# Muller’s ratchet

Following Gordo & Charlesworth (2000).

This Wright-Fisher model starts with a haploid asexual population at a mutation-selection balance. The population size is \(N\), the mutation rate is \(u\) and the selection coefficient is \(s\).

Denote the frequency of the best class by \(x\) and its initial value \(x_0 = e^{-u/s}\). The variance of \(x\) due to binomial sampling of \(N\) individuals from the previous generations is \[ b(x) = \frac{x(1-x)}{N} \approx \frac{x}{N}, \]

assuming \(x \ll 1\). This follows from the variance of a binomial distribution: \(Var(Bin(n,p)) = np(1-p)\).

The expected change in \(x\) due to mutation and selection is, assuming \(\bar{\omega}\) is the current population mean fitness and \(\Delta \bar{\omega}\) is the difference between the mutation selection balance (MSB) and the current population mean fitness: \[ a(x) = \frac{x(e^{-u} - \bar{\omega})}{\bar{\omega}} = x \frac{\Delta \bar{\omega}}{\bar{\omega}}, \] where the MSB mean fitness is \(e^{-u}\).

This follows from the standard difference equation of the frequency of type \(z\) after one generation: \[ f'(z) = f(z) Pr(no \; mutation) \frac{\omega(z)}{\bar{\omega}} \]

assuming the number of mutations is Poisson distributed with mean \(u\) and that the best class has fitness 1.

Next, assume that throughout the process the mean fitness is close enough to the MSB value so that \(\Delta \bar{\omega}\) can be modeled as a perturbation. Furthermore, the perturbations are assumed to be a result of fluctuations in the frequency of the best class, \(x\).

The mean fitness close to the MSB as a function of \(x/x_0\) is expressed as a Taylor expansion around 1: \[ \bar{\omega} (\frac{x}{x_0}) \approx \bar{\omega}_{eq} + [\frac{\partial \bar{\omega}}{\partial \frac{x}{x_0}}]_{eq} (\frac{x}{x_0}-1) + O((\frac{x}{x_0}-1)^2). \]

This gives a linear approximation for \(\Delta \bar{\omega}\) when it is small, that is, when \(x\approx x_0\): \[ \Delta \bar{\omega} \approx K (1-\frac{x}{x_0}), \; K = x_0 [\frac{\partial \bar{\omega}}{\partial x}]_{eq}. \]

Set \(K=0.6 s e^{-u}\) and use this for \(a(x)\): \[ a(x) \approx 0.6 s (1-\frac{x}{x_0}) x. \]

Using \(a(x)\) and \(b(x)\) as the drift and diffusion coefficients in a diffusion equation, the time spent in the frequency interval \([0,x_0]\) is: \[ T_{0,x_0} = \int_0^{x_0}{\frac{2N}{x G(x)} {\int_0^x{G(x')d x'}} dx}, \]

and the time spent in the interval \([x_0,1]\) is: \[ T_{x_0,1} = \int_{x_0}^1{\frac{2N}{x G(x)} \int_0^{x_0}{G(x') dx'} dx }, \] where \[ G(\xi) = exp[-2 \int_0^{\xi} {\frac{a(z)}{b(z)}dz}] = \\ exp[\frac{2N 0.6s}{x_0} \xi (\frac{\xi}{2}-x_0)]. \]

The time to the loss of the best class is then \[ T(N,u,s) = T_{0,x_0} + T_{x_0,1}. \]

## References

Gordo, Isabel, and Brian Charlesworth. 2000. “The degeneration of asexual haploid populations and the speed of Muller’s ratchet.” *Genetics* 154 (3): 1379–87. http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1460994.