Base-base interactions

For the classification of base-base interactions, we consider nucleotide pairs that involve at least one of the known chemical stabilizing forces, that is, covalent binding of adjacent bases, H-bond base pairing and base stacking. Base-base interactions are thus of five distinct types: adjacent, adjacent-stacked, adjacent-paired, non-adjacent-stacked and non-adjacent-paired (see Table 3.2). The non-adjacent-non-paired-non-stacked nucleotide pairs are the most frequent, but they were not considered since they do not involve actual chemical interactions.

Traditionnal encodings of adjacent base-base interactions uses the six backbone torsion angles $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ and $\zeta$ [24] or, more recently, the two pseudotorsion angles $\eta$ and $\theta$ [6]. These parameters accurately describe the relative placement of nucleotides linked by a phosphodiester bond. However, it has already been observed that distinct torsion angle combinations can result in similar backbone directions and base orientations. This phenomenon is known as the ``crankshaft effect'' [11,22]. Also, non-adjacent base-base interactions, like base pairings that are stabilized by H-bonds and non-adjacent base-base stacking, cannot be accurately parameterized using these angles. Rather, a plethora of rotation and translation parameters have been used to describe these interactions [2,15,17]. A simplified and unified encoding scheme for any type of base-base interactions that emerged from the introduction of HTMs is introduced. In order to allow us to effectively compare base-base interactions, a distance metric between between two HTMs, $\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}$ and $\mathbf{N}_{\mathbf{b}_1' \rightarrow \mathbf{b}_2'}$, should possess the following properties:

$$d(\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}, \mathbf{N}_{\mathbf{b}_1' \rightarrow \mathbf{b}_2'}) = d(\mathbf{N}_{\mathbf{b}_1' \rightarrow \mathbf{b}_2'}, \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2})$$ (3.1)

$$d(\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}, \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}^{-1}) = 0 \iff \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2} = \mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}^{-1}$$ (3.2)

$$d(\mathbf{M}_{\mathbf{b}_1 \rightarrow \mathbf{b}_2}, \mathbf{N}_{\mathbf{b}_1' \rightarrow \mathbf{b}_2'}) = d(\mathbf{M}_{\mathbf{b}_1\rightarrow \mathbf{b}_2}^{-1}, \mathbf{N}_{\mathbf{b}_1' \rightarrow \mathbf{b}_2'}^{-1})$$ (3.3)

Equation 3.1 states that the distance metric should obviously be commutative. Equation 3.2 states that a relation should have a null distance with itself, but not with it's inverse unless they are equal. Equation 3.3 states that the distance metric should not depend on the direction of application, implicit in the HTM representation. The metric should allow us to discriminate non-directional nucleotide relations.

The simple Euclidean distance in the 16 dimensional space of HTMs does not satisfy the above properties since HTMs embed a combination of translation and rotation terms that need to be considered separately. A HTM can be decomposed in the product of two HTMs, $\mathbf{M} = \mathbf{TR}$, where $\mathbf{T}$ contains the translation and $\mathbf{R}$ contains the rotation embedded in the original HTM. Paul [23] showed how to extract the length of the translation, $l$, as well as the angle $\theta$ and the axis of rotation $k$ from matrices $\mathbf{T}$ and $\mathbf{R}$. The strength of a transformation, $S(\mathbf{M})$, regardless of the axis of rotation, is defined by:

$$S(\mathbf{M}) = \sqrt{l^2 + (\frac{\theta}{\alpha})^2}$$ (3.4)

where $\alpha$ represents a scaling factor between the translation and rotation contributions. A scaling factor of 30$^\circ$/Å yields to a nice correlation with the RMSD metric, and means that a rotation of 30$^\circ$ around any axis is equivalent to a displacement of 1Å between two nucleotides' local referentials. Using this expression, the distance between two base-base interactions, $d(\mathbf{M},\mathbf{N})$, can be defined by:

$$d(\mathbf{M},\mathbf{N}) = \frac{[S(\mathbf{M}\mathbf{N}^{-1}) + S(\mathbf{M}^{-1}\mathbf{N})]}{2}$$ (3.5)

which satisfies the requirements of equations 3.1 to 3.3. In equation 3.5, the composition of transformation $\mathbf{MN}^{-1}$ can be seen as the necessary transformation needed to align the local referential $\mathbf{R}_{\mathbf{b}_2}'$ with $\mathbf{R}_{\mathbf{b}_2}$ when $\mathbf{R}_{\mathbf{b}_1}'$ and $\mathbf{R}_{\mathbf{b}_1}$ are aligned with the global referential. Similarly, $\mathbf{M}^{-1}\mathbf{N}$ can be interpreted as the transformation required to align $\mathbf{R}_{\mathbf{b}_1}'$ with $\mathbf{R}_{\mathbf{b}_1}$ when $\mathbf{R}_{\mathbf{b}_2}'$ and $\mathbf{R}_{\mathbf{b}_2}$ are aligned with the global referential.

Although HTMs are perfectly suited to uniformely encode base-base interactions, the information they contain is too compact to identify the type of relations they encode without reproducing the relation in 3-D space, and evaluating other parameters. For this reason, the symbolic annotations of base-base interactions are determined from atomic coordinates.