Chirality in a quaternionic representation of the genetic code

Abstract

A quaternionic representation of the genetic code, previously reported by the authors, is updated in order to incorporate chirality of nucleotide bases and amino acids. The original representation assigns to each nucleotide base a prime integer quaternion of norm 7 and involves a function that associates with each codon, represented by three of these quaternions, another integer quaternion (amino acid type quaternion) in such a way that the essentials of the standard genetic code (particulaty its degeneration) are preserved. To show the advantages of such a quaternionic representation we have, in turn, associated with each amino acid of a given protein, besides of the type quaternion, another real one according to its order along the protein (order quaternion) and have designed an algorithm to go from the primary to the tertiary structure of the protein by using type and order quaternions. In this context, we incorporate chirality in our representation by observing that the set of eight integer quaternions of norm 7 can be partitioned into a pair of subsets of cardinality four each with their elements mutually conjugates and by putting they in correspondence one to one with the two sets of enantiomers (D and L) of the four nucleotide bases adenine, cytosine, guanine and uracil, respectively. Thus, guided by two diagrams -specifically proposed to describe the hypothetical evolution of the genetic codes corresponding to both of the chiral systems of affinities: D-nucleotide bases/Lamino acids and L-nucleotide bases/D-amino acids at reading frames 5´→ 3´and 3´→ 5´, respectively-we define functions that in each case assign a L-(D-) amino acid type integer quaternion to the triplets of D-(L-) bases. The assignation is such that for a given D-amino acid, the associated integer quaternion is the conjugate of that one corresponding to the enantiomer L. The chiral type quaternions obtained for the amino acids are used, together with a common set of order quaternions, to describe the folding of the two classes, L and D, of homochiral proteins.