Transition and Transversion on the Common Trinucleotide Circular Code

In 1996, a trinucleotide circular code which is maximum, self-complementary, and , called , was identified statistically on a large gene population of eukaryotes and prokaryotes (Arquès and Michel (1996)). Transition and transversions I and II are classical molecular evolution processes. A comprehensive computer analysis of these three evolution processes in the code shows some new results; in particular (i) transversion I on the 2nd position of any subset of trinucleotides of generates trinucleotide circular codes which are always and (ii) transversion II on the three positions of any subset of trinucleotides of yields no trinucleotide circular codes. These new results extend our theory of circular code in genes to its evolution under transition and transversion. 1. Introduction We continue our study of properties of trinucleotide circular codes [1–5], trinucleotide comma-free codes [1, 6], strong trinucleotide circular codes [7], and the common trinucleotide circular code identified in genes [8] (see also the recent statistical analysis by [9]) which could be a translation code [10]. A trinucleotide is a word of three letters (triletter) on the genetic alphabet . The set of trinucleotides is a code (called genetic code), more precisely a uniform code but not a circular code (see Remark 2). In the past years, codes, comma-free codes, and circular codes have been mathematical objects studied in theoretical biology, mainly to understand the structure and the origin of the genetic code as well as the reading frame (construction) of genes, for example, [11–13]. In order to have an intuitive meaning of these notions, codes are written on a straight line while comma-free codes and circular codes are written on a circle, but in both cases, unique decipherability is required. Circular codes only belong to some subsets of the trinucleotide set while comma-free codes are even more constrained subsets of circular codes [1]. Before the discovery of the genetic code, Crick et al. [11] proposed a maximum comma-free code of trinucleotides for coding the amino acids. This comma-free code turned out to be invalid (see, e.g., [14]). In , a maximum circular code of trinucleotides was identified statistically on a large gene population of eukaryotes and also on a large gene population of prokaryotes [8] This code has remarkable mathematical properties as it is a self-complementary maximum circular code (see the following). Since , its properties have been studied in detail by different authors, for example, [9, 15–21]. Transition and transversions I and II are classical