Fig. 2

Derivation of structural features for an amino acid residue. a The orientation matrix for residue \(i\) consists of 3 orthogonal unit vectors: \({{\varvec{u}}}_{i}\), \({{\varvec{v}}}_{i}\), and \({{\varvec{w}}}_{i}\). The first basis vector of the orientation matrix is the normalised vector from the backbone amide nitrogen to the backbone carbonyl carbon of residue \(i\) in the PDB crystal structure, i.e., \({{\varvec{u}}}_{i}=\Vert {{\varvec{c}}}_{i}-{{\varvec{n}}}_{i}\Vert\). If \({{\varvec{a}}}_{i}\) is the vector from the amide nitrogen to the α-carbon \({{\varvec{c}}}_{\boldsymbol{\alpha }i}\), \({{\varvec{a}}}_{i}={{\varvec{n}}}_{i}-{{\varvec{c}}}_{\boldsymbol{\alpha }i}\), then the second basis vector \({{\varvec{v}}}_{i}\) is the normalised component of \({{\varvec{a}}}_{i}\) that is orthogonal to \({{\varvec{u}}}_{i}\). This vector is derived as the difference between \({{\varvec{a}}}_{i}\) and \({{\varvec{b}}}_{i}\), the projection of \({{\varvec{a}}}_{i}\) onto \({{\varvec{u}}}_{i}\) such that \({{\varvec{b}}}_{i}=({{\varvec{a}}}_{i}\bullet {{\varvec{u}}}_{i}) {{\varvec{u}}}_{i}\) and \({{\varvec{v}}}_{i}=\Vert {{\varvec{a}}}_{i}-{{\varvec{b}}}_{i}\Vert\). The third basis vector \({{\varvec{w}}}_{i}={{\varvec{u}}}_{i}\times {{\varvec{v}}}_{i}\). b The local structural environment of residue \(i\) is represented by its relation to nearby residues \(j\). The translation vectors \({{\varvec{t}}}_{i,j}\) are the vectors to the α-carbons of residues \(j\), in the reference frame \({{\varvec{O}}}_{i}\) with basis vectors \({{\varvec{u}}}_{i}\), \({{\varvec{v}}}_{i}\) and \({{\varvec{w}}}_{i}\) with the origin on \({{\varvec{c}}}_{\boldsymbol{\alpha }i}\). Thus \({{\varvec{t}}}_{i,j}={{\varvec{O}}}_{i}\bullet \left({{\varvec{c}}}_{\boldsymbol{\alpha }j}-{{\varvec{c}}}_{\boldsymbol{\alpha }i}\right)\). The relative orientations of proximal residues are represented by rotation quaternions \({{\varvec{r}}}_{i,j}\), such that \({{\varvec{r}}}_{i,j}^{-1}{{\varvec{O}}}_{i}{{\varvec{r}}}_{i,j}= {{\varvec{O}}}_{j}\). The \(\phi\) and \(\psi\) dihedral angles of residues \(i\) and \(j\) are encoded as \(\mathrm{sin}\phi\), \(\mathrm{cos}\phi\), \(\mathrm{sin}\psi\) and \(\mathrm{sin}\psi\) values