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Abstract: For modeling evolution of DNA sequences (string of letters A, C, T and G, such as "GATTACA"), Markov chains have become the computational biologist's preferred tool. The object of our study a reversible, infinite-state, continuous-time Markov chain introduced by Thorne, Kishino and Felsenstein in 1991, brings to the forefront three possible modes of evolution -- insertion, deletion and letter-switch (e.g., "A" -> "T"). Working with a simplified version of this model and drawing inspiration from Aldous's work on the hypercube graph, we use coupling to prove exponential ergodicity. This research has applications to reconstructing phylogenies, where, for Markov chain models, convergence rate and reconstructibility seem to be intimately linked (cf. Kesten--Stigum threshold). There are two main lines of future research: first, to obtain stronger results, such as L2 exponential convergence (i.e., a Poincare inequality) and a spectral decomposition for the transition function (cf. Karlin--McGregor theorem for birth-death processes); and second, to consider other models, such as the Poisson indel model introduced by Bouchard-Cote and Jordan. Joint work with Wai-Tong Fan (IU Bloomington) and Graham White (IU Bloomington).