Degenerate diffusions arising from gene duplication models

Details Degenerate diffusions arising from gene duplication models Degenerate diffusions arising from gene duplication models

2009-02-27

arxiv.org/abs/0902.4780v1

Annals of Applied Probability 2009, Vol. 19, No. 1, 15-48

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP530 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP530

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.

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