AI 中文总结
本研究在强选择机制下，通过分别以最终固定和丢失为条件，结合确定性轨迹与波动，提出了等位基因频率统计的近似方法，准确预测了均值和方差。
AI 中文摘要
在许多生物学背景下，选择相对于遗传漂变是强的。在扩散近似下，我们定义 $R = 2N_{e}|s|$，其中 $N_{e}$ 是有效种群大小，$s$ 是与焦点等位基因相关的选择系数。强选择对应于 $R \gg 1$，并且可能发生在相对适中的参数值下。例如，$N_e = 10^3$ 和 $s = 10^{-2}$ 产生 $R = 20$。本工作的重点是强选择机制下等位基因频率分布的统计。在这种机制下，标准近似常常失效。例如，在强正选择下（$R\gg1$ 且 $s>0$），走向固定的轨迹似乎主导等位基因频率统计。然而，忽略最终实现丢失的轨迹可能导致大的误差。在突变可以忽略的时间尺度上，所有等位基因频率轨迹分为两类：那些最终实现固定的和那些最终实现丢失的。我们通过分别以焦点等位基因的最终固定和最终丢失为条件，确保两类轨迹都对时间相关统计有贡献。对于大的 $R$，我们在小噪声近似下确定等位基因频率统计的近似，该近似从两个确定性轨迹（一个实现固定，另一个实现丢失）以及围绕这两个轨迹的波动中导出贡献。这种方法给出了平均等位基因频率及其方差的正确长时间值。与Wright-Fisher模型的数值比较表明，仅确定性轨迹对统计的贡献可能并不总是足以获得良好的精度，但加入波动后，可获得合理的精度。
英文摘要
In many biological contexts selection is strong relative to genetic drift. Working under the diffusion approximation, we define $R = 2N_{e}|s|$, where $N_{e}$ is the effective population size and $s$ is the selection coefficient associated with a focal allele. Strong selection corresponds to $R \gg1$ and can occur for relatively modest parameter values. For example, $N_e = 10^3$ and $s = 10^{-2}$ yield $R = 20$. The focus of this work is on statistics of the allele frequency distribution in the strong selection regime. In this regime, standard approximations often break down. For instance, under strong positive selection ($R\gg1$ and $s>0$), trajectories that proceed to fixation seem to dominate allele frequency statistics. However, omission of trajectories that ultimately achieve loss can lead to large errors. Over timescales where mutation can be neglected, all allele frequency trajectories fall into one of two classes: those that eventually achieve fixation and those that eventually achieve loss. We ensure that both types of trajectory contribute to time-dependent statistics, by separately conditioning on the eventual fixation and eventual loss of the focal allele. For large $R$, we determine approximations for allele-frequency statistics under a small noise approximation that derives contributions from two deterministic trajectories, one achieving fixation the other loss, along with fluctuations around these two trajectories. Such an approach yields the correct long time values of the mean allele frequency and its variance. Numerical comparisons with the Wright-Fisher model indicate that the contribution of the deterministic trajectories alone, to a statistic, may not always be sufficient for good accuracy, but with the inclusion of fluctuations, reasonable accuracy is obtained.
Comments34 pages, 3 figures
