Indirect Fourier Transformation - EMBL Hamburg · Indirect Fourier Transformation Introduction Pair Distance Distribution Function PDDF Indirect Fourier Transformation ... The phase

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Institute of Chemistry, University of Graz,Austria

Indirect Fourier Transformation

IntroductionPair Distance Distribution Function PDDF

Indirect Fourier TransformationSymmetries, Polydispersity

ExamplesDeconvolution of the PDDF

Institute of Chemistry, University of Graz, Austria

The Scattered Field Es(q)

ϕϕϕϕ

The scattering amplitudes of allcoherently scattered waves haveto be added according to theiramplitude and relative phase ÿ.

The phase difference dependson the relative location of thescattering centers.

Es(q)

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The Scattered Field Es(q)

( ) ( ) is

V

E const e dρ −= ÿ qrq r r

In order to find the total scattered field we have to integrate over the wholeilluminated scattering volume V

We can now express the density ÿ (r) by its mean and its fluctuations �ÿ (r):ρ

( ) ( )ρ ρ ρ= + ∆r r

( ) ( )i is

V V

E const e d e dρ ρ−� ��

= ⋅ + ∆ÿ ÿqr qrq r r r

( ) ( ) is

V

E const e dρ= ∆ÿ qrq r r

The Fourier integral is linear, so we can rewrite the above equation:

Taking into account the large dimension of the scattering volume we get:


From Scattering Amplitudes to Scattering Intensities

For monodisperse dilute systems we can write:

( ) ( )2| ( ) |sI q N F NI q= < > =q

We have introduced the particle scattering amplitude F(q) which is the scatteredfield resulting from integration over the particle volume only.

( ) ( ) i

V

F e dρ −= ∆ÿ qrq r r

( ) 1 2( )21 2 1 2| ( ) | ( ) ( ) ( ) i r

V

F F F e d dρ ρ − −∗= ⋅ = ∆ ∆ÿ ÿ q rq q q r r r r

We put r1 - r2 = r and use r2 = r1 - r and introduce the convolution square of thedensity fluctuations:

21 1 1( ) ( ) ( ) ( )

V

dγ ρ ρ ρ≡ ∆ = ∆ ∆ −ÿr r r r r rÿ

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The Convolution Square of the Density Fluctuations ÿ(r)and ÿ(r):

The function ÿ(r) is calculated by shifting the “ghost”particle a vector r and integrating the overlappingvolume.

This function is also called spatial autocorrelationfunction (ACF).

The spatially averaged convolution square ÿ(r) resultsfrom the same process, the ghost is shifted by adistance r = |r |, but we have to average over all possibledirections in space.

2 2 21 1 1( ) ( ) ( ) ( ) ( ) ( )

V

r r V dγ ρ ρ ρ ρ ρ= − = < ∆ > = < ∆ ∆ − >ÿr r r r rÿ ÿ


Spatially Averaged Intensity I(q)

( )2 2( ) | ( ) | i

V

I q F e dρ −= < > = < ∆ >ÿ qrq r rÿ

The spatially averaged intensity I(q) is given by:

( ) 2

0

sin4

qrr r dr

qrπ γ

∞

= ÿ

by introducing the pair distance distribution function (PDDF) p(r) with

( ) ( ) ( )2 2 2p r r r r rγ ρ= ⋅ = ∆ ⋅ÿ

we finally get

( )0

sin ( )( ) 4

qrI q p r dr

qrπ

∞

= ÿ

4



Definition of the Pair Distance Distribution Function (PDDF)p(r)

We can relate the meaning of a distance histogramto the PDDF p(r) if the particles are homogeneous.The height of p(r) is proportional to the number ofdistances that can be found inside the particle withinthe interval r and r+dr

The p(r) function of inhomogeneous particles isproportional to the product of the differencescattering lengths nink [ ] of twovolume elements i and k with a center-to-centerdistance between r and r+dr and we sum over allpairs with this distance.

i i in = ( )dV( )ρ∆ r r


The Scattering Problem and the Inverse Scattering Problem

For the solution of the inverse Problem it is essential to be able to calculate the PDDFform the experimental scattering curve with minimum termination effect.

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SAXS Cameras - Slit Collimation („Kratky Camera“)

The block camera, designed by O. Kratky, uses blocks to define the size of theprimary beam. Contrary to the slit system it does not allow measurements aboveand below the direct beam. The system is built by a U-shaped middle part M, abridge B and an entrance slit (or block) E. The main idea is to allow full parasiticscattering below the primary beam but to have negligible parasitic scatteringabove the beam, the half-plane used for the measurement.


SAXS Cameras - Slit Collimation with X-ray mirror

X-rayTube

GoebelMirrorPSD

Image

Plate

or

X-rayTube

GoebelMirrorPSD

Image

Plate

or

In this new, modified slit collimationsystem the divergent primary beam iscollimated by a Goebel mirror increasingthe flux by a factor of 5. At the same timethe radiation becomes monochromatic.

ParabolicGoebel mirror

X-ray tube

Collimationsystem

Pos

ition

sens

itive

dete

ctor

Sample

Beam stop

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The intensity can be increased by another factor of 4 with a focusingoptic , the total increase in intensity a factor of 20 , at the same timehaving monochromatic radiation !



0 1 2 3 4 50.1

1

10

Auflösung: 2 ππππ/qmin=125 nmqmin=0.05 nm -1

Hydroxy Nitril Lyase (64mg/mL)PufferHNL-Puffer

Inte

nsitä

t[P

SL/

s]

q [nm -1]0 1 2 3 4 5

0.1

1

10

Auflösung: 2 ππππ/qmin=125 nmqmin

=0.05 nm -1

Hydroxy Nitril Lyase (64mg/mL)PufferHNL-Puffer

Inte

nsitä

t[P

SL/

s]

q [nm -1]Typical scan of an image plate

Typical result for a protein solution. The blue curve isthe difference pattern after subtraction of the buffer. (A.Bergmann, Thesis).

7



Absolute Intensity - Calibration with Water

The horizontal part of the larger q-range corresponds to the isothermal compressibilityof water, therefore the constant scattering intensity of water is 1.648*10-2 cm-1 at 20°C.

Orthaber, D., Bergmann, A. and Glatter, O. J. Appl. Cryst. (2000) 33, 218-225. “SAXS experimentson absolute scale with Kratky systems using water as a secondary standard”


Application Absolute Intensity - Lysozyme

It is possible to put the scattering of any sample in relation to the water scatteringand bring the sample scattering data on absolute scale, the forward intensity I(0) oflysozyme is 0.202 cm-1. With this value of I(0) it is possible to estimate the molecularweight for lysozyme to 13300 g/mol. The effect of the finit concentration of 20 mg/mL(decreasing of the forward intensity) is taken into account.

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Inverse Problem in Scattering – Artists View*

primary beam

sample design of theexperiment

result inq-space

?structure

of the scattering particle

* “Asterix in Belgium”

associated by Anna Stradner & Gerhard Fritz


The Scattering Problem and the Inverse Scattering Problem

For the solution of the inverse Problem it is essential to be able to calculate the PDDFform the experimental scattering curve with minimum termination effect.

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From experimental data to the PDDF

All Transformations T1 to T4 are linear and are mathematically well defined, this doesnot hold for their inverse transformations.


The Principles of the Indirect Fourier Transformation I

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The Principles of the Indirect Fourier Transformation II

We start with the following “Ansatz”:

( ) ( )1

N

a i ii

p r c rϕ=

= � for 0 ≤ r ≤ Dmax

Here we have used the essential assumption that we can estimate amaximum dimension Dmax for the particle.

Now we transform this series into the reciprocal space using the lineartransformation T1:

1 1 11

N N N

a i i ia i i ii=1 i=1 i=

(q)= T (r)= T [ (r)] = T (r)= (q)p c c cI ϕ ϕ ψ� � �

Here we have introduced the functions ψi(q) defined by:

i 1 i(q)= T (r)ψ ϕ


The Principles of the Indirect Fourier Transformation III

Now we transform according to the instrumental broadening effects T2 - T4 (some of themmay be negligible) and get:

4 3 2( ) ( ) ( )N

a a i ii l

I q T T T I q c qχ=

= = �where we find again the same coefficients ci and the set of functions χi(q) in theexperimental space

( ) ( )4 3 2 4 3 2 1( )i i iq T T T q T T T T rχ ψ ϕ= =

With this operation we have created the three systems of functions ϕi(r), ψi(q) andχi(q) which are optimized for the representation of the scattering functions from aparticle (scattering object) with maximum dimension smaller or equal to Dmax. Inorder find these functions we must calculate the expansion coefficients ci by aweighted least squares operation:

expexp

N2

i2 k i kM Mak k i=1

2 2k=1 k=1k k

[ ( ) ( )q q ]cI[ ( ) ( )q q ]I IL= =

( ) ( )q q

χ

σ σ

−− �

� �

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The Stability Problem

If we apply the basic idea described above we find a good fitto the data, but a solution in real space which shows strongoscillations around the correct solution.

We can reduce or even eliminate these artificial oscillationsby adding the following condition to the least squarescondition:

( )1

2

' 11

N

c i ii

N c c−

+=

= −�This condition is coupled to the least squares condition Lby a so-called Lagrange-Multiplier λ:

This figure: x-x-x-x: exact solution, ——— unstable solution.

c(L + N )= Minλ ′

Here we are left with the problem to find the right value for � !


Stability Plot - Selection of Lagrange Multiplier

Stability plot – point of inflectionTypical influence of � on the solution inreal space

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Selection of Parameters

With the point of inflection method we can find the optimum value for � . Still open is theproblem how to select Dmax and the number of spline functions N.

Choice of Dmax:

The sampling theorem of Fourier transformation gives a clear answer to the question oflargest possible particle size. If the scattering curve is sampled at increments ∆q ≤ qminstarting at qmin, the scattering data contain full information for all particles with maximumdimension Dmax

In practice on will try to stay below this limit, i.e.

Number of Spline Functions N:

The stabilizing routine makes the procedure nearly independent of N, in practice N is chosenin the limits 10 ≤≤≤≤ N ≤≤≤≤ 40 depending on the measured q – range and on the statistical accuracyof the data

maxmin

Dq

π=

min minandq q qD

π< ∆ �


Application of IFT to simulated slit collimation data

Fit and desmeared scattering curve ofa sphere simulated with slit collimation

Example of an extreme cut-off

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Other IFT Applications - Overview

The IFT technique can also be applied to data from cylindrical or lamellar particles aswell as to polydisperse systems


Other IFT Applications - Equations

Summary of the different transforms T1 used in IFT:

Arbitrary shape:

Cylindrical Symmetry: Lamellar Symmetry:

Polydisperse Systemsnumber (volume distribution):

The structure is the same for all equations, just the kernels of the integrals differ!

( )0

sin ( )( ) 4

qrI q p r dr

qrπ

∞

= ÿ

( ) ( ) ( )2

0

0

2c

LI q p r J qr dr

q

π ∞

= ÿ ( ) ( ) ( )20

4cosplane t

AI q p r qr dr

q

π ∞

= ÿ

( ) ( ) ( )30

0

,v vI q c D R R P q R dR∞

= ⋅ ⋅ÿ

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Deconvolution of the PDDF (Convolution Square Root)

We have seen how inhomogeneities influence the scattering functions I(q) andp(r), but until now we have no solution for the inverse problem, i.e. how todetermine the structure of the particles from these functions.

There is no general solution for this problem for arbitrary three-dimensionalstructures.

Such methods do exist, however, for:

•spherical symmetry,

•circular cylinders with centro-symmetric radial density distributions (noangular or axial dependence of the density) and for

•centro-symmetric lamellae without in-plane inhomogeneities.

It is obvious that we do not lose any information by spatial averaging in thecase of spherical symmetry. So we may hope that there is a chance todetermine the one-dimensional information ∆ρ(r) from the measured one-dimensional function I(q). We discuss the solution of this problem in thefollowing section.


Deconvolution of the PDDF – The Magic Square

The Magic square of small-angle scattering: The correlations between the radialdensity ∆ρ(r) and the PDDF p(r) and their Fourier transforms, the scattering amplitudeF(q) and scattering intensity I(q) under the assumption of spherical symmetry.

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Deconvolution of the PDDF – Principles I

Here we are facing a similar situation as in the IFT method: for a given densitydistribution ρ(r) we can calculate the exact p(r)-function for all three cases(spherical, cylindrical and lamellar symmetry) by a convolution squareoperation but we do not have a useful description of the inverse problem, theso-called convolution square root.

As an additional problem we have to keep in mind the fact, that theconvolution square operation is a nonlinear transformation which will notallow an inversion by the solution of a simple linear least squares techniquelike in the case of the indirect Fourier transformation.

We start again with a series expansion of the radial density function ρ(r) in theusual way:

( ) ( )1

N

i ii

r c rρ ϕ=

= �


Deconvolution of the PDDF – Principles II

The approximation for the density profile corresponds to an approximation to the PDDF:

( ) ( ) ( )2

1

2N

ii i ik i ki i k

p r V r c V r c c= >

= +� �

The overlap integrals Vik(r) describe the overlapping of the i-th with the k-th step or shellwhere one function has been shifted an arbitrary distance r. These overlap orconvolution integrals are very simple for the planar case (one-dimensional convolutionof two step function leads simply to a triangle) but are a bit more complicated for thecylindrical and spherical case:

Illustration of the five sub-regions for thecalculation of the overlap integrals Vik(r).

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Deconvolution of the PDDF – Iterative Solution

The above equation for the PDDF is nonlinear in its coefficients ci. Thecorresponding least squares problem has to be linearized by a series expansionwhere higher order terms are omitted.

Such linearized systems must be solved iteratively. In addition one needsstarting values ci

(0) for the first iteration. Here we set all coefficients equal to aconstant.

We then calculate the difference function

( ) ( ) ( ) ( )op r p r p r∆ = −

which would be zero only if we would know the exact coefficients ci.

Now we calculate correction terms ∆ci in order to minimize ∆p(r) in a least squaresense.

( ) ( ) ( ) ( )( ) ( )2

1

2N

ii i i ik i i k k i ki i k

V r c c V r c c c c c c p r= >

� �+ ∆ + + ∆ + ∆ − = ∆� ��


Deconvolution of the PDDF – Iterative Solution II

We linearize this equation by omitting the second order terms ∆ci2 and ∆ci∆ck

and we get

( ) ( )1 1

2N N

i ik j k jk i

c V r c p r= =

∆ = ∆� �for j = 1,2,3,... M and M > N. These equations can be written in matrix notation

where the matrix elements Ajk are given by

This system is solved with a weighted least squares condition considering thestandard deviations of the function ∆p(r) and we get the correction terms ∆c.

( ) ( )orjk k jA c p∆ = ∆ ∆ = ∆0 0A c p

( )1

2N

jk i ik ji

A c V r=

= �

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Deconvolution of the PDDF – Iterative Solution III

( ) ( )1 0i i ic c c= + ∆

and with these coefficients we start the next iteration, get further improvementsand if this iterative procedure converges we have solved the problem.

This problem is, however, again an ill-posed problem so that we have to add againa stabilization criterion and we have to solve the nonlinear problem by iteration forevery Lagrange multiplier.

Many applications performed in the meantime have shown that the deconvolutiontechnique works well in combination with the indirect transformation method, alsoin cases where the conditions of symmetry are not perfectly fulfilled.

They allow the calculation of improved coefficients ci(1):


Deconvolution of the PDDF – Application

Example of an inhomogeneous oblate spheroid with an axial ratio of 1:1.2:1.2.

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Surfactant Systems - Spherical micelles

Scattering pattern of spherical aggregates with radius R. The scattering curve I(q) is theFourier transform of the PDDF p(r). This PDDF is the convolution square of the radialdensity distribution ∆ρ(r).


Surfactant Systems - Rod-like micelles I

Scattering function I(q) and PDDF p(r) for cylindrical aggregateswith radius R and length L.

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Surfactant Systems - Rod-like micelles. II: Cross-section

Cross-section functions I(q)q, pc(r) and ∆ρc(r) for cylindrical aggregates.


Surfactant Systems - Lamellar Systems I

Scattering function I(q) and PDDF p(r) for vesicles and planar aggregates (L3 phase).Full lines: typical results for lamellar structures; dashed lines: functions formonodisperse vesicles.

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Surfactant Systems - Lamellar Systems II

Thickness functions I(q)q2, pt(r) and ∆ρt(r) other planar aggregates with thethickness T.


IFT Application: Lipid IVA Vesicles

Lipid IVA is a bioactive precursor ofLipid IV, which is most important for thephysiological activity of the cells. LipidIVA has a disaccharide as hydrophilichead group and four hydrophobic sidechains.

Lipid IVA: experimental data points (ooo) andfitting function (——).

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PDDF of the Lipid IVA vesicles. Thickness Guinier plot for thedesmeared data from Lipid IVA.



Scattering curve I(q) desmeared underthe assumption of an extended lamellarstructure.

Thickness-PDDF, calculated for Dmax of 6nm (——) and 9 nm (-----).

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Electron density distribution ρ(r) calculatedwith 7 equidistant steps (——) and theoptimized 2-step model (-----).

Thickness PDDF pt(r) calculated fromLipid IVA data (ooo) and fit by theconvolution square root technique (—)model with 7 steps.

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Indirect Fourier Transformation - EMBL Hamburg · Indirect Fourier Transformation Introduction Pair Distance Distribution Function PDDF Indirect Fourier Transformation ... The phase