Fixation when N and s vary: classic approaches give elegant new results.

IN 1922, an article entitled “On the dominance ratio” appeared in the Proceedings of the Royal Society of Edinburgh. The author was R. A. Fisher, a 29-year-old statistician studying crop variation at the Rothamsted Experimental Station in Harpenden, England. The article was a follow-up to Fisher’s 1918 article on the statistical effects of Mendelian inheritance, but it is in the 1922 article that Fisher reveals most clearly his uncanny prescience of the questions and techniques that would dominate theoretical population genetics over the next century (Fisher 1918, 1922). In section 2 of that article, Fisher considers the survival of rare “mutant genes” and introduces what we now call a branching process to address this question. The succinctness with which Fisher explains his approach was typical of the author and does not seem overly concise to today’s reader. Fisher’s brevity, however, is astounding when we consider that his 1922 exposition was the first application of branching processes to any field of science after Galton and Watson’s development of these techniques to explain the extinction of English surnames (Watson and Galton 1874). [The underlying mathematics was later rediscovered for application to nuclear chain reactions; this rediscovery was necessary, according to Fisher, because physicists considered him “an ignorant country bumpkin” (Gale 1990, p. 114, citing personal communication with Fisher).] Similarly, in section 3 of the same article (Fisher 1922), a half-page derivation is provided for a heat equation, that is, a constant-coefficient diffusion equation, describing the time evolution of the gene frequency distribution. This was the first application of a diffusion process to population genetics, and Fisher, known for his distrust of methods borrowed from other disciplines, derives the approach from first principles without appealing to parallel work in stochastic processes (see Feller 1951). In the decades of work that followed Fisher’s 1922 article, branching processes and diffusion approximations became the two classic approaches for estimating fixation probabilities, that is, the probability that a segregating allele is ultimately carried by all individuals in a population. Most notably, eccentric English biologist J. B. S. Haldane, presumably during breaks from his studies of human physiology through dangerous self-experimentation (Crow 1992), used branching processes to derive the well-known probability that a rare, slightly beneficial allele will fix (Haldane 1927). Later, Motoo Kimura applied the diffusion approach to derive a more general expression for the fixation probability (Kimura 1955). In the second appendix of his well-known chapter (Feller 1951), Feller provides a formal “passage” between the two approaches by recasting the generating function f(x) as a characteristic function, f(eiz). How have both of these “rival” approaches remained current as the field has advanced over decades? Diffusion approximations are far more powerful: a diffusion equation can predict the dynamic frequencies, and eventual fixation, of beneficial, neutral, or deleterious alleles, starting from any initial frequency in the population. Yet, the assumptions underlying diffusion approximations are more difficult to understand, and the technique nearly always requires a numerical approach (number-crunching by computer) to give predictive results. In contrast, branching processes are limited in their applicability: they can be used only to explore situations in which the mutation of interest is beneficial and the mutant allele is initially rare. In these limited Copyright © 2011 by the Genetics Society of America doi: 10.1534/genetics.111.131748 Address for correspondence: Department of Applied Mathematics, Middlesex 255, University of Western Ontario, London, ON, Canada N6A 5B7. E-mail: [email protected]