Scaffold S9 — Mutation–selection balance

What you just did has a name

The general mutation–selection balance for a deleterious allele in a diploid population is: q̂ ≈ (−hs + √((hs)² + 4μs)) / (2s). Two limiting cases cover most biology:

• When h = 0 (fully recessive), q̂ = √(μ/s). The carrier frequency 2q̂ can reach 0.5–1%.
• When hs ≫ √(μs), the allele is exposed in heterozygotes and q̂ = μ/(hs). Selection sees heterozygotes; the allele is hammered out.

Rounds 1 and 2 differed only in h (0 vs. 0.5) — and the equilibrium frequency changed by a factor of 158. Round 4 sat near the crossover h ≈ √(μ/s), where the two approximations give different answers; the exact formula is required.

Round 5 was cystic fibrosis: predicted q̂ ≈ 0.0014 from mutation–selection balance, observed q ≈ 0.02 — an order of magnitude higher than the balance equation predicts. The discrepancy is real. Proposed explanations include heterozygote advantage (CF carriers may have been resistant to cholera or typhoid), lower effective s in historical populations, founder effects, and population-specific mutation rate. This is a genuine open question. When a balance equation fails by 10×, the biology is more complicated than the equation.