Neutral mutations
Most new neutral mutations are lost within a few generations, and the few that fix do so with probability 1/2Ne.
Read first: Genetic drift
One new copy at a time
A neutral mutation changes the DNA but not the fitness of its carrier. Drift is the only force that acts on it. The module adds one new mutation every Interval generations to a population of Ne diploid individuals and follows each mutation separately.
Each mutation starts as a single copy among 2Ne gene copies, at frequency 1/(2Ne). With Ne = 50 that is 1/100 = 0.01. Each generation the number of copies is a binomial draw, X(t+1) ~ Binomial(2Ne, p(t)), and p(t+1) = X(t+1)/2Ne. The values 0 and 1 are absorbing. A mutation at 0 is lost and a mutation at 1 is fixed, and neither changes again.
Each coloured line on the chart is one mutation, from its arrival to its loss, its fixation or the end of the run. The run is random, so the figures below use a fixed seed.
Eleven mutations in a population of 50
Most mutations are lost at once
A single copy is lost in its first generation if none of the 2Ne draws picks it. Each draw picks it with probability 1/2Ne, so the loss probability is (1 − 1/2Ne)^(2Ne), close to e^(−1) = 0.37. For Ne = 50 it is 0.366 and for Ne = 10 it is 0.358. About 37 percent of new neutral mutations are gone after one generation, whatever the population size.
The survivors are not safe. A mutation at frequency 0.02 or 0.03 is still lost most of the time. In the run above all nine lost mutations were gone within 34 generations of arising.
The chance that a mutation eventually fixes equals its starting frequency, P(fix) = 1/2Ne. Look far enough ahead and every copy in the population descends from one copy alive today. Each of the 2Ne copies is equally likely to be that ancestor, and the new mutation is one of them. For Ne = 50 the chance is 1 in 100. For Ne = 10 it is 1 in 20, or 0.05.
A hundred and one mutations in a population of 10
Exercise
Count fixations at two sizes
Set Ne 10, Gen 1000 and Interval 10, then run it several times and count the lines that end at 1. Set Ne to 50 and repeat.
What to expect. Ne = 10 gives about 5 fixations per run, from roughly 1 to 10. Ne = 50 gives under 1 per run on average, so most runs show none.
Open the walkthrough step →How long fixation takes, and the molecular clock
A mutation that fixes spends a long time drifting. The mean time to fixation, for the mutations that fix, is about 4Ne generations. For Ne = 10 that is 40. The six fixations in the Ne = 10 run took 15, 66, 30, 15, 70 and 40 generations, a mean of 39.3. For Ne = 50 it is 200 generations.
Population size therefore acts on two quantities in the same direction. A larger Ne makes fixation rarer (1/2Ne) and slower (4Ne).
A fixation is a substitution: the new allele has replaced the old one in the whole population. If the mutation rate is μ per copy per generation, the population gains 2Neμ new mutations each generation and each fixes with probability 1/2Ne. The substitution rate is 2Neμ × 1/2Ne = μ. Ne cancels.
Take μ = 0.00001. A population of Ne = 50 gains 0.001 new mutations per generation, and each fixes with probability 0.01. A population of Ne = 5,000 gains 0.1 per generation, and each fixes with probability 0.0001. Both substitute at 0.00001 per generation. Neutral divergence therefore grows with time at the mutation rate, in small and large populations alike. This is the molecular clock.
Exercise
Time to fixation
Set Ne 10, Gen 1000 and Interval 10, run, and hover a line that ends at 1. Subtract the generation where it starts from the generation where it reaches 1. Repeat for a few lines, then set Ne to 50.
What to expect. At Ne = 10 the differences average about 36 generations, near 4Ne = 40. At Ne = 50 they average about 176, a little under 4Ne = 200, because only fixations that finish within 1,000 generations are visible.
Open the walkthrough step →What to remember
- A new neutral mutation starts at 1/2Ne, and about 37 percent are lost in the first generation.
- P(fix) = 1/2Ne: 0.01 for Ne = 50 and 0.05 for Ne = 10.
- Mutations that fix take about 4Ne generations: 40 for Ne = 10 (39.3 in the run above) and 200 for Ne = 50.
- The substitution rate is 2Neμ × 1/2Ne = μ, independent of Ne. This is the basis of the molecular clock.