Applications of reversible Markov chains to biological evolution on the fitness landscape

Abstract: Biological evolution is sometimes thought of in terms of a population exploring a "fitness landscape". In this talk, I will discuss a population-genetic regime where this exploration can be modeled as a reversible Markov chain. In particular, I will apply two ideas from the theory of finite-state, reversible Markov chains to explore these evolutionary dynamics. First, I will show how Laplacian eigenmaps can be used to construct low-dimensional representations of the genetic state space that succinctly summarize the evolutionary dynamics at long time scales. Second, I will expand on some comments by Aldous and Fill on a natural ordering of the states of a reversible Markov chain in order to propose a general measure for the evolutionary accessibility of particular genetic states.