Modeling Macromolecular Complexes: A Journey Across Scales - Modeling in Computational Biology and Biomedicine - page 27

• TAP data can be used to enforce proximity, i.e., a distance relationship between

proteins participating in a given complex. As a protein is modeled by balls, a
distance restraint entails that two balls from the two proteins must be within a
distance threshold.

• Positional information gathered using immuno-EM can be used to enforce the

location of specific protein instances within a prescribed region of the model.

That is to say, using one restraint R exp i for each type of experimental data, the
reconstruction process aims at finding the model M minimizing the following
penalty function:

F ( M )= R exp i ( M ) .

(1.6)

All experiments exp i

For the NPC, which consists of n = 456 instances of p =30protein types, a
maximum of 1,848 balls (of fixed radius) have been used, whence an optimization
problem in a space of dimension 3 × 1 , 848 = 5 , 544. This problem being non
convex, local minima were sought using an optimization strategy based on simulated
annealing and coarse-grain molecular dynamics, from which N =1 , 000 plausible
configurations were singled out [ 3 ].

Output of the reconstruction. To interpret these N structures selected, a prob-
ability density map was created per protein type, by collecting all instances of
that protein type across the N models and blending the corresponding balls.
The probability density map is a 3D matrix, each voxel being endowed with the
probability of it being contained within an instance of that type. (Note that such a
map should not be confused with a cryoEM map which encodes a density of matter.)
Merging back all the probability densities yields a probabilistic model of the whole
NPC, which is illustrated by the contour plot of Fig. 1.8 a.
In a sense, the uncertainties in the various input data account for ambiguities
in the shape and position of the proteins encoded in the density map(s). In what
follows, we present a panoply of tools allowing one to make quantitative statements
from such ambiguous models.

1.3.2 Toleranced Models and Curved Voronoı Diagrams

We wish to accommodate ambiguous protein models within a probability density
map. To see which difficulties are faced, consider the fictitious map of Fig. 1.9 ,
which corresponds to a fictitious complex involving three molecules of three balls
each. The color coding clearly indicates that some regions of the map are much
more likely than others. Using balls to model the proteins contained within such a
map would face ambiguities regarding the locations and the radii of the balls. To
ease the process, we introduce toleranced models and explain their connexion to
curved Vorono ¨ ıdiagrams[ 12 ]. Recalling that the NPC consists of about 456 protein
instances of p =30protein types, we shall in particular use toleranced models in