1. No category

Form and Structure Factors:
Modeling and Interactions
SAXS lab
Jan Skov Pedersen,
iNANO Center and Department of Chemistry
University of Aarhus
Denmark
1
New SAXS instrument in ultimo Nov 2014:
Funding from:
Research Council for Independent Research:
Natural Science
The Carlsberg Foundation
The Novonordisk Foundation
2
Liquid metal jet X-ray sourcec
Gallium Kα X-rays (λ = 1.34 Å)
3 x 109 photons/sec!!!
Special Optics
3
Scatterless slits
+ optimized geometry:
3 x 1010 ph/sec!!!
Stopped-flow for rapid mixing
High Performance 2D
Detector
Sample robot
4
Outline
• 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
• Concentration effects and structure factors
Zimm approach
Spherical particles
Elongated particles (approximations)
Polymers
5
Motivation for ‘modelling’
- not to replace shape reconstruction and crystal-structure based
modeling – we use the methods extensively
- alternative approaches to reduce the number of degrees of
freedom in SAS data structural analysis
(might make you aware of the limited information
content of your data !!!)
- provide polymer-theory based modeling of flexible chains
- describe and correct for concentration effects
6
Literature
Jan 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 Pedersen
Monte Carlo Simulation Techniques Applied in the Analysis of Small-Angle
Scattering Data from Colloids and Polymer Systems
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 381
Jan Skov Pedersen
Modelling of Small-Angle Scattering Data from Colloids and Polymer Systems
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 391
Rudolf Klein
Interacting Colloidal Suspensions
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 351
7
MC applications in modelling
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.
8
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!
9
I will outline some calculations to show
that it is not black magic !
10
Input data: Azimuthally averaged data
qi , I (qi ),  I (qi ) i  1,2,3,...N
qi calibrated
I (qi ) calibrated, i.e. on absolute scale
- noisy, (smeared), truncated
 I (qi )
Statistical standard errors: Calculated from counting
statistics by error propagation
- do not contain information on systematic error !!!!
11
Least-squared methods
Measured data:
Model:
Chi-square:
Reduced Chi-squared:
= goodness of fit (GoF)
Note that
for
corresponds to
i.e. statistical agreement between model and data
12
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 particles
dσ(q)
dΩ
= I(q) = n ρ2V 2P(q)S(q) =
c M ρm2P(q)S(q)
n
ρ
is the number density of particles,
is the excess scattering length density,
given by electron density differences
V
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
13
Form factors of geometrical objects
14
Form factors I
Homogenous rigid particles
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 Lens
10. Cube with terraces:
11. Cylinder:
12. Cylinder with elliptical cross section:
13. Cylinder with hemi-spherical end-caps:
13x Cylinder with ‘half lens’ end caps
14. Toroid:
15. Infinitely thin rod:
16. Infinitely thin circular disk:
17. Fractal aggregates:
15
Form factors II
’Polymer models’
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: (Block copolymer micelle)
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. 16
Form factors III
P(q) = Pcross-section(q) Plarge(q)
40. 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
17
From factor of a solid sphere
1
(r)
0

A(q)  4   (r )
0
r
R
R
sin(qr ) 2
sin( qr ) 2
r dr  4 
r dr
qr
qr
0
4
sin(qr ) rdr 
q 0
R

 f ' g dx   fg    fg ' dx 
(partial integration)…

4
q
 R cos qR  sin qr 




q
 q 0


4
q
 R cos qR sin qr 

 

q
q 2 

R




4
sin qR  qR cos qR 
q3
4
3[sin(qR)  qR cos(qR)]
  R3
3
(qR) 3

spherical Bessel function
18
Form factor of sphere
P(q) = A(q)2/V2
1/R
q-4
Porod
19
C. Glinka
Ellipsoid
Prolates (R,R,R)

 /2
P(q)  
0
Oblates (R,R,R)

2
(qR' )
sin  d
R’ = R(sin2 + 2cos2)1/2
 ( x) 
3sin( x)  x cos( x)
x3
20
P(q): Ellipsoid of revolution
Glatter
21
Lysosyme
Lysozyme 7 mg/mL
100
Ellipsoid of revolution + background
R = 15.48 Å
2=2.4 (=1.55)
 = 1.61 (prolate)
-1
I(q) [cm ]
10-1
10-2
10-3
0.0
0.1
0.2
q [Å-1]
0.3
0.4
22
SDS micelle
Hydrocarbon core
Headgroup/counterions
20 Å
www.psc.edu/.../charmm/tutorial/ mackerell/membrane.html
23
mrsec.wisc.edu/edetc/cineplex/ micelle.html
Core-shell particles:
=
Rout
Rin


Acore shell (q)   shellVout (qRout )  ( shell   core )Vin  (qRin )
where
Vout= 43/3 and Vin= 4Rin3/3.
core is the excess scattering length density of the core,
shell is the excess scattering length density of the shell and:
 ( x) 
3sin x  x cos x 
x3
24
101
100
P(q)
10-1
10-2
10-3
10-4
10-5
0.0
0.1
0.2
0.3
0.4
0.5
q [Å-1]
Rout = 30 Å
Rcore = 15 Å
shell = 1
core  1.
25
SDS micelles:
prolate ellispoid with shell of constant thickness
0.1
I(q) [cm-1]
2 = 2.3
I(0) =
0.0323  0.0005 1/cm
Rcore =
13.5  2.6 Å
=
1.9  0.10
dhead =
7.1  4.4 Å
head/core =
 1.7  1.5
backgr = 0.00045  0.00010 1/cm
0.01
0.001
0.0
0.1
0.2
q [Å-1]
0.3
0.4
26
Molecular constraints:
e
 tail


e
head
e
Z tail
Ze
 H 2O
Vtail V H 2O
Vcore  N agg Vtail
e
 nZ H 2O Z He 2O
Z head


Vhead  nVH 2O VH 2O
nmicelles 
n water molecules in headgroup shell
Vshell  N agg (Vhead  nVH 2O )
c
N agg M surfactant
27
Cylinder
2R
L
 /2
P(q) 

0
 2 J 1 qR sin   sin( qL cos  / 2 
sin  d
 qR sin 
qL cos  / 2 

2
J1(x) is the Bessel function of first order and first kind.
28
P(q) cylinder
P(q)  Pcross-section(q) Plarge(q)
L>> R
q
L = 60 Å
120 Å
300 Å
3000 Å
R = 30 Å
29
Glucagon Fibrils
R=29Å
R=16 Å
Cristiano Luis Pinto Oliveira, Manja A. Behrens, Jesper Søndergaard Pedersen, Kurt Erlacher, 30
Daniel Otzen and Jan Skov Pedersen J. Mol. Biol. (2009) 387, 147–161
Primus
31
Form factor for particles with arbitrary shape
Spherical monodisperse particles
n
n
I q   Psphere (q, R )
i 1 j 1
sin qrij 
qrij
32
Monte Carlo integration
in calculation of
form factors for
complex structures
•Generate points in
space by Monte
Carlo simulations
•Select subsets by
geometric constraints
x(i)2+y(i)2+z(i)2 R2
•Caclulate histograms
p(r) functions
•(Include polydispersity)
Sphere
•Fourier transform
MC points
Protein
Micelle
33
ISCOM
vaccine
particle
34
Immune-stimulating complexes: ISCOMs
- Self-assembled vaccine nano-particles
Quil-A
Stained TEM
ISCOMs typically:
60–70wt% Quil-A,
10–15wt% Phospholipids
10-15wt% cholesterol
M T Sanders et al.
Immunology and Cell Biology (2005) 83, 119–128
35
Investigation of the structure of
SAXS data
ISCOM particles by SAXS
S. Manniche, H.B. Madsen, L. Arleth, H.M. Nielsen,
C.L.P. Oliveira, J.S. Pedersen, in preparation.
ISCOMs without toxoid
100
Oscillations  relatively monodisperse
Bump at high q  core-headgroup structure
I(q)
10-1
10-2
10-3
0.01
0.1
q [Å-1]
36
Combination of icosahedrons, tennis balls and footballs
100
Fit result:
Icosah: m M = 0.057
Tennis: m M = 0.770
Football: m M= 0.173
10-1
I(q)
From volumes of cores:
Icosah: M = 0.557 a.u.
Tennis: M = 1.000 a.u.
Football M= 1.616 a.u
10-2
10-3
0.01
0.1
q [Å-1]
Mass distribution:
Icosah: m = 0.104
Tennis: m = 0.786
Football: m = 0.110
37
Icosahedrons, tennis ball, football
Hydrophobic core:
(2 x) 11 Å
Hydrophilic headgroup region: Width 13 Å
242 Å
335 Å
440 Å
38
Pure SDS micelles in buffer:
C > CMC ( 5 mM)
Nagg= 66±1
oblate ellipsoids
R=20.3±0.3 Å
ɛ=0.663±0.005
C  CMC
39
OmpA in DDM
PDB structure for
protein
Monte Carlo points
for DDM
(absolute scale)
40
Full length monomer modelling
using BUNCH program of ATSAS package
9% DDM micelles
background
41
Homologue of periplasmic domain
Flexibility of subunits
42
Using homologue structure of periplasmic domian
Ensemble
Optimization
Method (EOM)
- In ATSAS
Micelles included as
a component
- Always selected
43
Polymer chains in solution
Gigantic ensemble of 3D random flights
- all with different configurations
44
Gaussian polymer chain
Look at two points: Contour separation: L
Spatial separation: r

r
Contribution to scattering:
L
I 2 (q) 
sin( qr )
qr

exp  q 2 Lb / 6
For an ensemble of polymers, points with
L has <r2>=Lb

and r has a Gaussian distribution:
D(r )  exp 
3r 2
2
 2r

 
Add scattering from all pair of points
’Density of points’: (Lo- L)
Lo
L
L
45
Gaussian chains: The calculation

L

L
L
o
o
1
sin(qr )
P(q )  2  dr  dL1  dL2 D(r , | L2  L1 |)
r
qr
Lo 0 0
0
o
1
sin( qr ) 2

dr  dL( Lo  L) D(r , L)
r

Lo 0 0
qr
1

Lo

Lo
 dL( L
o

 L) exp  q 2 Lb / 6

0
2[exp( x)  1  x]
x2
x  R g2 q 2
Rg2 = Lb/6
How does this function look?
46

Polymer scattering
Form factor of Gaussian chain
10
1
q  1 / Rg
P(q)
0.1
q
1

 q 2
0.01
0.001
0.0001
0.1
1
10
100
47
qRg
Gaussian chain+ background
Rg = 21.3 Å
2
 =1.8
-synuclein
Lysozyme in Urea 10 mg/mL
10-1
I(q) [cm-1]
Gaussian chain+ background
Rg = 21.3 Å
10-2
2=1.8
10-3
0.01
0.1
1
-1
q [Å ]
is low!
48
Self avoidence
No excluded volume
=> expansion
No excluded volume
P( x  R g2 q 2 ) 
2[exp( x)  1  x]
x2
no analytical solution!
49
Monte Carlo simulation approach
Pedersen and Schurtenberger 1996
(1) Choose model
(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
(5) Fit experimental data using numerical expressions
for P(q)
P(q,L,b)
L = contour length
b = Kuhn (‘step’) length
50
Expansion = 1.5
Form factor polymer chains
10
q  1 / Rg (Gaussian )
1
Excluded volume chains:
q  1 / Rg (excl. vol.)
q
P(q)
0.1
1 /
 q 1.70
  0.588
0.01
Gaussian chains
q
1

 q 2
q-1 local
stiffness
  1/ 2
0.001
0.0001
0.1
1
10
100
51
qRg (Gaussian)
C16E6 micelles with ‘C16-’SDS
Sommer C, Pedersen JS, Egelhaaf SU,
Cannavaciuolo L, Kohlbrecher J,
Schurtenberger P.
Variation of persistence
length with ionic strength
works also for
polyelectrolytes
like unfolded proteins
52
Models II
Polydispersity: Spherical particles as e.g. vesicles
No interaction effects: Size distribution D(R)
Oligomeric mixture: Discrete particles
d ( q )
 c  2  M i f i Pi (q )
d
i
fi = mass fraction
f
i
i
1
Application to insulin:
Pedersen, Hansen, Bauer (1994).
European Biophysics Journal 23, 379-389.
(Erratum). ibid 23, 227-229.
Used in PRIMUS
‘Oligomers’
53
Concentration effects
54
Concentration effects in protein solutions
-crystallin
eye lens protein
= q/2
55
p(r) by Indirect Fourier Transformation (IFT)
- as you have done by GNOM !
I. Pilz, 1982
At high concentration, the neighborhood is different from the
average further away!
(1) Simple approach: Exclude low q data.
(2) Glatter: Use Generalized Indirect Fourier Transformation (GIFT)
56
Low concentrations
P’(q) =
-
P’(q) = P(q) -  P(q)2
= P(q)[1 -  P(q)]
Zimm 1948 – originally for light scattering
Subtract overlapping configuration
 ~ concentration
Higher order terms:
P’(q) = P(q)[1 -  P(q){1 -  P(q)}]
= P(q)[1 -  P(q)+  2P(q)2]
…
= P(q)[1 -  P(q )+  2P(q)2 -  3P(q)3…..]

P(q)
1  P ( q )
= Random-Phase Approximation (RPA)
57
Zimm approach
I (q)  K
P(q)
1  P ( q )
I (q ) 1  K 1
With P ( q ) 
 1

1  P ( q )
 K 1 

P(q)
 P(q)

1
1  q 2 R g2 / 3

I (q ) 1  K 1 1  q 2 Rg2 / 3  

Plot I(q)1 versus q2 + c and extrapolate to q=0 and c=0 !
58
Kirste and Wunderlich
PS in toluene
Zimm plot
c=0
I(q)-1
P(q)
q=0
q2 + c
59
My suggestion:
( - which includes also information from what follows)
• Minimum 3 concentrations for same system.
• Fit data simultaneously all data sets
I (qi )
Pi

c
1  ca1 exp(qi2 a22 / 3)
with a1, a2, and Pi as fit parameters
60
Osteopontin (M=35 kDa )in water and 10 mM NaCl
2.5, 5 and 10 mg/mL
1.0000
-1
I(q) (cm )
0.1000
0.0100
0.0010
0.0001
0.01
0.1
1
-1
q (A )
61
Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)
Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)
62
63
Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)
64
Shipovskov, Oliveira, Hoffmann, Schauser, Sutherland, Besenbacher, Pedersen (2012)
But now we look at the
information content related to
these effects…
Understand the fundamental processes and principles
governing aggregation and crystallization
Why is the eye lens transparent despite a protein concentration
of 30-40% ?
65
Structure factor
Spherical monodisperse particles
I (q )  V 2  2 P (q )
sin qrjk
qrjk
 V 2  2 P (q ) S (q )
 V 2  2 P (q ) h(rj )
sin qrj
qrj
S(q) is related to the probability distribution function of
inter-particles distances, i.e. the pair correlation function g(r)
66
Correlation function g(r)
g(r) =
Average of the
normalized
density of atoms
in a shell [r ; r+ dr]
from the center of
a particle
67
g(r) and S(q)
q
S (q) 1  n 4  ( g (r )  1)
sin(qr ) 2
r dr
qr
68
GIFT
Glatter: Generalized Indirect Fourier Transformation (GIFT)
I (q)  4  p (r )
sin(qr )
dr
qr
With concentration effects
I (q)  S eff (q, , R,  ( R ), Z , csalt )4  p(r )
sin(qr )
dr
qr
Optimized by constrained non-linear least-squares method
- works well for globular models and provides p(r)
69
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
Home-written
software
6.4 mg/ml
p(r) [arb. u.]
I(q) [arb. u.]
10.7 mg/ml
Hexamers
5.1 mg/ml
trimers
monomers
2.4 mg/ml
1.0 mg/ml
0
0,01
0,1
10
20
30
40
50
60
70
r [Å]
-1
q [Å ]
70
Osmometry (Second virial coeff A2)
/c
Ideal gas A2 = 0
c
71
S(q), virial expansion and Zimm
From statistical mechanics…:
1
1
  
S (q  0)  RT 
 
1  2cMA2  3c 2 MA3  ...
 c 
 In Zimm approach  = 2cMA2
72
Isoelectric point
A2 in lysozyme solutions
A2
O. D. Velev, E. W. Kaler,
and A. M. Lenhoff
73
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 sphere
Debye-Hückel
screened Coulomb potential
74
Theory for colloidal stability
DLVO theory: (Derjaguin-Landau-Vewey-Overbeek)
Debye-Hückel
screened Coulomb potential
+ attractive interaction
75
Depletion interactions


Asakura & Oosawa, 5876
Integral equation theory
Relate g(r) [or S(q)] to V(r)
At low concentration g(r) = exp( V (r) / kBT) 1  V (r) / kBT
Boltzmann approximation (weak interactions)
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]
c(r) = direct correlation function
S(q) = 1 n c(q) – n2 c(q)2 – n3 c(q)3  …. =
1
1  nc(q )
 but we still need to relate c(r) to V(r) !!!
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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)
78
-crystallin
S. Finet, A. Tardieu
DLVO potential
HNC and numerical solution
79
-crystallin + PEG 8000
S. Finet, A. Tardieu
Depletion interactions:
DLVO potential
HNC and numerical solution
40 mg/ml -crystallin
solution pH 6.8,
150 mM ionic strength.
80
Anisotropy
Small: decoupling approximation (Kotlarchyk and Chen,1984) :
Measured structure factor:
 (q )

S meas (q ) 
 1   (q )S (q )  1  S (q ) !!!!!
2
n P(q )
81
Large anisotropy
Polymers, cylinders…
Anisotropy, large: Random-Phase Approximation (RPA):
d
P(q)
(q )  n 2V 2
d
1  P ( q )
 ~ concentration
Anisotropy, large: Polymer Reference Interaction Site Model (PRISM)
Integral equation theory – equivalent site approximation
d
P(q)
(q) n 2V 2
d
1  nc(q) P(q)
c(q) direct correlation function
related to FT of V(r)
82
Empirical PRISM expression
SDS micelle in 1 M NaBr
d
P(q)
(q)  n 2V 2
d
1  nc(q) P(q)
c(q) = rod formfactor
- empirical from MC simulation
Arleth, Bergström and Pedersen
83
Overview: Available Structure factors
1) Hard-sphere potential: Percus-Yevick approximation
2) Sticky hard-sphere potential: Percus-Yevick approximation
3) 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 approximation
5) Sticky hard-sphere potential, polydisperse system: Percus-Yevick approximation
6) 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’:
84
Summary
• 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
• Concentration effects and structure factors
Zimm approach
Spherical particles
Elongated particles (approximations)
Polymers
85
Literature
Jan 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 Pedersen
Monte Carlo Simulation Techniques Applied in the Analysis of Small-Angle
Scattering Data from Colloids and Polymer Systems
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 381
Jan Skov Pedersen
Modelling of Small-Angle Scattering Data from Colloids and Polymer Systems
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 391
Rudolf Klein
Interacting Colloidal Suspensions
in Neutrons, X-Rays and Light
P. Lindner and Th. Zemb (Editors) 2002 Elsevier Science B.V.
p. 351
86
MC applications in modelling
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.
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