Degenerate diffusions arising from gene duplication models
Abstract
We consider two processes that have been used to study gene duplication, Watterson's [Genetics 105 (1983) 745--766] double recessive null model and Lynch and Force's [Genetics 154 (2000) 459--473] subfunctionalization model. Though the state spaces of these diffusions are two and six-dimensional, respectively, we show in each case that the diffusion stays close to a curve. Using ideas of Katzenberger [Ann. Probab. 19 (1991) 1587--1628] we show that one-dimensional projections converge to diffusion processes, and we obtain asymptotics for the time to loss of one gene copy. As a corollary we find that the probability of subfunctionalization decreases exponentially fast as the population size increases. This rigorously confirms a result Ward and Durrett [Theor. Pop. Biol. 66 (2004) 93--100] found by simulation that the likelihood of subfunctionalization for gene duplicates decays exponentially fast as the population size increases.
Cite
@article{arxiv.0902.4780,
title = {Degenerate diffusions arising from gene duplication models},
author = {Rick Durrett and Lea Popovic},
journal= {arXiv preprint arXiv:0902.4780},
year = {2009}
}
Comments
Published in at http://dx.doi.org/10.1214/08-AAP530 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)