markovian - senecio7の日記

McCandlish DM & Stoltzfus A 2014 Modeling evolution using the probability of fixation: history and implications. Q Rev Biol 89:225-252.

early origin-fixation models were used to calculate an instantaneous rate of evolution across a large number of independently evolving loci
in the 1980s and 1990s, a second wave of origin-fixation models emerged to address a sequence of fixation events at a single locus
origin-fixation models constitute a coherent theory of mutation-limited evolution that contrasts sharply with theories of evolution that rely on the presence of standing genetic variation

two general classes may be discerned:
aggregate-rate models that consider the total rate of evolution over a large number of independently evolving loci
sequential fixation models that treat a series of substitutions at a single locus

true origin-fixation models did not appear until 1969, in the context of the neutral theory, taking the form of aggregate-rate models of neutral or beneficial changes (King and Jukes 1969; see also Kimura and Maruyama 1969)
over time, such models became fundamental to the analysis of molecular evolution

origin-fixation models should be treated as hypotheses to be tested rather than merely a method to be applied

we consider an infinite collection of independently evolving loci where the total mutation rate across this infinite collection is finite
the typical goal of such a model is to calculate a total rate of evolution across this infinite collection of loci
we will call these models "aggregate-rate" models

models in our second major class instead assume that mutations at any one locus occur sufficiently infrequently that each new mutation is lost or goes to fixation prior to the appearance of the next mutation
it is still the case that each new mutation occurs at a locus that is otherwise monomorphic
such models typically are presented as one-locus, multiallelic models where the questions of interest concern the trajectory of a population through some kind of explicit genotypic space (e.g., sequence space)
we will call these models "sequential fixation" models
this class of models, although very important (see below), is perhaps less well known than the aggregate-rate models

more formally, sequential fixations models treat evolution as a Markov chain where the states are the possible alleles at a single locus

in an aggregate-rate model, one assumes that the loci in question evolve independently from each other
this requires that the loci be unlinked (i.e., there is free recombination between them), that the fitness effects of a mutation at one locus do not depend on segregating variants at another locus (i.e., an absence of epistasis), and that selection is weak in the sense that the variance in fitness within the population is small
one must also assume that each new mutation happens at a location in the genome where no variation is currently segregating
this second assumption is known in the literature as the "infinite-sites" assumption (Kimura 1969)
aggregate-rate models are a subset of infinite-sites, free-recombination models
a different, but still common, class of models makes the infinite-sites assumption but also assumes complete linkage between sites, e.g., in Watterson 1975

in sequential fixation models, mutations must occur sufficiently infrequently that each new mutation is lost or becomes fixed in the population before the next new mutation enters the population
whereas in aggregate-rate models, the total rate at which mutations at all loci enter the population, 2Nu, can be arbitrarily large
the assumptions necessary for an origin-fixation model to faithfully capture the true evolutionary dynamics will in general depend on other factors in addition to the population size and mutation rate

Fisher never takes the next step of writing the rate of evolution, K, as the product of 2Nu and the probability of fixation
the first paper to describe the general relationship between K, 2Nu , and the probability of fixation is Wright's well-known paper
(Wright 1938)
K is the instantaneous equilibrium rate of fixations at a single site
we do not find a single article that cites Wright for an origin-fixation formalism or for the formula K = 4Nus

in general, Wright, Fisher, and Haldane believed that evolutionary change is due largely to natural selection acting on segregating (i.e., standing) variation (Wright 1960, 1980:829), and were impressed both with the speed of this process, and with the ability of the joint action of natural selection and recombination to make genotypes common that otherwise would have been vanishingly rare
Fisher and Wright also placed a strong emphasis on fitness interactions (epistasis) among segregating alleles, which precluded the possibility that an allele would have a constant selection coefficient during its sojourn in a population
for Fisher and Wright, formulas assuming an independent fixation probability for each new mutation had little relevance to evolution in actual populations

in contrast to the ambiguity in Kimura's presentation, King and Jukes (1969) derive both K = u and K = 4Nus using a fully explicit, and clearly explained, origin-fixation argument as part of their parallel proposal of the neutral theory
origin-fixation models used during this period are best understood as aggregate-rate models, even though the assumptions of infinite sites and free recombination underlying such models are not always stated explicitly

the assumption of vanishingly small mutation sizes in adaptive dynamics gives these models a deterministic character that is qualitatively different from other origin-fixation models

Wilke (2004) put the pieces together and pointed out that the earlier efforts by Bastolla and Takahata could be construed in origin-fixation terms using the framework of van Nimwegen et al. (1999)

implications of origin-fixation models are not shared by all theories of evolution
the mid-20th-century orthodoxy of population genetics rejected each of the above implications on the grounds that evolution is intrinsically polygenic and epistatic

when the architects of this orthodoxy were arguing for their distinctive "shifting gene frequencies" view throughout the 1960s and 1970s, they were arguing against both the earlier Mendelian mutationist view (Stoltzfus and Cable 2014) and the emerging "molecular evolution" view