Form and Structure Factors: Modeling and Interactions Jan Skov Pedersen, iNANO Center and Department of Chemistry SAXS lab University of Aarhus Denmark 1 New SAXS instrument in ultimo Nov 2014: Funding from: Research Council for Independent Research: Natural Science The Carlsberg Foundation The Novonordisk Foundation 2 Liquid metal jet X-ray sourcec Gallium Kα X-rays (λ = 1.34 Å) 3 x 109 photons/sec!!! 3 Special Optics Scatterless slits + optimized geometry: High Performance 2D 10 3 x 10 ph/sec!!! Detector Stopped-flow for rapid mixing Sample robot 4 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 5 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 6 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 7 MC applications in modelling K.K. Andersen, C.L.P. Oliveira, K.L. Larsen, F.M. Poulsen, T.H. Callisen, P. Westh, J. Skov Pedersen, D.E. Otzen (2009) The Role of Decorated SDS Micelles in Sub-CMC Protein Denaturation and Association. Journal of Molecular Biology, 391(1), 207-226. J. Skov Pedersen ; C.L.P. Oliveira. H.B Hubschmann, L. Arelth, S. Manniche, N. Kirkby, H.M. Nielsen (2012). Structure of Immune Stimulating Complex Matrices and Immune Stimulating Complexes in Suspension Determined by Small-Angle X-Ray Scattering. Biophysical Journal 102, 2372-2380. J.D. Kaspersen, C.M. Jessen, B.S. Vad, E.S. Sorensen, K.K. Andersen, M. Glasius, C.L.P. Oliveira, D.E. Otzen, J.S. Pedersen, (2014). Low-Resolution Structures of OmpA-DDM Protein-Detergent Complexes. ChemBioChem 15(14), 2113-2124. 8 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! 9 I will outline some calculations to show that it is not black magic ! 10 Input data: Azimuthally averaged data qi , I(qi ), I(qi ) i 1,2,3,...N qi calibrated I(qi ) calibrated, i.e. on absolute scale - noisy, (smeared), truncated I(qi ) Statistical standard errors: Calculated from counting statistics by error propagation - do not contain information on systematic error !!!! 11 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 12 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 13 Form factors of geometrical objects 14 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: 15 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 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: (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. 16 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 17 From factor of a solid sphere (r) 1 0 R r sin(qr) R sin(qr) A(q) 4 (r) r 2 dr 4 r 2 dr 0 qr 0 qr 4 R sin(qr) rdr q 0 f ' g dx fg fg'dx (partial integration)… R 4 R cos qR sin qr q q q 0 4 R cos qR sin qr 2 q q q 4 sin qR qR cos qR q 3 4 3[sin(qR) qR cos(qR)] R 3 18 3 (qR)3 spherical Bessel function Form factor of sphere P(q) = A(q)2/V2 1/R q-4 Porod 19 C. Glinka Ellipsoid Prolates (R,R,R) Oblates (R,R,R) / 2 2 P(q) (qR') sin d 0 R’ = R(sin2 + 2cos2)1/2 3sin(x) x cos(x) (x) x 3 20 P(q): Ellipsoid of revolution Glatter 21 Lysosyme Lysozyme 7 mg/mL 100 Ellipsoid of revolution + background R = 15.48 Å 2 = 1.61 (prolate) =2.4 (=1.55) 10-1 ] -1 I(q) [cm 10-2 10-3 0.0 0.1 0.2 0.3 0.4 q [Å-1] 22 SDS micelle Hydrocarbon core Headgroup/counterions 20 Å www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html mrsec.wisc.edu/edetc/cineplex/ micelle.html 23 Core-shell particles: Rout Rin = Acoreshell (q) shellVout(qRout ) (shell core )Vin(qRin ) where 3 3 Vout= 4 /3 and Vin= 4Rin /3. core is the excess scattering length density of the core, shell is the excess scattering length density of the shell and: 3sin x x cos x (x) x3 24 101 100 10-1 10-2 P(q) 10-3 10-4 10-5 0.0 0.1 0.2 0.3 0.4 0.5 q [Å-1] Rout =30Å Rcore =15Å shell =1 core 1. 25 SDS micelles: prolate ellispoid with shell of constant thickness 0.1 2 =2.3 I(0) = 0.0323 0.0005 1/cm Rcore = 13.5 2.6 Å =1.9 0.10 ] 0.01 dhead = 7.1 4.4 Å -1 head/core = 1.7 1.5 backgr = 0.00045 0.00010 1/cm I(q) [cm 0.001 0.0 0.1 0.2 0.3 0.4 26 q [Å-1] Molecular constraints: Z e Z e e tail H 2O tail Vcore N aggVtail Vtail VH 2O e e n water molecules in headgroup shell e Z head nZ H 2O Z H 2O head Vhead nVH 2O VH 2O Vshell N agg (Vhead nVH 2O ) c nmicelles N agg M surfactant 27 Cylinder 2R L / 2 2 2J1 qR sin sin(qL cos / 2 P(q) sin d 0 qR sin qL cos / 2 J1(x) is the Bessel function of first order and first kind. 28 P(q) cylinder L>> R P(q) Pcross-section(q) Plarge(q) q L = 60 Å R = 30 Å 120 Å 300 Å 3000 Å 29 Glucagon Fibrils R=29Å R=16 Å Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, 30 Daniel Otzen and Jan Skov Pedersen J.