## The Annals of Applied Probability

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 20 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1235140331

**Digital Object Identifier**

doi:10.1214/08-AAP530

**Mathematical Reviews number (MathSciNet)**

MR2498670

**Zentralblatt MATH identifier**

1211.60038

**Subjects**

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

**Keywords**

Gene duplication subfunctionalization one-dimensional diffusions Lyapunov function

#### Citation

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. https://projecteuclid.org/euclid.aoap/1235140331