Bad takes #6: requires “drift in small populations”

Unfamiliar ideas are often mis-identified and mis-characterized. It takes time for a new idea to be sufficiently familiar that it can be debated meaningfully. We look forward to those more meaningful debates. Until then, fending off bad takes is the order of the day! See the Bad Takes Index.

Svensson (here or here) has repeatedly asserted that the effect of biases in the introduction process requires “drift in small populations,” citing Lynch (2007) as a source.

However, like the fake requirement for reciprocal sign epistasis, this is not an actual requirement, but emerges from Svensson’s tendentious misinterpretation of sources.

For instance, consider a large population with strongly beneficial variants that we will designate as “left” and “right” introduced at rare intervals, i.e., uN is small. Assume that fitness favors going right by virtue of a K-fold higher selection coefficient, but mutation favors going left with a bias of magnitude B. This is roughly the same set-up as the Yampolsky-Stoltzfus model, i.e., 1-step adaptation with 2 beneficial options, left (more mutationally likely) and right (more strongly beneficial).

First, suppose that there is drift. Drift only affects the chance of fixation vs. loss (not the mutational dynamics) and the probability of fixation for strongly beneficial alleles is hardly affected at all by population size. Using Kimura’s formula, the chance of fixation for an allele with fitness benefit s = 0.02 for haploid populations ranging from N = 103 to N = 109 is the same to six digits, namely 0.0392106. So, if we are in the origin-fixation regime, i.e., ignoring clonal interference, the evolutionary bias toward going left is still roughly B/K as shown by Yampolsky and Stoltzfus, and population size hardly matters.

Now, let us suppose that there is no drift: the population mutates stochastically but reproduces deterministically. This means that, ignoring clonal interference, the first mutation to occur is assured of fixation regardless of the size of the fitness benefit, and it will proceed deterministically to fixation. Because the mutationally favored option (the left option) has a B-fold chance of happening first, there is a B-fold bias favoring the left option, and there is no dependence on K because (in this artificial scenario) beneficial mutations are fixed deterministically regardless of the degree of beneficiality.

Clearly, this effect of biases in the introduction process does not depend on drift in small populations.

Apparently Svensson is confused by Lynch (2007), who presents a version of Bulmer’s mutation-selection-drift model and then uses this to make an overly broad claim about the conditions under which mutation will deflect the “direction” of evolution relative to the expectations of adaptation. The critical problem with Lynch’s Manichean view is that it imagines a universe in which evolution has only two possible directions, adaptive and non-adaptive. Thus, one must begin by recognizing that, for Lynch, the efficacy of mutation bias to influence the “direction” of evolution is a matter of finding conditions under which mutation assists in sustaining some non-adaptive state, a motivation utterly unlike that which stimulates the Yampolsky-Stoltzfus model. The dependency on small populations in Lynch’s argument arises from the fixation of a slightly deleterious allele that happens to be favored by mutation: for this reason, it is irrelevant to understanding the Yampolsky-Stoltzfus model, which has no deleterious fixations. For a lengthy explanation, see Bad Takes #2.

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