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The infinitesimal model

N H Barton, A M Etheridge, A Véber
doi: https://doi.org/10.1101/039768
N H Barton
1Institute of Science and Technology, Am Campus I, A-3400 Klosterneuberg, Austria, Email:
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  • For correspondence: nick.barton@ist-austria.ac.at
A M Etheridge
2Department of Statistics, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK
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A Véber
3Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France
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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 Embedded Image. Simulations suggest that in particular cases the convergence may be as fast as 1/M.

Footnotes

  • ↵* NHB supported in part by ERC Grant 250152

  • ↵† AME supported in part by EPSRC Grant EP/K034316/1

  • ↵‡ AV supported by the chaire Modélisation Mathématique et Biodiversité of Veolia Environment - École Polytechnique - Museum National d’Histoire Naturelle - Fondation X

Copyright 
The copyright holder for this preprint is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under a CC-BY-NC-ND 4.0 International license.
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Posted February 15, 2016.
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The infinitesimal model
N H Barton, A M Etheridge, A Véber
bioRxiv 039768; doi: https://doi.org/10.1101/039768
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The infinitesimal model
N H Barton, A M Etheridge, A Véber
bioRxiv 039768; doi: https://doi.org/10.1101/039768

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