Scattering of X-rays P. Vachette IBBMC (CNRS-Université Paris-Sud), Orsay, France EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
SAXS measurement Sample SAXS measuring cell EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
SAXS measurement Scattering experiment SAXS pattern 1000 ? ? X-ray beam 100 I(q) 10 1 Detector 0 0.1 0.2 0.3 0.4 0.5 q = 4 (sin )/ Å -1 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
SAXS measurement SAXS pattern 1000 Structural ? parameters: 100 R g , D max , … I(q) 10 1 0 0.1 0.2 0.3 0.4 0.5 q = 4 (sin )/ Å -1 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
EMBO Practical Course on Solution Scattering from Biological Macromolecules D. Svergun Hamburg, October 27 th – November 3 rd 2014
Outline Reminder of elementary tools and notions - X-ray Scattering by an electron - X-ray Scattering by assemblies of electrons - Fourier transform - Convolution Product X-ray scattering by particles in solution. - ideality and monodispersity - Guinier law. - p(r) calculation - Porod law - Debye law - Kratky plot EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Elastic scattering by a single electron r 2 O - elastic : interaction without exchange of energy. The scattered photon has the same energy (or wavelength) than the incident photon. The elastically scattered intensity by an electron placed at the origin is given by the Thomson formula below: 2 1 cos (2 ) 1 2 : scattering angle, 2 J. Thomson I (2 ) r I cos2 close to 1 at small-angles 0 0 2 2 r I 0 intensity (energy/unit area /s) of the 2 incident beam. e 12 r 0.282 10 cm 0 2 r 0 classical radius of the electron. mc EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
differential scattering cross-section of the electron 2 1 cos (2 ) 2 2 26 2 d / d r r 7.9510 cm 0 0 2 the scattering length of the electron b e 2 b d / d e EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Scattering factor The scattering factor f of an object is defined as the ratio between the amplitude of the scattering of the object and that of one electron in identical conditions. The scattering factor of a single electron f e 1. We therefore eliminate d /d from all expressions . EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Scattering by an electron at a position r r.u 0 u 1 u 0 M u 1 r source 2 u 0 r.u 1 O Path difference = r.u 1 -r.u 0 = r.(u 1 - u 0 ) corresponding to a phase difference 2 r.(u 1 - u 0 )/ for X-rays of wavelength . EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
momentum transfer scattered wavevector k 2 k k 0 1 q length 2 / k 1 q k k 0 1 2 k 0 4 sin q q O length 2 / q is the momentum transfer A q e r q ( ) i . The scattered amplitude by the electron at r is where A(q) is the scattered amplitude by an electron at the origin Phase difference f = q.r EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
scattering vector 2sin Phase difference f = 2 r.s s ! 4 sin D. Svergun and coll. s EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
scattering by assemblies of electrons the distance D between scatterers is fixed, e.g. atoms in a molecule : coherent scattering one adds up amplitudes N e r q q F( ) = Σ f i i i i=1 Use of a continuous electron density r ( r ): r r rq q r q q q i F( ) ( ) e dV I( ) F( ).F ( ) and r V D is not fixed, e.g. two atoms in two distant molecules in solution : incoherent scattering one adds up intensities . EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Fourier Transform F( q ) is the Fourier transform of the electron density r ( r ) describing the scattering object. r r rq F. T. q r ( ) i r ( r ) F( ) e dV r V Properties of the Fourier Transform - 1 – linearity FT ( 1 r 1 2 r 2 ) = 1 FT( r 1 ) 2 FT( r 2 ) - 2 – value at the origin r r r F(0) ( ) dV r V EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product A convolution is an integral that expresses the amount of overlap of one function B as it is shifted over another function A. u r r u r u A( ) B( ) A( )B( ) dV u V 1 B(r) A(r)*B(r) A(r) r B r A r r EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r-u) A(r)*B(r) A(u) r A r B u r A + r B u -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r-u) A(r)*B(r) A(u) r A r B u r A + r B u -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r-u) A(r)*B(r) A(u) r B r A u r A + r B u -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r-u) A(r)*B(r) - (r A -r B ) A(u) r A r B u r A + r B u -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r-u) A(r)*B(r) - (r A -r B ) A(u) r B r A u r A + r B u -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Convolution product B(r) A(r)*B(r) - (r A -r B ) r A - r B A(r) u r A r B r A + r B r -(r A + r B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Fourier transform of a convolution product FT(A B) FT(A) FT(B) FT(A B) FT(A) FT(B) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Autocorrelation function u r r r r r r r r u u ( ) ( ) ( ) ( ) ( ) dV u V r r ( r )= r ( uniform density ) particle ghost => ( r )= r 2 V ov ( r ) ( 0 )= r 2 V and r ( ) r ( ) spherical average 0 ( ) r ( ) r (0) 1 characteristic function 0 (r) 0 (r) : probability of finding a point within the particle at a distance r from a given point D max r EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Distance (pair) distribution function p(r) is the distribution of distances between all pairs of points within the particle weighted by the respective electron densities r ij j i p (r) - 0 (r) : probability of finding within the particle a point j D max at a distance r from a given point i - number of el. vol. i V r - number of el. vol. j r 2 number of pairs (i,j) separated by the distance r r 2 V 0 (r) r 2 2 2 p r ( ) ( ) r Vr r ( ) r 0 EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Solution of particles = * Solution Motif (protein) Lattice Dr ( r ) Dr p ( r ) * = d( r ) . d (c, q ) F( c ,q) F( 0 ,q) EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Solution of particles For spherically symmetrical particles I(c,q) = I(0,q) x S(c,q) i 1 (q) structure factor form factor of the solution of the particle Still valid for globular particles though over a restricted q-range Information on: Shape of the particle Interactions between particles Talk of J.S. Pedersen, Friday EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
Solution X-ray scattering Diagram of the experimental set-up Momentum transfer X-ray q = 4 sin / = 2 s scattering Modulus of the scattering vector curve s = 2sin / X-ray beam 2 Sample 10µl – 50µl 0.1mg/ml – (>)10mg/ml Detector EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg, October 27 th – November 3 rd 2014
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