Modeling transposable element dynamics with fragmentation equations

Transposable elements (TEs), segments of DNA capable of self-replication, are abundant in the genomes of most organisms and thus serve as a record of past mutational events. While some work suggests TEs may serve a regulatory function for the host, most empirical and theoretical studies have shown that TEs often have deleterious effects on a host. Because they are not essential, the host genome consists of both full-length (actively replicating) and partial length (inactive remnant) copies of TEs. We developed a novel mathematical formulation of TE dynamics by modeling the density of full and partial length copies resulting from mutations (insertions and deletions) and TE replication within the host genome. More specifically, we model the time-evolution of the complete TE length distribution (full and partial elements) in a genome using fragmentation equations in both a discrete and continuous framework under two models of TE replication. In the first case, we assume that full-length TEs replicate at a constant rate regardless of the number of full-length TEs present in the genome. While this assumption simplifies the underlying biological processes, it allows us to derive an explicit analytical form of the time-varying TE density, as well as the steady-state behavior, under both discrete and continuous formulations. Next, we take into account the potential deleterious effects of TEs by modeling TE replication with a logistic growth equation. Under this assumption, the number of actively replicating TEs in a genome is limited by a carrying capacity. While we are still able to derive to derive analytical forms for the time-varying TE density, for both discrete and continuous length formulations, these solutions are implicit. For all four proposed models, we prove existence and uniqueness of these solutions describing TE length distributions. We compare both models and note that the logistic and exponential models initially agree. Since most TEs have not reached carrying capacity, we use the explicit exponential solution to quantify rates of replication to mutations. We apply our model to present day annotated collections of TEs from the genomes of species of fruit-flies, birds, and primates to uncover quantitative relationships of TE dynamics. With the increasing availability of complete genomes, our method is likely to uncover relationships of biological drivers of genomic variation in many species.