Irreversible mutation
One-way mutation lowers an allele frequency by the same fraction every generation, so the half-life depends on the mutation rate alone.
Read first: Genotype frequencies and Hardy-Weinberg
Mutation in one direction
Allele A mutates to allele a at rate μ per generation. A copy of A that mutates becomes a and never changes back. This is irreversible mutation. The module has one locus, two alleles and an infinite population, so there is no drift. Mutation is the only force.
Each generation a fraction μ of the A copies turns into a. The frequency of A, written p, is multiplied by (1 − μ). With μ = 0.001, p loses 0.1 percent of its current value per generation. The recurrence is p(t+1) = p(t)(1 − μ), and after t generations p(t) = p₀(1 − μ)^t, where p₀ is the starting frequency. For small μ this is close to p₀e^(−μt), an exponential decay.
A rate of 0.001 means one copy in a thousand mutates each generation. Among 10,000 A copies, about 10 turn into a, and p goes from 0.9 to 0.8991 in the first generation. The run is deterministic. The same inputs give the same curve every time, so the module has no seed.
The defaults: p₀ = 0.9, μ = 0.001
The half-life
The time for p to halve is the half-life. Solving (1 − μ)^t = 1/2 gives t = ln 2 / −ln(1 − μ), about ln 2 / μ. For μ = 0.001 this is 693 generations. The starting frequency does not appear in it. Going from 0.9 to 0.45 takes 693 generations, and going from 0.45 to 0.225 takes another 693.
The exact half-life of 692.8 is not a whole number, so the curve crosses 0.45 between generations 692 and 693: p is 0.4504 at 692 and 0.4499 at 693. The curve flattens because the loss per generation is μp. At p = 0.9 it is 0.0009. At p = 0.45 it is 0.00045. The loss halves when p halves, so each halving takes the same time. A tenfold drop, from 0.9 to 0.09, takes ln 10 / −ln(1 − μ) = 2,301 generations at μ = 0.001, which is 3.3 half-lives.
The rate sets the clock. Doubling μ to 0.002 halves the half-life to 346 generations. Raising μ tenfold to 0.01 cuts it to 69 generations. The half-life scales as 1/μ.
A different start, a faster rate
Exercise
Double the rate
Open the module with the defaults and run it. Save the run from the Runs panel. Set μ to 0.002, run again and tick the saved run so both curves show.
What to expect. The new curve reaches 0.45 near generation 346, half the time of the saved curve.
Open the walkthrough step →Realistic rates and what the model leaves out
Per-locus mutation rates in nature are far below 0.001. At μ = 0.00001 the half-life is about 69,000 generations, and after 5,000 generations p has only fallen from 0.9 to 0.856. At μ = 0.0001 it falls to 0.546 over the same time. Mutation pressure alone changes a gene pool very slowly. At μ = 0.00001 and p = 0.9 the loss is 0.000009 per generation, so a visible change takes thousands of generations.
The model has no back mutation. The Two-Way Mutation module adds a rate ν for a to A, and p then settles at ν / (μ + ν) instead of falling to 0. The model has no drift, so p never reaches exactly 0 for μ < 1. It has no selection, and a real loss-of-function allele is usually selected as well. Over long spans all of these act at once, so pure exponential decay is rarely seen.
Rates of 10⁻⁶ per generation, common for single sites, give a half-life of about 693,000 generations. The extreme case is μ = 1. Every A copy mutates in one generation and p is 0 at generation 1.
Exercise
A realistic rate
Set P(A) 0.9, μ 0.00001 and Gen 5000, then run. Read p at the last generation from the hover panel.
What to expect. p is 0.856. The curve looks nearly flat, and the y axis zooms in to about 0.85 to 0.90.
Open the walkthrough step →What to remember
- One-way mutation multiplies p by (1 − μ) each generation, so p(t) = p₀(1 − μ)^t.
- The half-life is about ln 2 / μ and does not depend on p₀: 693 generations at μ = 0.001, 346 at 0.002, 69 at 0.01.
- At μ = 0.00001 the half-life is about 69,000 generations, so mutation alone moves p from 0.9 to only 0.856 in 5,000 generations.
- The model has no drift, back mutation or selection. p falls toward 0 and reaches it only when μ = 1.