The accumulation of mutations in asexual populations and the structure of genealogical trees in the presence of selection

We study the accumulation of unfavourable mutations in asexual populations by the process of Muller's ratchet, and the consequent inevitable decrease in fitness of the population. Simulations show that it is mutations with only moderate unfavourable effect that lead to the most rapid decrease in fitness. We measure the number of fixations as a function of time and show that the fixation rate must be equal to the ratchet rate once a steady state is reached. Large bursts of fixations are observed to occur simultaneously. We relate this to the structure of the genealogical tree. We derive equations relating the rate of the ratchet to the moments of the distribution of the number of mutations k per individual. These equations interpolate between the deterministic limit (an infinite population with selection present) and the neutral limit (a finite size population with no selection). Both these limits are exactly soluble. In the neutral case, the distribution of k is shown to be non-self-averaging, i.e. the fluctuations remain very large even for very large populations. We also consider the strong-selection limit in which only individuals in the fittest surviving class have offspring. This limit is again exactly soluble. We investigate the structure of the genealogical tree relating individuals in the same population, and consider the probability $$\bar P$$(T) that two individuals had their latest common ancestor T generations in the past. The function $$\bar P$$(T) is exactly calculable in the neutral limit and the strong-selection limit, and we obtain an empirical solution for intermediate selection strengths.