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Introduction to Small-Angle X-Ray Scattering

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Introduction to Small-Angle X-Ray Scattering Introduction to Small-Angle X-ray Scattering Thomas M. Weiss Stanford University, SSRL/SLAC, BioSAXS beamline BL 4-2 BioSAXS Workshop, March 28-30, 2016 Sizes and Techniques Diffraction and Scattering Scattering of X-rays from a single electron I e Thomson formula for the scattered intensity from a single electron 2 I 0 2 1 cos( 2 ) 1 I e r0 I 0 2 r 2 2 e 15 r0 2.817 10 m I : Intensity of incoming X - rays 2 0 mc 2 : angle of observatio n Classical electron radius I e : Intensity of scattered X - rays The Thomson formula plays a central role for all scattering calculations involving absolute intensities. Typically calculated intensities of a given sample will be expressed in terms of the scattering of an isolated electron substituted for the sample. In small angle scattering the slight angle dependence (the so-called polarization factor) in the Thomson formula can be neglected. Interference of waves • waves have and amplitude and phase • interference leads to fringe pattern (e.g. water waves) • the fringe pattern contains the information on the position of the sources (i.e. structure) • in X-ray diffraction the intensities (not the amplitudes) of the fringes are measured “phase problem” constructive destructive Scattering from two (and more) electrons scattering vector q k k 0 two electrons 2 4 sin with k s and q q 2 F (q ) f exp( iq r ) f (1 exp( iq r )) e i e i 1 Note: … generalized to N electrons F(q) is the Fourier N Transform of the F (q ) f exp( iq r ) spatial distribution e i i 1 of the electrons … averaged over all orientations N sin( qr ) F (q ) f e i 1 qr using the continuous (radial) distribution ( r ) of the electron cloud in an atom sin( qr ) F (q ) f (q ) dr (r )r 2 with f (0 ) Z qr 0 atomic scattering factor Scattering from Molecules The scattering amplitude or form factor, F(q), of an isolated molecule with N atoms can be determined in an analogous manner: N F (q ) f (q ) exp( iq r ) i.e. the Fourier Transform i i i 1 of the atomic distribution The scattered intensity from the isolated molecule is then N N 2 I (q ) F (q ) f (q ) f (q ) exp( iq (r r )) i j i j i 1 j 1 In solution: N N average over all orientations sin( qrij ) I (q ) f (q ) f (q ) sin( qr ) i j ij qr due to solution average only exp( iq (ri r j )) i 1 j 1 ij qrij interatomic distances are Debye formula measured, not atomic coordinates Scattering from Molecules Each atomic distance rij in the molecule adds a sinx/x like term to the scattering intensity • small distance low frequency in sinx/x dominate signal at high q • large distance N N sin( qr ) high frequenciesij in sinx/x I (q ) f (q ) f (q ) i j dominate the signal at low q i 1 j 1 qrij Scattering Intensity I (q ) F (q ) F (q ) FT [ (r )] FT [ ( r )] FT [ (r ) ( r )] (r ) Autocorrelation function The measured scattering intensity is the spherically averaged Fourier transform of the auocorrelation of the electron density of the particle Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u Autocorrelation (r ) (r ) ( r ) (r u ) (r )dV u V u For a homogeneous particle Spherical average Characteristic Function (r ) r V (r ) (r ) 0 (r ) (r ) / (0) 2 0 r V with (0) V “probability of finding a point Pair distance distribution function: within the particle at a distance r from a given point” 2 2 2 p (r ) r (r ) Vr 0 (r ) Pair distance distribution function p(r) The p(r) function represents the histogram of distances between pairs of points within the particle. Dmax is the maximum diameter in the particle. Measured scattering intensity Pair distance distribution Dmax p (r ) 4 I (q ) qr sin( qr ) dq 0 Scattering from model structures Adopted from Svergun & Koch, “SAS studies of biological macromolecules in solution”, Rep. Prog. Phys. 66 (2003) 1735-1782, Fig. 5 (c)I Particles in Solution For solution scattering we typically require the following characteristics: • Monodisperse, i.e. identical particles i (q ) i (q ) j 1 j • Uncorrelated, i.e. no inter-molecular interactions present I (q ) n i (q ) j j j I (q ) Ni 1 (q ) Background Scattering and X-ray Contrast I (q ) I solution (q ) I solvent (q ) particle • The solvent scattering background must be properly subtracted to obtain the signal from the particles • the contrast, that makes the particles “visible” for X-rays, is the difference in electron density of the particle versus the solvent 2 ( (r ) s ) Protein solution scattering data • weak level of scattering at small angles • drops off quickly for higher angles • due to low contrast scattering level of background and sample is very similar except for the lowest angles • background and sample scattering need to be measured with high accuracy • a 1mg/ml solution of a globular protein of the size of lysozyme (14kD) scatters on the order of: 1 out of 106 incident photons “…. one in a million!” X-ray Contrast and Contrast Variation • change contrast by adding salts (e.g. Substance Average Contrast CsBr), sucrose or glycerol to the solvent (x1010 cm-2) • but that changes the chemical Protein 2.5 environment for the particles • other possibility to change contrast is Nucleic Acid 6.7 anomalous scattering Fatty Acid -1.1 Carbohydrates 4.5 Note: Contrast variation is widely used in neutron scattering, due to the large scattering length difference of hydrogen and deuterium Introducing the Radius of Gyration R 2 is the average electron density g r 2 (r )d r weighted squared distance of the 2 scatters from the centre of the R g object (r )d r 2 1 • Solid sphere radius R: 1 3 1 2 Rg = √(3/5) R • Thin rod length L 2 2 2 2 2 2 2 Rg = √(1/12) L Rg =(1 + 1 + 1 + 2 + 2 + 3 )/6=20/6 Rg=√3.333 = 1.82 • Thin disk radius R: Rg = √(1/2) R The Guinier approximation The low-q region of the scattering curve is characteristic for the overall dimension of the particle. I(q) 1 2 lim I I exp( q 2 R ) region 0 g q 0 3 Guinier I0 is proportional to Mw Radius of gyration: 0.1 0.2 0.3 0.4 0.5 q size of the particle “The Guinier Plot” 2 Plot ln I against q →Straight line, slope –Rg/3 1 2 2 (I(q) ln I ln I 0 q R g Ln 3 Deviation from the straight line in the Guinier plot indicate intermolecular interaction or aggregation 0.002 0.004 0.006 0.008 q*q The Guinier approximation Plot ln I against q2 →Straight line “The Guinier Plot” 1 2 ln I ln I q 2 R 0 g (I(q) 3 Ln N N sin( qrij ) I (q ) f (q ) f (q ) Recall: i j qr i 1 j 1 ij 0.002 0.004 0.006 0.008 q*q 2 cMV 2 thus I 0 f i I 0 partial 0 i N a i.e. the number of (excess) electrons in the sample c: concentration : partial specific Vol. M: molecular mass : prot. e-density V: Volume 0: solvent e-density Radius of Gyration 2 r (r )d r 2 R g (r )d r Alternatively to using the Guinier plot to determine the Rg of the protein of can also use the following experession involving the P(r) function: D max 2 i.e. Rg equals the second moment r p (r )dr of the electron density distribution R 0 as well as half the second moment g D max of the distance distribution 2 p (r )dr function 0 This is often better than using the Guinier plot as it involves the whole scattering curve Radius of gyration for proteins and viruses Molecular Weight Rg (A) Ribonuclease 12700 14.8 Lysozyme 14800 14.5 B-lactoglobulin 36700 21.7 BSA 68000 29.5 Myosin 493000 468 Brome Mosaic Virus 4.6 106 134 TMV 3.9 106 924 Kratky analysis • Kratky plot: I*q2 vs. q • sensitive to morphology of particle • sensitive to the compactness of a protein • unfolded and folded states of proteins are easy to distinguish Hiller et al., Biomaterials, 24 (2003), Fig5 Example: folding of cytochrome C Putnam et al., Quat.Rev.Bioph 40,3 (2007), Fig24 Akiyama et al., PNAS, 99, (2002) ab-initio structure determination Envelop models • using spherical harmonics to produce molecular envelopes that fit the experimental scattering data Bead models • fitting the scattering data using bead as scattering centers • so-called dummy residues (scattering centers representing the Ca atoms of the residues) Can be extremely powerful particularly if combined with (partial) crystal structures if available! but be careful: Program packages: you will always get a structure from these • ATSAS from EMBL Hamburg (Svergun group) • IMP from UCSF (Sali group) programs, but it doesn’t mean they make • SAXS3D from Stanford (Doniach group) sense • SASTBX from LBL (Zwart group) • … Experimental setup 2 I (q ) N F (q ) S (q ) 4 sin guard slit q detector defining slit 2 sample q D isotropic scattering signal: the 2D detector image is integrated to yield I(q) vs q BioSAXS instrument at SSRL BL 4-2 flightpath: 0.2 – 3.5m detector sample incident beam • widely re-configurable instrument for Q-range Q= 0.003/Å ..
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