Heterozygosity decline
Drift removes heterozygosity at a fixed rate of 1/2Ne per generation, on average, and single populations scatter around that average.
Read first: Genetic drift
What H measures
Heterozygosity here is H = 2p(1 − p), written 2pq. It is the chance that two gene copies drawn at random from the population carry different alleles. At p = 0.5 it is 0.5, its maximum. At p = 0 or p = 1 it is 0, because no two copies can differ.
Drift lowers H. Each generation the next 2Ne gene copies are a random sample of the current ones, so p wanders and sooner or later reaches 0 or 1. The module plots H over time for a diploid population of effective size Ne, with no mutation, selection or migration.
The module draws two things. The first is one smooth curve, the expected H. The second is a few simulated populations. The expected curve is H₀(1 − 1/2Ne)^t, where H₀ = 2p₀(1 − p₀). With Ne = 50 and p₀ = 0.5, H falls by 1% each generation: 0.5 × 0.99^t.
Two numbers fix the curve. Ne sets the rate and p₀ sets the starting level. At Ne = 50 the first steps are 0.5, 0.495, 0.490. After 100 generations 0.99^100 = 0.366 of the diversity is left. At Ne = 10 the factor is 0.95, and 20 generations leave 0.358 of it.
Expected decay and four populations
Why the rate is 1/2Ne
The new frequency p′ is a binomial sample of 2Ne copies, so its variance is p(1 − p)/2Ne and its mean is p. Then E[2p′(1 − p′)] = 2(p − p² − Var) = 2p(1 − p)(1 − 1/2Ne). The expected H shrinks by the factor (1 − 1/2Ne) every generation.
There is a second way to see it. Pick two gene copies from the new generation. They descend from the same parental copy with probability 1/2Ne, and then they are identical. H is the chance they are not, so it loses 1/2Ne of its value each generation.
The half-life of the expected curve follows. It is ln 2 divided by −ln(1 − 1/2Ne), which is about 1.4 Ne generations. For Ne = 50 that is 68.97 generations, and the curve reads 0.250 at generation 69. The half-life does not depend on p₀. A start at p₀ = 0.1 has H₀ = 0.18 and halves at the same time.
Ne is the effective size. It is the size of an ideal Wright-Fisher population that drifts as fast as the real one. Unequal sex ratios, overlapping generations and uneven family sizes make it smaller than the head count, so real populations lose H faster than their census suggests.
Eight populations of ten
Five hundred keeps diversity
Exercise
Compare two sizes
Set Ne 50, p 0.5, Gen 200 and Drift 0, then run. Save the run in the Runs panel. Change Ne to 500, run again and tick the saved run to overlay it.
What to expect. The Ne = 50 curve ends at 0.067. The Ne = 500 curve ends at 0.409, six times higher. Hover the end of each curve to read the values. Then set Ne 10 and Gen 100 and watch H collapse at 5% per generation.
Exercise
Start lower, same rate
With Ne 50, Gen 200 and Drift 0, run p 0.5 and then p 0.1. Divide each final H by its starting H.
What to expect. The start is 0.50 for p = 0.5 and 0.18 for p = 0.1. The ends are 0.067 and 0.024. Both ratios are 0.134, which is 0.99^200. The starting frequency sets the level and Ne sets the rate.
Open the walkthrough step →Summary
- H = 2p(1 − p) is the chance that two random gene copies differ.
- Expected H falls by a factor (1 − 1/2Ne) per generation, so H_t = H₀(1 − 1/2Ne)^t.
- The half-life is about 1.4 Ne generations, 68.97 for Ne = 50, whatever the starting p.
- The factor comes from the binomial sampling variance p(1 − p)/2Ne, or from two copies sharing a parent copy with probability 1/2Ne.
- A single population is ragged and ends at H = 0. The smooth curve is the average over populations.