A stochastic model of gene evolution with chaotic mutations

Abstract

We develop here a new class of stochastic models of gene evolution in which the mutations are chaotic, i.e. a random subset of the 64 possible trinucleotides mutates at each evolutionary time t according to some substitution probabilities. Therefore, at each time t, the numbers and the types of mutable trinucleotides are unknown. Thus, the mutation matrix changes at each time t. The chaotic model developed generalizes the standard model in which all the trinucleotides mutate at each time t. It determines the occurrence probabilities at time t of trinucleotides which chaotically mutate according to three substitution parameters associated with the three trinucleotide sites. Two theorems prove that this chaotic model has a probability vector at each time t and that it converges to a uniform probability vector identical to that of the standard model. Furthermore, four applications of this chaotic model (with a uniform random strategy for the 64 trinucleotides and with a particular strategy for the three stop codons) allow an evolutionary study of the three circular codes identified in both eukaryotic and prokaryotic genes. A circular code is a particular set of trinucleotides whose main property is the retrieval of the frames in genes locally, i.e. anywhere in genes and particularly without start codons, and automatically with a window of a few nucleotides. After a certain evolutionary time and with particular values for the three substitution parameters, the chaotic models retrieve the main statistical properties of the three circular codes observed in genes. These applications also allow an evolutionary comparison between the standard and chaotic models.