Degenerate diffusions arising from gene duplication models
Rick Durrett and Lea Popovic
Abstract.
We consider two processes that have been used to study gene duplication,
Watterson's (1983) double recessive null model, and Lynch and Force's (2000)
subfunctionalization model. Though the state spaces of these diffusions
are 2 and 6 dimensional respectively, we show in each case that the
diffusion stays close to a curve. Using ideas of Katzenberger (1991)
we show that one dimensional projections converge to diffusion processes,
and we obtain asymptotics for the 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 confirms
a result Ward and Durrett (2004) found by simulation and casts doubt
on subfunctionalization as an explanation of the preservation of gene duplicates
in organisms such as Drosophila with large effective population sizes.
Preprint
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