Information encoded in gene-frequency trajectories.

In this work we present a systematic mathematical approximation scheme that exposes the way that information, about the evolutionary forces of selection and random genetic drift, is encoded within gene-frequency trajectories. We determine approximate, time-dependent, gene-frequency trajectory statistics, assuming additive selection. We use the probability of fixation to test and illustrate the approximation scheme introduced. For the case where the strength of selection and the effective population size have constant values, we show how a standard diffusion approximation result, for the probability of fixation, systematically emerges when increasing numbers of approximate trajectory statistics are taken into account. We then provide examples of how time-dependent parameters influence gene-frequency statistics.

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.