scattering of x-rays

EMBO Practical Course on Solution Scattering from Biological Macromolecules Hamburg 4 –11 September 2001

Scattering of X-rays


Books on SAS

- " The origins" (no recent edition) : Small Angle Scattering of X-raysA. Guinier and A. Fournet, (1955), in English, ed. Wiley, NY

- Small-Angle X-ray ScatteringO. Glatter and O. Kratky (1982), Academic Press.

- Structure Analysis by Small Angle X-ray and Neutron ScatteringL.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


Reminder - notations

Fourier Transform

2F( ) ( ) i

Ve dV

r

rsrs r(r)

F. T.

2( ) F( ) i

Ve dV

S

rsSr sF(s)

F. T.-1


Properties of the Fourier Transform

- 1 – linearity

FT (+ ) = FT(FT()

-2 – value at the origin

F(0) ( )V

dVr

rr

- 3 – translation

2F( ) ie Rss(r + R) F. T.


Fourier Transforms of simple functions

- 1 – delta function at the origin

(r)=(r)

-2 – Gaussian of width 1/K2

2F( )

sKs e

- 3 – rectangular function

2 2

( ) K rf r e

- 1 – constant unitary value

F(s)=1

-2 – Gaussian of width K

- 3 – sinc

sinF( )

sas a

sa

1/a-1/a

a

1

-a/2 a/2

f(x)

1)

o rs


r r

B(r)

A(r) A(r)*B(r)

rArB

1

Convolution product

A( ) B( ) A( )B( )V

dV u

ur r u r u


Convolution product

u u

B(r-u)

A(u) A(r)*B(r)

rArB rA + rB

rA - rB


Fundamental Propertyof the convolution product

FT(A B) FT(A) FT(B)

FT(A B) FT(A) FT(B)


Autocorrelation function

( ) ( ) ( ) ( ) ( )V

dV u

ur r r r u u

r

particle ghost

0 ( ) ( ) (0)r r characteristic function

r

(r)

spherical average ( ) ( )r r

(r)= (uniform density) => (0)= V

1


Distance distribution function

2 2 20( ) ( ) ( )p r r r r r V

rij ji

r

p(r)

Dmax

(r) : probability of finding a point at r from a given point

number of el. vol. i V - number of el. vol. j 4r2

number of pairs (i,j) separated by the

distance r 4r2V(r)=(4/2)p(r)


The elastically scattered intensity is given by the Thomson formula2

20 0 2

1 1 cos 2

2I r I

d

2 : scattering angle, cos2 close to 1 at small-angles

I0 intensity (energy/unit area /s) of the incident beam.The scattered beam has an intensity Ie/d2 at a distance d from the scattering electron. In what follows, is omitted and only the number of electronsis considered

2 26 20 7.95 10 cmeI r is the scattering cross-section of the electron

20r

Scattering by a free electron

X-ray scattering length of an electron 2

130 2

2.82 10 cme

rmc


Scattering by an electron at a position r

source

s0

s0

s1

s1

detector

r

r.s0

r.s1

Path difference = r.s1-r.s0 = r.(s1 - s0) or r.(s1 - s0)/ in cycles, for X-rays of wavelength

2

O


the scattering vector s

0 1

1

s s

2sins

s

s1

s0

O

s

length 1/

length 1/

2

scattered

1 0 s s ss is called the scattering vector

The scattered amplitude by the electron at r iswhere E(s) is the scattered amplitude by an electron at the origin

2 .( ) iE s e r s


scattering vector

2sin 2s

s

SAXS measurements are restricted to small-angles :2 5 deg = 85 mrad at = 1.5 ÅIn this case, the further approximation is valid :

4 sin" "q h s

!


Scattered amplitude

2F( ) ( ) i

Ve dV

r

rsrs r

2sin

s

Scattered intensity

I( ) F( ).F ( )s s s

Remark

0s F(0) ( )V

dVr

rr

For a particle I(0) = m2 m : number of electrons

F(s) is the Fourier transform from the electron density (r) describing the scattering object


Solution X-ray scattering

2X-ray beam

Sample

Detector

Diagram of the experimental set-up

s = 2 sin/

X-ray scattering curve


Particles in solution

The relevant quantity is now the

contrast of electron density

between the particle and the solvent

(r) = (r) - 0

with 0 = 0.3346 el. A-3 the electron density of water


Contrast of electron density

solvent

particle 0.43

0 0.335

el. A-3


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) = 0.3/2 (A3)

Protein : solvent

Example : 10 mg/ml solution of myoglobin v= 0.74 cm3.g-1 r=20A, = 1.5A

W=2 10-5 (x efficiency x geometrical factor)

30.43 .el A 30 0.335 .el A

3 10 20.1 . 2.8 10el A cm

2 2( ) 0.3( ) /W cv dr cvr

42.3 10 /W cvr

from H.B. StuhrmannSynchrotron Radiation ResearchH. Winick, S. Doniach Eds. (1980)


Solution of particles

=

=Solution

(r)

F(c,s)

Motif (protein)

p(r)

F(0,s)

Lattice

d(r)

(c,s)



For spherically symmetrical particles

I(c,s) = I(0,s) x S(c,s)

form factorof the particle

structure factorof the solution

Still valid for globular particles though over a restricted s-range



- 1 – monodispersity: identical particles

- 2 – size and shape polydispersity

- 3 – ideality : no intermolecular interactions

- 4 – non ideality : existence of interactionsbetween particles

In the following, we make the double assumption 1 and 3

2 and 4 dealt with at a later stage in the course


Ideal and monodisperse solution

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.

21F ( ) ( ) i

Ve dV

r

rsrs r

1 1 1 1( ) ( ) F ( ).F ( )i s i s s s

Ideality and monodispersity 1I( ) i ( )s sN

I( ) I( ) F( ).F ( )s s s s

time

particles


1( ) [ ( )]. [ ( )] [ ( )* ( )]i s FT FT FT r r r r

21( ) [ ( )] ( ) i

Vi s FT e dV

r

rsrr r

Let us use the properties of the Fourier transform and of the convolution product

1 0

sin(2 )( ) 4 ( )

2

rsi s p r dr

rs

2( ) ( )p r r rwith


Or one of the symmetrical expressions :

1 0( ) 2 ( )sin(2 )si s r r rs dr

0

1( ) I( )sin(2 )r r s s rs ds


Relationships real space – reciprocal space

(r)

F(s) I(s) I(s)

p(r) p(r)

FTFT FT

*

.

< >

< >


If the particle is described as a discrete sum of elementaryscatterers,(e.g. atoms) the scattered intensity is :

Debye formula

where the fi(s) are the atomic scattering factors, and the spherically averaged intensity is (Debye) :

N N2

11 1

( ) ( ) ( ) i ji

i ji j

i f f e

r r ss s s

N N

11 1

sin 2( ) ( ) ( )

2ij

i ji j ij

r si s f s f s

r s

where ij i jr r r

The Debye formula is widely used for model calculations


1 2

121 1 2 1 2

12

sin 2( ) ( ) ( )

2V V

r si s d d

r s

r r

r r r r

Debye formula

If the particle is described using a density distribution, the Debye formula is written :


Porod invariant

2

0

sin(2 )( ) 2 I( )

2

rsr s s ds

rs

2

0(0) 2 I( ) 2s s ds Q

Q is called the Porod invariant

Q depends on the mean square electron density contrast

For r=0 :

2 2

0(0) ( ) dV V

rrBy definition :

22

0( )

2

VQ s I s ds


Intensity at the origin

'1(0) ( ) ( )

V Vi dV dV

r rr r'r r'

2

2 21 0 0(0) ( ) ( )P

A

Mi m m m v

N

A

NMc

N V is the concentration (w/v), e.g. in mg.ml-1

2

0(0) ( )PA

cMVI v

N

idealmonodisperse


Intensity at the origin

If : the concentration c (w/v),the partial specific volume ,the intensity on an absolute scale, i.e. the number of incident photons

are known,

Then, the molecular weight of the particle can be determinedfrom the value of the intensity at the origin

Pv

idealmonodisperse


Intensity on an absolute scale

The experimental intensity Iexp(0) is expressed as :

20

exp 02(0) ( )e

PA

I I cMdI v

N a

d : thickness of the samplea : sample-detector distance

scattering cross-section of the electronI0 total energy of the incident beam

2 26 20 7.95 10eI r cm

idealmonodisperse


Intensity on an absolute scale

The determination of I0 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 inD. Orthaber, A. Bergmann & O. Glatter, J. Appl. Cryst. (2000), 33, 218-225 Thorough presentation of calculations on an absolute scale

P. Bösecke & O. Diat J. Appl. Cryst. (1997), 30, 867-871 Clear presentation of the geometrical calculations and all the procedures used on ID2 (ESRF) to put intensities on an absolute scale.


Guinier law

For small values of x, sinx/x can be expressed as :

idealmonodisperse

2 4sin 2 2 2

1 ...2 3! 5!

rs rs rs

rs

0

sin(2 )( ) 4 ( )

2

rsI s p r dr

rs

Hence, close to the origin:

22with and

4 (0) 4 ( ) = ( ) ( )

6I p r dr k r p r dr p r dr

22( ) (0) 1 (0) ksI s I ks I e

The scattering curve of a particle can be approximated by a Gaussian curve in the vicinity of the origin


1 2

121 2 1 2

12

sin 2( ) ( ) ( )

2V V

r sI s d d

r s

r r

r r r r

Guinier law

Let us introduce the expansion of sinx/x into the Debye formula :

We obtain the following expression :

1 2 1 2

2 22

1 2 1 2 1 2 1 2 1 2

4( ) ( ) ( ) ( ) ( )

6V V V V

sI s d d d d

r r r r

r r r r r r r r r r

1 2

1 2

22 1 2 1 2 1 2

1 2 1 2

hence ( ) ( )4

6

)

( ) (

V V

V V

d dk

d d

r r

r r

r r r r r r

r r r r

idealmonodisperse


Guinier law

Radius of gyration

idealmonodisperse

22 2

2it comes 2 ( )4 4

6 3( )

Vg

V

r dVk R

dV

r

r

r

r

r

r

1 2 1 0 0 2writing r r r r r r where r0 is the center of mass

Guinier law24

3

2 2gI(s) I(0)exp R s


Radius of gyration

Radius of gyration :

2

2( )

( )V

g

V

r dVR

dV

r

r

r

r

r

r

Rg is the quadratic mean of distances to the center of mass weighted by the contrast of electron density.Rg is an index of non sphericity.For a given volume the smallest Rg is that of a sphere :

Ellipsoïd of revolution (a, b) Cylinder (D, H)

3

5gR R

2 22

5g

a bR

2 2

8 12g

D HR

idealmonodisperse


Guinier plotexample

Validity range : 0 < 2Rgs<1.2For a sphere

The law is generally used under its log form :

24

3

2 2

gln I(s) ln I(0) R s

A linear regression yields two parameters : I(0) (y-intercept)

Rg from the slope

idealmonodisperse


Attractive Interactions

-crystallins c=160 mg/ml in 50mM Phosphate pH 7.0

I(s

)

0

1 104

2 104

3 104

4 104

5 104

0 0.01 0.02

30C

25C

20C

15C

10C

s = 2(sinA-1A. Tardieu et al., LMCP (Paris)


Repulsive Interactions

ATCase in 10mM borate buffer pH 8.3

0

500

1000

1500

2000

2500

3000

3500

4000

0 0.005 0.01 0.015 0.02

92,5mg/ml31,5 mg/ml13,3 mg/ml6,6 mg/ml3,3 mg/ml

I(s)

s = 2 (sin A-1


Virial coefficient

In the case of moderate interactions, the intensity at the origin varies withconcentration according to :

2

I(0)I(0, )

1 2 ...idealc

A Mc

Where A2 is the second viral coefficient which represents pair interactionsand I(0)ideal is to c.A2 is evaluated by performing experiments at various concentrations c. A2 is to the slope of c/I(0,c) vs c.


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 s2

2 2 2( ) exp( 2 )csI s R s

Finally, in the case of a platelet, a thickness parameter is derived from a plot of s2I(s) vs s2 :

2 2 2 2( ) exp( 4 )ts I s R s

12tR Twith T : thickness

Rods and platelets

idealmonodisperse


Volume of the particle

21 1(0) ( )i V and

Hence the expression of the volume for a particleof uniform density :

(Porod invariant)

11

(0)

2

iV

Q

21

2

VQ

Hypothesis : the particle has a uniform density

uniform density =>2 2

idealmonodisperse


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 :

1423 48 lim ( ) ( )s s i s S Bs

S is the area of the solute / solvent interfaceB is a correction term accounting for :

- short distance density fluctuations-uncertainties of i(s) at large s (weak signal)

idealmonodisperse



2 2

0

sin(2 )( ) 2 I( )

2

rsp r r s s ds

rs

In theory, the calculation of p(r) from I(s) is simple.Problem : I(s) - is only known over [smin, smax] : truncation

- is affected by experimental errors - might be affected by distorsions due to the beam-size and the bandwidth (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

idealmonodisperse


Calculation of p(r)

p(r) is calculated from i(s) using the indirect Fourier Transform methodBasic hypothesis : The particle has a finite size

1 0

sin(2 )( ) 4 ( )

2

MaxD rsi s p r dr

rs

p(r) is parameterized on [0, DMax] by a linear combination of orthogonal functions



The radius of gyration and the intensity at the origin can be derived from p(r)using the following expressions :

and

This alternative estimate of Rg makes use of the whole scattering curve, andis much less sensitive to interactions or to the presence of a small fractionof oligomers.Comparison of both estimates : useful cross-check

max

max

2

2 0

0

( )

2 ( )

D

g D

r p r drR

p r dr

max

0(0) 4 ( )

DI p r dr

idealmonodisperse


p(r) :example

Aspartate transcarbamylase from E.coli (ATCase)Heterododecamer (c3) 2(r2)3

quasi D3 symmetryMolecular weight : 306 kDaAllosteric enzyme

2 conformations :T : inactive, compactR : active, expanded

T Rcrys Rsolideal

monodisperse


ATCase : scattering patterns

10

100

1000

104

0 0.01 0.02 0.03 0.04 0.05

T-state

R-state

I(s)

(a.

u.)

s = 2sin A-1

idealmonodisperse


ATCase : p(r) curves

0

5

10

15

20

25

0 50 100 150

T-state

R-state

p(r)

(a.

u.)

r (A)

DMax

idealmonodisperse


Information content of a SAXS curve

Basic hypothesis : The particle has a finite size Dmax

Sampling theorem : the curve is entirely determined from the values of the intensity at the nodes of the lattice sk = k/2Dmax where k is an integer.

Number of independent structural parameters :

min2 ( )Max MaxJ D s s

Example : spherical protein with 0.3 hydration (w/w) v=0.73 cm3.g-1

Dmax = 1.5 M1/3 => J=3.0 M1/3(sMax – smin)

for smin –=0.002 A-1, sMax =0.05 A-1 and M = 105 => J = 7

idealmonodisperse


The case of the sphere

Unique case in which an analytical expression of the scattered intensity can be derived :

2

3

sin cos( ) 3

x x xI s x=2πRs

x

R : radius of the sphere

I(s) = 0 for values of s solutions of 2Rs = tan(2Rs)

first minima at Rs = 0.7158, 1.23, 1.73, etc.


0

0

0

0

0.01

0.1

1

0 0.5 1 1.5 2 2.5 3 3.5 4

I(s)

sph

ere

Rs

Scattering intensity of the sphere


F. Rey, V. Prasad et al., EMBO J. (2001)20, 1485-1497


1

10

100

1000

10000

100000

0 0.005 0.01 0.015 0.02

DLP

VLP 2/6

I(s)

s = (2sin Å-1

Rotavirus derivedparticles

DLP : double layer particlescontaining RNAVLP2/6: Virus-like particlesdouble-layer VP6 and VP2About 700 Å diameter


1

10

100

1000

10000

100000

0 0.02 0.04 0.06

DLP

VLP 2/6

I(s)

s = (2sin Å-1

Rotavirus derivedparticles

Icosahedral symmetry on thecapsid surface : strong modulation of the electron density => modulation of I(s)

around 25-30 Å :signal associated

with the RNA layers inside the VLP.


Rotavirus DLP

RNA

F. Rey et al., EMBO J. (2001), 20, 1485-1497


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).

Debye formula : where

I(s) depends on a single parameter, Rg .

Valid over a restricted s-range in the case of interacting monomers

Limit at large s :

2

( ) 2( 1 )

(0)xI s

x eI x

22 gx R s

22

2 2

1 1 (2 )lim [ ( )]

2g

sg

R ss I s

R

I(s) varies like s-2 instead of s -4 for a globular particle (Porod law).


Guinier plot : NCS heat unfolding

Neocarzinostatine : small (113

residue long) all- protein.

arrows : angular range used

for Rg determination

Pérez et al., J. Mol. Biol.(2001)308, 721-743


arrows : angular range used

for Rg determination

Debye law : NCS heat unfolding

Pérez et al., J. Mol. Biol.(2001)

308, 721-743


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

Kratky plot of s2I(s) vs s.

Globular particle : bell-shaped curve

Gaussian chain : plateau at large s-values

but beware: a plateau does not imply a Gaussian chain


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.(2001), 308, 721-743


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 five-fold 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.


Extended conformation of a modular protein :Hemocyanin from O. vulgaris

1

10

100

1000

0 0.05 0.1 0.15

pH 7.5pH 9.5

I/c (

a.u

.)

q (A-1)

pH 7.5pH 7.5

pH 9.5pH 9.5


Hemocyanin from O. vulgaris

pH 7.5pH 7.5

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

p(r

)

r (A)

Rg= 62.5 AMW=400 kDa

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

0 0.0002 0.0004 0.0006

ln(I

/c)

(a.u

.)

q2 (A-2)

Rg=61.7 AMW= 400 kDa

=

Guinier plot Distance distribution function



pH 9.5pH 9.5

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350

p(r

)r (A)

Rg=102.3 A

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

5 10-5

1 10-4

1.5 10-4

2 10-4

2.5 10-4

3 10-4

ln(I

/c)

(a.u

.)

q2 (A

-2)

Rg=90.6 A

Slight upward curvature

Guinier plot Distance distribution function



pH 9.5pH 9.5

=

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350

p(r

)

r (A)

Rg=102.3 A

q (A )


1

10

100

1000

0 0.05 0.1 0.15

exp.Debye fit

I/c

(a

.u.)

-1

Rg=103.15 A

Debye plot


Some experimental considerations


Practical aspects

Small-angle data : Guinier analysis, Rg 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


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.


Lysozyme

Lysozyme : 14 289 Da MW

5 mg/ml solution in Acetate buffer pH 4.5

Rg=15.2 Å in the air

Rg=15.0 Å under vacuum

1000

10000

100000

0 0,01 0,02 0,03 0,04 0,05

Air

[Tam

pon

lyso

l07.

sta/

(3.

07/1

.25)

]*1.

0498

Air

[lyso

l08.

sta

/(3.

07/1

.25)

]*1.

0498

vide

Tam

pon

lyso

s02

.sta

*1.

25

vid

e ly

so s

03.s

ta *

1.25

I(s)

A

s = 2 sin Å-1

1000

10000

0 0,5 1 1,5 2 2,5

I(s)

104 x s2 Å-2

smin


PGK

Yeast 3-phosphoglycerate kinase (PGK) : 44 570 Da MW

5 mg/ml solution in Tris buffer pH 7.5

Rg=24.7 Å in the air

Rg=24.4 Å under vacuum

1000

10000

100000

0 0.005 0.01 0.015 0.02 0.025

Z4

2.st

a B

uff

2 A

ir *

4.7

54

6 in

t 25

1-3

63 P

GK

Vid

e *

1.2

5 *

2.8

8

Z4

1.s

ta P

GK

2 A

ir *

4.7

54

6 in

t 25

1-3

63 P

GK

Vid

e *

1.2

5 *

2.8

8

*2

.88

nor

me

lys

vid

e V

ide

Tam

pon

Q1

7.S

HF

*1

.25

(co

rre

c di

am

CA

P)

*2.

88

no

rme

lys

vid

e

Vid

e P

GK

5m

g/m

l Q1

8.S

HF

*1

.25

(co

rre

c di

am

CA

P)

I(s)

B

s = 2 sin Å-1

1000

10000

0 0.4 0.8 1.2 1.6 2

(1/3

)[P

GK

1 S

ub a

ir *5

.448

1 in

t 251

-363

PG

KV

ide*

1.25

*2.8

8 1

/3: p

our

grap

hiqu

e G

uini

erre

glin

(1/3

)[P

GK

1 S

ub a

ir *5

.448

1 in

t 251

-363

PG

KV

ide*

1.25

*2.8

8 1

/3: p

our

grap

hiqu

e G

uini

er *

2.88

nor

me

lys

vide

Vid

e P

GK

- 1

.0 x

Buf

fer

*1.2

5 (c

orr

diam

. CA

P)

reg

lin*2

.88

norm

e ly

s vi

de V

ide

PG

K -

1.0

x B

uffe

r *1

.25

(cor

r di

am. C

AP

)

I(s)

104 x s2 Å-2

smin


Cell under vacuum

Ratio of buffer scattering

air and windows / under vacuum

5 cm air

25 µm Kapton window upstream

25 µm mica window downstream

0

4

8

12

16

20

0 0.01 0.02 0.03 0.04 0.05

Ra

ppo

rt B

uff

er1

*5

.44

81

/Vid

e*1

.25

ratio

Bu

ff A

ir(c

ol 2

)/V

ide

(co

l9 d

e ly

s-D

ef)

IB

uff

er(A

ir) /

I Buf

fer(

Vac

uum

)

s = 2 sin Å-1


S/B ratio

S/B ratio lysozyme 5mg/ml

0

1

2

3

4

5

6

0 0.005 0.01 0.015 0.02 0.025

PG

K1

-Buf

fer/

Bu

ffe

r (s

cale

d!)

Vid

e (

PG

K-1

.0*B

uff

er)

/Bu

ffe

r

I(P

GK

-Bu

ffer

) / I B

uff

ers = 2 sin Å-1

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05

Air

(Lys

-Bu

ff)/

Bu

ffV

ide

(L

ys-B

uff

)/B

uff

I(L

yso-

Bu

ffer

) / I B

uff

er

s = 2 sin Å-1

S/B ratio PGK 5mg/ml


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.


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 physico-chemical 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.


Background on the PC : 240,200,130But pbs with the HH projector :

Changed to 230,170,120

Background on the PC : 240,200,130But pbs with the HH projector :

Changed to 230,170,120



solvent

particle

0

el. A-3



solvent

0 el. A-3

particle


Contrast of electron densityFormulation of Stuhrmann & Kirste :

P(r) : shape function = 1 inside the particle = 0 outside

The intensity is decomposed in three basic scattering functions

0( ) ( ) ( ) ( ) ( ) ( )P i P i r r r r r

( ) ( )i r r

( ) ( ) ( )P iF F s s s

The shape is observed at infinite contrastOnly fluctuations are observable at vanishing contrastIs and Ii are positive functions which can be directly observedThe cross-term Isi can be positive or negative and cannot be measured directly.

2( ) ( ) ( ) ( )s si iI s I s I s I s



solvent

particle

0

el. A-3

i

( ) r


100

1000

104

105

106

107

0 0.01 0.02 0.03 0.04 0.05

compactswollen

s (Å-1)

Inte

nsi

ty

Kinetics of the Ca2+-dependent swelling

transition of Tomato Bushy Stunt Virus

D24 LURE quadrant gas detector, 1200 s


Kinetics of the Ca2+-dependent swelling transition of Tomato Bushy Stunt Virus

ID2 ESRF, XRII-CCD detector, 0.1s


Kinetics of the Ca2+-dependent swelling transition of Tomato Bushy Stunt Virus

Inte

nsi

ty

10.0

1.0

0.10.02 0.04 0.06

Q (A-1)

0.1 s ID2 (ESRF)


Optimal thickness of the sample

This corresponds to a transmission of = e-1=0.37

( )dI de

( )/ 1ddI dd e d

1/optd µ

0

0.1

0.2

0.3

0.4

0.5

8 9 10 11 12 13

d opt (

cm)

Photon energy (keV)



1/µ

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

I/I op

t

d* = d/dopt



1/µ

0.6

0.7

0.8

0.9

1

1.1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

I/I op

t

d* = d/dopt


Special case f(r)=g(r)

f( )g ( ) F( )G ( )d d r r r s s s

Parseval’s theorem

2 2f( ) F( )d d r r s s

scattering of x-rays

Documents

scattering electron

electron scattering

scattering angle

scattering objectparticles

light scattering

scattering vectorthe

scattering crosssection

small angle scattering