Quantitative Genetics; Chap 6 Population Genetics by J.H. Gillespie
Finally my reading welcomed my beloved statistical applications: law of large number and regression, in addition to the fact that nonrandom mating chapter had probability theory application that I enjoyed. Reading on this book has becoming easier and more fun. This chapter first formulated trait variation in the population then showed what happen when there is selection and how does it evolve.
Trait variation
Phenotypic variation was broken down into additive contributions of maternal, paternal and environmental components (P= Xm + Xp + e). As for the genetic contribution, Xm or Xp, it introduced an interesting representation of population trait value X, by the allele frequencies (pi) that corresponds to different phenotypes (xi), so that the probability of X being xi is pi. So the mean allelic contribution of this locus is the sum of xi * pi, which should be zero.
Heritability
The proportion of additive effect over the total phenotypic variation is heritability, h. Another way of getting at this heritability is to look at covariance of phenotypes between offspring and parent. The phenotype of the parent and offspring is formulated respectively: Pp= Xm +Xm’ + ep and Po= Xm +Xp +eo. The covariance, after removing the zero covariance terms (assuming alleles are independently inherited), it comes to Var {Xm} only. Var {Xm} is half of VA, additive genetic variance, assuming Var {Xm} and Var {Xp} are equal. So Cov{Pp, Po}= VA/2. Then the correlation of the Pp and Po is VA/(2*Vp), thus half heritability. To generalize this relation, the correlation between relatives are just r * h. This is also happen to be the slope of regression line between phenotypes of relative Y against relative X.
Does natural history traits tend to have lower heritability? It’s mentioned that such traits might have higher environmental variance, or they tend to be the under strong natural selection, thus have lower additive variance.
Selection
Clayton (1957) did a selection experiment in which he bred the extreme flies in terms of bristle number. Over 24 generations, there was indeed separation in trait distribution of the low bristle line versus high bristle line. However such divergence did’t continue but leveled off. It was suspected that there might be a optimal number of bristle number and natural selection counteract artificial selection bringing the population back around the optima.
There are a few key terms for selection: S, selection differential; R, selection response, i(p), intensity of selection, Kh, the rate of phenotypic evolution (Haldane).
R=h^2 * S = h^2 *i(p) * sqrt(Vp).
Kh= ln(x2/x1)/T, where x1 and x2 are measures of some traits for two species that are separated for T millions of years.