Rate of Fixation of Rare Variants in a Population

The process of molecular evolution has been dominated by the Kimura paradigm for nearly 60 years; mutations arise at a certain rate in the population and they go to fixation with a probability given by Kimura’s classic formula, which assumes there are no further mutations that interfere with the fixation process. An alternative view is that rare variants exist in the population in a mutation-drift-selection balance and rise to fixation through a combination of chance (genetic drift), selection and mutation. When mutations increase in strength, but still in the weak regime, we would expect the Kimura rate approximation to be an overestimate, as a rare variant which grows in frequency will suffer a greater backward flux of mutations, slowing progress to fixation. However, to date calculating important quantities for a general model of selection and mutation, like the rate of fixation of these rare variants has not been tractable in the conventional diffusion approximation of population genetics. Here, we use Fisher’s angular transformation to convert the frequency-dependent diffusion inherent in population genetics to simple diffusion in an effective potential, which describes the forces of selection, drift and mutation. Once this potential is defined it is simple to show that the mean first passage time is given by a double integral which relate to populations at the barrier. Exact numerical integration shows excellent agreement with discrete Wright-Fisher simulations, which do show a slowing down of the fixation of mutants at higher mutation rates and for strong positive selection, compared to the Kimura prediction. We then seek a closed-form analytical expression for the rate of fixation of mutants, by adapting Kramer’s approximation for the mean first passage time. This overall gives an accurate approximation, but however, does not improve on the Kimura rate.