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We investigate the behavior of a population genetics model introduced byWaxman and Peck incorporating mutation, selection, and pleiotropy. Thepopulation is infinite and continuous variation of genotype is allowed.Nonetheless, Waxman and Peck showed that if the degree of pleiotropy is largeenough,in this model a nonzero fraction of the population can have identicalalleles. This `condensed mode' behavior appears in the limit of infinite times.  This paper explores the time-dependence of the distribution of alleles inthis model. First, the model is analyzed using a recursion technique whichenables the distribution of alleles to be calculated at finite times as well asin Waxman and Peck's infinite-time limit. Second, both Waxman and Peck'soriginal model and a related model in which mutations occur continuously aremapped onto problems in quantum mechanics. In both cases, the long-timeanalysis for the biological model is equivalent to finding the nature of theeigenstates of the quantum problem. The condensed mode appears if and only ifthere is no bound state in the quantum problem.  We compare the behavior of the discrete- and continuous-time versions of themodel. The results for the two cases are qualitatively similar, though thereare some quantitative differences. We also discuss our attempts to correlatethe statistics of DNA sequence variations with the degree of pleiotropy ofvarious genes.Comment: 26 pages, 2 tables, 6 figures, submitted to J. Math. Biolog

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