Dynamics of unconditionally deleterious mutations: Gaussian approximation and soft selection.

This paper studies the influence of two opposite forces, unidirectional unconditionally deleterious mutations and directional selection against them, on an amphimictic population. Mutant alleles are assumed to be equally deleterious and rare, so that homozygous mutations can be ignored. Thus, a genotype is completely described by its value with respect to a quantitative trait chi, the number of mutations it carries, while a population is described by its distribution p(chi) with mean M[p] and variance V[p] = sigma(2)[p]. When mutations are only slightly deleterious, so that M > 1, before selection p(chi) is close to Gaussian with any mode of selection. I assume that selection is soft in the sense that the fitness of a genotype depends on the difference between its value of chi and M, in units of sigma. This leads to a simple system of equations connecting the values of M and V in successive generations. This system has a unique and stable equilibrium, M = U/delta)2(2--rho) and V = (U/delta)2, where U is the genomic deleterious mutation rate, delta is the selection differential for chi in units of sigma, and rho is the ratio of variances of p(chi) after and before selection. Both delta and rho are parameters of the mode of soft selection, and do not depend on M or V. In an equilibrium population, the selection coefficient against a mutant allele is s = delta2[U(2--rho)]-1. The mutation load can be tolerable only if the genome degradation rate v = U/sigma is below 2. Other features of mutation-selection equilibrium are also discussed.