Dominance and the speed of selection
The same harmful allele is removed in 175 generations or in more than 1000, depending on how dominant it is.
Read first: Natural selection at one locus
Two coefficients, two jobs
Allele a is the harmful one. The selection coefficient s says how much two copies cost: the fitness of aa is 1 − s. The dominance coefficient h says how much one copy costs: the fitness of Aa is 1 − hs. The fitness of AA is 1.
With s = 0.1 the three fitnesses (AA, Aa, aa) are 1, 1, 0.9 when h = 0, 1, 0.95, 0.9 when h = 0.5 and 1, 0.9, 0.9 when h = 1. At h = 0 the harm of a is recessive: it shows only in aa. At h = 1 it is dominant: one copy is as bad as two. The module follows the frequency p of the favoured allele A from p = 0.01, with random mating, no drift and no mutation.
A is always favoured here, since AA has the highest fitness. The question is how fast. The answer depends on h, because a rare allele sits mostly in heterozygotes. At p = 0.01 a fraction 2pq = 0.0198 of individuals are Aa and only p² = 0.0001 are AA.
Additive effects: h = 0.5
Why h changes the speed
The change in the frequency of a is Δq = −s·pq·[h(1 − 2q) + q] / w̄. Take a rare a, so q is small. For h > 0 the bracket is close to h, and Δq is about −hs·pq: selection acts on the heterozygotes, where most copies of a live. For h = 0 the bracket is q, and Δq is about −s·q². That is far smaller when q is small.
One generation from p = 0.01 shows it. With s = 0.1, p rises to 0.01109 when h = 0, to 0.01055 when h = 0.5 and to 0.01001 when h = 1. The gain is 0.00109, 0.00055 and 0.00001. An h of 1 shields A nearly completely, because A has to be in AA to gain anything.
The two ends of the range therefore behave in opposite ways. With h = 0 the allele a is removed quickly while common and slowly once rare. With h = 1 the allele A is held back while rare and then takes off.
A dominant, a recessive: h = 0
A recessive, a dominant: h = 1
Exercise
Compare all five dominance values
Set s to 0.1, P(A) to 0.01 and Gen to 500, tick Compare h values and run.
What to expect. The curves for h = 0, 0.25, 0.5 and 0.75 pass 0.5 at generations 54, 65, 86 and 136. The curve for h = 1 stays near 0.02 at generation 500 (0.0221). The h = 0 curve rises first but ends lowest of the four at generation 500, at 0.9769. The other three are within 10⁻⁴ of 1 by then.
Open the walkthrough step →Exercise
Start from p = 0.5
Set P(A) to 0.5 and Gen to 2000. Run with h = 0, then h = 0.5, then h = 1, and note when each passes 0.99.
What to expect. The order reverses. With h = 1 it takes 53 generations, with h = 0.5 it takes 89 and with h = 0 it takes 1022. From p = 0.5 neither allele is rare at first. The end of the run is the loss of a, so what matters is how well selection sees the last copies of a. A dominant a is seen in every carrier. A recessive a is seen only in aa.
Open the walkthrough step →Summary
- The fitnesses are 1, 1 − hs and 1 − s for AA, Aa and aa. The coefficient h sets how much one copy of a costs.
- A rare allele sits mostly in heterozygotes, so h decides how much selection sees it.
- With s = 0.1 and p starting at 0.01, A passes 0.5 at generations 54, 65, 86, 136 for h = 0, 0.25, 0.5, 0.75, and at 928 for h = 1.
- A recessive a (h = 0) is removed fast at first and slowly at the end: Δq is about −s·q². A passes 0.99 at generation 1075.
- Additive effects (h = 0.5) reach 0.99 first, at generation 175.
- Under recurrent mutation the same effect leaves a at about 0.01 for recessive a and about 0.0002 for h = 0.5, at μ = 0.00001 and s = 0.1.