Lesson 10 — Watching alleles wander in a finite population

BIO 202, Spring 2026, draft v1. Wright-Fisher drift. Allele frequencies stagger around at random until they hit zero or one and the variation is gone.

What you'll do

Four stages. Watch trajectories. Watch them hit absorbing boundaries. Watch the fixation-probability statistic equal the starting frequency for neutral alleles. End at the Buri flies — Wright-Fisher in a lab.

In the big population, one lightning strike had a 0.005% effect. The smaller population has a 0.5% effect. On its own, that one lightning strike is not going to change much. But it's not literally zero effect. And that one lightning strike is not the only random thing that's going to happen to my population of birds, whether it's 10,000 or 100. All the random effects in a generation get summed together at the next round of reproduction. Each generation doesn't start fresh. — 202_lec09_06

A — One trajectory, then 50

A single allele drifts in a Wright-Fisher population. Slide N, slide p₀, slide seed. Watch one realization. Toggle the fan to see 50.

Locked — confirm your name above to begin.

Scenario

Each generation, sample 2N alleles with replacement from the previous generation. That is the Wright-Fisher rule. The expected next-generation frequency is the current frequency; the variance is p(1−p)/(2N).

Allele frequency over generations

N: 100  |  p₀: 0.50  |  gens: 200  |  final p (this run): —

Prediction

  1. Q1. Across 50 independent replicates with N = 100, p₀ = 0.5, after 200 generations:
Try at least 5 (N, p₀, seed) combinations to unlock Stage B. 0/5 combos

Controls

100
0.50
200
42

Click two points on the canvas to mark the upper and lower bound of where you expect most replicate trajectories to land at generation 200. Then toggle the fan to check.

R code — Wright-Fisher one trajectory

set.seed(42)N <- 100p <- 0.5gens <- 200traj <- numeric(gens + 1); traj[1] <- pfor (g in 1:gens) {  p <- rbinom(1, 2*N, p) / (2*N)  traj[g + 1] <- p}plot(traj, type = "l", ylim = c(0,1))

B — Run to fixation

Let the simulation run until every replicate is at 0 or 1. Plot the distribution of times-to-fixation. Mean is around 4N for neutral alleles starting at p = 0.5.

Complete Stage A to unlock this section.

Scenario

500 replicates, run until each is fixed. Histogram the fixation times. The expected mean for a neutral allele at p = 0.5 is about 2.8N generations (Kimura's classical result); conditional on fixation, ~4N.

Time to fixation (across replicates)

N: 50  |  replicates: 500  |  mean time: —  |  % fixed A: —

Prediction

  1. Q1. With N = 50, p₀ = 0.5, the mean time to fixation across 500 replicates will be:
Try at least 4 N values to unlock Stage C. 0/4 N values

Controls

50
0.50
42

R code — time to fixation

set.seed(42)N <- 50; p0 <- 0.5; reps <- 500tfix <- replicate(reps, {  p <- p0; g <- 0  while (p > 0 && p < 1) { p <- rbinom(1,2*N,p)/(2*N); g <- g + 1 }  g})mean(tfix); hist(tfix)

C — Fix(p₀) = p₀

For a neutral allele, the probability of fixation equals the starting frequency. Departures from this line are the signature of selection.

Complete Stage B to unlock this section.

Scenario

For each of 8 starting frequencies (0.1, 0.2, …, 0.9), run 300 replicates. Plot fraction that fixed the allele against p₀. The diagonal is the neutral prediction; deviation is selection (or sampling noise).

Fixation probability vs starting frequency

N: 50  |  replicates/level: 300  |  selection coeff s: 0.00  |  RMSE from p₀ line: —

Prediction

  1. Q1. With s = 0 (no selection) and N = 50, the eight points will:
Try at least 3 s values (0, weak, strong) to unlock Stage D. 0/3 s values

Controls

50
0.00
42

R code — fixation probability scan

set.seed(42)N <- 50; s <- 0; reps <- 300p0s <- seq(0.1, 0.9, 0.1)pfix <- sapply(p0s, function(p0) {  mean(replicate(reps, {    p <- p0    while (p > 0 && p < 1) {      p_sel <- p*(1+s)/(p*(1+s)+(1-p))      p <- rbinom(1,2*N,p_sel)/(2*N)    }    p == 1  }))})plot(p0s, pfix); abline(0, 1)

D — Buri 1956 — 107 fly lines

Peter Buri tracked the bw75 allele in 107 Drosophila lines for 19 generations, with 16 flies per line per generation. The variance in allele frequency across lines should grow at the rate Wright-Fisher predicts.

Complete Stage C to unlock this section.

Scenario

Buri's empirical trajectories overlaid with a Wright-Fisher simulation at N=16 (the actual sample size each generation). The simulation matches the observed spread — drift accounts for the variance.

Observed Buri lines vs simulated WF lines

N (per line): 16  |  gens: 19  |  observed lines fixed: —  |  predicted lines fixed: —

Prediction

  1. Q1. After 19 generations with N = 16, what fraction of the 107 Buri lines do you expect to have fixed or lost the allele?
Resimulate at least 3 times. 0/3 sims

Controls

42

R code — Buri overlay

buri <- read.csv("data/clean/buri_fly.csv")set.seed(42); N <- 16; gens <- 19; lines <- 107; p0 <- 0.5sim <- replicate(lines, {  p <- p0; traj <- p  for (g in 1:gens) { p <- rbinom(1,2*N,p)/(2*N); traj <- c(traj, p) }  traj})matplot(t(sim), type = "l", col = rgb(0,0,0,0.1))