A diploid population model for copy number variation of genetic elements

TL;DR

This paper models the genetic variation of copy number elements in a diploid population, demonstrating convergence of the population distribution to a Poisson distribution driven by a critical branching process.

Contribution

It introduces a new population model for genetic elements, analyzing the convergence of the distribution to a Poisson form coupled with a branching process.

Findings

Population distribution converges to a Poisson distribution.

The total element count follows a critical Feller branching process.

Model extensions and limitations are discussed.

Abstract

We study the following model for a diploid population of constant size $N$: Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability $\tfrac 12$ on to the offspring. We study the process $X^N = (X^N(1), X^N(2),...)$, where $X_t^N(k)$ is the frequency of individuals at time $t$ that carry $k$ elements, and prove convergence (in some weak sense) of $X^N$ jointly with its empirical first moment $Z^N$ to the ``slow-fast'' system $(Z,X)$, where $X_t = \text{Poi}(Z_t)$ and $Z$ evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.