Biomedical Engineering Reference
In-Depth Information
1.3.4
Building a Toleranced Model from Probability Density
Maps
Now consider a density map as specified in Sect. 1.3.1.3 , and assume that we wish
to create a number of protein instances equal to the stoichiometry of that protein
type in the NPC. A simple solution consists of the following three-stage process.
First, we allocate occupancy volumes to the protein instances. This step consists
of collecting voxels in such a way that the volume covered by these voxels
matches the estimated volume of all instances, namely
Vol ref multiplied by the
stoichiometry. These voxels are collected by a greedy region-growing strategy
that favors large values of the probability. That is, starting from local minima, a
priority queue of candidate voxels based is maintained. These candidate voxels are
those incident to the voxels already recruited, and they are sorted by decreasing
probability. The voxel with top density is added, and the process halts when
the aforementioned volume criterion is met. Second, we compute a canonical
representation involving 18 toleranced balls for each instance. This number allows
the construction of the four regular morphologies represented on Fig. 1.13 . Consider
an occupancy volume to be covered with 18 toleranced balls of identical radius.
Using a principal components analysis, each volume is assigned one of the four
canonical arrangements of Fig. 1.13 , which correspond to a shape that is roughly
isotropic, flat, semi-linear or linear. Finally, we set the inner and outer radii. For a
given protein type, the inner radius is set so that the volume of the union of the 18
inner balls matches the estimated volume of the protein
Vol ref . The specification of
the outer radius relies on the fact that the probability density maps of large proteins
tend to be more accurate than those of small proteins, a feature likely related to the
higher mobility of the latter. Therefore, r i
is set such that the discrepancy r i
− r i
is proportional to α/r i :
α
r i
r i
+ r i .
=
(1.12)
This formula actually entails that the Hasse diagram representing the evolution of
skeleton graphs depends only on the inner radii
{r i }
, but not on the parameter α .
We arbitrarily set α =10and compute the whole λ -complex of the toleranced
model.
As discussed in Sect. 1.3.2 , the growth process is controlled by the volume ratio
of Eq. ( 1.11 ); that is, it is stopped at λ = λ max such that r λ max 5.
1.3.5
Success Stories
We now develop some insights provided by toleranced models of the NPC [ 13 ].
Qualifying the contact between protein types. Given two protein types p i and p j ,
and a stoichiometry k ≥ 1, we wish to qualify the contacts between the instances
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