Information Technology Reference
In-Depth Information
A monomeric triangle defines a Cartesian system where abscissa, or x-axis, goes
from
P
to
P
, ordinate or y-axis goes from
P
to
P
,
and altitude or z-axis is
orthogonal to the xy-plane and oriented according to a counterclockwise rotation
of x-axis toward y-axis. The pairing has a strictly dual nature: only two monomers
can be paired, because this relationship is exclusive, and when two monomers are
paired, both of them are unavailable to be paired with something else. In conclu-
sion, monomers need to be asymmetric with respect to two distinct directions, the
concatenation direction and the pairing direction. In fact, both these two relations
are intrinsically oriented. For this reason, monomers are
chiral
(from a Greek term
for hand), that is, each monomer defines univocally a three-dimensional Cartesian
coordinate system.
A monomeric triangle clearly defines a plane. Therefore, we can consider dif-
ferent possibilities for the planes where concatenated and paired monomeric
triangles lie.
Figure 2.5 shows some possible planar bilinear arrangements of monomers in the
different cases of right, acute, or obtuse monomeric triangles in parallel and antipar-
allel arrangements. Apart from the difficulty of keeping planar structures in a fluid
environment, the space occupancy of these structures would become prohibitive for
long DNA molecules. In all these cases, no rotation of paired monomeric triangles
is allowed around any axis lying in the plane of concatenation. Namely, as indicated
at the bottom of the figure (in the case of a right angle parallel arrangement) such a
kind of rotation would be in conflict with the parallelism (or anti-parallelism) of the
these concatenated structures.
If concatenated triangles, lying on the same or different planes, form an angle
along the concatenation line, then we can have the possibilities illustrated in Fig.
2.6. In both possibilities the concatenation angles vary along the concatenation line,
because in a spiral the curvature increases from the periphery to the center (a log-
arithmic spiral could avoid the angular variability, but its space occupancy would
be prohibitive). Therefore, this kind of arrangement is impossible, because it would
imply an angle between two concatenated monomers that depend on their positions
in concatenation line.
In the cases of acute or obtuse monomeric triangles, parallel arrangements are
impossible, as indicated in Fig. 2.7, because the bilinear structure cannot be realized.
When monomeric triangles are acute or obtuse, they can be arranged in antipar-
allel way (see Fig. 2.8), and the arrangement of both concatenation lines in the same
plane can be avoided by means of a rotation along the pairing line, as indicated at
the bottom of the figure. Of course, acute monomeric triangles realize more compact
arrangements than obtuse monomeric triangle, therefore are preferable.
