Haldane’s Probability of Mutant Survival is Not the Probability of Allele Establishment

Abstract

Haldane notably showed in 1927 that the probability of fixation for an advantageous allele is approximately 2s, for selective advantage s. This widely known result is variously interpreted as either the fixation probability or the establishment probability, where the latter is considered the likelihood that an allele will survive long enough to have effectively escaped loss by drift. While Haldane was concerned with escape from loss by drift in the same paper, in this short note we point out that: 1)Haldane's 'probability of survival' is analogous to the probability of fixation in a Wright-Fisher model (as also shown by others);and 2) This result is unrelated to Haldane's consideration of how common an allele must be to 'probably spread through the species'. We speculate that Haldane's survival probability may have become misunderstood over time due to a conflation of terminology about surviving drift and 'ultimately surviving' (i.e., fixing). Indeed, we find that the probability of establishment remarkably appears to have been overlooked all these years, perhaps as a consequence of this misunderstanding. Using straightforward diffusion and Markov chain methods, we show that under Haldane's assumptions, where establishment is defined by eventual fixation being more likely that extinction, the establishment probability is actually 4s when the fixation probability is 2s. Generalizing consideration to deleterious, neutral, and adaptive alleles in finite populations, if establishment is defined by the odds ratio between eventual fixation and extinction, k, the general establishment probability is (1 + k)/k times the fixation probability. It is therefore 4s when k = 1, or 3s when k = 2 for beneficial alleles in large populations. As k is made large, establishment becomes indistinguishable from fixation, and ceases to be a useful concept. As a result, we recommend establishment be generally defined as when the odds of ultimate fixation are greater than for extinction (k = 1, following Haldane), or when fixation is twice as likely as extinction (k = 2).