A Brief History of Muller’s Ratchet

H.J. Muller (1964, Mutation Research 1: 2-9) pointed out that

… an asexual population incorporates a kind of ratchet mechanism, such that it can never get to contain in any of its lines, a load of mutation smaller than that already existing in its present least-loaded lines…

Joseph Felsenstein (1974, Genetics 78: 737-756) introduced the term Muller’s ratchet, and drew attention to its connection with what he also called the Hill-Robertson effect (the mutual interference between linked sites subject to selection).

He simulated haploid populations with many equivalent, completely linked loci subject to mutation and selection, with each mutant allele causing the same reduction in fitness, and assuming multiplicative fitness effects across loci.

He contrasted the (analytical) results for free recombination between loci, and showed that the mean number of deleterious mutations present in an an individual increases much faster over time in the absence of recombination.

MUTATION-SELECTION BALANCE Assume a large population size, so that the loci under selection are approximately at deterministic equilibrium.

Assume a mean number of new deleterious mutations per genome of U, and a harmonic mean selection coefficient against mutations of s. Mutation is assumed to be one-way, from wild-type to deleterious alleles at each locus.

The equilibrium mean number of deleterious mutations per gamete is:

n = U/s

The equilibrium frequency of gametes carrying i deleterious mutations is Poisson-distributed with mean n = U/s.

The frequency of the mutation-free class is thus:

f0 = exp - n

e.g. with n = 5, f0 = 0.007, with n = 20, f0 = 2x10-9

Mutation

At the instigation of John Maynard Smith, John Haigh made the first attempt at an analytic model of Muller’s ratchet (1978, Theor. Pop. Biol. 14: 251-267).

He looked at the process in terms of the loss of the “least-loaded” class from the extreme left of the distribution of numbers of mutations per genome.

Mutation

Drift

He showed that, if the process is started at the deterministic equilibrium, the process of loss of the initial class (with 0 mutations) is similar to a branching process initiated with n0 = Nf0 individuals.

This assumes n0 > 1.

Once the zero class is lost, the distribution rearranges itself over a short time period, and the next class (class 1) approaches a size close to n1 = 1.6n0, before becoming extinct in turn.

Class 2 then behaves like class 1 and so on.

Haigh’s analysis suggested that, once the zero class has been lost, the ratchet process is simply a set of repetitions of losses of least-loaded classes, involving two phases.

Phase 1: the approach of the new least-loaded class to a size of 1.6n0

Phase 2: the loss of this class

Stephan et al. (1993, Genet. Res. 61: 225-231) applied diffusion equation theory to derive approximations for the expected times in phases 1 and 2, treating the frequency of the least-loaded class as the variable x subject to diffusion.

T1 for Phase 1 is approximated by the standard solution for the expected time for frequency x to move from f1 to 1.6f0, assuming Poisson variances for the ni:

T1 = s -1 ln {U/(1.6s)}

w weq + ((x/x

0)

w )eq (x0

x - 1)

Phase 2 is more tricky. They suggested the trick of using a Taylor series expansion of mean fitness around the equilibrium, considering only the effect of deviations in the least-loaded class:

This is used to calculate the drift coefficient for the diffusion,since the expected change in mean for the deterministic process forthe frequency of the least-loaded class is given by: x =

wx (e-U - w )

= w

x w

The problem with this it is hard to get the right value for the deviation in mean fitness, since the process does not start at equilibrium (x = f0).

If it were at equilibrium, then we could simply use s exp - U as an estimate of the change in mean fitness caused by a shift from x = f0 to x = 0, representing a reduction from a mean fitness of exp - U to (1 - s) exp - U.

In reality, x will be greater than f0, so the change is smaller.

Stephan et al. (1993) suggested halving it.

Gordo & Charlesworth (2000a, Genetics 154, 1379-1387) suggested using x = 1.6f0, given Haigh’s finding that the process stays near x = 1.6f0 for a long time.

This gives the drift coefficient as:

a(x) 0.6s (1 - x/f0) x

This seems to work quite well for some parameters, but use of 0.5 is more accurate for others (Stephan and Kim 2002, pp. 72-93, in Modern Developments in Population Genetics, eds. M. Slatkin & M. Veuille, Oxford UP).

Stephan et al. used the diffusion equation solution for the expected time to take the process from x = f0 to x = 0.

But, as pointed out by Charlesworth & Charlesworth (1997, Genet. Res. 70: 63-73), diffusion across the whole interval (0, 1) should be considered in computing T2, the expected time for phase 2.

Putting all of these considerations together, for n0 > 1 we can get a solution for the total expected time for loss of a least-loaded class as:

T = T1 + T2

(Gordo & Charlesworth 2000b, Genetics 156:2137-2140; Stephan and Kim 2002)

In a haploid population, each click of the ratchet is followed by the rapid fixation of a single haplotype in the least-loaded class, due to the small size of this class, so that all the mutations carried by this haplotype will become fixed in the least-loaded class.

Since the next class up is eventually derived from this class by mutation, these mutations will become fixed in this this class, then the next class, and eventually in the whole population.

This was first suggested verbally by Rice (1987, Evolution 116: 161-167), and modelled by Charlesworth & Charlesworth (1997).

1

2

3

4

5

6

7

8

500 1000 1500 2000 2500

Number of loci in least loaded genome

Number of loci fixed

Num

ber

of lo

ci

Generation

2 3 4 5 6 7Fixations inleast loaded class

Fixations in class next to least loaded class

How well does all this work in practice? Gordo and Charlesworth (2000a,b) and Loewe (Genetical Research, in press) have done Monte Carlo simulations that test the approximations.

SPEED OF MULLER’S RATCHET

N U s T Mean Fitness

Sims. Theory (at 5 x105gens)

275 0.06 0.015 63 87 10-52

5 x103 0.04 0.01 514 473 5.6 x10-5

5 x104 0.04 0.01 45922 38990 0.90

5 x105 0.04 0.004 371 573 0.38

5 x105 0.015 0.0015 732 1352 0.36

5 x105 0.015 0.0025 4924 7785 0.77

Gessler (1995, Genet. Res. 66: 241-253) studied the case when n0 < 1.

He pointed out that this condition means that the deterministic equilibrium cannot apply to the starting point, and suggested that the ratchet is then a quasi-deterministic process, driven by mutation pressure overcoming selection.

By truncating the equilibrium distribution at the left-hand end, and discarding classes for which Nfi < 1, he showed that the distribution of the number of mutations at a given point in time is approximated by a negative binomial, with known mean and variance:

n = s

U - k

Vn = n - 2k + b

where k is the smallest integer for which Nfi < 1, and b is the similar number for the truncated distribution (determined by an algorithm).

He then argued that, since the ratchet is quasi-deterministic in this parameter range,the change in mean copy number is given by the infinite population value, and corresponds to the rate of clicking of the ratchet.

This gives the expression for the time between clicks as:

T s(2k - b)

1

Gessler showed that this gives a good fit to the rate of the ratchet when n0 < 1.

There is clearly a question as to how this fits together with the diffusion equation results for n0 > 1.

This has been looked at by Laurence Loewe (Genet. Res., in press).

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Questions

1. What is the effect of a wide variation in selection coefficients on the rate of the ratchet?

2. How do back-mutation and low rates of recombination affect it?

3. Is it possible to have an analytical model that includes everything?

Given the level of polymorphism for amino-acid variants in D. miranda, we can estimate that an average fly is heterozygous for about 6000 amino-acid site mutations, of which about 5500 are probably subject to effective selection.

The mean selection coefficient associated with the latter is of the order of 10-5, so that the fitness of the population is reduced by about 5.5% below what it would be in the absence of these variants.

Mutation