Tree-valued Fleming–Viot dynamics with mutation and selection

Abstract

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions.The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces.To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.

Genealogical distances under low levels of selection

“…In the sequel, we will rely here on the possibility to pick a sample from the Moran population in the large population limit at equilibrium and describe its genealogical tree, which is given by (i) genealogical distances between any pair of individuals, resulting in an ultra-metric tree and (ii) marks on the tree which describe mutation events from • to • or from • to •; see also Figure 1. This possibility is implicitly made by the ancestral selection graph from Neuhauser and Krone (1997), and formally justified by some results obtained in Depperschmidt et al (2012); precisely, their Theorem 4 states that the genealogical tree under selection has a unique equilibrium.…”

“…In our manuscript, we will make use of this theory in order to compute an approximation for the total tree length under a general bi-allelic selection scheme, which is assumed to be weak; see Section 4. Our results are extensions of Theorem 5 of Depperschmidt et al (2012), where an approximation of the Laplace-transform of the genealogical distance of a pair of individuals under bi-allelic mutation and low levels of selection was computed.…”

“…Recently, genealogies under selection have been studied by Depperschmidt et al (2012) using Markov processes taking values in the space of trees, i.e. the genealogical tree is modelled as a stochastic process which is changing as the population evolves.…”

“…Section 4 gives some preliminaries for the proofs. In particular, we give a brief review of the construction of evolving genealogies from Depperschmidt et al (2012). Finally, Section 5 contains all proofs.…”

A model for Mendelian populations demogenetics

“…As noted in [11] (Remark 2.1), diploid additive and haploid selection are equivalent for populations with constant size. In our model, in an additive case for which β 2 −δ 2 = (β 1 −δ 1 )−s, β 3 −δ 3 = (β 1 −δ 1 )−2s, ρ ij = 0 and α ij = α for all i, j, the limiting diffusion (N, X) given in Equation (4) satisfies:…”