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Parallelism and Loss in the Evolution of Origin-Fixation Models
Since the "molecular revolution" of 1968-1971, evolutionary geneticists have become increasingly familiar with models that treat evolution as a series of events with origin-fixation dynamics, characterized by multiplying a rate of origin and a probability of fixation. For instance, this conceptions leads to the familiar approximation of k = 2Nu * 2s = 4Nus for the rate of evolution by selective fixation of beneficial mutations. This type of model is widely used in molecular evolution, including phylogenetic analysis software that uses codon-based models, and more recently has been explored in the form of so-called "mutational landscape" models of adaptation.
The curious origin of these models is being explored by David McCandlish (Duke) and Arlin Stoltzfus. They played no known role in the Modern Synthesis, which opposed the mutationist thinking behind origin-fixation models and instead was committed to a "shifting gene frequencies" view of evolution. Oddly, origin-fixation models seem to have emerged several times in the period of 1968 to 1971 (Kimura & Ohta; King and Jukes; Vogel & Zuckerkandl). The canonical reference, which provides a simple derivation, would seem to be Kimura & Ohta, 1971. We searched for an earlier origin, paying particular attention to the work of Wright. Interestingly, Wright (1938) derives an equation k = 4Nus, albeit with a (very) slightly different meaning, in a famous paper on the distribution of allele frequencies in a stochastic population under irreversible mutation. Wright's paper has been cited over 100 times, but it is never cited for this, oddly, even by Kimura, et al. In turn, some contemporary authors do not cite the molecular-evolutionary origins of the origin-fixation model.