ANALYTICAL EXPRESSION OF THE PURINE/PYRIMIDINE AUTOCORRELATION FUNCTION AFTER AND BEFORE RANDOM MUTATIONS

The mutation process is a classical evolutionary genetic process. The type of mutations studied here is the random substitutions of a purine base R (adenine or guanine) by a pyrimidine base Y (cytosine or thymine) and reciprocally (transversions). The analytical expressions derived allow us to analyze in genes the occurrence probabilities of motifs and d-motifs (two motifs separated by any d bases) on the R/Y alphabet under transversions. These motif probabilities can be obtained after transversions (in the evolutionary sense; from the past to the present) and, unexpectedly, also before transversions (after back transversions, in the inverse evolutionary sense, from the present to the past). This theoretical part in Section 2 is a first generalization of a particular formula recently derived. The application in Section 3 is based on the analytical expression giving the autocorrelation function (the d-motif probabilities) before transversions. It allows us to study primitive genes from actual genes. This approach solves a biological problem. The protein coding genes of chloroplasts and mitochondria have a preferential occurrence of the 6-motif YRY(N)(6)YRY (maximum of the autocorrelation function for d = 6, N = R or Y) with a periodicity module 3. The YRY(N)(6)YRY preferential occurrence without the periodicity module 3 is also observed in the RNA coding genes (ribosomal, transfer, and small nuclear RNA genes) and in the noncoding genes (introns and 5' regions of eukaryotic nuclei). However, there are two exceptions to this YRY(N)(6)YRY rule: the protein coding genes of eukaryotic nuclei, and prokaryotes, where YRY(N)(6)YRY has the second highest value after YRY(N)(0)YRY (YRWRY) with a periodicity module 3. When we go backward in time with the analytical expression, the protein coding genes of both eukaryotic nuclei and prokaryotes retrieve the YRY(N)(6)YRY preferential occurrence with a periodicity module 3 after 0.2 back transversions per base. In other words, the actual protein coding genes of chloroplasts and mitochondria are similar to the primitive protein coding genes of eukaryotic nuclei and prokaryotes. On the other hand, this application represents the first result concerning the mutation process in the model of DNA sequence evolution we recently proposed. According to this model, the actual genes on the R/Y alphabet derive from two successive evolutionary genetic processes: an independent mixing of a few nonrandom types of oligonucleotides leading to genes called primitive followed by a mutation process in these primitive genes. Indeed, the mutation process can simulate statistical properties identified in genes, e.g., the variations between YRY(N)(0)YRY and YRY(N)(6)YRY, which could not have been so far simulated with the mixing process.