Biomedical Engineering Reference
In-Depth Information
M
M
F i ( q ) F j ( q ) sin( q
·
r ij )
I ( q )=
(1)
q
·
r ij
i =1
j =1
where F i and F j are the scattering form factors of the individual particles i and j ,
and r ij is the Euclidean distance between them. The summations run over all the M
scattering particles.
2.2
Coarse-Grained Protein Models
If some scatterers are fixed relative to each other, they can be approximated by a single
large scattering body ( dummy body ). This approximation, more precise at low q , fades
with the progression of the scattering angle up to a resolution equal to the scattering
diameter of the dummy body. We found that the amino acids constituent to the pro-
tein chain can be approximated by one or two large bodies ( dummy atoms ), and that
this approximation holds up to scattering angles normally not measured in the current
experimental standards [15].
In the two body model, the amino acids are individually represented by two dummy
atoms; one representing the backbone, and the other representing the side chain. Glycine
and alanine, lacking a side chain with conformational freedom, are represented by a sin-
gle dummy atom. The dummy atoms are placed at the respective centers of mass (see
Fig. 1). A total of 21 form factors need to be estimated for the two body model: one
for alanine, one for glycine, one for the generic backbone and 18 for the remaining side
chains.
For the one body model, the single dummy atom is placed at the center of mass
of the amino acid. Hence, 20 form factors need to be estimated; one for each amino
acid type. For a given protein, the one body model allows to represent the molecule
with roughly half the number of scattering bodies employed in the two body model.
If the experimental data is recorded at low resolutions only, the former is thus clearly
preferable for reasons of computational efficiency.
2.3
Form Factor Descriptors
Due to the lack of publicly available high-quality experimental data needed for the
estimation of the form factors, artificial data curves were generated for known high-
resolution protein structures using the state-of-the-art program CRYSOL [21]. This pro-
gram computes the theoretical scattering curve from a given full-atom structure using
spherical harmonics expansions, therefore limiting its applications at studying compact
quasi-globular proteins. We can however use this input to feed a learning protocol, and
make use of the Debye formula in eq. 1 to overcome structural assumptions.
Therefore, a large scale Monte Carlo simulation has been conducted to estimate the
values of the form factors of the dummy atoms [15]. The resulting profiles for these
descriptors are shown in Fig. 2.
In Fig. 3 we show a SAXS curve generated with our method, and the theoretical
scattering computed by CRYSOL as a reference.
 
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