The local referential of a nucleotide, and thus of a nitrogen base, is
defined by a Cartesian coordinate system whose position, relative to
the base, can be computed from its atomic coordinates (see
Figure 3.1). The local referential of a nucleotide can be
defined arbitrarily, but must be identical for each type of
nucleotide. Let be the unit vector between
coordinates of atom N1 and C2 in pyrimidines, and N9 and C4 in
purines. Let
be the unit vector between
coordinates of atom N1 and C6 in pyrimidines, and N9 and C8 in
purines. Then, the unit vector
of the Cartesian
coordinate system lies in the direction given by the sum
, the unit vector
is oriented along the cross
product
, and the unit vector
, following the right hand rule for a Cartesian coordinate
system, is given by
. The relative
positions of local referentials can be expressed using homogeneous
transformation matrices (HTM), which were first developed in the field
of geometry [20], and later extensively used in computer
graphics and robotics. HTMs encode, in the form of a 4x4 matrix, the
geometric operations needed to transform objects in 3-D space from one
local referential to another. In the base-base interaction context, a
HTM describes the relation by a composition of a translation and a
rotation between the two local referentials of the involved nitrogen
bases.
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Let
and
be the
local referentials of nucleotides
and
as
expressed relative to the global referential centered at the origin,
. The spatial relation between
and
is then given by the HTM
(see
Figure 3.1). In a molecular modeling context such as
implemented in MC-Sym [19], this relation can be
reproduced and the atomic coordinates of nucleotide
relative to nucleotide
computed by applying the
transformation obtained by the matrix product
to the absolute atomic
coordinates of
. In a similar way, the atomic
coordinates of
relative to
can be
computed by applying the inverse transformation
to the absolute
coordinates of
. It is worth noting here that
, that is the
inverse of the transformation extracted between
and
, is
equivalent to the one that would have been extracted between
and
.