Biomedical Engineering Reference
In-Depth Information
P
3
P
3
P
1
P
1
P
2
P
2
Fig. 1.9
From probability density maps to toleranced models. (
a
) Three conformations of three
flexible molecules defining a fictitious assembly, and a probability density map whose color
indicates the probability of a given point to be covered by a random conformation—from low
(
black pixels
)tohigh(
white pixels
) probabilities. (
b
) A toleranced model, where each toleranced
molecule consists of a set of pairs of concentric balls, inner and outer. Note that the three molecules
have been assigned to two groups corresponding to
red
and
blue
molecules, thus defining a bicolor
toleranced model
the so-called bicolor setting, which consists of segregating the
p
protein types into
two families: the red and the blue groups. Typically, the red group will refer to the
protein types involved in a TAP experiment or to those involved in a sub-complex.
Toleranced models.
Let a
toleranced ball
B
i
(
c
i
;
r
i
,r
i
) be a pair of concentric
balls centered at
c
i
,the
inner
and
outer
balls of radii
r
i
>r
i
, respectively.
Inner and outer balls are respectively meant to encode high confidence regions and
uncertain regions in a map. In order to deal with balls of intermediate size, we
introduce a parameter
λ>
0 governing a
growth process
co
nsi
sting of linearly
interpolating and extrapolating the radii. That is, the
grown ball
B
i
[
λ
] stands for the
ball centered at
c
i
and of radius:
r
i
(
λ
)=
r
i
+
λ
(
r
i
− r
i
)
.
(1.7)
Note that for
λ
=0(resp.
λ
=1), the grown ball matches the inner (resp. outer) ball.
We d e fi n e a
toleranced protein
as a collection of toleranced balls, and a
toleranced
assembly
as a collection of t
ole
ranced proteins. For a given value of
λ
, a protein of
intermediate size is den
oted
P
j
[
λ
],
and
F
λ
denotes the domain of the space-filling
diagram, that is
F
λ
=
∪
i
B
i
[
λ
]=
∪
j
P
j
[
λ
].Forafixed
λ
, the topology of the domain
F
λ
is of utmost interest: a connected component of this domain is called a
complex
,
and the domain is called a
mixture
if it involves several complexes.
Similarlytothe
α
-shapes of Sect.
1.2.2
, a toleranced model defines a 1-parameter
family of shapes, except that the linear interpolation of the radius specified by
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