Analysis of a Population Genetics Model with Mutation, Selection, and Pleiotropy

We investigate a population genetics model introduced by Waxman and Peck⁽¹⁾ incorporating mutation, selection, and pleiotropy (one gene affecting several unrelated traits). The population is infinite, and continuous variation of genotype is allowed. Nonetheless, Waxman and Peck showed that if each gene affects three or more traits, then the steady-state solution of the model can have a nonzero fraction of the population with identical alleles. We use a recursion technique to calculate the distribution of alleles at finite times as well as in the infinite-time limit. We map Waxman and Peck's model into the mean-field theory for Bose condensation, and a variant of the model onto a bound-state problem in quantum theory. These mappings aid in delineating the region of parameter space in which the unique genotype occurs. We also discuss our attempts to correlate the statistics of DNA-sequence variation with the degree of pleiotropy of various genes.