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

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

There is no such restriction. This is an invention of Svensson, not found in any of the original theoretical works on this topic.

For instance, consider a large population with strongly beneficial variants that we will designate as “left” and “right” introduced at rare intervals. 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 chance that left happens before right is B, there is a B-fold bias favoring left for this artificial scenario, and there is no dependence on K because 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. In Lynch’s Manichean view, for mutation to influence the course of evolution means having a non-adaptive state. So, the dependency on small populations in Lynch’s argument involves the fixation of a slightly deleterious allele. This is why the dependency is irrelevant to understanding the Yampolsky-Stoltzfus model, which is not a model of deleterious fixations. For a more lengthy explanation, see Bad Takes #2.

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