Formal properties of the probability of fixation: Identities, inequalities and approximations

The formula for the probability of fixation of a new mutation is widely used in theoretical population genetics and molecular evolution. Here we derive a series of identities, inequalities and approximations for the exact probability of fixation of a new mutation under the Moran process (equivalent results hold for the approximate probability of fixation under the Wright-Fisher process, after an appropriate change of variables). We show that the logarithm of the fixation probability has particularly simple behavior when the selection coefficient is measured as a difference of Malthusian fitnesses, and we exploit this simplicity to derive inequalities and approximations. We also present a comprehensive comparison of both existing and new approximations for the fixation probability, highlighting those approximations that induce a reversible Markov chain when used to describe the dynamics of evolution under weak mutation. To demonstrate the power of these results, we consider the classical problem of determining the total substitution rate across an ensemble of biallelic loci and prove that, at equilibrium, a strict majority of substitutions are due to drift rather than selection.