Biomedical Engineering Reference
In-Depth Information
F
α
=
∪
i
B
i
[
α
] called the
space-filling
diagram
. It is easily checked that the Voronoı diagram of the grown balls
These grown balls define the domain
S
α
. The restriction of a grown ball is defined as
R
i
[
α
]=
B
i
[
α
]
∩
Vo r(
B
i
), and these restrictions also partition
S
matches that of
F
α
, as specified by Eq. (
1.3
) mutatis
mutandis. Restrictions can be used to define the analogue of Eq. (
1.2
), resulting
in a simplicial complex called the
α
-complex, which is a subset of the Delaunay
triangulation Del(
S
) [
23
]. More precisely, given a set of restrictions identified by
their indices
I
=
{i
0
,...,i
k
}
, one (generically) finds the corresponding
k
-simplex
in the
α
-complex
K
α
provided that the following holds:
∩
i∈I
=
{i
0
,...,i
k
}
R
i
[
α
] =
∅.
(1.5)
The domain covered by the simplices of
K
α
is called the
α
-shape
, and is denoted
W
α
. Note that in increasing
α
, the restrictions associated with a given ball are
nested, and so are the
α
-complexes. The finite set of distinct
α
-complexes is called
the
α
-complex
filtration
(Fig.
1.3
a, b).
Practically, the computation of this filtration is non-trivial, and the only robust
software we are aware of the Alpha shape 3 package of CGAL.
α
-complex and topological features: cavities and tunnels.
The quantities defi-
ned so far are atom-centric, in the sense where they provide information on a given
atom and its neighbors. Remarkably, selected features of the
α
-complex also encode
global features of the molecule. This is illustrated on Fig.
1.3
b, where the cavity in
the middle of the seven atoms is mirrored by the cavity delimited by the edges and
triangles of the
α
-complex—for
α
=0in this case since balls have not been grown.
To make a precise statement, the
α
-shape
W
α
and the space-filling diagram
F
α
have
the same homotopy type [
24
].
In molecular modeling, two global topological features of utmost interest
are cavities and tunnels. Mathematically, such features are defined in terms of
generators of so-called homology groups, and efficient algorithms exist to compute
them for collections of balls. Practically, cavities found in the interior of a macro-
molecule are important since they may contain small or solvent molecules. As for
tunnels, also called channels, they typically provide passages from the bulk solvent
to an active site.
α
-complex and multi-scale analysis.
As mentioned above, real data are often
plagued with uncertainties, and the question arises to decide whether a particular
feature is noise or not. This can be done with
α
-shapes as follows. Upon growing
atoms as specified by Eq. (
1.4
), topological features (connected components, tunnels
and cavities) appear and disappear. For example, in moving from the situation of
Fig.
1.3
btothatofFig.
1.3
c, the inner void disappears. In particular, one can define
the stability of a feature as its life-time interval, in
α
units, a piece of information
easily computed from the
α
-complex filtration. (Notice, though, that the growth
model consisting of adding
α
to the squared radius does not have a straightforward
physical motivation.)
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