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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

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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 Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, Department of Chemistry and iNANO Center University of Aarhus Denmark SAXS lab 1 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 3 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! 5 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 !!!! 7 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: 11 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 13 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: 17 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 25 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 31 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 <r 2>=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)
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