Strong Comma-Free Codes in Genetic Information.

Comma-free codes constitute a class of circular codes, which has been widely studied, in particular by Golomb et al. (Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab 23:1-34, 1958a, Can J Math 10:202-209, 1958b), Michel et al. (Comput Math Appl 55:989-996, 2008a, Theor Comput Sci 401:17-26, 2008b, Inf Comput 212:55-63, 2012), Michel and Pirillo (Int J Comb 2011:659567, 2011), and Fimmel and Strüngmann (J Theor Biol 389:206-213, 2016). Based on a recent approach using graph theory to study circular codes Fimmel et al. (Philos Trans R Soc 374:20150058, 2016), a new class of circular codes, called strong comma-free codes, is identified. These codes detect a frameshift during the translation process immediately after a reading window of at most two nucleotides. We describe several combinatorial properties of strong comma-free codes: enumeration, maximality, self-complementarity and [Formula: see text]-property (comma-free property in all the three possible frames). These combinatorial results also highlight some new properties of the genetic code and its evolution. Each amino acid in the standard genetic code is coded by at least one strong comma-free code of size 1. There are 9 amino acids [Formula: see text] among 20 such that for each amino acid from S, its synonymous trinucleotide set (excluding the necessary periodic trinucleotides [Formula: see text]) is a strong comma-free code. The primeval comma-free RNY code of Eigen and Schuster (Naturwissenschaften 65:341-369, 1978) is a self-complementary [Formula: see text]-code of size 16. Furthermore, it is the union of two strong comma-free codes of size 8 which are complementary to each other.