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=== Transposable elements === To model the dynamics of transposable elements (TEs) within a genome, one has to realize that the elements behave like a population within each genome, and they can jump from one haploid genome to another by horizontal transfer. The mathematics has to describe the rates and dependencies of these transfer events. It was observed early on that the rate of jumping of many TEs varies with copy number, and so the first models simply used an empirical function for the rate of transposition. This had the advantage that it could be measured by experiments in the lab, but it left open the question of why the rate differs among elements and differs with copy number. Stan Sawyer and Daniel L. Hartl<ref>{{cite journal | vauthors = Sawyer S, Hartl D | title = Distribution of transposable elements in prokaryotes | journal = Theoretical Population Biology | volume = 30 | issue = 1 | pages = 1β16 | date = August 1986 | pmid = 3018953 | doi = 10.1016/0040-5809(86)90021-3}}</ref> fitted models of this sort to a variety of bacterial TEs, and obtained quite good fits between copy number and transmission rate and the population-wide incidence of the TEs. TEs in higher organisms, like ''Drosophila'', have a very different dynamics because of sex, and [[Brian Charlesworth]], [[Deborah Charlesworth]], Charles Langley, John Brookfield and others<ref name=":13" /><ref>{{cite journal | vauthors = Brookfield JF, Badge RM | title = Population genetics models of transposable elements | journal = Genetica | volume = 100 | issue = 1β3 | pages = 281β94 | date = 1997 | pmid = 9440281 | doi = 10.1023/A:1018310418744| s2cid = 40644313 }}</ref><ref>{{cite journal | vauthors = Charlesworth B, Charlesworth D | title = The population dynamics of transposable elements. | journal = Genet. Res. | date = 1983 | volume = 42 | pages = 1β27 | doi = 10.1017/S0016672300021455 | doi-access = free }}</ref> modeled TE copy number evolution in ''Drosophila'' and other species. What is impressive about all these modeling efforts is how well they fitted empirical data, given that this was decades before discovery of the fact that the host fly has a powerful defense mechanism in the form of piRNAs. Incorporation of host defense along with TE dynamics into evolutionary models of TE regulation is still in its infancy.<ref>{{cite journal | vauthors = Lu J, Clark AG | title = Population dynamics of PIWI-interacting RNAs (piRNAs) and their targets in Drosophila | journal = Genome Research | volume = 20 | issue = 2 | pages = 212β27 | date = February 2010 | pmid = 19948818 | pmc = 2813477 | doi = 10.1101/gr.095406.109 }}</ref>
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