## Section: New Results

### Applications

#### Design of interferometric telescopes

Participants : Jean-Daniel Boissonnat, Trung Nguyen.

In collaboration with P. Blanc and F. Falzon (Alcatel Alenia Space)

Given a disk O in the plane called the objective, we want to find
n small disks P_{1}, ..., P_{n} called the pupils such that
, where denotes
the Minkowski difference operator, while minimizing the number of
pupils, the sum of the radii or the total area of the pupils. This
problem is motivated by the construction of very large telescopes from
several smaller ones by so-called Optical Aperture Synthesis. In this
paper [42] , we provide exact, approximation and heuristic solutions to
several variations of the problem.

#### Revisiting the Voronoi description of protein-protein interfaces

Participant : Frédéric Cazals.

In collaboration with F. Proust, R. Bahadur(CNRS, Orsay) and J. Janin (CNRS, Orsay).

This paper [23] develops a model of macromolecular interfaces based on the Voronoi diagram and the related alpha-complex, and we test its properties on a set of 96 protein-protein complexes taken from the Protein Data Bank. The Voronoi model provides a natural definition of the interfaces, and it yields values of the number of interface atoms and of the interface area that have excellent correlation coefficients with those of the classical model based on solvent accessibility. Nevertheless, some atoms that do not loose solvent accessibility are part of the interface defined by the Voronoi model. The Voronoi model provides robust definitions of the curvature and of the connectivity of the interfaces, and leads to estimates of these features that generally agree with other approaches. Our implementation of the model allows an analysis of protein-water contacts that highlights the contacts of structural water molecules at protein-protein interfaces.

#### Geometric, topological and contact analysis of interfaces in macro-molecular complexes

Participant : Frédéric Cazals.

Understanding the *sociology* of interactions between the proteins
encoded in a genome is a central question of structural biology, and
interface models between molecules forming a complex are instrumental
in this perspective.
Qualifying interface atoms as atoms loosing solvent accessibility in
the complex, or pairs of atoms within a distance threshold, several
interface models have been proposed. Yet, until recently, no interface
model existed to answer coherently (if at all) the following
questions:
can one bridge the gap from atoms loosing solvent accessibility to
interface pairs? is the interface flat or curvy? is it connected or
not (does it have a multi-patch structure)? is a connected component
of the interface simply connected or not (does it have a hole)? what
is precisely the role played by interface structural water?

Using the -complex of the Van der Waals balls, a construction
derived from the Voronoi diagram, we designed such an interface
model, and validated it on the usual database of co-crystallized
protein-protein complexes.
This paper [56] is a methodological contribution
aiming at easing the access of the interface model to structural
biologists. As such, the following topics are covered:
(i) the geometric principles underlying the interface model
(ii) the definitions of the interface and its extension to
accommodate structural water
(iii) the statistics one can compute from the interface model
(iv) the Software *Intervor* and the associated web site.
These presentations are accompanied by illustrations and insights
on protein - protein complexes.