Effective size and polymorphism of linked neutral loci in populations under directional selection

The general theory of the effective size (N-e) for populations under directional selection is extended to cover linkage. N-e is a function of the association between neutral and selected genes generated by finite sampling. This association is reduced by three factors: the recombination rate, the reduction of genetic variance due to drift, and the reduction of genetic variance of the selected genes due to selection. If the genetic size of the genome (L in Morgans) is not extremely small the equation for N-e is N-e = N exp(-C-2/(1-Z) L), where N is the number of reproductive individuals, C-2 is the genetic variance for fitness scaled by the squared mean fitness, (1 - Z) = V-m/C-2 is the rate of reduction of genetic variation per generation and V-m is the mutational input of genetic variation for fitness. The above predictive equation of N-e is valid for the infinitesimal model and for a model of detrimental mutations. The principles of the theory are also applicable to favorable mutation models if there is a continuous nux of advantageous mutations. The predictions are tested by simulation, and the connection with previous results is found and discussed. The reduction of effective size associated with a neutral mutation is progressive over generations until the asymptotic value (the above expression) is reached after a number of generations. The magnitude of the drift process is, therefore, smaller for recent neutral mutations than for old ones. This produces equilibrium values of average heterozygosity and proportion of segregating sites that cannot be formally predicted from the asymptotic N-e, but both parameters can still be predicted by following the drift along the lineage of genes. The spectrum of gene frequencies in a given generation can also be predicted by considering the overlapping of distributions corresponding to mutations that arose in different generations and with different associated effective sizes.