In population genetics, allele-frequency statistics are usually studied under weak selection or neutrality. However, in many biological scenarios, such as adaptation of viruses and bacteria or artificial selection, selection is strong relative to drift. Nearby to this work are studies that either focus on long-term outcomes under strong selection or use a single deterministic trajectory. This paper fills the gap: it examines transient (time-dependent) statistics in the strong-selection regime, where standard approximations produce large errors because they ignore trajectories leading to allele loss.
The author works within the diffusion approximation of the Wright-Fisher model. A parameter R = 2Ne|s| is introduced, where Ne is the effective population size and s is the selection coefficient. When R >> 1, selection is strong. The allele-frequency statistic is represented as a weighted sum of two conditional expectations: conditioned on fixation and conditioned on loss. Each conditional expectation is then rewritten in terms of a modified process with altered selection. Next, an expansion is applied in the small parameter ε = 1/R, where √ε is the noise (drift) amplitude. The approximation retains terms up to order ε, which captures the contribution from the two deterministic trajectories and the fluctuations around them. The equations for the deterministic trajectories and the fluctuation corrections are solved numerically.
Explicit expressions are obtained for the mean and variance of the allele frequency as functions of time. The mean is approximated by a sum of weighted deterministic trajectories plus a correction of order 1/R. The variance contains a term arising from the difference between the two deterministic trajectories and a term from fluctuations. Numerical comparison with exact results from the Wright-Fisher model shows that a purely deterministic approximation can produce errors of up to 100% for the variance, especially when the initial frequency is small. Including the fluctuation correction reduces the error to a few percent. For example, at R=50 and initial frequency y=0.001, the error in the mean drops from 10.6% to 3.2%, and that in the variance from 12.6% to 4.9%. The fluctuation contribution is especially important at early times, when the deterministic variance is nearly zero.
The work shows that even under strong selection, trajectories leading to allele loss cannot be neglected. Unlike neighboring studies that use a single deterministic trajectory or consider only long-term limits, this paper offers a systematic method for transient statistics. This fills an important gap: it is now possible to correctly compute the temporal dynamics of allele frequencies in the strong-selection regime, which was impossible with standard approaches. The method also reveals that fluctuations are not always small corrections: for the variance, they can dominate over certain time intervals.
The method can be extended to the case of multiple alleles, where summation over fixations of different alleles will be required. Incorporating mutations is also possible, though it will require a different approach, since the stationary distribution is not described by two deterministic trajectories. Generalization to a changing population size and variable selection is also of interest. In addition, one could study behavior in the boundary layers near frequencies 0 and 1, where asymptotic expansions may require separate treatment.
The method is applicable to analyzing time series of allele frequencies in experimental evolution, viral and bacterial populations, and artificial selection programs. It can improve the estimation of selection parameters from data and help predict the fate of beneficial alleles in small populations, which is important for conservation genetics.
The next step could be to apply the method to real time-series data on allele frequencies to test its effectiveness in estimating selection coefficients. It is also worth exploring the extension to multiple alleles and the inclusion of mutations.
In population genetics, the problem of accurately describing the transient dynamics of allele frequencies under strong selection remains open, especially for the variance and higher moments. This work directly addresses this problem for the two-allele case without mutations, offering a computationally efficient method. It also raises the question of the applicability of asymptotic expansions near the boundaries of the frequency range.
Even under strong selection, when the deterministic force is large, the variance of the allele frequency can be almost entirely determined by random fluctuations for a long time. For example, at R=50 and initial frequency y=0.001, the deterministic contribution to the variance is essentially zero up to time τ≈5, whereas the fluctuation contribution accurately describes the actual variance.