Degenerate diffusions arising from gene duplication models

TL;DR

This paper rigorously analyzes gene duplication models using degenerate diffusions, showing that the probability of subfunctionalization decreases exponentially with population size, confirming previous simulation results.

Contribution

It provides a mathematical proof that the probability of subfunctionalization declines exponentially as population size grows, using diffusion process convergence techniques.

Findings

Probability of subfunctionalization decreases exponentially with population size

Diffusions stay close to a curve despite high dimensionality

One-dimensional projections converge to simpler diffusion processes

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.