Muller's ratchet
In an asexual population drift can remove the least-loaded class of genomes, and each loss raises the minimum mutation load for good.
Read first: Genetic drift, Neutral mutations
Mutations that cannot be removed
Most new mutations that change fitness are harmful. With sex, recombination can combine the genomes of two parents and produce an offspring with fewer harmful mutations than either. An asexual population cannot. A genome passes to its offspring whole, so the only way to carry fewer mutations is to descend from a genome that carried fewer.
The module follows N haploid asexual genomes. Each genome is described by one number, k, its count of harmful mutations. All start at k = 0. Mutations are never reversed. Without selection the mean load would rise by U per generation, which is 30 mutations per genome in 300 generations at U = 0.1. Selection acts against that rise, and whether it wins depends on the size of the best class.
Each generation has three steps. Mutation: every genome gains a Poisson(U) number of new mutations, where U is the page's μ. Selection: a genome with k mutations has fitness (1 − s)^k. Reproduction: N offspring are drawn, each from a parent chosen in proportion to fitness.
The ratchet at the defaults
Why the best class is lost
The class with the fewest mutations, of size n₀, is the one that can be lost. Mutation moves genomes out of every class and selection removes the classes with more mutations. At balance the number of mutation-free genomes right after mutation is N e^(−U/s) (Haigh 1978). The mean load at balance is U/s. With U = 0.1 and s = 0.02 that is 5 mutations per genome, and the class holds 50 × e^(−5) = 0.34 genomes.
That is less than one genome. The best class is always about to vanish, and drift removes it within a few generations. Each loss is a click: the minimum load rises by one and cannot come down, because no genome can lose a mutation.
The plotted n₀ is counted after selection, which makes it larger than the value above by a factor e^U. With N = 200, U = 0.1 and s = 0.1 the expected class size is 73.6 after mutation and 81.3 after selection.
Strong selection, large population
Exercise
Weaken selection and shrink the population
Set N 200, μ 0.1, s 0.1 and Gen 300, then run. Set N 50 and s 0.02 (the defaults) and run again.
What to expect. With N = 200 the n₀ line stays near 80 and the chart title reports no clicks. With the defaults n₀ stays between 1 and about 20 and the title reports around 13 clicks.
Open the walkthrough step →Population size, mutation rate and selection
Three parameters set the class size N e^(−U/s). Raising N scales it up in proportion. Raising U or lowering s increases the exponent U/s, and the class shrinks exponentially.
The mean number of clicks in 300 generations over 40 seeded runs shows the effect. N = 50, U = 0.1 and s = 0.02 give a class of 0.34 and 13.0 clicks. With s = 0.05 the class is 6.8 and there are 6.1 clicks. Raising s from 0.02 to 0.05 enlarges the class about 20 times, because U/s falls from 5 to 2. A class of a few genomes is still lost now and then. With N = 500 and s = 0.05 the class is 67.7 and there are 0.4 clicks.
Population size alone cannot rescue the defaults. The largest N the module allows is 1,000, and at U = 0.1 and s = 0.02 that gives a class of only 6.7 genomes. Raising the mutation rate to U = 0.3 with N = 50 and s = 0.02 gives a class of 50 × e^(−15) = 0.000015 and 45.1 clicks, one every 7 generations.
Exercise
Scale up the population
Set N 50, μ 0.1, s 0.05 and Gen 300, and run it several times, reading the click count in the chart title. Then set N 500 and repeat.
What to expect. N = 50 gives about 6 clicks per run. N = 500 gives 0 or 1 click.
Open the walkthrough step →What to remember
- Without recombination, each loss of the least-loaded class raises the minimum load by one for good. This is a click.
- The class holds about N e^(−U/s) genomes: 0.34 for N = 50, U = 0.1, s = 0.02 and 73.6 for N = 200, U = 0.1, s = 0.1.
- Classes of under one genome click 13.0 times in 300 generations, classes of 6.8 click 6.1 times and classes of 67.7 click 0.4 times.
- Larger N, stronger s or lower U enlarge the class and stop the ratchet. Recombination removes the problem.