Arrangements of circles on a sphere : Algorithms and applications to molecular models represented by a union of balls

Sebastien Loriot 1, 2, 3
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
3 ABS - Algorithms, Biology, Structure
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Since the early work of Richard et al., geometric constructions have
been paramount for the description of macromolecules and macro-molecular
assemblies. In particular, Voronoï and related constructions have been
used to describe the packing properties of atoms, to compute molecular
surfaces, to find cavities. This thesis falls in this realm, and
after a brief introduction to protein structure, makes four
contributions.

First, using the sweep line paradigm of Bentley and Ottmann, we
present the first effective algorithm able to construct the exact
arrangement of circles on a sphere. Moreover, assuming the circles
stem from the intersection between spheres, we present a strategy to report
the covering list of a face of the arrangement---that is the list of
spheres covering it. Along the way, we ascertain the fact that
exactness of the arrangement can be achieved with a small
computational overhead.

Second, we develop the algebraic and geometric primitives required by the sweep
algorithm, so as to make it generic and robust. These primitives are integrated in
a broader context, namely the CGAL 3D Spherical Kernel.

Third, we use the aforementioned machinery to tackle a computational structural
biology problem,
namely the selection of diverse conformations from a large redundant set.
We propose to solve this selection problem by computing
representatives maximizing the surface area or the volume of the
selection. From a geometric standpoint, these questions can be handled
resorting to arrangements of circles and spheres.
The validation is carried out along two lines. On the geometric side,
we show that our selections match the molecular surface area of
selections output by standard strategies but using a smaller number
of conformers by one and two orders of magnitude. On
the docking side, we show that our selections can significantly
improve the results obtained for a flexible-loop docking algorithm.

Finally, we discuss the implementation issues and the design choices,
in the context of the best practices underlying the development of
CGAL.
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Sebastien Loriot. Arrangements of circles on a sphere : Algorithms and applications to molecular models represented by a union of balls. Mathematics [math]. Université de Bourgogne, 2008. English. ⟨tel-00345002⟩

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