Ergodicity of Some Dynamics of DNA Sequences

M. Falconnet, N. Gantert, E. Saada

2020, v.26, Issue 4, 567-612

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 (\hspace{-0.1pt}\cite{piau:solv}),
with three particular cases,
the models JC+$\cc$p$\gg$, T92+$\cc$p$\gg$, 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+$\cc$p$\gg$, T92+$\cc$p$\gg$
as well as the attractive RNc+YpR, all with an additional
cut-and-paste mechanism.

Keywords: dynamics of DNA sequences, Jukes\tire Cantor model, interacting particle systems, substitution processes, ergodicity, duality, coupling