A Continuous Analog of Run Length Distributions Reflecting Accumulated Fractionation Events

Abstract

We propose a new, continuous model of the fractionation process (duplicate gene deletion after polyploidization) on the real line. The aim is to infer how much DNA is deleted at a time, based on segment lengths for alternating deleted (invisible) and undeleted (visible) regions. After deriving a number of analytical results for "one-sided" fractionation, we undertake a series of simulations that help us identify the distribution of segment lengths as a gamma with shape and rate parameters evolving over time. This leads to an inference procedure based on observed length distributions for visible and invisible segments. We suggest extensions of this mathematical and simulation work to biologically realistic discrete models, including two-sided fractionation.