"The gene’s eye-view of quantitative genetics"Courau, PhilibertModelling the evolution of a continuous trait in a biological population is one of the oldest problems in evolutionary biology, which led to the birth of quantitative genetics. With the recent development of GWAS methods, it has become essential to link the evolution of the trait distribution to the underlying evolution of allelic frequencies at many loci, co-contributing to the trait value. The way most articles go about this is to make assumptions on the trait distribution, and use Wright's powerful formula to model how the evolution of the trait translates on each individual locus. Here, we take a gene's eye-view of the system, starting from an explicit finite-loci model with selection, drift, recombination and mutation, in which the trait value is a direct product of the genome. We let the number of loci go to infinity under the assumption of strong recombination, and characterize the limit behavior of a given locus with a McKean-Vlasov SDE and the corresponding Fokker-Planck PDE. In words, the selection on a typical locus depends on the mean behaviour of the other loci. This yields powerful results, including independence of two loci, explicit stationary distribution for allelic frequencies at a given locus (under some hypotheses on the fitness function), and normality of the trait distribution. We recover Wright's formula and the breeder's equation as special cases. |
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