The advance of Muller's ratchet in a haploid asexual population: approximate solutions based on diffusion theory

Asexual populations experiencing random genetic drift can accumulate an increasing number of deleterious mutations, a process called Muller's ratchet. We present here diffusion approximations for the rate at which Muller's ratchet advances in asexual haploid populations. The most important parameter of this process is n0 = N e-U/s, where N is population size, U the genomic mutation rate and s the selection coefficient. In a very large population, n0 is the equilibrium size of the mutation-free class. We examined the case n0 > 1 and developed one approximation for intermediate values of N and s and one for large values of N and s. For intermediate values, the expected time at which the ratchet advances increases linearly with n0. For large values, the time increases in a more or less exponential fashion with n0. In addition to n0, s is also an important determinant of the speed of the ratchet. If N and s are intermediate and n0 is fixed, we find that increasing s accelerates the ratchet. In contrast, for a given n0, but large N and s, increasing s slows the ratchet. Except when s is small, results based on our approximations fit well those from computer simulations.