A brief introduction to SAXS M.H.J. Koch 1 SAXS / ~0 r>>t - - PowerPoint PPT Presentation

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A brief introduction to SAXS

M.H.J. Koch

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SAXS

2 2 2 2 2

1 2 2 cos 1 2 I r r I r ) θ ( r ) θ ( Ie =         + =

classical electron radius = 2.82 10-15 m For SAXS this factor =1 sin2θ = 2θ cos 2θ = 1 I0=Iinexp(-µt)

r>>t

∆λ/λ~0 For a single electron:

very small!!

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Electrons in an electromagnetic field

are accelerated and therefore emit radiation: they scatter. The spatial distribution of the scattered intensity depends on the geometry of the

• experiment. For unpolarized incident radiation the spatial distribution
• n the equator is:

2Ө

(1) I ) ω (ω ω r 1 2 ) θ 2 ( cos 1 r ) I(2

2 2 2 4 2 2 2

− ⋅ ⋅ + ⋅ = θ

where 2θ is the angle between the incident and the scattered beam. corresponds to the natural frequency (ν0=ω0/2π) of the oscillator and ω to the frequency of the incident radiation. Obs.

m k = ω

4

) ω ω ( ω

2 2 2

− in the previous equation is the one describing the frequency

• dependence. For the AMPLITUDE (E with I= E·E*)

The natural frequency of the oscillator (ω0) corresponds to the binding strength of electrons in atoms and lies somewhere in the UV to X-ray region. If the incident radiation is visible light (λ ≈ 500 nm), ω << ω0 and the factor reduces to:

2 2

ω ω The amplitude of the scattered radiation at r is proportional to ω2 and in phase with the incident radiation. This is Rayleigh scattering. The scattered intensity is proportional to ω4 hence the blue sky.

The most interesting factor

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X-Rays

If the incident radiation is X-rays (λ ≈ 0.1 nm), ω0 << ω and the factor ) ω ω ( ω

2 2 2

− The scattering amplitude is independent of the frequency and its phase is shifted by 180 degrees relative to the incident radiation. This is

Thomson scattering

= -1

As it is independent of the frequency of the incident radiation, the world

• f X-rays is colorless with shades of gray (i.e. contrast) only and Eq.1

above simplifies to:

radius electron classical 10 817 . 2 mc e r

15 2 2

m

−

= =

2 2 2

I 1 2 ) 2 ( cos 1 r ) I(2 r ⋅ + ⋅ = θ θ

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Energy out /energy in

2 2 e

I r 1 r ) (0 I = For unpolarized X-rays at 2Ө = 0: I0 1m2 r Ie(2θ) r2dΩ Detector sample

2 2 e

b r I ) (2θ I d dσ = ⋅ =

b: scattering length = r0 = 2.810-15 m for the electron Energy scattered/unit solid angle/unit time Energy incident/unit area/unit time The differential scattering cross-section has the dimension of an area and represents

For one electron: the amplitude of scattering |Ie(0)|1/2= 2.810-15|I0|1/2 and as the scattering amplitude ≡ f = amplitude scattered by an object amplitude scattered by an electron in identical conditions

fe =1

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Particle size (D) and wavelength of the radiation

When λ >> D all N electrons in the particle are accelerated in phase the scattering amplitude is N times that of one electron. When λ < D the electrons in the particle are no longer moving in phase and one has to take the phase shift of the waves into account. Observer Observer λ >> D Light scattering λ < D X-ray scattering

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Waves and Interference

Interferences lead to fringe patterns. This is illustrated here with water waves. When “solving” a structure the problem is to go from the fringe pattern – in the case of X-ray diffraction from the intensities of the fringes – to the distribution of sources i.e.

• f scatterers.

Similar effects are observed with optical transforms obtained by shining coherent visible (laser) light through small apertures (see e.g. Cantor and Schimmel , Biophysical Chemistry, Part II, Ch. 13).

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Interference and coherent scattering

r

2Θ r.S r.S0

λ ) ( S S s − =

λ 2sin θ = s

S/λ S0/λ

The total amplitude from two centers (one at the origin and one at r) is thus:

) r s i exp(2π f f ) sr i exp(2π f F(s)

2 e e i 2 1 i e

⋅ + = = ∑

=

S r S r ⋅ − ⋅

Path difference = In Thomson (coherent scattering) the scattered wave is 180° out of phase with the incident wave. Phase difference =

) S r S (r ⋅ − ⋅ λ π 2

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The sum of amplitudes for N electrons:

F(s) is the Fourier transform of the distribution of electrons

) i exp(2π f ) F(

N 1 i e i

r s s ⋅ = ∑

=

The average of the exponential factor over all orientations of r relative to s for randomly oriented particles (e.g. in solution) is:

sr sr π π 2 ) 2 ( sin ) i π 2 ( exp = > ⋅ <

i

r s

As this is a real number there is no phase problem but one has lost most of the structural information.

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Scattering factor

For λ = 0.15 nm f’ = 0.3641 f” = 1.2855

dr sr π 2 ) sr π sin(2 r ρ(r) π 4 f(s) F(s)

2

∫

∞

= ≡

For an atom with a continuous radial electron density ρ(r): 1 sin lim =      

→ x

x x and since

f(0) = Z

( ) ( ) ( ) ( )

sR sR sR sR s f

sphere

π π π π 2 cos 2 2 sin 2 3 ) (

3

− =

For modeling purposes one often uses larger spherical subunits (beads, dummy residues, etc) for which:

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Anomalous scattering

The scattering factor fe must be modified to take anomalous scattering into account

fe = 1 + fe’ + ife”

(f” is always π/2 ahead of the phase of the real part) Note that for most practical purposes, f’ and f” are independent of s = 2sinθ/λ

Kronig Kramers ' dω ω' ω ) ' (ω " f ω' π 2 ) (ω ' f weight atomic A number, s Avogadro' : N t, coefficien absorption tric) (photoelec linear : µ ) ω (

• cN

r 4π ωA 0) ω, ( " f

2 2 A A

− − = =

∫

Near an absorption edge, the dissipative effects due to the rearrangement

• f the electrons can no longer be neglected.

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Selenium f‘ and f‘‘

SeK ) ω (

• cN

r 4π ωA 0) ω, ( " f

A

= ' dω ω' ω ) ' (ω " f ω' π 2 ) (ω ' f

2 2

∫

− = Note the sign and absolute value of the corrections.

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Scattering from N spherical atoms:

F(s) is the Fourier transform of the distribution of the spherical atoms. Crystallographers call this the structure factor. Note that in SAXS the structure factor refers to structure of the solution. The intensity is, of course:

) i (s)exp(2π f ) F(

N 1 i i i

r s s ⋅ = ∑

= ij ij j

r s π 2 ) r s π 2 ( sin )

• (

i π 2 ( exp = > ⋅ < r r s

i

) )

• (

i (s)exp(2π (s)f f ) I(

1 i j 1 j j i

∑∑

= =

⋅ =

N N

r r s s

i

For random orientation

ij ij N 1 i N 1 j j i

sr 2π ) sr sin(2π ) (s (s)f f I(s) ∑∑

= =

= and Debye (1915) Crystal structure - atomic coordinates Solution – Distance distribution only! This is a real number!

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Short distances >> low frequencies dominate at high angles Large distances >> high frequencies contribute

• nly at low angles.

In isotropic systems, each distance d = rij contributes a sinx/x –like term to the intensity. A scattering pattern is a continuous function of s.

SAXS:

ij ij N 1 i N 1 j j i

sr 2π ) sr sin(2π ) (s (s)f f I(s) ∑∑

= =

=

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The wider a function in real space the narrower its transform in reciprocal space

1) The Fourier transform of the Dirac delta function is the 1(x) function (i.e. the function which has a constant value of 1

• ver the interval [-∞,∞].

2) Obviously, the Fourier transform of 1(x) is δ(x).

∫

∞ ∞ −

= ∞ = 1 δ(x)dx δ(0)

3) The Fourier transform of a Gaussian is also a Gaussian

) / exp( ) ( )) (exp(

2 2 2

a k a k F ax FT π π − = = −

Note the relationship between the widths. If the Gaussian has a width σR=(1/2a)½, its transform has a width σF=(a/2π2)½ and σRσF=1/2π. The δ-function is an infinitely narrow Gaussian.

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Intensity

10.0 1.0 0.1 0.02 0.04 0.06

Q (Å-1)

0.1 s ID2 (ESRF)

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

Perez et al.

Larger objects scatter at lower angles!

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In an ideal solution

The solute particles are randomly oriented and their positions are uncorrelated in space and time. Consequently their scattering in isotropic and incoherent. The total scattering intensity is the sum of the coherent scattering intensity of all molecules. It is a function of the scattering angle or modulus of the scattering vector only: I(s). Usually one plots log(I(s)) vs s, because the intensity falls off rapidly due to the interferences. i.e. for atoms one can neglect the s-dependence of fi If one uses a continuous density distribution ρ(r) this becomes

ij ij N 1 i N 1 j j i 1

sr 2π ) sr sin(2π f f (s) i

∑∑

= =

=

2 1 12 12 2 2 1 1 1

2 ) 2 sin( ) ( ) ( ) ( r r r r d d sr sr s i

V

π π ρ ρ

∫∫

=

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Interactions of X-rays with matter

Sample fluorescence Incident beam Transmitted beam Coherent scattering I0 I=I0 exp(-µt) Incoherent scattering

Structural information at the atomic/molecular level is in: coherent scattering and to a limited extent in absorption/fluorescence near edges (EXAFS, XANES) At lower resolution transmission/phase contrast imaging is also useful

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Choice of wavelength

Absorption!! Incoherent!!

Optimal thickness of the sample: topt =1/µ ≈ 1 mm for H2O @ 1.5Å

For oxygen

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Background

solution buffer with air gap vacuum

Background arises from the incoherent (Compton) scattering from the sample and from coherent and incoherent scattering due to air gaps, windows, solvent …. To obtain the coherent scattering of the solute normalize the intensities of solution and buffer to transmitted beam and subtract :

I(s)= [I(s)/I0T]solution – [I(s)/I0T]buffer

Divide by c to normalize for concentration Lysozyme : 14 289 Da MW 5 mg/ml solution in Acetate buffer pH 4.5

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CONTRAST: < ρparticle(r) > - ρbuffer

Homogeneous solvent Particle: Solvent: Only fluctuations in electron density contribute to the scattering:

r r s r s )d i exp(2π ) ( ρ ) ( F

V p p

⋅ = ∫ r r s s )d i exp(2π ρ ) ( F

V b b

⋅ =

∫

Ideally, a Dirac δ(0) and constant in practice not

Iobs(s) = Ip(s) –Ib(s)

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Isolution(s) Isolvent (s) Iparticle(s)

♦ To obtain scattering from the particles, solvent scattering

must be subtracted to yield the excess scattering ρp(r) = ρsolution(r) - ρb where ρb is the scattering density of the solvent

Solvent scattering and contrast

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Scattering arises from fluctuations

In a perfectly homogeneous body the contribution of each small scattering volume element to the scattering amplitude would always be cancelled out by that of another one which is out of phase. This is why perfect crystals do not scatter visible light. Solutions of macromolecules or colloidal suspensions strongly scatter visible light because of fluctuations in concentration. The fluctuations are also what links scattering to diffusion (e.g. DLS) and thermodynamics (compressibility).

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EXCESS SCATTERING DENSITY

(Stuhrmann and Kirste, 1965) ) ( ρ ρ

• )

( ρ ) ( ρ ρ ) ρ(

c b s c x

r r r r + = shape

internal structure

equivalent volume

• f solvent

) ( ρ ) ( ρ ) ρ

• ρ

( ) ρ(

s c b x

r r r + =

) ( ρ ) ( ρ ρ

s c

r r + =

contrast

(s) I (s) I ρ (s) I ρ I(s)

s cs c 2

+ + =

The corresponding intensity is: For a homogeneous particle: Is(s) =Ics(s) =0 ρc (r) has a value of 1 inside the particle and 0 outside and thus represents the shape. ρs(r), the internal structure, represents the fluctuations around the average electron density of the particle ρx.

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A communication problem

(s) I (s) I ρ (s) I ρ I(s)

s cs c 2

+ + = For a homogeneous particle (i.e. a shape): Is(s) =Ics(s) =0 Is(0) =Ics(0)=0 always! These two terms are the most important in SAXS This is essentially what crystallographers talk about

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Scattered intensity

) F( ) F( ) ( F ) F( ) ( i

* 1

s s s s s − ⋅ = ⋅ =

) ( ) ( )] ( [ ) ( )] ( [

* s

s r s r F F FT F FT = − = − = ρ ρ

ρ(r) is the excess electron density Hence

)] ( ) ( [ )] ( [ )] ( [ ) (

* 1

r r r r s − = − ⋅ = ρ ρ ρ ρ FT FT FT i

γ(r): Autocorrelation function of the excess electron density and: γ (r)

∫

⋅ = =

r

V r

dV i FT s i ) 2 exp( ) ( )) ( ( ) (

1

r s r r π γ γ dr rs rs r p s i

∫

∞

=

1

2 ) 2 sin( ) ( 4 ) ( π π π ) ( ) (

2

r r r p γ = After spherical averaging: with

( ) ( ) r γ γ = r

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Autocorrelation: Shift-Multiply-Integrate

∫

+ = = − =

u

u

u u r r r r r r

V

dV ) ( ) ( ) ( ) ( ) ( ) ( ) (

*

ρ ρ ρ ρ ρ ρ γ

• r

u Dmax 2Dmax r ρ(u) γ(r) p(r) r

× r2

Dmax

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2 2 2

( ) ( ) ( ) p r r r r r V γ γ ρ = =

rij j i r p(r) Dmax

For a homogeneous body ρ(r) = ρ, γ(0) = ρ2V

γ(r) : probability of finding a point at r from a given point (i). Number of volume elements i ∝ V; Number of volume elements j ∝ 4πr2. Number of pairs (i,j) separated by the distance r ∝ 4πr2Vγ0(r)=(4π/ρ2) p(r).

0( )

( ) (0) r r γ γ γ =

Characteristic function Distance distribution function

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I(s) and p(r)

dr sr sr r p s I

D

π π π 2 ) 2 sin( ) ( 4 ) (

max

∫

=

dr sr sr s I r r p

∫

∞

=

2

2 ) 2 sin( ) ( 2 ) ( π π

For a homogeneous particle p(r) represents the histogram of distances between pairs of points within the particle.

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At low angles

( ) ( )

• −

+ − = 120 2 6 2 1 2 ) 2 sin(

4 2

sr sr sr sr π π π π

ij ij N 1 i N 1 j j i 1

sr 2π ) sr sin(2π f f (s) i

∑∑

= =

=

At very low angles one can use the approximation: exp(x) = 1+ x + x2/2 +… which yields the Guinier formula ) s R π 3 4 (1 V ρ (s) i

2 2 g 2 2 2 1

• +

− = mass Molecular ~ V ρ ) ( ) s R π 3 4 exp( I(0) ) (

2 2 2 2 g 2

= − = I with s I Since

2

     ∑

i i

f Rg is the mean squared distance to the centre of scattering mass weighted by the excess electron density.

32

Valid for a sphere for 0 < 2π π π πRgs<1.2 The plot of ln[I(s)] vs s2

[ ] [ ]

2

4 3 π ≅ −

2 2 g

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

yields two parameters : y-intercept: I(0) Rg: from the slope

Guinier plot for ideal solutions

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Forward scattering I(0) and molar mass (M)

2 ´

´

´) ( ) ( ) (

excess i i r V V r

f dV dV r r I

r r

      = =

∑ ∫ ∫

ρ ρ

2 2

) ( ) ( ) (       − = − = ρ ρ ν ρ

A

N M m m I

V N NM c

A

=

[ ]2

0)

( ) ( ρ ρ ν ρ − =

A

N cMV I

concentration (w/v) Molar mass Avogadro’s number Number of electrons contrast Partial specific volume

∫

=

max

) ( 4 ) (

D

dr r p I π

• r

number of molecules Tip: Use a sample of similar contrast e.g. a known protein ( BSA) as standard to calibrate the measurements: MMsample= MMstand.I(0)/c)sample/(I(0)/c)stand.

34

Radius of gyration

2 1 2 1 2 1 2 2 1 2 1 2

) ( ) ( 2 ) ( ) ( r r r r r r r r r r d d d d Rg

∫∫ ∫∫

− = ρ ρ ρ ρ

If one places the origin of the coordinates at r0,,the centre of scattering mass of the particles, this yields the expression for the radius of gyration which is obtained from a Guinier plot, Or even better, calculated from the whole experimental scattering pattern:

∫ ∫

=

V V g

d d r R r r r r ) ( ) (

2 2

ρ ρ

∫ ∫

=

max max

2 2

) ( 2 ) (

D D g

dr r p dr r p r R

Rg is the second moment (standard deviation) of the electron density distribution.

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Guinier’s formula and the contrast

1 s R π 3 4 for s R π 3 4 exp V ρ ) s R π 3 4 (1 V ρ I(s)

2 2 2 2 2 2 2 2 2 2 2 2 2

≤      − = + − =

• where R is the radius of gyration and I(0) the forward scattering is:

2 2V

ρ If the contrast changes so do I(0) and R:

2 2 c 2

ρ β ρ α R R − + = second moment of internal structure displacement of centre

• f scattering mass with

contrast α = 0 α = 0 α = 0 α = 0 α > 0 α > 0 α > 0 α > 0 β=0 β=0 β=0 β=0 β>0 β>0 β>0 β>0 α α α α < 0 radius of gyration at infinite contrast

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Contrast: X-rays

Substance X- rays Proteins 2.5 Nucleic acids 6.7 Fatty acids

• 1.1

Carbohydrates 4.5 Average contrast (X 10

10 cm

• 2) of biological macromolecular assemblies in water.

One can change the contrast e.g. by adding salts like CsBr but this has disadvantages like increasing absorption, fluorescence and changing ionic strength. The alternative is to use anomalous scattering (see e.g Stuhrmann HB. Acta Crystallogr A. 2008, 64:181-91) but beware of radiation damage. TIP: If you need to change the contrast USE NEUTRONS!

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Typical values of radii of gyration

Mw Rg(nm) Ribonuclease 12700 1.48 Lysozyme 14800 1.45 B-lactoglobulin 36700 2.17 Bovine serum albumin 68000 2.95 Myosin 493000 46.8 Brome mosaic virus 4.6 106 13.4 Tobacco mosaic virus 3.9 107 92.4 For a sphere of radius R:

2 2

5 3 R Rg =

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For an infinite rod

) ( 1 ) ( ) ( ) , , ( z y x f δ δ = ∞

) ( ) ( 1 ) ( 1 ) , , ( Z Y X F δ = ∞ ∞

• r a long fiber with its axis along z, the transform is limited to the

X,Y plane (i.e. the cross-section) z x y Z X Y

FT

Remember δ(x)! In such a case the radius of gyration of the cross-section and the mass/unit length can be derived using a representation analogous to the Guinier plot with a plot of sI(s) vs s2 to obtain

) 2 exp( ) (

2 2 2

s R L m s sI

c

π − =

For a circular cross-section Rc=R/√2

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For a flat object like a membrane

) ( ) ( 1 ) ( 1 ) , , ( z y x f δ = ∞ ∞

) ( 1 ) ( ) ( ) , , ( Z Y X F δ δ = ∞

With a small thickness (T) along z, the transform is limited to the Z-axis (i.e. the thickness) z x y Z X Y

FT

Remember δ(x)! In such a case the radius of gyration of the thickness and the mass/unit area are obtained from a plot of s2I(s) vs s2 to obtain

) 4 exp( ) (

2 2 2 2

s R A m s I s

T

π − = 12 / T RT =

with

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Shape scattering

At larger s-values the scattering of a particle with ρs ≠ 0 oscillates around a straight line given by POROD’s law:

s4I(s)= Bs4 +A

Subtract a constant equal to the slope of the Porod plot to

• btain an approximation to the

SHAPE scattering

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Mixtures

) ( / ) ( ) ( ) (

N 1 i 2 2 1

I R I n R s I n s I

i i i N i i i

∑ ∑

= =

= =

ni: number concentration of the ith species, with forward scattering Ii(0) and radius of gyration Ri. These formula illustrate that a small quantity of high molecular mass or hight contrast particles will have a large influence on the scattering curve. For modeling it is thus indispensable to make sure that the solutions do not contain aggregates!

42

Proteins at low resolution

Hydration shell Crowding max. conc. 300-500mg/ml IONS: Kosmotropes e.g. Na+ Chaotropes e.g. K+ OSMOLYTES e.g. free amino acids polyhydroxy alcohols methylated ammonium and sulfonium compounds urea. solvent Interactions/ stability/activity modulated by

FOLDING

Coupled equilibria Non-contact interactions

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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)

x

I s x e I x

−

= − +

( )

2

2

g

x R s π =

2 2 2 2

1 1 (2 ) lim [ ( )] 2

g s g

R s s I s R π π

→∞

− =

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

Scattering by an extended chain

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arrows : angular range used for Rg determination

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

Debye law for extended chains: NCS unfolding

2

( ) 2 ( 1 ) (0)

x

I s x e I x

−

= − +

( )

2

2

g

x R s π =

Neocarzinostatine : small (113 residue long) all-β protein.

45

Is sensitive to the degree of compactness of a protein. Globular particle : bell-shaped curve Gaussian chain : plateau at large s-values but a plateau does not imply a Gaussian chain!

Kratky plot: s2I(s) vs s

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

In thermal unfolding of NCS the Kratky plot has a plateau although unfolded NCS is not a Gaussian chain when unfolded. A thick persistence chain is a better model in this case.

46

Protein folding: cytochrome c

Akiyama, S. et al. (2002) Proc. Natl. Acad. Sci. USA, 99, 1329-34

47

Protein folding: cytochrome c

Akiyama, S. et al. (2002) Proc. Natl. Acad. Sci. USA 99, 1329-34

?

Initial Ensemble ?

expect >900 Å2 for random chain !

48

• S. Akiyama et al. (2002), PNAS, (2002), 99, 1329-1334.

160 µs after mixing 44 ms after mixing

cytochrome c folding

49

Labeling

D L1 L2 D X X X L1 X X X L2 X X X D L1 L2 D X 0 0 L1 0 0 0 L2 0 0 0 D L1 L2 D X X 0 L1 X X 0 L2 0 0 0 D L1 L2 D X 0 X L1 0 0 0 L2 X 0 X

+ +

• L1

D L2

L1-D or L2-D L1-D-L2 D Interference FT Distance between L1 and L2

Matthew-Fenn R.S. et al. (2008) Science 232, 446-9

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35 30 25 20 15 10bp

Note the difference in position and width (variance) of the distribution of end to end distances as a function of the number of base pairs between labels. In absence of applied force DNA is at least 1000 softer than in single molecule stretching experiments. Stretching is cooperative over more than two turns of the double helix. DNA is not an elastic rod.

Interference

FT

51

What SAXS/SANS have to offer

SAXS SANS Electron microscopy NMR Modelling Light scattering Molecular Dynamics ultracentrifugation viscosity Electric or flow dichroism chromatography Perturbation methods Labeling methods