The Annals of Applied Probability

Degenerate diffusions arising from gene duplication models

Rick Durrett and Lea Popovic

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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.

Article information

Ann. Appl. Probab., Volume 19, Number 1 (2009), 15-48.

First available in Project Euclid: 20 February 2009

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D15: Problems related to evolution 92D20: Protein sequences, DNA sequences

Gene duplication subfunctionalization one-dimensional diffusions Lyapunov function


Durrett, Rick; Popovic, Lea. Degenerate diffusions arising from gene duplication models. Ann. Appl. Probab. 19 (2009), no. 1, 15--48. doi:10.1214/08-AAP530.

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