Fig. 4

Birth and death process for populations of transferable genes: Circles represent microbial populations, each of which hosts a population of transferable genes with indispensability and connectivity \(\left({y}_{1},{z}_{1}\right)\) for transferable gene type \({T}_{1}\) and \(\left({y}_{2},{z}_{2}\right)\) for transferable gene type \({T}_{2}\). It is assumed that \({y}_{1}>{y}_{2}\) and\({z}_{1}>{z}_{2}\), meaning that \({T}_{1}\) is more sessile and \({T}_{2}\) is more transient. The ancestral metapopulation contained one population of type \({T}_{1}\) and two populations of type\({T}_{2}\). In this imaginary scenario, the type \({T}_{1}\) population, being more sessile, survived into the descendant metapopulation of transferable genes, its probability of doing so being \({w}_{1}^{p}=1-{\delta p}_{D}\left({y}_{1}\right)\). It did not produce any new populations, however. One population of type \({T}_{2}\) survived into the descendant metapopulation (probability\({w}_{2}^{p}=1-{\delta p}_{D}\left({y}_{2}\right)\)) but failed to multiply by HGT. The other type \({T}_{2}\) population was eliminated by gene loss (probability \({\delta p}_{D}\left({y}_{2}\right)\)), but managed to generate two new populations before doing so, the expected number of such new populations being \({w}_{2}^{m}=\beta \left(1-N{/N}_{max}\right){p}_{B}\left({z}_{2}\right)\). Note that one ancestor–descendant mapping corresponds to some billions of generations, which is assumed to be long enough for genetic novelties generated by mutation or HGT to be fixed or eliminated