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The infinitesimal model: Definition, derivation, and implications

Author

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  • Barton, N.H.
  • Etheridge, A.M.
  • Véber, A.

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 within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. Thus, the variance that segregates within families is not perturbed by selection, and can be predicted from the variance components. This does not necessarily imply that the trait distribution across the whole population should be Gaussian, and indeed selection or population structure may have a substantial effect on the overall trait distribution. One of our main aims is to identify some general conditions on the allelic effects for the infinitesimal model to be accurate. We first review the long history of the infinitesimal model in quantitative genetics. Then we formulate 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. We prove in particular that, within each family, 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 1∕M. Simulations suggest that in some cases the convergence may be as fast as 1∕M.

Suggested Citation

  • Barton, N.H. & Etheridge, A.M. & Véber, A., 2017. "The infinitesimal model: Definition, derivation, and implications," Theoretical Population Biology, Elsevier, vol. 118(C), pages 50-73.
    • Handle: RePEc:eee:thpobi:v:118:y:2017:i:c:p:50-73
    DOI: 10.1016/j.tpb.2017.06.001
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    References listed on IDEAS

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    1. de Jong, Peter, 1990. "A central limit theorem for generalized multilinear forms," Journal of Multivariate Analysis, Elsevier, vol. 34(2), pages 275-289, August.
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    Cited by:

    1. Steiner, Ulrich K. & Tuljapurkar, Shripad, 2020. "Drivers of diversity in individual life courses: Sensitivity of the population entropy of a Markov chain," Theoretical Population Biology, Elsevier, vol. 133(C), pages 159-167.
    2. Olivier David & Arnaud Le Rouzic & Christine Dillmann, 2022. "Optimization of sampling designs for pedigrees and association studies," Biometrics, The International Biometric Society, vol. 78(3), pages 1056-1066, September.
    3. Dekens, L. & Otto, S.P. & Calvez, V., 2022. "The best of both worlds: Combining population genetic and quantitative genetic models," Theoretical Population Biology, Elsevier, vol. 148(C), pages 49-75.
    4. Yengo, Loic & Visscher, Peter M., 2018. "Assortative mating on complex traits revisited: Double first cousins and the X-chromosome," Theoretical Population Biology, Elsevier, vol. 124(C), pages 51-60.
    5. Manuel Plate & Richard Bernstein & Andreas Hoppe & Kaspar Bienefeld, 2019. "Comparison of infinitesimal and finite locus models for long-term breeding simulations with direct and maternal effects at the example of honeybees," PLOS ONE, Public Library of Science, vol. 14(3), pages 1-22, March.
    6. David, Olivier & van Frank, Gaëlle & Goldringer, Isabelle & Rivière, Pierre & Turbet Delof, Michel, 2020. "Bayesian inference of natural selection from spatiotemporal phenotypic data," Theoretical Population Biology, Elsevier, vol. 131(C), pages 100-109.
    7. Barton, N.H. & Etheridge, A.M., 2018. "Establishment in a new habitat by polygenic adaptation," Theoretical Population Biology, Elsevier, vol. 122(C), pages 110-127.

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