A synoptic solution to a Wright-Fisher model with recurrent mutation, drift and selection
The time to fixation (conditional upon fixation) and probability of fixation are key summary statistics of classical population genetics. Here, they are integrated to solve the time to fixation unconditional on fixation in a Wright-Fisher model with recurrent mutation, drift and selection. In its derivation, minor improvements to the probability of fixation and the probability distributions of emergence and spread (or passage) are made. The solution is an exponentially modified Gaussian (ex-Gaussian) distribution with a mean and variance that is the sum of the mean and variance of the underlying exponential and normal (i.e. Gaussian) distributions. To demonstrate its value as a synoptic solution to the Wright-Fisher model, the ex-Gaussian distribution is used to partition the effects of the evolutionary forces of mutation, drift, selection and (to some extent) migration on chosen points of the probability distribution of fixation times. Further, the solution is used to derive (and show the quantitative meaning of) the stochastic drift barrier to nearly neutral mutations with very low selection coefficients, and its consequences on the probability distribution of phenotypic and/or fitness effects that contribute to adaptation. In this way, the probability distribution of the time to fixation unconditional on fixation provides an elegant summary of some key results from classical population genetics, and is also likely to have many other theoretical and applied uses.