A diploid population model for copy number variation of genetic elements

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 $\frac{1}{2}$ on to the offspring. We study the process $X^N=(X^N(1),X^N(2),\mathrm{...})$, 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.