Solvable models of neighbor-dependent substitution processes.
We prove that a wide class of Markov models of neighbor-dependent substitution processes on the integer line is solvable. This class contains some models of nucleotidic substitutions recently introduced and studied empirically by molecular biologists. We show that the polynucleotidic frequencies at equilibrium solve some finite-size linear systems. This provides, for the first time up to our knowledge, explicit and algebraic formulas for the stationary frequencies of non-degenerate neighbor-dependent models of DNA substitutions. Furthermore, we show that the dynamics of these stochastic processes and their distribution at equilibrium exhibit some stringent, rather unexpected, independence properties. For example, nucleotidic sites at distance at least three evolve independently, and all the sites, when encoded as purines and pyrimidines, evolve independently.
Bérard
Jean
J
Institut Camille Jordan - UMR 5208, Université Claude Bernard Lyon 1, 69622, Villeurbanne, France. Jean.Berard@univ-lyon1.fr
Gouéré
Jean-Baptiste
JB
Piau
Didier
D
D000465:Algorithms; D001482:Base Composition; D001483:Base Sequence; D019143:Evolution, Molecular; D008390:Markov Chains; D008957:Models, Genetic; D017354:Point Mutation / Q000235:genetics*; D016012:Poisson Distribution; D011685:Purine Nucleotides / Q000235:genetics; D011742:Pyrimidine Nucleotides / Q000235:genetics
D011685:Purine Nucleotides; D011742:Pyrimidine Nucleotides