Local referentials and homogeneous transformation matrices

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 $\mathbf{u}$ be the unit vector between coordinates of atom N1 and C2 in pyrimidines, and N9 and C4 in purines. Let $\mathbf{v}$ be the unit vector between coordinates of atom N1 and C6 in pyrimidines, and N9 and C8 in purines. Then, the unit vector $\mathbf{y}$ of the Cartesian coordinate system lies in the direction given by the sum $\mathbf{u} + \mathbf{v}$, the unit vector $\mathbf{z}$ is oriented along the cross product $\mathbf{u} \times \mathbf{v}$, and the unit vector $\mathbf{x}$, following the right hand rule for a Cartesian coordinate system, is given by $\mathbf{y} \times \mathbf{z}$. 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.

Figure: Local referentials and base-base relations. $\mathbf{R}_{\mathbf{b}_1}$ and $\mathbf{R}_{\mathbf{b}_2}$ are the HTMs representing the local referentials of two nucleotides, $\mathbf{b}_1$ and $\mathbf{b}_2$. $\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}$ encodes the relation between $\mathbf{R}_{\mathbf{b}_1}$ and $\mathbf{R}_{\mathbf{b}_2}$, that is the position of $\mathbf{R}_{\mathbf{b}_2}$ relative to $\mathbf{R}_{\mathbf{b}_1}$.

Let $\mathbf{R}_{\mathbf{b}_1}$ and $\mathbf{R}_{\mathbf{b}_2}$ be the local referentials of nucleotides $\mathbf{b}_1$ and $\mathbf{b}_2$ as expressed relative to the global referential centered at the origin, $(0,0,0)$. The spatial relation between $\mathbf{R}_{\mathbf{b}_1}$ and $\mathbf{R}_{\mathbf{b}_2}$ is then given by the HTM $\mathbf{M}_{\mathbf{b_1} \rightarrow \mathbf{b_2}} = \mathbf{R}_{\mathbf{b}_1}^{-1} \mathbf{R}_{\mathbf{b}_2}$ (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 $\mathbf{b}_2'$ relative to nucleotide $\mathbf{b}_1'$ computed by applying the transformation obtained by the matrix product $\mathbf{R}_{\mathbf{b}_1'} \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2} \mathbf{R}_{\mathbf{b}_2'}^{-1}$ to the absolute atomic coordinates of $\mathbf{b}_2'$. In a similar way, the atomic coordinates of $\mathbf{b}_1'$ relative to $\mathbf{b}_2'$ can be computed by applying the inverse transformation $\mathbf{R}_{\mathbf{b}_2'} \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}^{-1} \mathbf{R}_{\mathbf{b}_1'}^{-1}$ to the absolute coordinates of $\mathbf{b}_1'$. It is worth noting here that $\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}^{-1} = \mathbf{M}_{\mathbf{b}_2 \rightarrow \mathbf{b}_1}$, that is the inverse of the transformation extracted between $\mathbf{R}_{\mathbf{b}_1}$ and $\mathbf{R}_{\mathbf{b}_2}$, is equivalent to the one that would have been extracted between $\mathbf{R}_{\mathbf{b}_2}$ and $\mathbf{R}_{\mathbf{b}_1}$.