Biomedical Engineering Reference
In-Depth Information
1.2.3
Molecular Surfaces and Volumes
Assume that the balls in
represent a van der Waals model. The van der Waals
surface is clearly defined as the boundary
∂F
S
of the union of these balls. From a
modeling perspective, this surface raises two difficulties. The first is related to the
fact that most macromolecules are found in an aqueous environment, so that a vdW
model generally delimits a number of tiny cavities which are not accessible to sol-
vent molecules—recall our discussion of electrostatics in Sect.
1.2.1.2
. The second
stems from the fact that non covalent interactions account for the structure of macro-
molecular complexes. Given that the distance between non-covalently bonded atoms
is strictly larger than the sum of their vdW radii, a vdW model does not inform
us about such contacts in a complex—atoms from two chains in contact do not
intersect.
Both problems are solved resorting in the Solvent Accessible Model, which
consists of expanding the atomic radii, thus mimicking a continuous layer of solvent
molecules. This fills meaningless cavities and recovers contacts between interacting
atoms in a complex. More precisely, let
W
be a water probe, i.e. a ball representing
a water molecule, and denote
r
w
its radius. (Note that this is a coarse representation
focused on the oxygen atom of the water molecule; neither the hydrogen atoms nor
the polarity of the molecule are represented.) To define a SAS model from a vdW
model, one rolls the probe
W
over the vdW surface, tracing the locii of points visited
by the center of
W
. Equivalently, the SAS surface is defined as the boundary of the
union of the expanded balls
,seeFig.
1.2
a. An atom contributing
to this surface is called
accessible
,and
buried
if not. A typical value for
r
w
is 1.4 A.
As just discussed, both the vdW and the SAS surfaces are defined as the boundary
of a collection of balls. Moreover, as already seen with Eq. (
1.3
), the description
of such a surface as well as its enclosed volume only require computing the
restrictions of balls. The information to compute the geometry of restriction is
actually contained in the
α
-complex for
α
=0[
2
].
For the boundary
∂F
{B
i
(
a
i
,r
i
+
r
w
)
}
of the union, one actually builds from the 0-complex a
representation of
∂F
which is a cell complex. Its 2-cells are spherical polygons,
also called caps; if two such cells intersect, they share a circle arc supported by the
intersection circle of their defining spheres. Its 1-cells are circle arcs; if two such
arcs intersect, they share one vertex defined by the intersection of (generically) three
spheres. To represent the volume
using its partitioning into restrictions, following
Eq. (
1.3
), one actually resorts to a tiling of each restriction
R
i
using two types of
pyramids depicted on Fig.
1.2
b. In particular, adding up the (signed) volumes of such
pyramids allows one to compute the volume of
R
i
and thus of
F
with a controlled
accuracy. The proofs can be found in [
14
], and the corresponding program, Vorlume,
is available at
http://cgal.inria.fr/abs/Vorlume/
.
F
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