Abstract :
A new method is presented for analysing the dynamics of a classical model where infinitesimally rare alleles segregate at an infinite number of unlinked loci, and where the alleles at different loci have equivalent effects. The dynamics of the distribution of the number of rare alleles per individual (the “phenotypic distribution”) can be followed without knowing the frequencies of individual genotypes. Meiosis and random union of gametes have a very simple effect on the factorial cumulants of the phenotypic distribution which are consequently the natural set of variables to follow. An exact solution is presented for the dynamics of rare alleles under mutation and multiplicative selection. This solution has a simple representation in terms of the factorial cumulants. Unlike the QLE (quasi-linkage equilibrium) solution, this solution applies even when the population is far from linkage equilibrium. This approach is extended to analyse the joint dynamics of infinitesimally rare alleles at an infinite number of unlinked loci, together with a locus (with arbitrary allele frequencies) with which they interact. This more general method is used to investigate (1) the joint dynamics of a modifier of the mutation rate, together with deleterious alleles under mutation and multiplicative selection, and (2) the fate of an allele that ameliorates or exacerbates the fitness effects of deleterious alleles. When a new modifier allele causes a large change in the mutation rate, strong linkage disequilibrium is generated during its progress. However, using this new approach based on factorial cumulants, it is found that a remarkably simple invasion condition applies to alleles at the modifier locus, even when strong linkage disequilibrium is generated.
