Specifically, of all the possible evolutionary paths a population can take, the analysis of Lynch and Abegg considers only those special paths that lead directly to the desired end—the complex adaptation. This is best illustrated with an example. Suppose a population carries an allele that confers no selective benefit in its current state (e.g., a pseudogene or a gene duplicate) but which would confer a benefit if it were to acquire five specific nucleotide changes relative to that initial state, which we will again refer to as stage 0. Lynch and Abegg assign a waiting time of (5u)-1 for a stage-1 allele to become fixed in this situation, which is valid only if we can safely assume that the population remains at stage 0 during this wait. But this cannot be assumed. A stage-0 allele of kilobase length, for example, would have about 200-fold more correct bases than incorrect ones (with respect to the complex adaptation), which means the rate of degradation (i.e., fixation of changes that make the complex adaptation more remote) would be about 600-fold higher³ than the rate of progression to stage 1. It is therefore very unlikely in such a case that the population will wait at stage 0 long enough to reach stage 1, and the situation becomes progressively worse as we consider higher stages.

It is possible to adjust the problem to some extent in order to achieve a more favorable result. For example, if we suppose that all changes except the five desired ones are highly maladaptive, then fixation becomes restricted to changes at the five sites. But even under these artificial restrictions, Equation 2 is incorrect in that it ignores back mutations [6]. Since the aim is to acquire the correct bases at all d sites, and there are more incorrect possibilities than correct ones at each site, counterproductive changes must substantially outnumber productive changes as the number of correct bases increases. So even in this highly favorable case, the analysis suffers from neglect of counterproductive competing paths. Productive changes cannot be ‘banked’, whereas Equation 2 presupposes that they can.

Lynch and Abegg’s treatment of stochastic tunneling with neutral intermediates is also problematic. To derive an expression for the waiting time when the population is large enough to preclude fixation of intermediate stages (Equation 5b of reference 6), they approximate the frequency of stage-d alleles at t generations as (ut)^d. While they note that this approximation is valid only if ut << 1, they overlook the fact that this restricts their analysis to exceedingly small values (ut)^d. Specifically, they equate this term with the substantial allele frequency at which fixation becomes likely⁴, and then proceed to solve for t, taking the result to be valid for arbitrarily large values of d. This leads them to the unexpected conclusion that “in very large populations with neutral intermediates, as d→∞, the time to establishment converges on the reciprocal of the per-site mutation rate, becoming independent of the number of mutations required for the adaptation” [6]. But since they have in this way neglected the effect of d, it should be no surprise that they find d to have little effect.

A primary reason for this behavior is that a multistage route to adaptation increases the number of paths to the final adaptive state by magnifying the numbers of possible mutations in the early stages of the process. At large d, this increase in the number of possible paths comes close to compensating for the increase in the number of steps required to reach the final adaptive state.