Loss and fixation of strongly favoured new variants: Understanding and extending Haldane’s result via the Wright–Fisher model

The implications of Haldane’s analysis for the fixation of beneficial alleles lies at the heart of much of ‘population genetic thinking’ and underlies many approaches that have been tailored to the detection of positive selection. Within the framework of a branching process, Haldane gave an approximation for the probability that fixation ultimately occurs when the selective advantage of a beneficial allele is small (≪1). Here, we make no use of branching processes. Rather, we work solely within a finite-population Wright-Fisher framework. We use this framework to analyse where Haldane’s result applies, and extend Haldane’s analysis. In particular, we present results for: (i) the domain of applicability of Haldane’s analysis; (ii) the probability that loss occurs up to a given time; (iii) the probabilities that loss and fixation ultimately occur; (iv) an analytic approximation associated with the probability of loss and fixation ultimately occurring; (v) quantification of the crossover from weak to strong selection; (vi) determination of the number of invasive alleles that have a significant probability (>0.95) of invading a novel population. We note that the results obtained for (ii), (iii) and (iv) hold for an arbitrary initial number of mutations, and for selection that can be arbitrarily strong. Our results have fundamental implications for population and conservation genetics, and open up new avenues to identify traces of historically beneficial alleles through comparative genomics.