The Wright-Fisher Model

In a diploid population of constant size $N$, the number of copies of allele $A$ in generation $t+1$ follows a binomial draw:

\[X_{t+1} \mid X_t = k \;\sim\; \text{Binomial}\!\left(2N,\, \frac{k}{2N}\right)\]

Allele frequency $p_t = X_t/(2N)$ is a bounded martingale. Both 0 and 1 are absorbing states; the probability of ultimate fixation starting at frequency $p_0$ is simply $p_0$.

Genetic Drift and Effective Population Size

Variance in allele frequency change per generation under pure drift:

\[\text{Var}(\Delta p) = \frac{p(1-p)}{2N_e}\]

The effective population size $N_e$ absorbs deviations from the idealised WF model (unequal sex ratios, variance in offspring number, population size fluctuations). For fluctuating sizes across $g$ generations: $1/N_e = (1/g)\sum_{i=1}^g 1/N_i$ (harmonic mean).

The Coalescent

Looking backward in time, any two gene copies coalesce (find a common ancestor) in a given generation with probability $1/(2N_e)$. Waiting time to coalescence $T_2$ has a geometric distribution; for large $N_e$ it is approximately:

\[T_2 \sim \text{Exponential}\!\left(\frac{1}{2N_e}\right) \quad \text{(in generations)}\]

Nucleotide diversity $\pi = 4N_e\mu$ where $\mu$ is the per-site mutation rate, linking observable variation to demographic parameters.

Tajima’s D

Tajima’s $D$ compares two estimators of $\theta = 4N_e\mu$:

\[D = \frac{\hat{\pi} - \hat{\theta}_W}{\sqrt{\text{Var}(\hat{\pi} - \hat{\theta}_W)}}\]

where $\hat{\pi}$ is the average pairwise difference and $\hat{\theta}W = S/a_1$ (Watterson) with $S$ segregating sites and $a_1 = \sum{i=1}^{n-1} 1/i$.

$D$	Interpretation
$D < 0$	Excess rare variants — purifying selection or expansion
$D \approx 0$	Neutral expectation
$D > 0$	Excess intermediate variants — balancing selection or bottleneck