SAXS/ SANS data processing and overall parameters Petr V. Konarev - PowerPoint PPT Presentation

EMBO Global Exchange Lecture Course 30 November 2012 Hyderabad India SAXS/ SANS data processing and overall parameters Petr V. Konarev European Molecular Biology Laboratory, Hamburg Outstation BioSAXS group

Small-angle scattering: experiment Detector Sample Log (Intensity) Monochromatic beam 2 1 0 2 θ -1 Wave vector k , k=2 π / λ 0 1 2 3 s=4 π sin θ/λ, nm- 1 k 1 Radiation sources: X-ray generator ( λ = 0.1 - 0.2 nm) Scattering vector s=k 1 -k, Synchrotron ( λ = 0.03 - 0.35 nm) s=4 π sin θ / λ Thermal neutrons ( λ = 0.2 - 1 nm)

BIOSAXS beamline P12 (Petra-3) Pilatus 2M 2D Raw Data Iron nanoparticles (June 2011)

PILATUS Pixel X-ray Detector at P12 PILATUS 2M (24*100K modules) Silver behenate Active area 250*290 mm 2 , pixel size: 172 μ m Axis calibration standard Readout time: 3.6ms, framing rate: 50Hz

Raw data reduction steps Radial integration of 2D image into 1D curve • Exact coordinates of the beam center are required for • integration (determined from AgBeh data) Mask file is used to eliminate beamstop and inactive • detector area Associated errors in the data points are computed from the • numbers of counts using Poisson statistics Data are normalized to the pindiode value (intensity of the • transmitted beam) and exposure time Data are transferred into ASCII format containing 3 • columns: s I(s) Er(s)

Small Angle Scattering Normalization against: Radial averaging • data collection time, • transmitted sample intensity. Log I(s), a.u. s, nm -1 |s| = 4 π sin θ / λ

Scattering by matter • X-rays are scattered mostly by electrons • Thermal neutrons are scattered mostly by nuclei • Scattering amplitude from an ensemble of atoms A( s ) is the Fourier transform of the scattering length density distribution in the sample ρ ( r ) • Experimentally, scattering intensity I( s ) = [A( s )] 2 is measured.

Small-angle scattering: contrast I sample (s) I matrix (s) I particle (s) ♦ To obtain scattering from the particles, matrix scattering must be subtracted, which also permits to significantly reduce contribution from parasitic background (slits, sample holder etc) ♦ Contrast Δρ = < ρ (r) - ρ s > , where ρ s is the scattering density of the matrix, may be very small for biological samples

X-rays neutrons • X-rays: scattering factor increases with atomic number, no difference between H and D • Neutrons: scattering factor is irregular, may be negative, huge difference between H and D Element H D C N O P S Au At. Weight 1 2 12 14 16 30 32 197 N electrons 1 1 6 7 8 15 16 79 b X ,10 -12 cm 0.282 0.282 1.69 1.97 2.16 3.23 4.51 22.3 b N ,10 -12 cm -0.374 0.667 0.665 0.940 0.580 0.510 0.280 0.760

Sample and buffer scattering Looking for protein signals less than 5% above background level…

Sample and buffer scattering

Sample and buffer scattering [ ] − − − T I (s) T I (s) T T I (s) ( ) = I(s) m s s m m s e cT T Det s ( ) s m Here, subscripts s, m and e denote the scattering from sample, matrix (e.g. solvent) and empty cell (camera background), T stands for transmission, c for sample concentration and Det ( n ) is the detector response function. For solution scattering studies T s usually equals to T m and the third term vanishes.

Analysis of biological SAS data Overall 3 Parameters lg I, relative R g 2 D max Scattering curve I ( s) MM exp Excluded 1 Volume 0 2 4 6 8 s, nm -1

Overall parameters Radius of gyration R g (Guinier, 1939) 1 ≅ − I(s) I( ) R s ) 2 2 0 exp( g 3 Maximum size D max : p(r)=0 for r> Dmax Maximum size D max : p(r)=0 for r> Dmax Excluded particle volume (Porod, 1952) ∞ ∫ = π = Q s I s ds 2 2 V 2 I(0)/Q; ( ) 0

Program PRI MUS- graphical package for data manipulations and analysis ♦ data manipulations (averaging, background subtraction, merging of data in different angular ranges, extrapolation to infinite dilution ) ♦ evaluation of radius of gyration and forward intensity (Guinier plot, module AUTORG), estimation of Porod volume ♦ calculation of distance/size distribution function p(r)/V(r) (module GNOM) ♦ data fitting using the parameters of simple geometrical bodies (ellipsoid, elliptic/hollow cylinder, rectangular prism) (module BODIES) ♦ data analysis for polydisperse and interacting systems, mixtures and partially ordered systems (modules OLIGOMER, SVDPLOT, MIXTURE and PEAK) P.V. Konarev, V.V. Volkov, A.V. Sokolova, M.H.J. Koch, D.I. Svergun J.Appl. Cryst. (2003) 36, 1277-1282

PRIMUS: graphical user interface

Data quality Radiation damage Log I(s) , a.u. sample s, nm -1

Data quality Radiation damage Log I(s) , a.u. sample same sample again RADIATION DAMAGE! s, nm -1

Merging data Low and High Concentration Log I(s) 1 mg/ml 10 mg/ml s, nm -1

s, nm -1 Low and High Concentration Merging data Log I(s)

Merging data Low and High Concentration Log I(s) s, nm -1

Merging data Low and High Concentration Log I(s) s, nm -1

Merging data Low and High Concentration Log I(s) s, nm -1

E xtrapolation to zero concentration Infinite dilution Log I(s) 10 mg/ml 1 mg/ml 0 mg/ml? s, nm -1

Shape and size Log I(s) a.u. lysozyme apoferritin s, nm -1

The scattering is related to the shape Solid sphere lg I(s), relative lg I(s), relative lg I(s), relative lg I(s), relative lg I(s), relative 0 0 0 0 0 -1 -1 -1 -1 -1 Hollow sphere -2 -2 -2 -2 -2 -3 -3 -3 -3 -3 -4 -4 -4 -4 -4 -5 -5 -5 -5 -5 -6 -6 -6 -6 -6 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.0 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 s, nm -1 Dumbbell s, nm -1 s, nm -1 s, nm -1 s, nm -1 Flat disc Long rod

Guinier law ∞ sr sin( ) ∫ = π I s p r ds ( ) 4 ( ) sr 0 sr sr sr 2 4 sin( ) ( ) ( ) For small values of x, sinx/x can be expressed as : = − + − 1 .. sr 3 ! 5 ! Hence, close to the origin: I(s) = I(0)[1-ks 2 +…] ≈ I(0)exp(-ks 2 ) The scattering curve of a particle can be approximated by a Gaussian curve in the vicinity of the origin ≅ − 2 I s I s R 2 ( ) ( 0 ) exp( / 3 ) g This is a classical formula derived by Andre Guinier in 1938, in his first SAXS application (to defects in metals)

Radius of gyration ∫ Δ ρ r dV 2 ( ) r r = V R 2 Radius of gyration : r ∫ g Δ ρ dV ( ) r r V r R g is the quadratic mean of distances to the center of mass weighted by the contrast of electron density. R g is an index of non sphericity . 3 = R R For a given volume the smallest R g is that of a sphere : g 5 Ellipsoïd of revolution (a, b) Cylinder (D, H) + a b 2 2 D H 2 2 2 = R R = + g g 5 8 12 ideal monodisperse

Guinier plot example The law is generally used under its log form : ≅ − 2 I s I s R 2 ln[ ( )] ln[ ( 0 )] / 3 g A linear regression yields two parameters : I(0) (y-intercept) R g from the slope Validity range : 0 < sR g <1.3

PRIMUS: Guinier plot 2 Rg = ⋅ − ⋅ I s I s 2 ( ) ( 0 ) exp( 3 ) Rg – radius of gyration Rg = 2.68 +- 1.11e-2 I0 = 271.07 +- 0.605 M ≈ M BSA *(I(0)/I BSA (0)) Guinier

PRIMUS: AutoRG module AutoRg Petoukhov, M.V., Konarev, P.V., Kikhney, A.G. & Svergun, D.I. (2007) J. Appl. Cryst. , 40 , s223-s228.

Rods and platelets In the case of very elongated particles, the radius of gyration of the cross-section can be derived using a similar representation, plotting this time sI(s) vs s 2 ∝ − sI s s R 2 2 ( ) exp( / 2 ) c Finally, in the case of a platelet, a thickness parameter is derived from a plot of s 2 I(s) vs s 2 : ∝ − s I s s R 2 2 2 ( ) exp( ) t R t = T / 12 with T : thickness

Porod law and excluded particle Volume Intensity decay is proportional to s -4 at higher I(s) ~ s -4 angles (for globular particles of uniform density) π K is a constant determined to ensure the I 2 2 ( 0 ) = V P asymptotical intensity decay proportional to s -4 at higher angles following the Porod's law ∞ ∫ − for homogeneous particles I s K s ds 2 [ ( ) ] 0 V p is excluded volume of the hydrated partcile, for globular macromolecultes its value in nm 3 is approximately twice (1.7 times) of the molecular mass in kDa V p =120 nm 3 MM exp =(70±5) kDa

ds ) K − V – excluded volume of particle ) s ( Porod I ( 2 s ∞ ∫ 0 ) 0 ( I 2 PRIMUS: Porod plot π 2 V = 92.37 = Q ) 0 ( I 2 π 2 = V

Real/ reciprocal Real/ reciprocal space transformation space transformation p(r)=r 2 γ (r) distance distribution function γ 0 (r)= γ (r)/ γ (0) i Probability to find a point at r ij distance r from a given point j inside the particle

Distance distribution function from simple shapes

function of helix helix Distance distribution function of Distance distribution

PRIMUS: GNOM menu Gnom Indirect Fourier Transform Run

PRIMUS: P(R) function Gnom Indirect Fourier Transform