Statistical genetics and evolution of quantitative traits

The distribution and heritability of many traits depends on numerous loci in the genome. In general, the astronomical number of possible genotypes makes the system with large numbers of loci difficult to describe. Multilocus evolution, however, greatly simplifies in the limit of weak selection and frequent recombination. In this limit, populations rapidly reach quasilinkage equilibrium (QLE) in which the dynamics of the full genotype distribution, including correlations between alleles at different loci, can be parametrized by the allele frequencies. This review provides a simplified exposition of the concept and mathematics of QLE which is central to the statistical description of genotypes in sexual populations. Key results of quantitative genetics such as the generalized Fisher’s “fundamental theorem,” along with Wright’s adaptive landscape, are shown to emerge within QLE from the dynamics of the genotype distribution. This is followed by a discussion under what circumstances QLE is applicable, and what the breakdown of QLE implies for the population structure and the dynamics of selection. Understanding the fundamental aspects of multilocus evolution obtained through simplified models may be helpful in providing conceptual and computational tools to address the challenges arising in the studies of complex quantitative phenotypes of practical interest.

Figure 1

Time scales in sexual populations. A population is described by the distribution of $2^L$ genotypes, on which selection, recombination, and mutation acts. In sexual populations, mating and recombination is the fastest process, so that different loci are only weakly correlated (close to linkage equilibrium) and its dynamics can be approximately described via $L$ allele frequencies, a number much smaller than the $2^L$ possible genotypes that would have to be tracked otherwise. Allele frequencies change slowly and means of quantitative traits follow the laws of quantitative genetics. Over the much longer time scale of ${\mu }^{-1}$, allele frequencies themselves tend to an equilibrium between selection, mutation, and genetic drift (assuming a constant environment).

Figure 2

Genetic interactions and the breakdown of QLE. (A) The range of validity of QLE as a function of $\overline{\sigma }/r$ and the heritability, i.e., the ratio of additive variance to the total fitness variance. Below the transition line, strong linkage equilibrium is expected and selection operates on genotypes rather than alleles. (B) An illustration of a possible “interaction graph” of polymorphisms on a block of chromosome. Loci interact with nearby loci, as well as with distant loci outside the block. The epistatic fitness variance solely within the block, i.e., averaged over the rest of chromosome, is proportional to the number of interaction terms (arcs in the figure).