Ergodicity of some dynamics of DNA sequences

TL;DR

This paper studies the ergodicity of DNA sequence dynamics modeled by interacting particle systems, combining substitution processes with a cut-and-paste mechanism to simulate insertions and deletions, providing conditions for ergodicity.

Contribution

It introduces a combined model of DNA sequence evolution with substitution and cut-and-paste dynamics and establishes ergodicity conditions using duality and attractiveness methods.

Findings

Ergodicity conditions depend only on substitution rates.

Models JC+cpg and T92+cpg remain ergodic with cut-and-paste.

RNc+YpR model is ergodic under additional mechanisms.

Abstract

We define interacting particle systems on configurations of the integer lattice (with values in some finite alphabet) by the superimposition of two dynamics: a substitution process with finite range rates, and a circular permutation mechanism(called "cut-and-paste") with possibly unbounded range. The model is motivated by the dynamics of DNA sequences: we consider an ergodic model for substitutions, the RN+YpR model ([BGP08]), with three particular cases, the models JC+cpg,T92+cpg, and RNc+YpR. We investigate whether they remain ergodic with the additional cut-and-paste mechanism, which models insertions and deletions of nucleotides. Using either duality or attractiveness techniques, we provide various sets of sufficient conditions, concerning only the substitution rates, for ergodicity of the superimposed process. They imply ergodicity of the models JC+cpg, T92+cpg as well as the attractive RNc+YpR, all with an additional cut-and-paste mechanism.