Fixation probabilities of additive alleles in diploid populations

The calculation of the survival probability of a selectively advantageous allele is a central part of the quantitative theory of genetic evolution. However, several areas of investigation in population genetics theory, including the generalized neutrality theory, the concept of Muller's ratchet, and the risk of extinction of sexually reproducing populations due to the accumulation of deleterious mutations, rely on the calculation of the survival probability of selectively disadvantageous mutant genes. The calculation of these probabilities in the standard Wright-Fisher model of genetic evolution appears to be intractable, and yet is a key element in the above investigations. In this paper we find bounds for the fixation probability of deleterious and advantageous additive mutants, as well as finding close approximations for these probabilities. In addition, we derive analytical estimates for the relative error of our approximations and compare our results with those from numerical computation. Our results justify the diffusion approximation for the fixation probability of a single mutant.