Scattering of X-rays. EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 11 September 2001

Transcription

1 Scattering of X-rays

2 Books on SAS - " The origins" (no recent edition) : Small Angle Scattering of X-rays A. Guinier and A. Fournet, (1955), in English, ed. Wiley, NY - Small-Angle X-ray Scattering O. Glatter and O. Kratky (198), Academic Press. - Structure Analysis by Small Angle X-ray and Neutron Scattering L.A. Feigin and D.I. Svergun (1987), Plenum Press. - Neutron, X-ray and Light scattering: Introduction to an investigative Tool for Colloidal and Polymeric Systems (1991), P. Lindner and Th. Zemb (ed.), North-Holland -The Proceedings of the SAS Conferences held every three years are usually published in the Journal of Applied Crystallography. -The latest proceedings are in the J. Appl. Cryst., 30, (1997), next one soon to come out

3 Reminder - notations Fourier Transform ρ(r) F. T. = ρ r e iπ rs V r F( s) ( ) dv r F(s) F. T. -1 = s e iπ rs V S ρ() r F() dv S

4 Properties of the Fourier Transform - 1 linearity FT (λ 1 ρ 1 + λ ρ ) = λ 1 FT(ρ 1) + λ FT(ρ ) - value at the origin - 3 translation ρ(r + R) F. T. F(0) = ρ( r) dvr r V F( s) i e π Rs

5 Fourier Transforms of simple functions - 1 delta function at the origin δ(ρ) ρ(r)=δ(r) o r - Gaussian of width 1/K f() r = e π K r - 1 constant unitary value F(s)=1 - Gaussian of width K F( s) = e π s K s 1-3 rectangular function -3 sinc 1 f(x) -1/a a 1/a F( s) = a sinπ sa π sa -a/ a/

6 Convolution product A( r) B( r) = A( u)b( r u) dv V u u 1 B(r) A(r) A(r)*B(r) r B r A r r

7 Convolution product B(r-u) r A -r B A(u) A(r)*B(r) r B r A u r A +r B u

8 Fundamental Property of the convolution product FT(A B) = FT(A) FT(B) FT(A B) = FT(A) FT(B)

9 Autocorrelation function γ() r = ρ() r ρ( r) = ρ( r+ u) ρ() u dv spherical average γ () r = γ () r V u u ρ(r)=ρ (uniform density) => γ(0)= ρ V characteristic function particle ghost r γ () r = γ () r γ (0) 0 γ 0 (r) 1 r

10 Distance distribution function p() r = r γ() r = r γ () r Vρ 0 γ 0 (r) : probability of finding a point at r from a given point number of el. vol. i V - number of el. vol. j 4πr number of pairs (i,j) separated by the distance r 4πr Vγ 0 (r)=(4π/ρ )p(r) i r ij j p(r) D max r

11 The elastically scattered intensity is given by the Thomson formula 11+ cosϑ I = r0 I0 d e 13 r0 = =.8 10 cm mc Ie Scattering by a free electron θ : scattering angle, cosθ close to 1 at small-angles 6 = r = is the scattering cross-section of the electron cm X-ray scattering length of an electron I 0 intensity (energy/unit area /s) of the incident beam. The scattered beam has an intensity I e /d at a distance d from the scattering electron. In what follows, r 0 is omitted and only the number of electrons is considered

12 Scattering by an electron at a position r detector s 0 r.s 0 s 1 source r s 1 s 0 θ r.s O 1 Path difference = r.s 1 -r.s 0 = r.(s 1 -s 0 ) or r.(s 1 -s 0 )/λ in cycles, for X-rays of wavelength λ

13 s s = s = 0 1 s= s1 s0 = s = the scattering vector s 1 λ s is called the scattering vector sinϑ λ s 0 s length 1/λ s 1 O θ scattered i. Ese () π rs length 1/λ The scattered amplitude by the electron at r is where E(s) is the scattered amplitude by an electron at the origin

14 scattering vector! q = h= " s" = 4πsinϑ λ SAXS measurements are restricted to small-angles : θ 5 deg = 85 mrad at λ = 1.5 Å In this case, the further approximation is valid : s = s = sinϑ θ λ λ

15 Scattered amplitude F( s) ( ) iπ = rs ρ r e V r dv r s = sinϑ λ F(s) is the Fourier transform from the electron density ρ(r) describing the scattering object I( s) = F( s).f ( s) Scattered intensity Remark s = 0 F(0) = ρ( r) dvr r For a particle I(0) = m m : number of electrons V

16 Solution X-ray scattering Diagram of the experimental set-up X-ray beam θ X-ray scattering curve Sample s = sinθ /λ Detector

17 Particles in solution The relevant quantity is now the contrast of electron density between the particle and the solvent ρ(r) = ρ(r) - ρ 0 with ρ 0 = el. A -3 the electron density of water

18 Contrast of electron density ρ el. A -3 ρ = 0.43 ρ 0 = particle solvent ρ

19 X-ray scattering power of a protein solution Let W be the probability of an incident photon being scattered by a solution of spherical proteins : using the expression for the optimal thickness d(cm) = 0.3/λ (A 3 ) Protein : ρ = W = ( ρ) cvλdr = 0.3( ρ) cvr/ λ 0.43 el. A 3 ρ = 0.1 el. A =.8 10 solvent cm 3 10 ρ0 = el. A Example : 10 mg/ml solution of myoglobin v= 0.74 cm 3.g -1 r=0a, λ = 1.5A from H.B. Stuhrmann Synchrotron Radiation Research H. Winick, S. Doniach Eds. (1980) W W= 10-5 (x efficiency x geometrical factor) = / cvr λ

20 Solution of particles = Solution Motif (protein). Lattice ρ(r) = ρ p (r) d(r) F(c,s) F(0,s) δ(c,s)

21 Solution of particles For spherically symmetrical particles I(c,s) = I(0,s) x S(c,s) form factor of the particle structure factor of the solution Still valid for globular particles though over a restricted s-range

22 Solution of particles -1 monodispersity: identical particles - size and shape polydispersity -3 ideality : no intermolecular interactions - 4 non ideality : existence of interactions between particles In the following, we make the double assumption 1 and 3 and 4 dealt with at a later stage in the course

23 Ideal and monodisperse solution F( s) = ρ( r) e 1 V r iπ rs dv Particles in solution => thermal motion => particles have a random orientation / X-ray beam. The sample is isotropic. Therefore, only the spherical average of the scattered intensity is experimentally accessible. time particles 1 = i1 = 1 1 i () s () s F().F() s s I( s) = I( s) = F( s).f ( s) Ideality and monodispersity 1 r I( s) = Ni ( s)

24 i () s = FT[ ρ ()]. r FT[ ρ ( r )] = FT[ ρ ()* r ρ ( r )] 1 Let us use the properties of the Fourier transform and of the convolution product iπ i1( s) = FT[()] γ = rs r γ() r e dv V r r sin( π rs) i () s = 4 π p() r dr 1 0 π rs with pr rγ r () = ()

25 Or one of the symmetrical expressions : si () s = rγ()sin( r πrs) dr rγ() r = si()sin( s πrs) ds π 0

26 Relationships real space reciprocal space ρ(r) * p(r) < > p(r) FT FT FT. < > F(s) I(s) I(s)

27 Debye formula If the particle is described as a discrete sum of elementary scatterers,(e.g. atoms) the scattered intensity is : N N i ( i j) i () s f () s f () s e π 1 = i= 1 j= 1 where the f i (s) are the atomic scattering factors, and the spherically averaged intensity is (Debye) : N N sin π rs ij i1() s f () s f () s rs i j i= 1 j= 1 π ij i j r r s = where r ij = ri rj The Debye formula is widely used for model calculations

28 Debye formula If the particle is described using a density distribution, the Debye formula is written : sin ( πrs) () = ρ( r) ρ( r) 1 drdr V V π rs r1 r 1 i s

29 Porod invariant sin( π rs) γ () r = s I() s ds 0 π rs For r=0 : By definition : γ Q is called the Porod invariant Q depends on the mean square electron density contrast (0) = I( ) = 0 s sds Q (0) = ( r) dvr = V 0 γ ρ ρ ρ V Q= = 0 s I s ds ()

30 Intensity at the origin i1(0) = ρ( r) ρ( r' ) dvrdv V r V r' M i (0) = m = ( m m ) = v ( ρ ρ ) 1 0 P 0 N A NM c = is the concentration (w/v), e.g. in mg.ml -1 NV A cmv I(0) = v ( ρ ρ ) P 0 N A r' ideal monodisperse

31 Intensity at the origin If : the concentration c (w/v), the partial specific volume v P, the intensity on an absolute scale, i.e. the number of incident photons are known, Then, the molecular weight of the particle can be determined from the value of the intensity at the origin ideal monodisperse

32 Intensity on an absolute scale The experimental intensity I exp (0) is expressed as : I IIcMd (0) ( ) = 0 e v ρ ρ exp P 0 Na A d : thickness of the sample a : sample-detector distance 6 Ie = r0 = cm scattering cross-section of the electron I 0 total energy of the incident beam ideal monodisperse

33 Intensity on an absolute scale The determination of I 0 is performed using: - attenuation of the direct beam by calibrated filters (valid in the case of monochromatic radiation, beware of harmonics) - by using a well-known reference sample like the Lupolen (Graz) - using the scattering by water as proposed by O. Glatter in D. Orthaber, A. Bergmann & O. Glatter, J. Appl. Cryst. (000), 33, 18-5 Thorough presentation of calculations on an absolute scale P. Bösecke & O. Diat J. Appl. Cryst. (1997), 30, Clear presentation of the geometrical calculations and all the procedures used on ID (ESRF) to put intensities on an absolute scale.

34 Guinier law sin( π rs) I () s = 4 π p() r dr 0 π rs For small values of x, sinx/x can be expressed as : Hence, close to the origin: with ( πrs) ( πrs) ( πrs) 4 sin = π rs 3! 5! I() s I(0)1 ks I(0) e ks 4 I(0) = 4 π prdr ( ) and k= π r prdr ( ) prdr ( ) 6 The scattering curve of a particle can be approximated by a Gaussian curve in the vicinity of the origin ideal monodisperse

35 sin ( π rs) Is () = ρ( r) ρ( r) dd r r V V r1 r Guinier law Let us introduce the expansion of sinx/x into the Debye formula : We obtain the following expression : π rs 1 4 s Is ( ) = ρ( r ) ρ( r ) dd r r π ρ( ) ρ( ) dd r1 r 6 r r r r r r r1 r V V 1 1 V V hence k = 4π V V r1 r ρ( r) ρ( r ) r r dd r r 6 ρ( r) ρ( r) dd r r V V r1 r 1 1 ideal monodisperse

36 Guinier law writing r1 r = r1 r0+ r0 r where r 0 is the center of mass it comes 4π ρ ( r) rdv 4 r k= = 6 ρ() r dv 3 r V π V r r R g Radius of gyration Guinier law 4π I(s) I(0)exp R s 3 g ideal monodisperse

37 Radius of gyration Radius of gyration : R g = V r V ρ( r) r rdvr ρ( r) dv 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. For a given volume the smallest R g is that of a sphere : Ellipsoïd of revolution (a, b) Cylinder (D, H) a + b Rg = D H R g = r Rg = 3 5 R ideal monodisperse

38 Guinier plot example The law is generally used under its log form : 4π ln[ I(s) ] ln[ I(0) ] Rg s 3 A linear regression yields two parameters : I(0) (y-intercept) R g from the slope Validity range : 0 < πr g s<1. For a sphere ideal monodisperse

39 Attractive Interactions C 5C 0C I(s) C 10C 10 4 γ-crystallins c=160 mg/ml in 50mM Phosphate ph EMBO Practical Course on Solution Scattering 0 from A. Tardieu et al., LMCP (Paris) s = (sin θ)/λ A -1

40 Repulsive Interactions ,5mg/ml 31,5 mg/ml 13,3 mg/ml 6,6 mg/ml 3,3 mg/ml I(s) 000 ATCase in 10mM borate buffer ph s = (sin θ)/λ A -1

41 Virial coefficient In the case of moderate interactions, the intensity at the origin varies with concentration according to : I(0, c) I(0) ideal = 1 + AMc +... Where A is the second viral coefficient which represents pair interactions and I(0) ideal is to c. A is evaluated by performing experiments at various concentrations c. A is to the slope of c/i(0,c) vs c.

42 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 si( s) exp( π R s ) Finally, in the case of a platelet, a thickness parameter is derived from a plot of s I(s) vs s : s I( s) exp( 4 π R s ) t c with Rt = T 1 T : thickness ideal monodisperse

43 Volume of the particle Hypothesis : the particle has a uniform density i V ρ 1(0) = ( 1 ) and uniform density => ρ V Q = ρ = ρ 1 (Porod invariant) Hence the expression of the volume for a particle of uniform density : V 1 = i 1 (0) Q ideal monodisperse

44 The asymptotic regime : Porod law Hypothesis : the particule has a sharp interface with the solvent with a uniform electron density Porod showed that the asymptotic behaviour of the scattering intensity is given by : s si1 s S ρ = + Bs 8π lim ( ) ( ) S is the area of the solute / solvent interface B is a correction term accounting for : - short distance density fluctuations -uncertainties of i(s) at large s (weak signal) ideal monodisperse

45 Distance distribution function sin( π rs) p() r = r s I() s ds 0 π rs In theory, the calculation of p(r) from I(s) is simple. Problem : I(s) - is only known over [s min, s max ] : truncation - is affected by experimental errors - might be affected by distorsions due to the beam-size and the bandwidth λ/λ (neutrons) Calculation of the Fourier transform of incomplete and noisy data, which is an ill-posed problem. Solution : Indirect Fourier Transform. See lectures by O. Glatter and D. Svergun ideal monodisperse

46 Calculation of p(r) p(r) is calculated from i(s) using the indirect Fourier Transform method Basic hypothesis : The particle has a finite size D sin( π rs) Max i () s = 4 π p() r dr 1 0 π rs p(r) is parameterized on [0, D Max ] by a linear combination of orthogonal functions

47 Distance distribution function The radius of gyration and the intensity at the origin can be derived from p(r) using the following expressions : R 0 g D = max r p() r dr max ( ) 0 D p rdr and I max (0) 4 π ( ) = 0 D p r dr This alternative estimate of R g makes use of the whole scattering curve, and is much less sensitive to interactions or to the presence of a small fraction of oligomers. Comparison of both estimates : useful cross-check ideal monodisperse

48 p(r) :example Aspartate transcarbamylase from E.coli (ATCase) Heterododecamer (c 3 ) (r ) 3 quasi D 3 symmetry Molecular weight : 306 kda Allosteric enzyme conformations : T : inactive, compact R : active, expanded T R crys R sol ideal monodisperse

49 ATCase : scattering patterns T-state R-state I(s) (a.u.) (θ)/λ A -1 s = sin (θ)/λ ideal monodisperse

50 ATCase : p(r) curves 5 0 T-state R-state p(r) (a.u.) D Max r (A) ideal monodisperse

51 Information content of a SAXS curve Basic hypothesis : The particle has a finite size D max Sampling theorem : the curve is entirely determined from the values of the intensity at the nodes of the lattice s k = k/d max where k is an integer. Number of independent structural parameters : J = D ( s s ) Max Max min Example : spherical protein with 0.3 hydration (w/w) v=0.73 cm 3.g -1 D ma x = 1.5 M 1/3 => J=3.0 M 1/3 (s Max s min ) for s min =0.00 A -1, s Max =0.05 A -1 and M = 10 5 => J = 7 ideal monodisperse

52 The case of the sphere Unique case in which an analytical expression of the scattered intensity can be derived : sin x xcos x I() s = 3 x=πrs 3 x R : radius of the sphere I(s) = 0 for values of s solutions of πrs = tan(πrs) first minima at Rs = , 1.3, 1.73, etc.

53 Scattering intensity of the sphere I(s) sphere Rs

54 F. Rey, V. Prasad et al., EMBO J. (001) 0,

55 Rotavirus derived particles DLP VLP /6 DLP : double layer particles containing RNA VLP/6: Virus-like particles double-layer VP6 and VP About 700 Å diameter I(s) s = (sin θ) / λ Å -1

56 Rotavirus derived particles Icosahedral symmetry on the capsid surface : strong modulation of the electron density => modulation of I(s) around 5-30 Å :signal associated with the RNA layers inside the VLP. I(s) DLP VLP / s = (sin θ) / λ Å -1

57 Rotavirus DLP RNA F. Rey et al., EMBO J. (001), 0, Biological Macromolecules Hamburg 4 11 September 001

58 Scattering by an extended chain In the case of an unfolded protein : models developed for polymers Gaussian chain : linear association of N monomers of length l with no persistence length (no rigidity due to short range interactions between monomers) and no excluded volume (i.e. no long-range interactions). Is () Debye formula : where ( 1 x = x + e ) x= ( π R ) gs I(0) x I(s) depends on a single parameter, R g. Valid over a restricted s-range in the case of interacting monomers 1 1 ( π R g s) Limit at large s : lim s [ sis ( )] = π R I(s) varies like s - instead of s -4 for a globular particle (Porod law). g

59 Guinier plot : NCS heat unfolding Neocarzinostatine : small (113 residue long) all-β protein. arrows : angular range used for R g determination Pérez et al., J. Mol. Biol.(001) 308,

60 Debye law : NCS heat unfolding arrows : angular range used for R g determination Pérez et al., J. Mol. Biol.(001) 308,

61 Kratky plot This provides a sensitive means of monitoring the degree of compactness of a protein as a function of a given parameter. This is most conveniently represented using the so-called Globular particle : bell-shaped curve Kratky plot of s I(s) vs s. Gaussian chain : plateau at large s-values but beware: a plateau does not imply a Gaussian chain

62 Kratky plot : NCS heat unfolding In spite of the plateau, not a Gaussian chain when unfolded. Can be fit by a thick persistent chain Pérez et al., J. Mol. Biol.(001), 308,

63 Extended conformation of a modular protein :Hemocyanin from O. vulgaris O. Vulgaris hemocyanin (Hc), a dioxygen binding protein, is a decameric structure having the shape of a hollow cylinder with a fivefold symmetry axis. At alkaline ph, or at neutral ph in the presence of a divalent cation chelator (EDTA), the protein dissociates into its constitutive subunits. Each subunit comprises seven functional units of ca 50kDa joined by flexible linkers. Study of the dissociation product (subunit) at ph 7.5 and 9.5.

64 Extended conformation of a modular protein :Hemocyanin from O. vulgaris 1000 I/c (a.u.) ph 7.5 ph 9.5 ph 7.5 ph q (A -1 )

65 Hemocyanin from O. vulgaris ph 7.5 Guinier plot Distance distribution function ln(i/c) (a.u.) Rg=61.7 A MW= 400 kda = p(r) Rg= 6.5 A MW=400 kda q (A - ) r (A)

66 Hemocyanin from O. vulgaris ph 9.5 Guinier plot Distance distribution function ln(i/c) (a.u.) Slight upward curvature Rg=90.6 A p(r) Rg=10.3 A q (A - ) r (A)

67 Hemocyanin from O. vulgaris ph 9.5 Distance distribution function Debye plot 1. 1 Rg=10.3 A 1000 Rg= A p(r) = I/c (a.u.) exp. Debye fit r (A) q (A )

68 Some experimental considerations

69 Practical aspects Small-angle data : Guinier analysis, R g and I(0) - requirements : monodisperse and ideal solution to estimate the MW form the I(0) value, the partial specific volume must be known, and the intensity calibration performed with a reference sample (Lupolen, other protein). monodispersity : previous check by SEC-HPLC or LS ideality : extrapolation to infinite dilution by recording experiments at several, precisely determined, concentrations in the range 1mg/ml (or less) 5 mg/ml according to the MW of the particle. volume : less than 100µl for each sample (down to 30µl) - total amount : less than 1mg/ml

70 Practical aspects Wider-angle data : p(r), modeling - requirements : 5mg/ml <c<10mg/ml with an area detector can provide a reasonable curve with adequate statisitics. If available, a higher concentration will give better data (no stringent monodispersity and ideality requirements) Note : case of a special sensitivity to X-ray irradiation : - addition of radical scavengers and/or -sample circulation through the beam cross-section (a few µls in the beam), with a correlative increase in volume.

71 Lysozyme A I(s) Lysozyme : Da MW 5 mg/ml solution in Acetate buffer ph 4.5 I(s) s min R g =15. Å in the air R g =15.0 Å under vacuum ,5 1 1,5, x s Å ,01 0,0 0,03 0,04 0,05 s = sin θ/λ Å -1

72 PGK B Yeast 3-phosphoglycerate kinase (PGK) : Da MW I(s) I(s) s mi n 5 mg/ml solution in Tris buffer ph 7.5 R g =4.7 Å in the air x s Å - R g =4.4 Å under vacuum s = sin θ/λ Å -1

73 Cell under vacuum Ratio of buffer scattering air and windows / under vacuum 5 cm air 5 µm Kapton window upstream 5 µm mica window downstream I Buffer(Air) / I Buffer(Vacuum) s = sin θ/λ Å -1

74 S/B ratio 3 S/B ratio lysozyme 5mg/ml 6 S/B ratio PGK 5mg/ml 5 I (Lyso-Buffer) / I Buffer I (PGK-Buffer) / I Buffer s = sin θ/λ Å -1 s = sin θ/λ Å -1

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78 Limits and drawbacks SAXS provides spherically averaged intensity, which entails a loss of structural information. The intensity strongly decreases with the scattering angle, thereby restricting the resolution of the available information. The method is therefore not sensitive to details, to potentially crucial changes in the structure involving e.g. side-chain movements. The weak signal restrict the characteristic times of kinetics to typically the ms range as compared to the ns or less with Laue TR- PX.

79 Advantages of the method SAXS provides time-resolved structural information as well as information on the interactions between particles. A solution method, SAXS allows one to vary at will most physicochemical parameters (T, P, ph, ionic strength, substrate, etc.) and assess their effect on the conformation of the particle under investigation. It can be used as a stand-alone experimental approach (structural parameters, ab initio modeling) or, often most advantageously, coupled to other techniques : structural (EM, PX), spectroscopic (XR or UV-vis absorption, fluorescence, NMR), hydrodynamic, biochemical, molecular biology, etc.

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81 Background on the PC : 40,00,130 But pbs with the HH projector : Changed to to 30,170,10

82 Contrast of electron density ρ el. A -3 particle ρ 0 = ρ solvent

83 Contrast of electron density ρ 0 ρ ρ el. A -3 particle solvent

84 Contrast of electron density Formulation of Stuhrmann & Kirste : ρ() r = ( ρ ρ0) ϕp() r + ρi() r = ρϕp() r + ρi() r φ P (r) : shape function = 1 inside the particle = 0 outside ρ () r = ρ() r ρ i F() s = ρφ () s + F() s P i The intensity is decomposed in three basic scattering functions Is () = ρ I() s+ ρi() s+ I() s s si i The shape is observed at infinite contrast Only fluctuations are observable at vanishing contrast I s and I i are positive functions which can be directly observed The cross-term I si can be positive or negative and cannot be measured directly.

85 Contrast of electron density ρ el. A -3 ρ ρ 0 ρ( r) ρi particle ρ solvent

86 Kinetics of the Ca + -dependent swelling transition of Tomato Bushy Stunt Virus 10 7 D4 LURE quadrant gas detector, 100 s Intensity compact swollen s (Å -1 )

87 Kinetics of the Ca + -dependent swelling transition of Tomato Bushy Stunt Virus ID ESRF, XRII-CCD detector, 0.1s

88 Kinetics of the Ca + -dependent swelling transition of Tomato Bushy Stunt Virus s ID (ESRF) Intensity Q (A -1 )

89 Optimal thickness of the sample I de µd ( ) [ µ ] ( µ d ) di / dd = e 1 d d = 1/ µ opt This corresponds to a transmission of τ = e -1 =0.37 d opt (cm) Photon energy (kev)

90 Optimal thickness of the sample I/I opt /µ d* = d/d opt

91 Optimal thickness of the sample I/I opt /µ d* = d/d opt

92 Parseval s theorem f( r)g ( r) dr= F( s)g ( s) ds Special case f(r)=g(r) f( r) dr = F( s) ds