Chemistry Reference
In-Depth Information
Other geometrical descriptors are the volume and the area of the parallelepiped,
sphere, and the ellipsoid circumscribed to the analyzed molecule. The area of the
ellipsoid can be calculated only by numerical methods because there is no analytical
formula for its calculation.
The molecular volume and area of molecular surface are two geometrical descrip-
tors used by all programs for QSAR calculations.
Molecules could be considered ensembles of rigid spheres that overlap, and in this
conceptualization, the calculation of the volume of the molecules using geometrical
methods would be simple. The atomic radius that should be considered is the van
der Waals atomic radius. After geometry optimization, the position of the center of
the spheres is known, and the van der Waals radii of the atoms can be determined
using physical methods (Bondi 1966). Consequently, the volume of the region of
interpenetration of the atomic spheres could be calculated. This volume would be
subtracted later from the total volume of the spheres. But the atomic spheres are
elastic and deformable, and their overlapping is negligible. Therefore, the values of
molecular volumes, calculated through geometrical methods, are significantly dif-
ferent from those obtained by physical measurements. This being the case, they can
be considered wrong.
To calculate the molecular volume, statistical methods that involve the following
steps are usually employed:
•
An easily calculated virtual body (such as parallelepiped, sphere, ellipsoid)
circumscribed around the analyzed molecule is generated, and a body with
volume V is estimated.
•
Inside the circumscribed body, a large number of N points are generated
ra ndom ly.
•
n
points within the molecule are identified and counted (within a certain
atomic sphere).
•
The volume
v
of the molecule is calculated with formula (4.21).
v
=
V
·
n
/
N
(4.21)
The molecular surface is calculated through similar methods. Points are randomly
generated on the surface of each atomic sphere. A certain percentage of the points
generated on the surface of a certain atomic sphere is situated outside of any other
atomic spheres. Thereby, for each atomic sphere, the “exposed” van de Waals sur-
face is determined. The total van der Waals surface will be the sum of the exposed
surfaces. The PCModel program computes for the guanine isomer A in
Figure 4.4
,
Surface Area = 175.4 Å
2
and Volume = 156.1 Å
3
, and for guanine isomer B, Surface
Area = 173.9 Å
2
, Volume = 155.2 Å
3
.
The shadow areas of molecules (Rohrbaugh and Jurs 1987) are the projections
of the molecules on the three planes obtained after the rotation of the molecule in a
standard position. Because of the irregular shape of these projections, their surface
is calculated using numerical methods. This descriptor is a measure of the size of
the molecule as seen from a certain direction. The comparison of the surfaces of the
projections is another way of assessing the deviation from the spherical shape.
