Analytical expression of the purine/pyrimidine codon probability after and before random mutations.

Recently, we proposed a new model of DNA sequence evolution (Arquès and Michel. 1990b. Bull. math. Biol. 52, 741-772) according to which actual genes on the purine/pyrimidine (R/Y) alphabet (R = purine = adenine or guanine, Y = pyrimidine = cytosine or thymine) are the result of two successive evolutionary genetic processes: (i) a mixing (independent) process of non-random oligonucleotides (words of base length less than 10: YRY(N)6, YRYRYR and YRYYRY are so far identified; N = R or Y) leading to primitive genes (words of several hundreds of base length) and followed by (ii) a random mutation process, i.e., transformations of a base R (respectively Y) into the base Y (respectively R) at random sites in these primitive genes. Following this model the problem investigated here is the study of the variation of the 8 R/Y codon probabilities RRR, ..., YYY under random mutations. Two analytical expressions solved here allow analysis of this variation in the classical evolutionary sense (from the past to the present, i.e., after random mutations), but also in the inverted evolutionary sense (from the present to the past, i.e., before random mutations). Different properties are also derived from these formulae. Finally, a few applications of these formulae are presented. They prove the proposition in Arquès and Michel (1990b. Bull. math. Biol. 52, 741-772), Section 3.3.2, with the existence of a maximal mean number of random mutations per base of the order 0.3 in the protein coding genes. They also confirm the mixing process of oligonucleotides by excluding the purine/pyrimidine contiguous and alternating tracts from the formation process of primitive genes.