In 2010, Lynch and Abegg claimed in a widely cited journal article that neutral evolutionary processes could generate complex adaptations much more rapidly than was previously believed. Their article contained a mathematical flaw, which was pointed out by Dr. Douglas Axe, but Axe’s critique continues to be ignored by senior evolutionary biologists, including the article’s authors, Professor Joe Felsenstein and Professor Larry Moran. Want proof? Read on.

For those readers who don’t know him, Michael Lynch is an eminent scientist: he is Distinguished Professor of Evolution, Population Genetics and Genomics at Indiana University, Bloomington, Indiana. He has also written a two-volume textbook with Bruce Walsh, which is widely regarded as the “Bible” of quantitative genetics. In 2009, he was elected to the National Academy of Sciences. His co-author, Adam Abegg, was an undergraduate student at the time when the paper was published. He is now a Research Assistant at University of Southern California.

For his part, Dr. Douglas Axe is the director of the Biologic Institute. His research uses both experiments and computer simulations to examine the functional and structural constraints on the evolution of proteins and protein systems. After obtaining a Caltech Ph.D., he held postdoctoral and research scientist positions at the University of Cambridge, the Cambridge Medical Research Council Centre, and the Babraham Institute in Cambridge. He has also written two articles for the Journal of Molecular Biology (see here and here for abstracts). He has also co-authored an article published in the Proceedings of the National Academy of Sciences, an article in Biochemistry and an article published in PLoS ONE.

### Academic cover-up? You decide

On October 8, 2015, I addressed the following comment to Professor Larry Moran, over on his *Sandwalk* blog:

Hi Professor Moran,

Frankly, I’d be a lot more impressed with the claims of evolutionists (of whatever stripe) of they were willing to take criticisms by Intelligent Design scientists seriously.

Case in point: consider the 2010 paper by Lynch and Abegg, titled, “The Rate of Establishment of Complex Adaptations” (

Molecular Biology and Evolution27:1404-1414. doi:10.1093/molbev/msq020, available online at http://mbe.oxfordjournals.org/content/27/6/1404.full ), purporting to show that under the neutral theory, complex adaptations could get established in a population a lot faster than previously assumed.Dr. Doug Axe wrote a response to that paper (“The limits of complex adaptation: An analysis based on a simple model of structured bacterial populations,” BIO-Complexity 2010(4):1-10.doi:10.5048/BIO-C.2010.4, available online at http://bio-complexity.org/ojs/index.php/main/article/view/BIO-C.2010.4 ). Lynch did not reply.

I emailed Lynch a couple of months ago, inviting him to comment on Axe’s paper. To my surprise, he said he hadn’t seen it before.

But when I asked him what was wrong with Axe’s criticisms, he declined to be drawn into the discussion.

I also invited you to point out what was wrong with Dr. Axe’s criticisms of Lynch, in an online post on Uncommon Descent at https://uncommondesc.wpengine.com/intelligent-design/id-like-a-straight-yes-or-a-straight-no-professor-moran/ .I’m still waiting.

I also wrote several times to Professor Joe Felsenstein, inviting him to say what was wrong with the paper.At first he promised he would respond, but he hasn’t gotten back to me, after several weeks. I’ve given up sending him reminders.

Something very, very fishy is going on here.It sounds like the evolutionists have given up debating Intelligent Design advocates, and are preferring to either ignore them or lampoon their views, instead…

### Professor Moran’s excuse: The math is over my head

In response, Professor Moran wrote a brief comment dated October 8, 2015 (1:51 p.m.), in which he excused himself on the grounds that he wasn’t qualified to assess the mathematical modeling (I’ve highlighted key sentences in bold type – VJT):

Vincent Torely (

sic) says,

I also invited you to point out what was wrong with Dr. Axe’s criticisms of Lynch, in an online post on Uncommon Descent. I’m still waiting.Unlike you and most of your creationist friends, I’m not an expert on everything.

In this particular case it requires expertise in mathematical modeling of evolution and I know very little about that subject.Presumably you don’t either yet you imply that Doug Axe has challenged one of the world’s leading experts on the subject. Why should I take you seriously?

Professor Moran’s disclaimer is very odd, considering that he was confident enough to answer my question (which I posed to him in a May 2015 post), **“Are you claiming that Dr. Axe and Dr. Meyer have misconstrued the nature of random genetic drift?”** with a simple, unqualified, “Yes.” How could he be so sure of that, if he didn’t understand the math in Dr. Axe’s paper?

### Professor Felsenstein’s response: “I’m on it!” (That was three months ago.)

Professor Felsenstein (who is, I have to say, more civil than Larry Moran) then promised to review Axe’s paper, in a brief follow-up comment on the same thread, dated October 8, 2015 (11:36 p.m.):

Vincent Torley recently reminded me of his request (and my excessively optimistic promise) that I look over the math of Lynch and Abegg vs. Axe.

I will get that done.

Three months have passed since he made that promise. I might add that I first contacted him about Dr. Axe’s critique of Lynch and Abegg (which is just nine pages long, excluding references), way back in August 2015. That’s five months ago. I realize that he is a busy man, but I have to say that the lack of a response appears rather suspicious to me. Could it be that Professor Felsenstein hasn’t managed to find any errors or gaps in Dr. Axe’s mathematical reasoning? It’s beginning to look that way.

In an email to me, Professor Felsenstein also mentioned that he did not understand why, in Lynch and Abegg’s paper, the rate of occurrence of alleles with multiple mutations would depend on a linear function of the mutation rate, rather than a higher order of the mutation rate. In regard to Dr. Axe’s paper (which he had yet to read), Professor Felsenstein noted that someone putting forward an “impossibility” argument has to close off *all* available options, and not just some. Otherwise, the argument fails.

### Professor Lynch’s response: feigned interest, followed by brusque dismissal

When I first contacted Professor Lynch last year about Dr. Axe’s critique of his 2010 paper, he kindly thanked me for alerting him to it, but claimed he’d never heard of it before, and added it would take some time to get to it (which I took to mean: read and respond to it). When I contacted him subsequently, his reply this time was rather dismissive: he said he didn’t think it was a worthwhile use of his time. And what was his excuse? He said he doubted whether any serious biologist took Dr. Axe’s paper, let alone his journal (*BIO-Complexity*), seriously.

**I have to ask my readers: doesn’t it sound a little suspicious** when someone writes a detailed mathematical critique of a paper a scientist has written, and that scientist is initially *thankful* when his attention is drawn to the critique, but then later on, he *refuses to respond* to it, simply because he doesn’t think any *other* scientist would take the paper seriously?

I should add that I bent over backwards to make matters as easy as I could for Professor Lynch to review Dr. Axe’s paper, by cutting and pasting the relevant sections from the paper into my email, to save Professor Lynch the trouble of reading through the entire paper to locate them. I shall reproduce them below, and invite readers to form their own judgments.

### What Lynch and Abegg claimed in their paper

In their 2010 paper, Lynch and Abegg argued for “the plausibility of the relatively rapid emergence of specific complex adaptations by conventional population genetic mechanisms.”

After some detailed calculations, Lynch and Abegg concluded (bolding below is mine – VJT):

Noting that realistic population sizes, mutation rates, and selection coefficients have been applied throughout, these results suggest that quite

complex alleles, with multiple neutral or deleterious intermediate states, can readily emerge in populations on time scales of 10^3 – 10^8 generations.Thus, for microbes with generation lengths of hours to days and very large population sizes, the mean time to establishment can easily be on the order ofa few weeks to several monthsdepending on the complexity of the final allelic state.Even multicellular species, with effective population sizes in the vicinity of 10^6 (Lynch 2007),are capable of establishing fairly complex adaptations on time scales of a few tens of millions of generations, the exact time span depending on the magnitude of the selective (dis)advantages of the intermediate and end states.

A few tens of millions of generations would mean a few tens of millions of years, for most animals. If Lynch and Abegg are correct, then evolutionists all around the world can heave a huge sigh of relief. After all, even a span of 80 million years would represent a mere 2% of the 4-billion-year history of life on Earth, so the origin of complex adaptations within the allotted time-span no longer appears to be much of a problem.

The concluding paragraph of Lynch and Abegg’s paper reads as follows:

Ultimately, the paths most frequently taken in the origins of evolutionary novelties will also depend on the rates at which neutral versus deleterious intermediate mutations arise. However, assuming that the distributions of selection coefficients for de novo mutations are not radically different among organisms, the preceding observations strongly suggest that

the paths to adaptation may deviate strongly among organisms from different domains of life.Relative to multicellular eukaryotes,prokaryotes are expected to acquire adaptive alleles by paths involving neutral intermediates several times more rapidly than eukaryotes on a per-generation basis.In contrast,when there are multiple deleterious intermediate steps, multicellular species have much greater expected rates of acquisitionof adaptive alleles on a per-generation basis. Of course, because the generation lengths of multicellular species can easily be 10^3 – 10^5 times greater than those for microbes, on an absolute time scale, rates of microbial adaptation may be comparable to or even greater than those for multicellular species even when intermediate allelic states are deleterious. However,the message to be gained from the preceding results is that the elevated power of both random genetic drift and mutation may enable the acquisition of complex adaptations in multicellular species at rates that are not greatly different from those achievable in enormous microbial populations.

### The meat of Dr. Axe’s criticisms

In the passages quoted below, I have used ^ for superscripts and _ for subscripts. All bolding is mine – VJT.

First of all, here’s the abstract of Dr. Axe’s paper:

AbstractTo explain life’s current level of complexity, we must first explain genetic innovation. Recognition of this fact has generated interest in the evolutionary feasibility of complex adaptations—adaptations requiring multiple mutations, with all intermediates being non-adaptive. Intuitively, one expects the waiting time for arrival and fixation of these adaptations to have exponential dependence on d, the number of specific base changes they require. Counter to this expectation, Lynch and Abegg have recently concluded that in the case of selectively neutral intermediates, the waiting time becomes independent of d as d becomes large. Here, I confirm the intuitive expectation by showing where the analysis of Lynch and Abegg erred and by developing new treatments of the two cases of complex adaptation — the case where intermediates are selectively maladaptive and the case where they are selectively neutral. In particular, I use an explicit model of a structured bacterial population, similar to the island model of Maruyama and Kimura, to examine the limits on complex adaptations during the evolution of paralogous genes—genes related by duplication of an ancestral gene. Although substantial functional innovation is thought to be possible within paralogous families, the tight limits on the value of d found here (d ≤ 2 for the maladaptive case, and d ≤ 6 for the neutral case) mean that the mutational jumps in this process cannot have been very large. Whether the functional divergence commonly attributed to paralogs is feasible within such tight limits is far from certain, judging by various experimental attempts to interconvert the functions of supposed paralogs. This study provides a mathematical framework for interpreting experiments of that kind, more of which will needed before the limits to functional divergence become clear.

Next, here’s a short passage which will help readers understand the background to Lynch and Abegg’s paper:

Three potential routes to the fixation of complex adaptations have been recognized.

The simplest is the, which for large innovations is tantamount tode novoappearance in one organism of all necessary changesmolecular saltation. This route has the advantage of avoiding non-adaptive intermediates but the disadvantage of requiringa very rare convergence of mutations.The second potential route issequential fixation, whereby point mutations become fixed successively, ultimately producing the full set needed for the innovation. By this route, the rate of appearance of each successive intermediate en route to the complex adaptation is boosted by allowing the prior intermediate to become fixed.But because these fixation events have to occur without the assistance of natural selection(or, in the case of maladaptive intermediates, even against natural selection)they are in themselves improbable events.The third potential route isstochastic tunneling, which differs from sequential fixation only in that itdepends on each successive point mutation appearing without the prior one having become fixed.Here fixation occurs only after all the mutations needed for the innovation are in place. This route therefore benefits from anabsence of improbable fixation events, but it must instead rely on the necessary mutations appearing withinmuch smaller subpopulations.

Molecular saltation seems incompatible with Darwinian evolutionfor the same reason all forms of saltation do — namely,the apparent inability of ordinary processes to accomplish extraordinary changes in one step.If specific nucleotide substitutions occur spontaneously at an average rate of u per nucleotide site per cell, and a particular innovation requires d specific substitutions, then the rate of appearance of the innovation by molecular saltation (i.e., constructionde novoin a single cell) is simply u^d per cell. This means that the expected waiting time (in generations) for appearance and fixation of the innovation scales as u^-d. But since u has to be a very small fraction in order for a genome to be faithfully replicated (the upper bound being roughly the inverse of the working genome length in bases), u^-d becomes exceedingly large even for modest values of d, resulting in exceedingly long waiting times.Because of this,

sequential fixation and stochastic tunneling are thought to be the primary ways that complex adaptations become fixed.

Finally, here’s the meaty mathematical critique which I emailed to Professor Lynch:

Among the many treatments of complex adaptation by sequential fixation and/or stochastic tunneling (e.g., references 2–6),

one recently offered by Lynch and Abegg [6] is of particular interestbecause it claims that the above limitation vanishes in situations where the genetic intermediates en route to a complex adaptation are selectively neutral. For this case, they report that“regardless of the complexity of the adaptation, the time to establishment is inversely proportional to the rate at which mutations arise at single sites.”In other words, they find the waiting time for appearance and fixation of a complex adaptation requiring d base changes to scale as u^(-1) rather than the commonly assumed u^(-d).Because this should apply not only in the case of strict neutrality (which may seldom exist) but also in the more realistic case of approximate neutrality, and because it represents a striking departure from common probabilistic intuitions, it is important for this result to be examined carefully.…ANALYSIS

Assessing Lynch and Abegg’s treatment of the neutral case

Innovation by sequential fixation has to work against natural selectionSequential fixation.if any of the genetic intermediates are maladaptive, which is apt to be the case in many evolutionary scenarios. In such cases,sequential fixation is conceivable only in small populations because the efficiency of natural selection in large populations makes maladaptive fixation nearly impossible[7]. Focusing therefore on small populations, and initially on the limiting case of selectively neutral intermediates,Lynch and Abegg calculate the mean waiting time for arrival of an allele carrying a particular complex adaptationthat is destined to be fixed within a diploid population as (1):w_seq = [d.u]^(-1) + [(d-1).u]^(-1) + [(d-2).u]^(-1) + … + [2u]^(-1) + [2u. N_e. phi_2]^(-1), (2)

where d is the number of specific base substitutions needed to produce the complex adaptation, N_e is the effective population size (2), u is the mean rate of specific base substitution (per site per gamete), and phi_2 is the probability that an instance of producing the adaptive allele will result in fixation (given by Equation 1 of reference 6).

When written in the above expanded form, we see that the waiting time is being equated with a sum of d terms, each of these terms being the inverse of a rate. As explained by Lynch and Abegg [6], the individual rates are simply the rates of appearance within the whole population (per generation) of instances of the successive genotypic stages that are destined to become fixed, given a population in which the prior stage is fixed. So, for example, in a case where the complex adaptation requires five specific base substitutions (d = 5) and the predominant genotype in the initial population lacks all five, there are five possible ways for an allele of the initial type (call it stage 0) to progress by mutation to the next stage (stage 1). The first term enclosed in square brackets is 5u in this case, which is the per-gamete rate of appearance of stage-1 alleles in a stage-0 population. Because each individual allele in a population of neutral variants is expected to become fixed with a probability equal to the inverse of the total allele count [8], the rate of appearance of stage-1 alleles that are destined to become fixed in a diploid population of N individuals is equal to the total rate of appearance (2N × 5u per generation) divided by 2N, which equals the per-gamete rate of mutation to stage 1, namely 5u per generation.

When a stage-1 allele is fixed, there are now four possible ways for mutation to produce a stage-2 allele. The second term in square brackets, 4u, is again the rate of appearance of stage-2 alleles that are destined to become fixed within a population where a stage-1 allele has become fixed, and so on for the third and fourth terms. The final term differs because the complete complex adaptation (stage 5) has an enhanced fixation probability resulting from its selective advantage.

Since the mean wait for a stochastic event to occur is just the inverse of its mean rate of occurrence, the right-hand side of Equation 2 is now seen to be the sum of waiting times, specifically the mean waiting times for appearance of destined-to-be-fixed alleles at each successive stage within populations in which the prior stage has become fixed.

The actual process of fixation (after a destined-to-be-fixed allele appears) is relatively fast in small populations, taking an average of (4.N_e) generations for neutral alleles[8]. Assuming this to be negligible, Equation 2 may seem at first glance to represent precisely the intended quantity — the overall waiting time for fixation of the complete complex adaptation from a stage-0 starting point. But if the implications of this equation are as implausible as has been argued, thenthere must be a mistake in the calculation.

This indeed appears to be the case.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 higher(3) than the rate of progression to stage 1.It is therefore very unlikelyin 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,Stochastic tunneling.they overlook the fact that this restricts their analysis to exceedingly small values [of] (ut)^d.Specifically, they equate this term with the substantial allele frequency at which fixation becomes likely(4), 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.

Footnotes(parentheses)

1 See Equation 5a of reference 6, which uses t-bar_e, s to represent the same quantity.

2 Much of the genetic drift in real populations results from non-uniform population structure and dynamics. In essence, the effective size of a real population is the size of an ideal population lacking these non-uniformities that has the same level of genetic

drift. A detailed discussion of N_e is found in the results and Discussion section.

3 Of the three possible changes to an incorrect base in this example, only one corrects it.

4 This being (4N_e .s_2)^-1 where s_2 is the fractional advantage conferred by the stage-d allele.

References[square brackets]2. Behe MJ, Snoke DW (2004) Simulating evolution by gene duplication of protein features that require multiple amino acid residues. Protein Sci 13: 2651-2664. doi:10.1110/ps.04802904

3. Lynch M (2005) Simple evolutionary pathways to complex proteins. Protein Sci 14: 2217-2225. doi:10.1110/ps.041171805

4. Durrett R, Schmidt D (2008) Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics 180: 1501-1509. doi:10.1534/genetics.107.082610

5. Orr HA (2002) The population genetics of adaptation: The adaptation of DNA sequences. Evolution 56: 1317-1330. doi:10.1111/j.0014-3820.2002.tb01446.x

6. Lynch M, Abegg A (2010) The rate of establishment of complex adaptations.

7. Kimura M (1980) Average time until fixation of a mutant allele in a finite population under continued mutation pressure: Studies by analytical, numerical, and pseudo-sampling methods. P Natl Acad Sci USA 77: 522-526. doi:10.1073/pnas.77.1.522

8. Kimura M, Ohta T (1969) The average number of generations until fixation of a mutant gene in a finite population. Genetics 61: 763-771.

### Conclusion

The passages above represent a very brief selection from Dr. Axe’s paper, but hopefully, they are sufficient to expose the faulty mathematical reasoning in Lynch and Abegg’s 2010 paper.

I shall now throw the discussion open to readers. Am I the only one who thinks there’s something very funny going on here, when leading evolutionary biologists refuse to respond to the trenchant criticisms put forward by Dr. Axe in his paper?

What do readers think?