Math @ Duke

Publications [#243538] of Richard T. Durrett
Papers Published
 Durrett, R; Popovic, L, Degenerate diffusions arising from gene duplication models,
The Annals of Applied Probability, vol. 19 no. 1
(2009),
pp. 1548, ISSN 10505164 [MR2521876 (2010g:60213)], [doi]
(last updated on 2018/10/22)
Abstract: We consider two processes that have been used to study gene duplication, Watterson's [Genetics 105 (1983) 745766] double recessive null model and Lynch and Force's [Genetics 154 (2000) 459473] subfunctionalization model. Though the state spaces of these diffusions are two and sixdimensional, respectively, we show in each case that the diffusion stays close to a curve. Using ideas of Katzenberger [Ann. Probab. 19 (1991) 15871628] we show that onedimensional 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) 93100] found by simulation that the likelihood of subfunctionalization for gene duplicates decays exponentially fast as the population size increases. © Institute of Mathematical Statistics, 2009.


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