Trees within trees: simple nested coalescents

Abstract

We consider the compact space of pairs of nested partitions of N, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of Λ-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo Λ-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. Λ-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure ν s on (0, 1] × M 1 ([0, 1]) providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

Coagulation-transport equations and the nested coalescents

“…This model can be enriched by embedding gene lineages into species (each gene belongs to a living organism which in turn belongs to some species). Nested coalescents were recently introduced [8] to model jointly the genealogy of the genes and the genealogy of the species. The nested Kingman coalescent is the simplest example of a nested coalescent, where both the gene tree and the species tree are given by (non-independent) Kingman coalescents:…”

Particle systems with coordination

“…We note that in Blancas et al (2018), an equivalent condition for coming down from infinity for the nested coalescent was found, where the marginal coalescent of species can be an arbitrary Λ-coalescent instead of a Kingman one. Here, the only assumption on the measures corresponding to the Λ-coalescent of individual ancestral lines (called 'marginal gene coalescent' in that paper) and the one of species is that they have no mass at one, i.e., the probability that all species or all individuals of a given species merge simultaneously is zero.…”

“…The nested coalescent and its dual. The nested coalescent is an object introduced recently Blancas et al (2018), which has already received some attention Blancas et al (2019); Duchamps (2020); Lambert and Schertzer (2020). Its purpose is to integrate speciation events and individual reproduction in the same model, in order to be able to trace ancestry at the level of species.…”

“…We discuss a number of examples of processes in the (recent) mathematical literature, and construct their duals, some of which seem to be new. Via duality, we also provide some results on the long time behaviour of these models, such as a criterion for coming down from infinity for the so-called nested coalescent (Blancas et al, 2018) and for models exhibiting death but no coalescence, and almost sure fixation in a variant of the peripatric coalescent (Lambert and Ma, 2015) with non-coordinated selection and coordinated migration. Examples extend to situations not generally looked at from the point of view of population models, such as the famous Parabolic Anderson Model (PAM).…”

Analysing Emergent Dynamics of Evolving Computation in 2D Cellular Automata

“…The study of inheritance in evolution has included genealogies and their statistics as an essential component in reconstructing the past, most recently and successfully in the form of coalescence theory [53][54][55], describing the statistics of the number of generations back to a common ancestor, applied to simple models of evolution [56] and the construction of evolutionary trees, the quality of which has been evaluated with statistical geometry [57]. Limited samples of sequences from current populations can be used to construct coalescence statistics and evolutionary trees.…”