The infinitesimal model Journal Article

Author(s): Barton, Nicholas H; Etheridge, Alison M; Veneer, Amandine
Article Title: The infinitesimal model
Affiliation IST Austria
Abstract: Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. We first review the long history of the infinitesimal model in quantitative genetics. Then we provide a definition of the model at the phenotypic level in terms of individual trait values and relationships between individuals, but including different evolutionary processes: genetic drift, recombination, selection, mutation, population structure, ... We give a range of examples of its application to evolutionary questions related to stabilising selection, assortative mating, effective population size and response to selection, habitat preference and speciation. We provide a mathematical justification of the model as the limit as the number M of underlying loci tends to infinity of a model with Mendelian inheritance, mutation and environmental noise, when the genetic component of the trait is purely additive. We also show how the model generalises to include epistatic effects. In each case, by conditioning on the pedigree relating individuals in the population, we incorporate arbitrary selection and population structure. We suppose that we can observe the pedigree up to the present generation, together with all the ancestral traits, and we show, in particular, that the genetic components of the individual trait values in the current generation are indeed normally distributed with a variance independent of ancestral traits, up to an error of order M^{-1/2}. Simulations suggest that in particular cases the convergence may be as fast as 1/M.
Keywords: Selection; Epistasis; quantitative genetics; Infinitesimal model
Journal Title: Theoretical Population Biology
ISSN: 0040-5809
Publisher: Academic Press  
Date Published: 2017-07-11
Start Page: 1
End Page: 54
Copyright Statement: The copyright holder for this preprint is the author/funder. It is made available under a CC-BY-NC-ND 4.0 International license
Sponsor: NHB supported in part by ERC Grant 250152.
DOI: 10.1016/j.tpb.2017.06.001
Notes: The authors thank Bill Hill for valuable comments on a previous draft of the paper. A.V. would like to thank Olivier Martin and the members of a working group on the infinitesimal model (in particular Patrick Phillips, Gaƫl Raoul and Henrique Teotonio) for interesting discussions on this topic.
Open access: yes (OA journal)