Modeling Macromolecular Complexes: A Journey Across Scales - Modeling in Computational Biology and Biomedicine - page 13

1.2.1.3 Geometrical Modeling

Many molecular dynamics simulations can be performed without any light being
shed on questions such as why one type of conformational change predominates
over another or why two flexible proteins associate in a particular way. But the
results of these blind calculations can of course be used to arrive at new hypotheses.
Geometrical reasoning becomes important once again as we wish to model the
nature of the macromolecular complex interfaces and their organization from a dif-
ferent perspective, that of the 3D shapes involved. Such modeling does not require
intensive conformational sampling, but complements the simulation methods that
provide it. It is usually based on an experimental structure, which is used as the de
facto reference point for subsequent geometrical and biophysical reasoning.
Most analyses start with a space-filling representation of the macromolecule
itself. High resolution, atomic-level models can be made using the van der Waals
model. While an atom nucleus, composed of protons and neutrons, is only a few
femtometers across, the atom dimensions are on the order of A. This reflects the
region of space around the nucleus in which there is a high probability for finding
the electrons. In the van der Waals model this region is represented by a ball. Two
non-bonded atoms can only get so close to each other before the interaction of their
electron clouds engenders a prohibitive repulsive energy. This can be used to define
their atomic radii (vdW radii), which can be measured experimentally by deviations
from ideal gas behavior or from actual atom separations seen in small-molecule
crystals [ 52 ].
Lower resolution representations are also used in many studies, including
embedding the molecule in a grid [ 47 ] or defining its shape by a sum of spherical
harmonics [ 57 ]. Methods based on the Voronoı diagrams offer alternatives that
provide a different attribution of regions of space to the atoms or residues of the
protein. They can be useful in describing the protein's 3D properties, and can be
extended to the characterisation of macromolecular interfaces themselves.
We change speed now as we explore Voronoı diagrams and related constructions
in detail.

1.2.2 Affine Voronoı Diagrams and α -Shapes

Vo r o n oı diagrams and spatial partitions. Consider a set of sites (points, spheres,
polygons, etc) in 3D, and a generalized distance to these sites. In simple terms,
the Voronoı diagram is the partition of the space into Voronoı regions, defined as
follows: the Voronoı region of a site consists of the points in space having this site
as nearest neighbor for the distance considered. The most classical Vorono¨ıdiagram
is certainly that of points under the Euclidean distance. But atoms have unequal size,
as the van der Waals radii of the atoms found in bio-molecules range from 1 ˚ Afor
hydrogen to 2 A for sulfur. (We note in passing that since hydrogen atoms are not
often reported in crystal structures, modeling may be carried out using the so-called