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 1 2 on to the offspring. We study the process XN=(XN(1),XN(2),...), where XtN(k) is the frequency of individuals at time t that carry k elements, and prove convergence (in some weak sense) of XN jointly with its empirical first moment ZN to the “slow-fast” system (Z,X), where Xt=Poi(Zt) and Z evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.