A symmetry model for genetic coding via a wallpaper group composed of the traditional four bases and an imaginary base E: Towards category theory-like systematization of molecular/genetic biology

BACKGROUND: Previously, we suggested prototypal models that describe some clinical states based on group postulates. Here, we demonstrate a group/category theory-like model for molecular/genetic biology as an alternative application of our previous model. Specifically, we focus on deoxyribonucleic acid (DNA) base sequences. RESULTS: We construct a wallpaper pattern based on a five-letter cruciform motif with letters C, A, T, G, and E. Whereas the first four letters represent the standard DNA bases, the fifth is introduced for ease in formulating group operations that reproduce insertions and deletions of DNA base sequences. A basic group Z(5) = {r, u, d, l, n} of operations is defined for the wallpaper pattern, with which a sequence of points can be generated corresponding to changes of a base in a DNA sequence by following the orbit of a point of the pattern under operations in group Z(5). Other manipulations of DNA sequence can be treated using a vector-like notation ‘D(j)’ corresponding to a DNA sequence but based on the five-letter base set; also, ‘D(j)’s are expressed graphically. Insertions and deletions of a series of letters ‘E’ are admitted to assist in describing DNA recombination. Likewise, a vector-like notation R(j) can be constructed for sequences of ribonucleic acid (RNA). The wallpaper group B = {Z(5)(×∞), ●} (an ∞-fold Cartesian product of Z(5)) acts on D(j) (or R(j)) yielding changes to D(j) (or R(j)) denoted by ‘D(j)◦B((j→k)) = D(k)’ (or ‘R(j)◦B((j→k)) = R(k)’). Based on the operations of this group, two types of groups—a modulo 5 linear group and a rotational group over the Gaussian plane, acting on the five bases—are linked as parts of the wallpaper group for broader applications. As a result, changes, insertions/deletions and DNA (RNA) recombination (partial/total conversion) are described. As an exploratory study, a notation for the canonical “central dogma” via a category theory-like way is presented for future developments. CONCLUSIONS: Despite the large incompleteness of our methodology, there is fertile ground to consider a symmetry model for genetic coding based on our specific wallpaper group. A more integrated formulation containing “central dogma” for future molecular/genetic biology remains to be explored.