A path integral approach for allele frequency dynamics under polygenic selection.

Many phenotypic traits have a polygenic genetic basis, making it challenging to learn their genetic architectures and predict individual phenotypes. One promising avenue to resolve the genetic basis of complex traits is through evolve-and-resequence (E&R) experiments, in which laboratory populations are exposed to some selective pressure and trait-contributing loci are identified by extreme frequency changes over the course of the experiment. However, small laboratory populations will experience substantial random genetic drift, and it is difficult to determine whether selection played a role in a given allele frequency change (AFC). Predicting AFCs under drift and selection, even for alleles contributing to simple, monogenic traits, has remained a challenging problem. Recently, there have been efforts to apply the path integral, a method borrowed from physics, to solve this problem. So far, this approach has been limited to genic selection, and is therefore inadequate to capture the complexity of quantitative, highly polygenic traits that are commonly studied. Here, we extend one of these path integral methods, the perturbation approximation, to selection scenarios that are of interest to quantitative genetics. We derive analytic expressions for the transition probability (i.e. the probability that an allele will change in frequency from x to y in time t) of an allele contributing to a trait subject to stabilizing selection, as well as that of an allele contributing to a trait rapidly adapting to a new phenotypic optimum. We use these expressions to characterize the use of AFC to test for selection, as well as explore optimal design choices for E&R experiments to uncover the genetic architecture of polygenic traits under selection.