A population genetics theory for piRNA-regulated transposable elements

Transposable elements (TEs) are self-reproducing selfish DNA sequences that can invade the genome of virtually all living species. Population genetics models have shown that TE copy numbers generally reach a limit, either because the transposition rate decreases with the number of copies (transposition regulation) or because TE copies are deleterious, and thus purged by natural selection. Yet, recent empirical discoveries suggest that TE regulation may mostly rely on piRNAs, which require a specific mutational event (the insertion of a TE copy in a piRNA cluster) to be activated — the so-called TE regulation “trap model”. We derived new population genetics models accounting for this trap mechanism, and showed that the resulting equilibria differ substantially from previous expectations based on a transposition–selection equilibrium. We proposed three sub-models, depending on whether or not genomic TE copies and piRNA cluster TE copies are selectively neutral or deleterious, and we provide analytical expressions for maximum and equilibrium copy numbers, as well as cluster frequencies for all of them. In the full neutral model, the equilibrium is achieved when transposition is completely silenced, and this equilibrium does not depend on the transposition rate. When genomic TE copies are deleterious but not cluster TE copies, no long-term equilibrium is possible, and active TEs are eventually eliminated after an active incomplete invasion stage. When all TE copies are deleterious, a transposition–selection equilibrium exists, but the invasion dynamics is not monotonic, and the copy number peaks before decreasing. Mathematical predictions were in good agreement with numerical simulations, except when genetic drift and/or linkage disequilibrium dominates. Overall, the trap-model dynamics appeared to be substantially more stochastic and less repeatable than traditional regulation models.