The dynamics of infinitesimally rare alleles, applied to the evolution of mutation rates and the expression of deleterious mutations
A classical model of Kimura and Maruyama [see, e.g., \textit{T. Maruyama} and \textit{M. Kimura}, Proc. Natl. Acad. Sci., U.S.A. 72, 1602-1604 (1975)] is investigated, in which infinitesimally rare deleterious alleles segregate at an infinite number of unlinked loci. It has been long known that under multiplicative selection a population that initially is in linkage equilibrium (i.e., the allele frequencies are stochastically independent across loci) remains in linkage equilibrium. However, in practice, linkage equilibrium will often be distorted. In this article, the author provides an exact and explicit solution for the dynamics of this model with arbitrary initial conditions. He achieves this by describing the frequency distribution of types in terms of the factorial cumulant generating function and the factorial cumulants. The latter are most useful in describing departures from the Poisson distribution. Thus, he obtains an explicit representation for the decay of the linkage disequilibria. Furthermore, he applies this method to determine the joint dynamics of a modifier of the mutation rate together with deleterious alleles, whose mutation rate he is altering, and derives a simple invasion criterion for such a modifier.
scientific article; zbMATH DE number 1271888
scientific article