  GWAS in humans are revealing the genetic architecture of biomedical and
anthropomorphic traits, i.e., the frequencies and effect sizes of variants that
contribute to heritable variation in a trait. To interpret these findings, we
need to understand how genetic architecture is shaped by basic population
genetics processes-notably, by mutation, natural selection and genetic drift.
Because many quantitative traits are subject to stabilizing selection and
genetic variation that affects one trait often affects many others, we model
the genetic architecture of a focal trait that arises under stabilizing
selection in a multi-dimensional trait space. We solve the model for the
phenotypic distribution and allelic dynamics at steady state and derive robust,
closed form solutions for summary statistics of the genetic architecture. Our
results provide a simple interpretation for missing heritability and why it
varies among traits. They also predict that the distribution of variances
contributed by loci identified in GWAS is well approximated by a simple
functional form that depends on a single parameter: the expected contribution
to genetic variance of a strongly selected site affecting the trait. We test
this prediction against the results of GWAS for height and BMI and find that it
fits the data well, allowing us to make inferences about the degree of
pleiotropy and mutational target size for these traits. Our findings help to
explain why the GWAS for height explains more of the heritable variance than
similarly-sized GWAS for BMI, and to predict the increase in explained
heritability with study sample size. Considering the demographic history of
European populations, in which these GWAS were performed, we further find that
most of the associations they identified likely involve mutations that arose
during the out of Africa bottleneck at sites with selection coefficients around
$s=10^{-3}$.
