← all lessons

Scaffold S5 — Time to fixation scales with Ne

Five rounds of Wright–Fisher drift. You are given p0 and Ne; 500 replicate populations run forward until each one fixes (p = 1) or is lost (p = 0). Predict the median number of generations it takes to resolve. Rounds 1–3 fix p0 = 0.5 and sweep Ne across 10× steps. Rounds 4–5 change p0 at fixed Ne.

Locked — answer the pretest above first.

Running tally — median resolution time vs. Ne

What you just did has a name

Rounds 1–3 fixed the starting frequency at 0.5 and varied only Ne: 20, 50, 200. The median time to resolution scaled almost perfectly linearly with Ne. A 10× larger population takes roughly 10× as long to lose or fix an allele. If you plot median time on the y-axis against Ne on the x-axis (in the tally above), the three symmetric points fall on a line through the origin with slope ≈ 2.

The theoretical prediction under neutral drift is E[time to resolution | p₀ = 0.5] ≈ 2.77 · Ne generations (Kimura 1955). The more general result is the Kimura–Ohta formula: E[t | fix] = −4Ne(1−p)/p · ln(1−p), E[t | loss] = −4Ne·p/(1−p) · ln(p).

Rounds 4 and 5 varied p0 at fixed Ne = 50. Starting at 0.1 or 0.9 (symmetric cases) gave much shorter median times — around 30 generations instead of 120. Most populations lose a rare allele (or fix a common one) within a few dozen generations; the asymmetry of the starting frequency is what speeds things up. The tall tail of "fixed a rare allele slowly" is still there, but by definition it sits past the median.

The bottom line: Ne sets the clock; p0 sets the starting hand. Both change what you see.