Computation of solution
scattering from atomic models
D.I. Svergun
C.Barberato, M.Malfois, M.Koch
Plan of the talk
• Introduction
• Basic ideas and formulae
• Computational programs
• Influence of hydration shell
• Conclusions
The use of high resolution models
Theoretical model available
Validation
Crystal structure available
Structure of subunits available
The same in solution?
Model of the complex
Computation of solution scattering
from atomic models permits:
• To validate theoretically predicted models
• To analyse similarities between macromolecules in
solution and in the crystal
• To model the quaternary structure of multisubunit
particles by rigid body refinement against
scattering data
Solvent scattering and contrast
Isolution(s)
Isolvent (s)
Iparticle(s)
♦
To obtain scattering from the particles, solvent
scattering must be subtracted to yield effective density
distribution ∆ρ = <ρ(r) - ρs>, where ρs is the scattering
density of the solvent
♦
Further, the bound solvent density may differ from
that of the bulk
Scattering from a macromolecule in solution
I(s) = A( s)
2
Ω
= A( s) − ρs E(s) + δρ b B(s)
2
Ω
♦ A(s): atomic scattering in vacuum
♦ E(s): scattering from the excluded
volume
♦B(s): scattering from the hydration
shell
The use of multipole expansion
I(s) = A( s)
2
Ω
= Aa (s) − ρs E(s) + δρ b B(s )
2
Ω
If the intensity of each contribution is
represented using spherical harmonics
I ( s) = 2π
2
∞
l
å å
l =0
m=− l
Alm ( s)
2
the average is performed analytically:
I ( s) = 2π
L
2
l
ååA
lm
( s) − ρ0 Elm ( s) + δρBlm ( s )
2
l =0 m =− l
This approach permits to further use rapid
algorithms for rigid body refinement
Atomic scattering in vacuum
I a ( s) = 2π
2
∞
l
å å
l =0
m=− l
Alm ( s )
2
• The partial amplitudes of the atomic scattering are
computed by summing up scattering contributions from
atoms or atomic groups
∞
l
é
ù
l
*
Alm ( s ) = 4π i å ê f j ( s )å å jl ( sr)Ylm (ω )ú
j ë
l =0 m=−l
û
• For the models taken from PDB, the atomic groups are
the form factors of the atoms having covalently bound
hydrogens (CH, CH2, NH etc)
f j ( s) =
N
N
åå fi ( s ) f k ( s )
i =1 k =1
sin d ik s
d ik s
Notes on atomic scattering
• One should never use Debye-Waller
factors [exp(-Bs2/16π2)] contained
in the PDB files!
lg I
8
7
• One should use crystallographic
waters with caution – better not
use them!
Atomic scattering
Simulated/averaged
Using Debye factor
6
5
4
3
2
0.0
0.2
0.4
0.6
0.8
1.0
s
1.2
1.4
• For neutrons in D2O-containing
solvents, H/D exchange must be
taken into account. Usually, protons
in hydrophilic groups are assumed
to be substituted by deuterons with
the probability P=FD2O
where
P=FD2O is the D2O concentration.
• Scattering from lysozyme with thermal
displacements of atoms 0.1 nm (B=6.2 nm2)
Scattering from the excluded volume
I e ( s ) = 2π
2
∞
l
å å
l =0
m=−l
Elm ( s )
2
• The partial amplitudes of the excluded volume scattering are
computed by summing up contributions from dummy solvent
atoms, positioned at the centers of the atomic groups (socalled effective atomic factors method)
∞
l
é
ù
*
l
Elm ( s ) = 4π i å ê g j ( s )å å jl ( sr )Ylm (ω )ú
û
j ë
l = 0 m= − l
• The form factors are those of the gaussian spheres
g j ( s) = G( s)V j exp( −πs 2V j2 / 3 )
• where Vj =(4π/3) r3j is the solvent volume displaced by the
atom or atomic group (may be varied to allow for uncertainty
in the total excluded volume)
Notes on excluded volume scattering
• There are many different methods to compute
scattering from the excluded volume.
• The ‘cube method’ (Luzzati et al, 1972; Fedorov and
Pavlov, 1983; Müller, 1983) is claimed to be superior
over the effective atomic factors method at higher
angles. The argument is that the cube method
ensures uniform filling of the excluded volume. This
argument does not hold as the solvent is not a
uniform continuum!
• The effective atomic factors method seems to be a
reasonable representation of the solvent scattering
Scattering from the hydration shell
I b ( s ) = 2π
2
∞
l
å å
l =0
m=− l
Blm ( s )
2
• The hydration shell is represented by an angular
envelope function parameterised with spherical
harmonics
L
l
F (ω ) = å å f lmYlm (ω )
l =0
m= − l
ì0, 0 < r < F (ω )
ï
ρ (r ) = í1, F (ω ) ≤ r < F (ω ) + ∆
ï0 , r ≥ F (ω ) + ∆
î
∆ = 0.3 nm
• The partial amplitudes Blm(s) are computed as in
shape determination routines
Notes on scattering from the hydration shell
• The use of an angular envelope effectively
represents the outer envelope of the particle only.
The water in internal cavities is thus assumed to
have the same density as the solvent. This is, of
course, a simplification.
• There are other methods to add the hydration
layer, e.g. by generating virtual water atoms using
MD simulations or by adding a layer on the surface
(see program CRYDAM).
• In all cases, one considers only most ordered
waters (the first shell with thickness about 0.3
nm).
CRYSOL and CRYSON:: X-ray and neutron
scattering from proteins
♦ Atomic scattering in vacuum: accounts
for covalently bound hydrogens
♦ Scattering from the excluded volume:
effective atomic factors
♦ Hydration shell: particle is surrounded
by an angular envelope
♦ A 0.3 nm thick border layer is built
around the envelope
Svergun, D.I., Barberato, C. & Koch, M.H.J. (1995). J. Appl. Crystallogr.
28, 768-773.
Svergun, D.I., Richards, S., Koch, M.H.J., Sayers, Z., Kuprin, S. &
Zaccai, G. (1998). Proc. Natl. Acad. Sci. USA, 95, 2267-2272.
CRYSOL and CRYSON:: X-ray and neutron
scattering from macromolecules
I(s) = A(s) − ρs E(s) + δρ b B(s)
• The programs:
2
Ω
– either fit the experimental data by varying the density
of the hydration layer δρ (affects the third term) and
the total excluded volume (affects the second term)
– or predict the scattering from the atomic structure
using default parameters (theoretical excluded volume
and bound solvent density of 1.1 g/cm3 )
– provide output files (scattering amplitudes) for rigid
body refinement routines
– compute particle envelope function F(ω )
Scattering components (lysozyme)
1)
2)
3)
4)
Atomic
Shape
Border
Difference
lg I, relative
Effect of the hydration shell, X-rays
Experimental data
Fit with shell
Fit without shell
3
Lysozyme
2
Hexokinase
1
EPT
0
PPase
-1
0
1
2
3
s, nm
-1
4
Denser shell or floppy chains:
X-rays versus neutrons
Scattering length density, 1010cm-2
12
solvent density
denser solvent layer
10
floppy side chains
protein density
8
♦ Floppy chains would in
all cases increase the
apparent particle size
♦ Neutrons in H2O (shell):
particle would appear
nearly unchanged
6
4
♦ Neutrons in D2O (shell):
particle would appear
smaller than the atomic
model
2
0
SAXS
-2
♦ For X-rays: both lead to
similar effect (particle
appears larger)
SANS in H2O
SANS in D2O
X-rays versus neutrons: experiment
lg I, relative
lg I, relative
Neutrons, D 2O
Neutrons, H 2O
X-rays
1
-1
0
X-rays
-1
Neutrons, H2O
-2
-2
Neutrons, D2O
-3
0
1
2
3
s, nm -1
Lysozyme: appears larger for X-rays
and smaller for neutrons in D2O
0
2
s, nm-1
Thioredoxine reductase : CRYSOL
and CRYSON fits with denser shell
4
Program CRYDAM
lg I, relative
♦ Represents hydration
shell by dummy water
atoms
♦ Computes X-ray and
neutron scattering profiles
2
♦ Handles proteins,
carbohydrates, nucleic
acids and their complexes
X-ray data, lysozyme
Fit by CRYSOL
Fit by CRYDAM
1
♦ Is applicable for wide
angle scattering range
Malfois, M. & Svergun, D.I.
(2001), to be submitted
0
5
10
s, nm
-1
Conclusions
• X-rays and neutrons “see” particles in solution
along with their hydration shells
• Accounting for hydration shell is indispensable
in computation of scattering from atomic
models, and the density of the bound solvent
is on average about 10 larger than that of the
bulk
• Some of the crystallographic notions (like
Debye-Waller factor) are not applicable in
solution scattering