1

Jan Skov Pedersen, iNANO Center and Department of Chemistry

University of AarhusDenmark

Form and Structure Factors:Modeling and Interactions

SAXS lab

New SAXS instrument in ultimo Nov 2014:

Funding from:

Research Council for Independent Research: Natural Science

The Carlsberg Foundation The Novonordisk Foundation

2

3

Gallium Kα X-rays (λ = 1.34 Å)

Liquid metal jet X-ray sourcec

3 x 109 photons/sec!!!

Special Optics Scatterless slits

High Performance 2D Detector

Stopped-flow for rapid mixing Sample robot

+ optimized geometry: 3 x 1010 ph/sec!!!

4

5

Outline

• Concentration effects and structure factorsZimm approachSpherical particles Elongated particles (approximations)Polymers

• Model fitting and least-squares methods• Available form factors

ex: sphere, ellipsoid, cylinder, spherical subunits…ex: polymer chain

• Monte Carlo integration for form factors of complex structures

• Monte Carlo simulations for form factors of polymer models

6

Motivation for ‘modelling’

- not to replace shape reconstruction and crystal-structure based modeling – we use the methods extensively

- describe and correct for concentration effects

- alternative approaches to reduce the number of degrees of freedom in SAS data structural analysis

- provide polymer-theory based modeling of flexible chains

(might make you aware of the limited information content of your data !!!)

7

LiteratureJan Skov Pedersen, Analysis of Small-Angle Scattering Data from Colloids and Polymer Solutions: Modeling and Least-squares Fitting (1997). Adv. Colloid Interface Sci., 70, 171-210.

Jan Skov PedersenMonte Carlo Simulation Techniques Applied in the Analysis of Small-Angle Scattering Data from Colloids and Polymer Systemsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 381

Jan Skov PedersenModelling of Small-Angle Scattering Data from Colloids and Polymer Systemsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 391

Rudolf KleinInteracting Colloidal Suspensionsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 351

MC applications in modelling

8

K.K. Andersen, C.L.P. Oliveira, K.L. Larsen, F.M. Poulsen, T.H. Callisen, P. Westh, J. Skov Pedersen, D.E. Otzen (2009) The Role of Decorated SDS Micelles in Sub-CMC Protein Denaturation and Association.Journal of Molecular Biology, 391(1), 207-226.

J. Skov Pedersen ; C.L.P. Oliveira. H.B Hubschmann, L. Arelth, S. Manniche, N. Kirkby, H.M. Nielsen (2012). Structure of Immune Stimulating Complex Matrices and Immune Stimulating Complexes in Suspension Determined by Small-Angle X-Ray Scattering.Biophysical Journal 102, 2372-2380.

J.D. Kaspersen, C.M. Jessen, B.S. Vad, E.S. Sorensen, K.K. Andersen, M. Glasius, C.L.P. Oliveira, D.E. Otzen, J.S. Pedersen, (2014). Low-Resolution Structures of OmpA-DDM Protein-Detergent Complexes.ChemBioChem 15(14), 2113-2124.

9

Form factors and structure factors

Warning 1:Scattering theory – lots of equations!= mathematics, Fourier transformations

Warning 2:Structure factors:Particle interactions = statistical mechanics

Not all details given - but hope to give you an impression!

10

I will outline some calculations to show that it is not black magic !

11

Input data: Azimuthally averaged data

NiqIqIq iii ,...3,2,1)(),(,

)( iqI

)( iqI

iq calibrated

calibrated, i.e. on absolute scale - noisy, (smeared), truncated

Statistical standard errors: Calculated from countingstatistics by error propagation- do not contain information on systematic error !!!!

12

Least-squared methods

Measured data:

Model:

Chi-square:

Reduced Chi-squared: = goodness of fit (GoF)

Note that for corresponds toi.e. statistical agreement between model and data

13

Cross section

dσ(q)/dΩ : number of scattered neutrons or photons per unit time, relative to the incident flux of neutron or photons, per unit solid angle at q per unit volume of the sample.

For system of monodisperse particlesdσ(q)dΩ = I(q) = n ρ2V 2P(q)S(q)

n is the number density of particles, ρ is the excess scattering length density,

given by electron density differencesV is the volume of the particles, P(q) is the particle form factor, P(q=0)=1

S(q) is the particle structure factor, S(q=)=1

• V M• n = c/M• ρ can be calculated from partial specific density, composition

= c M ρm2P(q)S(q)

14

Form factors of geometrical objects

15

Form factors I

1. Homogeneous sphere 2. Spherical shell: 3. Spherical concentric shells: 4. Particles consisting of spherical subunits: 5. Ellipsoid of revolution: 6. Tri-axial ellipsoid: 7. Cube and rectangular parallelepipedons: 8. Truncated octahedra: 9. Faceted Sphere: 9x Lens10. Cube with terraces: 11. Cylinder: 12. Cylinder with elliptical cross section: 13. Cylinder with hemi-spherical end-caps: 13x Cylinder with ‘half lens’ end caps14. Toroid: 15. Infinitely thin rod: 16. Infinitely thin circular disk: 17. Fractal aggregates:

Homogenous rigid particles

16

Form factors II

18. Flexible polymers with Gaussian statistics: 19. Polydisperse flexible polymers with Gaussian statistics: 20. Flexible ring polymers with Gaussian statistics: 21. Flexible self-avoiding polymers: 22. Polydisperse flexible self-avoiding polymers: 23. Semi-flexible polymers without self-avoidance:24. Semi-flexible polymers with self-avoidance: 24x Polyelectrolyte Semi-flexible polymers with self-avoidance: 25. Star polymer with Gaussian statistics: 26. Polydisperse star polymer with Gaussian statistics: 27. Regular star-burst polymer (dendrimer) with Gaussian statistics: 28. Polycondensates of Af monomers: 29. Polycondensates of ABf monomers: 30. Polycondensates of ABC monomers: 31. Regular comb polymer with Gaussian statistics: 32. Arbitrarily branched polymers with Gaussian statistics: 33. Arbitrarily branched semi-flexible polymers: 34. Arbitrarily branched self-avoiding polymers: 35. Sphere with Gaussian chains attached: 36. Ellipsoid with Gaussian chains attached: 37. Cylinder with Gaussian chains attached: 38. Polydisperse thin cylinder with polydisperse Gaussian chains attached to the ends: 39. Sphere with corona of semi-flexible interacting self-avoiding chains of a corona chain.

’Polymer models’

(Block copolymer micelle)

17

Form factors III40. Very anisotropic particles with local planar geometry: Cross section:(a) Homogeneous cross section (b) Two infinitely thin planes(c) A layered centro symmetric cross-section (d) Gaussian chains attached to the surface

Overall shape:(a) Infinitely thin spherical shell(b) Elliptical shell (c) Cylindrical shell(d) Infinitely thin disk

41. Very anisotropic particles with local cylindrical geometry: Cross section:(a) Homogeneous circular cross-section (b) Concentric circular shells (c) Elliptical Homogeneous cross section. (d) Elliptical concentric shells (e) Gaussian chains attached to the surface

Overall shape:(a) Infinitely thin rod(b) Semi-flexible polymer chain with or without excluded volume

P(q) = Pcross-section(q) Plarge(q)

18

(partial integration)…

From factor of a solid sphere

R r

(r)1

0

33

3

2

0

0

2

0

2

0

)(

)]cos()[sin(3

3

4

cossin4

sincos4

sincos4

''

)sin(4

)sin(4

)sin()(4)(

qR

qRqRqRR

qRqRqRq

q

qr

q

qRR

q

q

qr

q

qRR

q

dxfgfgdxgf

rdrqrq

drrqr

qrdrr

qr

qrrqA

R

R

R

spherical Bessel function

19

q-4

1/R

Form factor of sphere

C. Glinka

Porod

P(q) = A(q)2/V2

20

Ellipsoid

Prolates (R,R,R)

Oblates (R,R,R)

dqRqP sin)'()(22/

0

R’ = R(sin2 + 2cos2)1/2

3

)cos()sin(3)(

x

xxxx

21Glatter

P(q): Ellipsoid of revolution

22

Lysosyme

Lysozyme 7 mg/mL

q [Å-1]

0.0 0.1 0.2 0.3 0.4

I(q)

[cm

-1]

10-3

10-2

10-1

100

Ellipsoid of revolution + backgroundR = 15.48 Å = 1.61 (prolate) 2=2.4 (=1.55)

23www.psc.edu/.../charmm/tutorial/ mackerell/membrane.htmlmrsec.wisc.edu/edetc/cineplex/ micelle.html

SDS micelle

20 Å

Hydrocarbon core

Headgroup/counterions

24

Core-shell particles:

)()()()( inincoreshelloutoutshellshellcore qRVqRVqA

where Vout= 43/3 and Vin= 4Rin

3/3. core is the excess scattering length density of the core, shell is the excess scattering length density of the shell and:

3

cossin3)(

xxxx

x

=

Rout Rin

25

q [Å-1]

0.0 0.1 0.2 0.3 0.4 0.5

P(q

)

10-5

10-4

10-3

10-2

10-1

100

101

Rout = 30 ÅRcore = 15 Åshell = 1core 1.

26q [Å-1]

0.0 0.1 0.2 0.3 0.4

I(q)

[cm

-1]

0.001

0.01

0.1

2 = 2.3

I(0) = 0.0323 0.0005 1/cmRcore = 13.5 2.6 Å = 1.9 0.10 dhead = 7.1 4.4 Åhead/core = 1.7 1.5 backgr = 0.00045 0.00010 1/cm

SDS micelles:prolate ellispoid with shell of constant thickness

27

Molecular constraints:

OH

eOH

tail

etaile

tail V

Z

V

Z

2

2tailaggcore VNV

OH

eOH

OHhead

OHeheade

head V

Z

nVV

nZZ

2

2

2

2

n water molecules in headgroup shell

)( 2OHheadaggshell nVVNV

surfactantMN

cn

aggmicelles

28

Cylinder

L

2R

dqL

qL

qR

qRJqP sin

2/cos

2/cossin(

sin

sin2)(

2

12/

0

J1(x) is the Bessel function of first order and first kind.

29

P(q) cylinder

q

R = 30 ÅL = 60 Å120 Å300 Å

3000 Å

L>> R P(q) Pcross-section(q) Plarge(q)

30

Glucagon Fibrils

Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387, 147–161

R=29Å

R=16 Å

31

Primus

32

Form factor for particles with arbitrary shape

Spherical monodisperse particles

n

i

n

j ij

ijsphere qr

qrRqPqI

1 1

sin),(

33

•Generate points in space by Monte Carlo simulations

•Select subsets by geometric constraints

•Caclulate histogramsp(r) functions

•(Include polydispersity)

•Fourier transformMC points

Sphere

x(i)2+y(i)2+z(i)2 R2

Monte Carlo integration in calculation ofform factors for complex structures

34

Protein Micelle

ISCOMvaccineparticle

35

Immune-stimulating complexes: ISCOMs- Self-assembled vaccine nano-particles

M T Sanders et al. Immunology and Cell Biology (2005) 83, 119–128

Stained TEM

ISCOMs typically:

60–70wt% Quil-A, 10–15wt% Phospholipids10-15wt% cholesterol

Quil-A

36

SAXS data

q [Å-1]

0.01 0.1

I(q)

10-3

10-2

10-1

100

ISCOMs without toxoid

Oscillations relatively monodisperseBump at high q core-headgroup structure

Investigation of the structure of

ISCOM particles by SAXSS. Manniche, H.B. Madsen, L. Arleth, H.M. Nielsen,

C.L.P. Oliveira, J.S. Pedersen, in preparation.

37

Combination of icosahedrons, tennis balls and footballs

q [Å-1]

0.01 0.1

I(q)

10-3

10-2

10-1

100Fit result:Icosah: m M = 0.057Tennis: m M = 0.770Football: m M= 0.173

From volumes of cores:Icosah: M = 0.557 a.u.Tennis: M = 1.000 a.u.Football M= 1.616 a.u

Mass distribution:Icosah: m = 0.104Tennis: m = 0.786Football: m = 0.110

38

Icosahedrons, tennis ball, football

242 Å 440 Å335 Å

Hydrophobic core: (2 x) 11 ÅHydrophilic headgroup region: Width 13 Å

39

Pure SDS micelles in buffer:

C > CMC ( 5 mM)

C CMC

Nagg= 66±1 oblate ellipsoidsR=20.3±0.3 Å ɛ=0.663±0.005

OmpA in DDM

PDB structure forprotein

Monte Carlo pointsfor DDM (absolute scale)

40

Full length monomer modellingusing BUNCH program of ATSAS package

9% DDM micellesbackground

41

Homologue of periplasmic domain

Flexibility of subunits

42

Using homologue structure of periplasmic domian

Ensemble OptimizationMethod (EOM)- In ATSAS

Micelles included asa component- Always selected

43

44

Polymer chains in solutionGigantic ensemble of 3D random flights

- all with different configurations

45

Gaussian polymer chain

Look at two points: Contour separation: LSpatial separation: r

Contribution to scattering:rL qr

qrqI

)sin()(2

2

2

23

exp)(rr

rD

For an ensemble of polymers, points withL has <r2>=Lband r has a Gaussian distribution:

’Density of points’: (Lo- L) Lo

LL

Add scattering from all pair of points

6/exp 2 Lbq

46

Gaussian chains: The calculation

222

0

2

2

00

0

122

0

1

02

]1)[exp(2

6/exp)(1

)sin(),()(

1

)sin(|)|,(

1)(

qRxx

xx

LbqLLdLL

rqr

qrLrDLLdLdr

L

rqr

qrLLrDdLdLdr

LqP

g

L

oo

L

oo

LL

o

o

o

oo

How does this function look?

Rg2 = Lb/6

47

Polymer scattering

Form factor of Gaussian chain

qRg

0.1 1 10 100

P(q

)

0.0001

0.001

0.01

0.1

1

10

gRq /1

21 qq

48

Lysozyme in Urea 10 mg/mL

q [Å-1]

0.01 0.1 1

I(q)

[cm

-1]

10-3

10-2

10-1

Gaussian chain+ backgroundRg = 21.3 Å

2=1.8

-synuclein

is low!

Gaussian chain+ backgroundRg = 21.3 Å

2=1.8

49

Self avoidence

No excluded volume No excluded volume=> expansion

222 ]1)[exp(2)(

x

xxqRxP g

no analytical solution!

50

Monte Carlo simulation approach

(1) Choose model

(5) Fit experimental data using numerical expressionsfor P(q)

(2) Vary parameters in a broad range Generate configs., sample P(q)

(3) Analyze P(q) using physical insight

(4) Parameterize P(q) using physical insight

Pedersen and Schurtenberger 1996

P(q,L,b)

L = contour length

b = Kuhn (‘step’) length

51

Expansion = 1.5

Form factor polymer chains

qRg (Gaussian)

0.1 1 10 100

P(q

)

0.0001

0.001

0.01

0.1

1

10

2/1

21

qq

588.0

70.1/1

qq

Excluded volume chains:

Gaussian chains

.).(/1 volexclRq g

)(/1 GaussianRq g

q-1 local stiffness

52

C16E6 micelles with ‘C16-’SDS

Sommer C, Pedersen JS, Egelhaaf SU, Cannavaciuolo L, Kohlbrecher J, Schurtenberger P.

Variation of persistencelength with ionic strength

works also for polyelectrolyteslike unfolded proteins

53

Models IIPolydispersity: Spherical particles as e.g. vesicles

No interaction effects: Size distribution D(R)

i

iii qPfMcd

qd)(

)( 2Oligomeric mixture: Discrete particles

fi = mass fraction i

if 1

Application to insulin:

Pedersen, Hansen, Bauer (1994). European Biophysics Journal 23, 379-389.

(Erratum). ibid 23, 227-229.

Used in PRIMUS‘Oligomers’

54

Concentration effects

55

Concentration effects in protein solutions

= q/2

-crystallineye lens protein

56

p(r) by Indirect Fourier Transformation (IFT)

At high concentration, the neighborhood is different from theaverage further away!

(1) Simple approach: Exclude low q data.(2) Glatter: Use Generalized Indirect Fourier Transformation (GIFT)

I. Pilz, 1982

- as you have done by GNOM !

57

Low concentrations

= Random-Phase Approximation (RPA)

P’(q) = -

Zimm 1948 – originally for light scattering

Subtract overlapping configuration

~ concentrationP’(q) = P(q) - P(q)2

= P(q)[1 - P(q)]

)(1

)(

qP

qP

…= P(q)[1 - P(q )+ 2P(q)2 - 3P(q)3…..]

Higher order terms:P’(q) = P(q)[1 - P(q)1 - P(q)]

= P(q)[1 - P(q)+ 2P(q)2]

58

Zimm approach

)(1)(

)(qP

qPKqI

)(1

)()(1

)( 111

qPK

qPqP

KqI

3/1

1)(

22gRq

qP

3/1)( 2211gRqKqIWith

Plot I(q)1 versus q2 + c and extrapolate to q=0 and c=0 !

59

Zimm plot

I(q)-1

q2 + c

Kirste and WunderlichPS in toluene

q = 0

P(q)

c = 0

60

My suggestion:

)3/exp(1)(

22

21 aqca

PcqI

i

ii

with a1, a2, and Pi as fit parameters

( - which includes also information from what follows)

• Minimum 3 concentrations for same system.• Fit data simultaneously all data sets

61

Osteopontin (M=35 kDa )in water and 10 mM NaCl

q (A-1)

0.01 0.1 1

I(q)

(cm

-1)

0.0001

0.0010

0.0100

0.1000

1.0000

2.5, 5 and 10 mg/mL

Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

62Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

63

Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

64

Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)

65

But now we look at the information content related to

these effects…

Understand the fundamental processes and principlesgoverning aggregation and crystallization

Why is the eye lens transparent despite a protein concentrationof 30-40% ?

66

Structure factor

)()(

sin)()(

22

22

qSqPV

qr

qrqPVqI

jk

jk

Spherical monodisperse particles

S(q) is related to the probability distribution function of inter-particles distances, i.e. the pair correlation function g(r)

j

jj qr

qrrhqPV

sin)()(22

67

Correlation function g(r)

g(r) =Average of the normalized density of atoms in a shell [r ; r+ dr]from the center ofa particle

68

g(r) and S(q)

q

drrqr

qrrgnqS 2)sin(

)1)((41)(

69

GIFTGlatter: Generalized Indirect Fourier Transformation (GIFT)

drqr

qrrpqI

)sin()(4)(

drqr

qrrpcZRRqSqI salteff

)sin()(4),),(,,,()(

With concentration effects

Optimized by constrained non-linear least-squares method

- works well for globular models and provides p(r)

70

A SAXS study of the small hormone glucagon: equilibrium aggregation and fibrillation

29 residue hormone, with a net charge of +5 at pH~2-3

0 10 20 30 40 50 60 70

p(r

) [a

rb.

u.]

r [Å]0,01 0,1

10.7 mg/ml

6.4 mg/ml

5.1 mg/ml

2.4 mg/ml

I(q

) [a

rb.

u.]

q [Å-1]

1.0 mg/ml

Hexamers

trimersmonomers

Home-written software

71

Osmometry (Second virial coeff A2)

c

/c

Ideal gas A2 = 0

72

S(q), virial expansion and Zimm

...3211

)0(3

22

1

MAccMAcRTqS

In Zimm approach = 2cMA2

From statistical mechanics…:

73

A2 in lysozyme solutions

O. D. Velev, E. W. Kaler, and A. M. Lenhoff

A2

Isoelectric point

74

Colloidal interactions• Excluded volume ‘repulsive’ interactions (‘hard-sphere’)• Short range attractive van der Waals interaction (‘stickiness’) • Short range attractive hydrophobic interactions

(solvent mediated ‘stickiness’) •Electrostatic repulsive interaction (or attractive for patchy charge distribution!)

(effective Debye-Hückel potential)• Attractive depletion interactions (co-solute (polymer) mediated )

hard sphere sticky hard sphereDebye-Hückel screened Coulomb potential

75

Theory for colloidal stability

Debye-Hückel screened Coulomb potential+ attractive interaction

DLVO theory: (Derjaguin-Landau-Vewey-Overbeek)

76

Depletion interactions

Asakura & Oosawa, 58

77

Integral equation theory

Relate g(r) [or S(q)] to V(r)

At low concentration g(r) = exp( V (r) / kBT)Boltzmann approximation

c(r) = direct correlation function

Make expansion around uniform state [Ornstein-Zernike eq.]

g(r) = 1 – n c(r) –[3 particle] – [4 particle] - …= 1 – n c(r) – n2 c(r) * c(r) – n3 c(r) * c(r) * c(r) …

[* = convolution]

1 V (r) / kBT(weak interactions)

S(q) = 1 n c(q) – n2 c(q)2 – n3 c(q)3 …. =

but we still need to relate c(r) to V(r) !!!

)(1

1

qnc

78

Closure relations

Systematic density expansion….

Mean-spherical approximation MSA: c(r) = V(r) / kBT(analytical solution for screened Coulomb potential

- but not accurate for low densities)

Percus-Yevick approximation PY: c(r) = g(r) [exp(V (r) / kBT)1](analytical solution for hard-sphere potential + sticky HS)

Hypernetted chain approximation HNC: c(r) = V(r) / kBT +g(r) 1 – ln (g(r) )

(Only numerical solution - but quite accurate for Coulomb potential)

Rogers and Young closure RY:Combines PY and HNC in a self-consistent way(Only numerical solution

- but very accurate for Coulomb potential)

79

-crystallin S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

80

-crystallin + PEG 8000 S. Finet, A. Tardieu

DLVO potential

HNC and numerical solution

Depletion interactions:

40 mg/ml -crystallin solution pH 6.8, 150 mM ionic strength.

81

Anisotropy

Small: decoupling approximation (Kotlarchyk and Chen,1984) :

Measured structure factor:

)(1)()(1)(

)(

)( 2 qSqSqqPn

q

qSmeas

!!!!!

82

Large anisotropy

Anisotropy, large: Random-Phase Approximation (RPA):

)(1)(

)( 22

qPqP

Vnqdd

~ concentration

Anisotropy, large: Polymer Reference Interaction Site Model (PRISM)Integral equation theory – equivalent site approximation

)()(1)(

)( 22

qPqncqP

Vnqdd

c(q) direct correlation function

related to FT of V(r)

Polymers, cylinders…

83

Empirical PRISM expression

Arleth, Bergström and Pedersen

)()(1)(

)( 22

qPqncqP

Vnqdd

c(q) = rod formfactor- empirical from MC simulation

SDS micelle in 1 M NaBr

84

Overview: Available Structure factors

1) Hard-sphere potential: Percus-Yevick approximation 2) Sticky hard-sphere potential: Percus-Yevick approximation3) Screened Coulomb potential:

Mean-Spherical Approximation (MSA).Rescaled MSA (RMSA). Thermodynamically self-consistent approaches (Rogers and Young closure)

4) Hard-sphere potential, polydisperse system: Percus-Yevick approximation5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation6) Screened Coulomb potential, polydisperse system: MSA, RMSA, 7) Cylinders, RPA 8) Cylinders, `PRISM': 9) Solutions of flexible polymers, RPA:10) Solutions of semi-flexible polymers, `PRISM': 11) Solutions of polyelectrolyte chains ’PRISM’:

85

Summary

• Concentration effects and structure factorsZimm approachSpherical particles Elongated particles (approximations)Polymers

• Model fitting and least-squares methods• Available form factors

ex: sphere, ellipsoid, cylinder, spherical subunits…ex: polymer chain

• Monte Carlo integration for form factors of complex structures

• Monte Carlo simulations for •form factors of polymer models

86

LiteratureJan Skov Pedersen, Analysis of Small-Angle Scattering Data from Colloids and Polymer Solutions: Modeling and Least-squares Fitting (1997). Adv. Colloid Interface Sci., 70, 171-210.

Jan Skov PedersenMonte Carlo Simulation Techniques Applied in the Analysis of Small-Angle Scattering Data from Colloids and Polymer Systemsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 381

Jan Skov PedersenModelling of Small-Angle Scattering Data from Colloids and Polymer Systemsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 391

Rudolf KleinInteracting Colloidal Suspensionsin Neutrons, X-Rays and LightP. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.p. 351

MC applications in modelling

87

K.K. Andersen, C.L.P. Oliveira, K.L. Larsen, F.M. Poulsen, T.H. Callisen, P. Westh, J. Skov Pedersen, D.E. Otzen (2009) The Role of Decorated SDS Micelles in Sub-CMC Protein Denaturation and Association.Journal of Molecular Biology, 391(1), 207-226.

J. Skov Pedersen ; C.L.P. Oliveira. H.B Hubschmann, L. Arelth, S. Manniche, N. Kirkby, H.M. Nielsen (2012). Structure of Immune Stimulating Complex Matrices and Immune Stimulating Complexes in Suspension Determined by Small-Angle X-Ray Scattering.Biophysical Journal 102, 2372-2380.

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