Temporal Allele-Frequency Change Under Genetic Drift

Why Allele Frequencies Change Without Selection

An allele can become more or less common even when it gives its carriers no reproductive advantage. In a finite population, the alleles passed to the next generation are a random sample of the current gene pool. This generation-to-generation randomness is genetic drift. Its expected direction is not toward higher or lower frequency: under a neutral model, the next-generation frequency has the current frequency as its conditional mean. But its variance is positive, and that variance accumulates over generations.

A second source of variation arises when researchers observe only a sample of individuals or gene copies. A sample estimate may differ from the population frequency even if the population frequency is known to be unchanged. These two sources must not be conflated: drift describes a random change in the population itself, while observation error describes uncertainty about the population frequency from finite data.

The Wright–Fisher model is an idealization. \(N_e\) is the effective size governing drift, not necessarily the census number of individuals. The model also assumes a single neutral allele-frequency process; departures such as selection or migration can create systematic changes that are not represented by the drift expectation derived here.

One-Generation Expected Change and Variance

Write the one-generation change as \(\Delta p_{t+1}=p_{t+1}-p_t\). Conditional on the current population frequency, \(X_{t+1}\) is binomial with mean \(Mp_t\) and variance \(Mp_t(1-p_t)\). Dividing the count by \(M\) gives the frequency. Therefore:

Subtracting the fixed, given value \(p_t\) does not change the conditional variance, so \(E(\Delta p_{t+1}\mid p_t)=0\) and \(\operatorname{Var}(\Delta p_{t+1}\mid p_t)=p_t(1-p_t)/(2N_e)\). The conditional standard deviation of the change is thus \(\sqrt{p_t(1-p_t)/(2N_e)}\). Drift is more variable when the population is smaller and, for a fixed size, when the allele is at an intermediate frequency. The variance is zero at frequency 0 or 1, where the allele is absent or fixed and cannot change under this model.

An expected change of zero does not mean the allele frequency will remain exactly constant. It means that, across repeated populations starting at the same frequency and following this model, positive and negative changes balance in expectation. A particular population follows one random trajectory.

Worked Change Versus Model-Based Variation

An observed difference can be compared with the model's one-generation standard deviation to describe its scale relative to drift. That comparison is not automatically a formal hypothesis test: a standard deviation alone is not a p-value, and the exact distribution of the change is discrete and binomial. The next example shows the arithmetic and its limited interpretation.

How Variance Accumulates Across Generations

Even though the conditional expected change is zero in each generation, the population frequency becomes more variable across replicate histories as generations pass. Let the initial frequency \(p_0\) be fixed, and let \(V_t=\operatorname{Var}(p_t)\) across replicate populations at generation \(t\). The law of total variance gives a recurrence for \(V_t\).

The result also implies \(E(p_t)=p_0\) for every \(t\): the population frequency remains a martingale under this neutral model. But its variance increases toward \(p_0(1-p_0)\). As a related summary, expected heterozygosity for a biallelic locus, \(2p_t(1-p_t)\), declines over time in expectation. In fact, \(E\{2p_t(1-p_t)\}=2p_0(1-p_0)(1-1/(2N_e))^t\). These results describe the effect of finite-population sampling alone, under the assumptions stated above.

Separating Drift From Observation-Sampling Noise

Suppose researchers sample \(m_t\) gene copies from the population at time \(t\), with each sampled copy modeled as an independent draw from that generation's population. Let \(\hat p_t\) be the sample allele frequency. Conditional on \(p_t\), the estimate has mean \(p_t\) and variance \(p_t(1-p_t)/m_t\). This sampling variation is distinct from the random change in \(p_t\) caused by reproduction.

For two independent samples taken in consecutive generations, the conditional variance of their difference includes all three contributions: drift between generations, sampling at the later time, and sampling at the earlier time. Under the model and independent sampling assumptions:

For the later sample, the second term averages its binomial sampling variance over the random next-generation frequency. Specifically, \(E\{p_{t+1}(1-p_{t+1})\mid p_t\}=p_t(1-p_t)(1-1/(2N_e))\). The equation assumes the research samples are independent of each other and of the reproduction process. Sampling gene copies without replacement from a non-negligible fraction of a finite population requires a finite-population correction instead of the independent-binomial sampling variance used here.

Common Mistake: Treating a Sample Difference as the Population Change

A common error is to observe \(\hat p_{t+1}-\hat p_t\) and treat it as if it were exactly \(p_{t+1}-p_t\). That ignores uncertainty in both estimates. The correction is to distinguish the population frequencies from their sample estimates, then include the sampling variances when assessing how surprising the observed difference is under a model. A second error is to interpret \(E(\Delta p_{t+1}\mid p_t)=0\) as “there is no drift.” In fact, drift is precisely the random variability around that zero expected change.

Temporal data can therefore help characterize allele-frequency dynamics, but conclusions depend on the model and on how frequencies were measured. A departure from a neutral drift expectation could reflect selection, migration, mutation, population-size misspecification, sampling design, or other violations of the assumptions. The calculation alone does not distinguish among those explanations.