ∆ρ head = − Vhead + nV H 2O VH 2O Vshell = N agg (Vhead + nV H 2O )

c nmicelles = N agg M surfactant

24 Cylinder

P(q) cylinder

26 E. Gilbert Glucagon Fibrils

R=29Å

R=16 Å

Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, 27 Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387, 147–161

Primus

28 Collection of particles with spherical symmetry center of sphere r r r =r'+s

r r r r r r A(q) = ∑b j exp( − qi ⋅ rj ) = ∑bij exp( − qi ⋅ (ri '+s j )) r r r r r I(q) =∑bij bkl exp( − qi ⋅ (ri '+s j − rk '−sl )) r r r r r r r = ∑bij exp( − qi ⋅ ri )' bkl exp( qi ⋅ rk )' exp( − qi ⋅ (s j '−sk ))'

sin qd jk =∑V j ∆ρ j Φ(qR j )Vk ∆ρ k Φ(qR k ) qd jk

2 2 sin qd jk = ∑V j ∆ρ j P(qR j ) + ∑V j ∆ρ j Φ(qR j )Vk ∆ρ k Φ(qR k ) j≠k qd jk

self-term interference term phase factor 29 Debye, 1915

Monte Carlo integration in calculation of form factors for complex structures •Generate points in space by Monte Carlo simulations

•Select subsets by x(i) 2+y(i) 2+z(i) 2 ≤ R2 geometric constraints

•Caclulate histograms p(r) functions

•(Include polydispersity)

•Fourier transform Sphere 30 MC points Pure SDS micelles in buffer:

C > CMC ( ∼ 5 mM)

Nagg = 66±1 oblate ellipsoids R=20.3±0.3 Å ɛɛɛ=0.663±0.005

C ≤ CMC

Polymer chains in solution Gigantic ensemble of 3D random flights - all with different configurations

32 Gaussian polymer chain

Look at two points: Contour separation: L Spatial separation: r r sin( qr ) r Contribution to scattering: I (q) = L 2 qr

exp [− q 2 Lb 6/ ] For an ensemble of polymers, points with L has =Lb  3r 2  and r has a Gaussian distribution: D(r) ∝ exp −   2 < r 2 > Add scattering from all pair of points

’Density of points’: (L -L) o Lo L L 33

Gaussian chains: The calculation

1 ∞ Lo Lo sin( qr ) P(q) = 2 ∫ dr ∫ dL 1 ∫ dL 2 D(r |, L2 − L1 |) r Lo 0 0 0 qr

∞ Lo 1 sin( qr ) 2 = ∫ dr ∫ dL (Lo − L)D(r, L) r Lo 0 0 qr

Lo 1 2 = ∫ dL (Lo − L)exp []− q Lb 6/ Lo 0 2[exp( −x) −1+ x] 2 2 2 = x = Rg q R = Lb /6 x 2 g

How does this function look?

34 Polymer scattering

Form factor of Gaussian chain 10

q ≈ /1 Rg 0.1 1 − ν −2 P(q) q = q 0.01

0.001

0.0001 0.1 1 10 100 35 qR g

Lysozyme in urea Lysozyme in Urea 10 mg/mL 10 -1 Gaussian chain+ background

Rg = 21.3 Å χ2=1.8

-2

] 10 -1 I(q) [cm I(q)

10 -3

0.01 0.1 1 36 q [Å -1 ] Self avoidence

No excluded volume No excluded volume => expansion 2[exp( −x) −1+ x] 2 2 no analytical solution! P(x = Rg q ) = 2 x 37

Monte Carlo simulation approach Pedersen and Schurtenberger 1996 (1) Choose model

(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 l (5) Fit experimental data using numerical expressions for P(q)

P(q,L,b) L = contour length

b = Kuhn (‘step’) length 38 Expansion = 1.5

Form factor polymer chains

q ≈ /1 Rg (Gaussian ) 1

Excluded volume chains: q ≈ /1 R (excl . vol .) g /1 ν 0.1 q− = q− 70.1 .0 588

P(q) ν = 0.01 Gaussian chains 1 q− ν = q−2 q-1 local 0.001 ν = 2/1 stiffness

0.0001 0.1 1 10 100

qR g (Gaussian) 39

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

40 Models II Polydispersity: Spherical particles as e.g. vesicles

No interaction effects: Size distribution D(R)

Oligomeric mixture: Discrete particles

dσ (q) 2 Application to insulin: = c ∆ρ ∑ M i fi Pi (q) dΩ Pedersen, Hansen, Bauer (1994). i European Biophysics Journal 23, 379-389. (Erratum). ibid 23, 227-229.

fi = mass fraction ∑ f i = 1 Used in PRIMUS i ‘Oligomers’ 41

Concentration effects

42 Concentration effects in protein solutions

α-crystallin eye lens protein

= q/2 π

p(r) by Indirect Fourier Transformation (IFT)

- as you have done by GNOM !

I. Pilz, 1982

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)44 Low concentraions Zimm 1948 – originally for light scattering

P’(q) = - Subtract overlapping configuration

P’(q) = P(q) - υ P(q)2 υ ~ concentration = P(q)[1 - υ P(q)] 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] … = P(q)[1 - υ P(q )+ υ 2P(q)2 - υ 3P(q)3…..] P(q) = 1+υP(q)

= Random-Phase Approximation (RPA) 45

Zimm approach P(q) I(q) = K 1+υP(q)

−1 −1 1+υP(q) −1  1  I(q) = K = K  +υ P(q)  P(q)  1 With P(q) ≈ −1 −1 2 2 2 2 I(q) = K [1+ q Rg 3/ +υ] 1+ q Rg 3/

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

46 Zimm plot Kirste and Wunderlich PS in toluene

c = 0 I(q)-1 P(q)

q = 0

q2 + c 47

My suggestion: ( - which includes also information from what follows)

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

I(qi ) Pi = 2 2 c 1+ ca 1 exp( −qi a2 )3/

With a1, a2, and Pi fit parameters

48 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 of 30-40% ? 49

Structure factor

Spherical monodisperse particles sin qr I(q) = V 2∆ρ 2 P(q)∑ jk qr jk = V 2∆ρ 2 P(q) S(q)

2 2 sin qr j = V ∆ρ P(q)∑ p(rj ) qr j

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

50 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

g(r) and S(q)

sin( qr ) S(q) =1+ n 4π (g(r) − )1 r 2dr ∫ qr

52 GIFT

Glatter: Generalized Indirect Fourier Transformation (GIFT)

sin( qr ) I(q) = 4π p(r) dr ∫ qr

With concentration effects

sin( qr ) I(q) = S (q,η, R,σ (R), Z,c 4) π p(r) dr eff salt ∫ qr

Optimized by constrained non-linear least-squares method

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

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

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

10.7 mg/ml

6.4 mg/ml Hexamers p(r)[arb. u.]

I(q)[arb. u.] 5.1 mg/ml

2.4 mg/ml trimers monomers 1.0 mg/ml 0 10 20 30 40 50 60 70 0,01 0,1 r [Å] q [Å -1 ]

54 Osmometry (Second virial coeff A2)

> 0 Π/c A 2 ion uls rep

Ideal gas A 2 = 0 attr acti on A 2 < 0

S(q), virial expansion and Zimm

From statistical mechanics…:

 ∂Π −1 1 S(q = )0 = RT   = 2  ∂c  1+ 2cMA 2 + 3c MA 3 +...

In Zimm approach ν = 2 cMA 2

56 Isoelectric point A2 in lysozyme solutions

O. D. Velev, E. W. Kaler, and A. M. Lenhoff 57

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 )

Debye-Hückel hard sphere sticky hard sphere screened Coulomb potential

58 Depletion interactions

π π

Asakura & Oosawa, 58 59

Theory for colloidal stability

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

Debye-Hückel screened Coulomb potential + attractive interaction

60 Integral equation theory

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

At low concentration g(r) = exp( − V (r) / kBT) ≈1 − V (r) / kBT Boltzmann approximation (weak interactions) 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] c(r) = direct correlation function 1 S(q) = 1 − n c (q) – n2 c(q)2 – n3 c(q)3 − …. = 1− nc (q) ⇒ but we still need to relate c(r) to V(r) !!! 61

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) 62 α-crystallin S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

α-crystallin + PEG 8000 S. Finet, A. Tardieu Depletion interactions: DLVO potential

HNC and numerical solution

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

64 Anisotropy

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

Measured structure factor: ∂σ (q) S (q) ≡ ∂Ω =1+ β (q)[]S(q) −1 ≠ S(q) !!!!! meas n∆ρ 2 P(q)

Large anisotropy

Polymers, cylinders…

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

dσ P(q) (q) = n∆ρ 2V 2 dΩ 1+νP(q) υ ~ concentration

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

dσ P(q) (q) =n∆ρ 2V 2 c(q) direct correlation function dΩ 1− nc (q)P(q) related to FT of V(r)

66 Empirical PRISM expression

SDS micelle in 1 M NaBr

dσ P(q) (q) = n∆ρ 2V 2 dΩ 1− nc (q)P(q)

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

Arleth, Bergström and Pedersen

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’:

68 Summary • 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

• Concentration effects and structure factors Zimm approach Spherical particles Elongated particles (approximations) Polymers 69

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 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 70