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

Influence of selection on the probability of fixation at a locus with multiple alleles

BACKGROUND

Genes exist in a population in a variety of forms (alleles), as a consequence of multiple mutation events that have arisen over the course of time. In this work we consider a locus that is subject to either multiplicative or additive selection, and has n alleles, where n can take the values 2, 3, 4, … . We focus on determining the probability of fixation of each of the n alleles. For n = 2 alleles, analytical results, that are 'exact', under the diffusion approximation, can be found for the fixation probability. However generally there are no equally exact results for n ≥ 3 alleles. In the absence of such exact results, we proceed by finding results for the fixation probability, under the diffusion approximation, as a power series in scaled strengths of selection such asR i , j = 2 N e ( s i - s j ) , where N e is the effective population size, while s i and s j are the selection coefficients associated with alleles i and j, respectively.

RESULTS

We determined the fixation probability when all terms up to second order in the R i , j are kept. The truncation of the power series requires that the R i , j cannot be indefinitely large. For magnitudes of the R i , j up to a value of approximately 1, numerical evidence suggests that the results work well. Additionally, results given for the particular case of n = 3 alleles illustrate a general feature that holds for n ≥ 3 alleles, that the fixation probability of a particular allele depends on that allele's initial frequency, but generally, this fixation probability also depends on the initial frequencies of other alleles at the locus, as well as their selective effects.

CONCLUSIONS

We have analytically exposed the leading way the probability of fixation, at a locus with multiple alleles, is affected by selection. This result may offer important insights into CDCV traits that have extreme phenotypic variance due to numerous, low-penetrance susceptibility alleles.

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.