Abstract
We develop here a new class of stochastic models of gene evolution in which a random subset of the 64 possible trinucleotides mutates at each evolutionary time t according to some time dependent 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. This pseudochaotic 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 pseudochaotically mutate according to 3 time dependent substitution parameters associated with the 3 trinucleotide sites. The main result proves that under suitable assumptions, this pseudochaotic model converges to a uniform probability vector identical to that of the standard model. Furthermore, an application of this pseudochaotic model allows an evolutionary study of the 3 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 time dependent functions for the 3 substitution parameters, precisely an exponential decrease in the 1st and 2nd trinucleotide sites and an exponential increase in the 3rd one, this pseudochaotic model retrieves the main statistical properties of the 3 circular codes observed in genes. Furthermore, it leads to a circular code asymmetry stronger than the standard model (nonpseudochaotic) and, therefore, to a better correlation with the genes.
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Bahi, J.M., Michel, C.J. A Stochastic Model of Gene Evolution with Time Dependent Pseudochaotic Mutations. Bull. Math. Biol. 71, 681–700 (2009). https://doi.org/10.1007/s11538-008-9376-4
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DOI: https://doi.org/10.1007/s11538-008-9376-4