Volume 1 Chapter 7 The Geometric Basis of 3D Searching
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In describing 3D molecular structures, it is natural to think in terms of the
standard geometrical parameters, viz. bond lengths, valence angles and torsion
angles, ie. parameters which are calculated directly from the coordinates of 2,
3 or 4 bonded atoms.
However, in describing the spatial arrangements of more general patterns of
atoms, we can extend these descriptors to include general distances, angles and
torsions derived from the coordinates of 2, 3 or 4 atoms which may not be
In these cases, the atom is the key geometric object from
which the parameters are calculated.
For many years, structural chemists (and crystallographers in particular) have
used many non-standard geometrical parameters to describe specific structural
features. Calculation of these parameters often requires the use of geometric
objects which are more complex than the individual atom positions.
Examples, illustrated below, are:
- distances, angles and torsions calculated using the coordinates of the
centroid of some group of atoms.
- distances, angles and torsions calculated using the coordinates of a
dummy atom, positioned at some point relative to the fragment,
perhaps along a non-bonded vector X-Y, at some specified distance from Y.
- angles between pairs of vectors, where the vectors are themselves
defined in terms of atomic positions.
- distances of specific atoms, centroids or dummy atoms from a mean
plane through several atoms.
- angles between vectors and normals to mean planes, or between two
plane normals (dihedral angles).
- special geometric parameters used to describe the geometry of specific
types of structural units: the puckering parameters used to
describe ring conformations are an example of special parameters. A number of
directionality parameters are also provided for examining the
angular orientations of non-bonded vectors with respect to special planes in
Ex. In the morphine example use of the following instructions:
7 8 9 10 11 12
SETUP P1 7 8 9 10 11 12
SETUP P2 1 2 3 4 5 6
DEFINE NANG 1 6 5
DEFINE N - C1 1 6
DEFINE N - C2 5 6
DEFINE DIST X1 6
DEFINE ?ANG P1 P2
has defined geometric objects as follows:
These geometric objects are then used to calculate geometrical parameters:
- the atoms of the fragment (atoms being regarded as "default" geometric
- the centroid X1 of the aromatic ring
- the mean planes P1 and P2 through the aromatic and heterocyclic rings
- the intra-annular valence angle NANG at the heterocyclic N atom
- the two N-C bond lengths N - C1 and N - C2
- the distance DIST between the centroid X1 and atom 6 of the fragment (the N
- the angle ?ANG between the normals to the two planes P1 and P2 (the
significance of ? will be noted later)
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Volume 1 Chapter 7 Composite Geometric Parameters.