Form and Structure Factors: Modeling and Interactions

Transcription

1 Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and inano Center University of Aarhus Denmark SAXS lab 1

2 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

3 Motivation for modelling - 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 3

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, 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

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! 5

6 I will outline some calculations to show that it is not black magic! 6

7 Input data: Azimuthally averaged data [ I( q )] i 1,2,3 N qi, I( qi ), σ i =,... qi calibrated I σ ( q i ) [ I q )] ( i 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!!!! 7

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 8

9 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) dω = I(q) = n Δρ 2 V 2 P(q)S(q) n is the number density of particles, Δρ 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 = c M Δρ m2 P(q)S(q) V M n = c/m Δρ can be calculated from partial specific density, composition 9

10 Form factors of geometrical objects 10

11 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: 11

12 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 A f 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: 35. Sphere with Gaussian chains attached: 36. Ellipsoid with Gaussian chains attached: 37. Cylinder with Gaussian chains attached: (Block copolymer micelle) 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

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 P(q) = P cross-section (q) P large (q) 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 13

14 14 From factor of a solid sphere R r ρ(r) 1 0 [ ] ( ) ( ) ) ( )] cos( ) 3[sin( 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 g dx 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 Form factor of sphere P(q) = A(q) 2 /V 2 1/R q -4 Porod 15 C. Glinka

16 Measured data from solid sphere (SANS) dσ dω [sin( qr) qr cos( qr)] ( q) = Δρ V 3 ( qr) 10 5 I(q) SANS from Latex spheres Smeared fit Ideal fit Instrumental smearing is routinely included in SANS data analysis Data from Wiggnal et al. q [Å -1 ] 16

17 Ellipsoid Prolates (R,R,εR) ε > 1 Oblates (R,R,εR) ε < 1 P( q) π / 2 = Φ 0 2 [ ( qr') ] sinα dα R = R(sin 2 α + ε 2 cos 2 α) 1/2 Φ( x) = [ x) x cos( x) ] 3 sin( x 3 17

18 P(q): Ellipsoid of revolution Glatter 18

19 Lysosyme Lysozyme 7 mg/ml 10 0 Ellipsoid of revolution + background 10-1 R = Å ε = 1.61 (prolate) χ 2 =2.4 (χ=1.55) I(q) [cm -1 ] q [Å -1 ] 19

20 Core-shell particles: R out R in = + A core shell [ Δρ V Φ( qr ) ( Δρ Δρ ) V Φ( qr )] ( q) = shell out out shell core in in where V out = 4π 3 /3 and V in = 4πR in3 /3. Δρ core is the excess scattering length density of the core, Δρ shell is the excess scattering length density of the shell and: Φ( x) = [ x x cos x] 3 sin x 3 20

21 P(q) R out =30Å R core =15Å Δρ shell =1 Δρ core = 1. q [Å -1 ] 21

22 Hydrocarbon core SDS micelle Headgroup/counterions 20 Å mackerell/membrane.html mrsec.wisc.edu/edetc/cineplex/ micelle.html 22

23 0.1 SDS micelles: prolate ellispoid with shell of constant thickness χ 2 =2.3 I(q) [cm -1 ] 0.01 I(0) = ± /cm R core = 13.5 ± 2.6 Å ε = 1.9 ± 0.10 d head = 7.1 ± 4.4 Å ρ head /ρ core = 1.7 ± 1.5 backgr = ± /cm q [Å -1 ] 23

24 Molecular constraints: Δρ e tail = Z V e tail tail Z V e H 2O H 2O V = core N agg V tail Δρ e head = Z V e head head + nz + nv H 2O H 2O Z V e H 2O H 2O n water molecules in headgroup shell V = N ( V + nv ) 2O shell agg head H n micelles = N c agg M surfactant 24

25 Cylinder L 2R P( q) π / 2 = 0 2J 1 ( qr sinα ) qr sinα sin( ql cosα / 2 ql cosα / 2 2 sinα dα J 1 (x) is the Bessel function of first order and first kind. 25

26 P(q) cylinder L>> R P(q) P cross-section (q) P large (q) q 1 L = 60 Å 120 Å 300 Å 3000 Å R = 30 Å 26

27 Glucagon Fibrils R=29Å R=16 Å Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387,

28 Primus 28

29 Form factor for particles with arbitrary shape Spherical monodisperse particles I ( q) = P sphere ( q, R) n n i= 1 j= 1 sin ( qr ) qr ij ij 29

30 Monte Carlo integration in calculation of form factors for complex structures Generate points in space by Monte Carlo simulations Select subsets by geometric constraints x(i) 2 +y(i) 2 +z(i) 2 R 2 Caclulate histograms p(r) functions (Include polydispersity) Fourier transform MC points Sphere 30

31 Protein Micelle ISCOM vaccine particle 31

32 Immune-stimulating complexes: ISCOMs - Self-assembled vaccine nano-particles Quil-A Stained TEM ISCOMs typically: 60 70wt% Quil-A, 10 15wt% Phospholipids 10-15wt% cholesterol M T Sanders et al. Immunology and Cell Biology (2005) 83,

33 SAXS data 10 0 ISCOMs without toxoid Investigation of the structure of ISCOM particles by SAXS S. Manniche, H.B. Madsen, L. Arleth, H.M. Nielsen, C.L.P. Oliveira, J.S. Pedersen, in preparation. Oscillations relatively monodisperse Bump at high q core-headgroup structure 10-1 I(q) q [Å -1 ] 33

34 Combination of icosahedrons, tennis balls and footballs 10 0 Fit result: Icosah: ϕ m M = Tennis: ϕ m M = Football: ϕ m M= I(q) 10-2 From volumes of cores: Icosah: M = a.u. Tennis: M = a.u. Football M= a.u q [Å -1 ] Mass distribution: Icosah: ϕ m = Tennis: ϕ m = Football: ϕ m =

35 Icosahedrons, tennis ball, football Hydrophobic core: (2 x) 11 Å Hydrophilic headgroup region: Width 13 Å 242 Å 335 Å 440 Å 35

36 Pure SDS micelles in buffer: C > CMC ( 5 mm) N agg = 66±1 oblate ellipsoids R=20.3±0.3 Å ɛ=0.663±0.005 C CMC 36

37 Polymer chains in solution Gigantic ensemble of 3D random flights - all with different configurations 37

38 Gaussian polymer chain Look at two points: Contour separation: L Spatial separation: r r Contribution to scattering: sin( qr) I2 ( q) = L qr For an ensemble of polymers, points with L has <r 2 >=Lb and r has a Gaussian distribution: D( r) 3r exp 2 < r 2 2 > exp [ q 2 Lb / 6] Add scattering from all pair of points Density of points : (L o -L) L o L L 38

39 39 Gaussian chains: The calculation [ ] ] 1 ) 2[exp( 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 o o L o o L L o o o o o = + = = = = How does this function look? R g2 = Lb/6

40 Polymer scattering 10 Form factor of Gaussian chain q 1/ R g P(q) q ν = q qr g 40

41 Gaussian chain+ background R g = 21.3 Å χ 2 = Lysozyme in Urea 10 mg/ml α-synuclein Gaussian chain+ background R g = 21.3 Å I(q) [cm -1 ] 10-2 χ 2 = q [Å -1 ] Δρ is low! 41

42 Self avoidence No excluded volume P( x R 2 2 2[exp( x) 1+ x] g q ) = 2 x No excluded volume => expansion = no analytical solution! 42

43 Monte Carlo simulation approach (1) Choose model Pedersen and Schurtenberger 1996 (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 (5) Fit experimental data using numerical expressions for P(q) P(q,L,b) L = contour length b = Kuhn ( step ) length 43

44 Expansion = 1.5 Form factor polymer chains 10 1 q 1/ R ( Gaussian) g 0.1 q 1/ R ( excl. vol.) g Excluded volume chains: q 1/ ν 1.70 = q P(q) ν = Gaussian chains q 1 ν 2 = q ν = 1/ 2 q -1 local stiffness qr g (Gaussian) 44

45 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 45

46 Models II Polydispersity: Spherical particles as e.g. vesicles No interaction effects: Size distribution D(R) Oligomeric mixture: Discrete particles dσ q) dω ( 2 = c Δρ i M i f i P ( q) f i = mass fraction f = 1 i i i Application to insulin: Pedersen, Hansen, Bauer (1994). European Biophysics Journal 23, (Erratum). ibid 23, Used in PRIMUS Oligomers 46

47 Concentration effects 47

48 Concentration effects in protein solutions α-crystallin eye lens protein = q/2π 48

49 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) 49

50 Low concentrations Zimm 1948 originally for light scattering P (q) = - Subtract overlapping configuration P (q) = P(q) - υ P(q) 2 = P(q)[1 - υ P(q)] υ ~ concentration Higher order terms: P (q) = P(q)[1 - υ P(q){1 - υ P(q)}] = P(q)[1 - υ P(q)+ υ 2 P(q) 2 ] = P(q)[1 - υ P(q )+ υ 2 P(q) 2 - υ 3 P(q) 3..] = P( q) 1+υP( q) = Random-Phase Approximation (RPA) 50

51 51 Zimm approach ) ( 1 ) ( ) ( q P q P K q I +υ = + = + = υ υ ) ( 1 ) ( ) ( 1 ) ( q P K q P q P K q I 3 / 1 1 ) ( 2 2 R g q q P + [ ] +υ + = 3 / 1 ) ( R g q K q I With Plot I(q) 1 versus q 2 + c and extrapolate to q=0 and c=0!

52 Zimm plot Kirste and Wunderlich PS in toluene I(q) -1 P(q) c = 0 q = 0 q 2 + c 52

53 My suggestion: ( - which includes also information from what follows) Minimum 3 concentrations for same system. Fit data simultaneously all data sets I( q c i ) = 1+ Pi 2 ca1 exp( q a i 2 2 / 3) with a 1, a 2, and P i as fit parameters 53

54 Osteopontin (M=35 kda )in water and 10 mm NaCl , 5 and 10 mg/ml I(q) (cm -1 ) q (A -1 ) Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 54

55 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 55

56 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 56

57 Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012) 57

58 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%? 58

59 Structure factor Spherical monodisperse particles 2 2 I( q) = V Δρ P( q) = V 2 2 Δρ P( q) S( q) sin qr qr jk jk 2 2 = V Δρ P( q) p( r j ) sin qr qr j j S(q) is related to the probability distribution function of inter-particles distances, i.e. the pair correlation function g(r) 59

60 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 60

61 g(r) and S(q) q S( q) = 1+ n 4π ( g( r) 1) sin( qr) r qr 2 dr 61

62 GIFT Glatter: Generalized Indirect Fourier Transformation (GIFT) sin( qr) I( q) = 4π p( r) dr qr With concentration effects sin( qr) I( q) = Seff ( q, η, R, σ ( R), Z, csalt )4π p( r) dr qr Optimized by constrained non-linear least-squares method - works well for globular models and provides p(r) 62

63 A SAXS study of the small hormone glucagon: equilibrium aggregation and fibrillation 29 residue hormone, with Home-written a net charge software of +5 at ph~ mg/ml I(q) [arb. u.] 6.4 mg/ml 5.1 mg/ml p(r) [arb. u.] Hexamers 0,01 0,1 q [Å -1 ] 2.4 mg/ml 1.0 mg/ml r [Å] trimers monomers 63

64 Osmometry (Second virial coeff A 2 ) Π/c Ideal gas A 2 = 0 64 c

65 S(q), virial expansion and Zimm From statistical mechanics : S( q = 0) = RT Π c 1 = 1+ 2cMA c 2 MA In Zimm approach ν = 2cMA 2 65

66 A 2 in lysozyme solutions Isoelectric point A 2 O. D. Velev, E. W. Kaler, and A. M. Lenhoff 66

67 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 67

68 Theory for colloidal stability DLVO theory: (Derjaguin-Landau-Vewey-Overbeek) Debye-Hückel screened Coulomb potential + attractive interaction 68

69 Depletion interactions π π Asakura & Oosawa, 58 69

70 Integral equation theory Relate g(r) [or S(q)] to V(r) At low concentration g(r) = exp( V (r) / k B T) Boltzmann approximation 1 V (r) / k B T (weak interactions) Make expansion around uniform state [Ornstein-Zernike eq.] g(r) = 1 n c(r) [3 particle] [4 particle] - = 1 n c(r) n 2 c(r) * c(r) n 3 c(r) * c(r) * c(r) [* = convolution] c(r) = direct correlation function S(q) = 1 n c(q) n 2 c(q) 2 n 3 c(q) 3. = but we still need to relate c(r) to V(r)!!! 1 1 nc( q) 70

71 Closure relations Systematic density expansion. Mean-spherical approximation MSA: c(r) = V(r) / k B T (analytical solution for screened Coulomb potential - but not accurate for low densities) Percus-Yevick approximation PY: c(r) = g(r) [exp(v (r) / k B T) 1] (analytical solution for hard-sphere potential + sticky HS) Hypernetted chain approximation HNC: c(r) = V(r) / k B T +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) 71

72 α-crystallin S. Finet, A. Tardieu DLVO potential HNC and numerical solution 72

73 α-crystallin + PEG 8000 Depletion interactions: DLVO potential S. Finet, A. Tardieu HNC and numerical solution 40 mg/ml α-crystallin solution ph 6.8, 150 mm ionic strength. 73

74 Anisotropy Small: decoupling approximation (Kotlarchyk and Chen,1984) : Measured structure factor: S meas σ ( q) ( q) Ω = 1+ β ( q) q 2 nδρ P( q) [ S( q) 1] S( )!!!!! 74

75 75 Large anisotropy Anisotropy, large: Random-Phase Approximation (RPA): ) ( 1 ) ( ) ( 2 2 q P q P V n q d d ν ρ σ + Δ = 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 Δ = Ω ρ σ c(q) direct correlation function related to FT of V(r) Polymers, cylinders

76 Empirical PRISM expression SDS micelle in 1 M NaBr dσ ( q) dω = 2 nδρ V 2 P( q) 1 nc( q) P( q) c(q) = rod formfactor - empirical from MC simulation Arleth, Bergström and Pedersen 76

77 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 : 77

78 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 78

79 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, 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