Another month, another livestream between @GutsickGibbon and Will Duffy.
Duffy’s recap presentation lasted for 37 minutes [10:52 - 48:00]. I am not going to cover everything Will said. Only the parts that are related to what I wrote in PART I regarding a version of the ‘waiting-time argument’ which is practically copy-pasted from Vox Day.
Summary: The argument proposes that the fixation rate of mutations is too slow to account for the observed mutational differences we see between humans and chimps if they diverged from their last common ancestors around 6 million years ago. I addressed these arguments in PART I. See that for more background information.
1. Factor “d”
Skipping to section [29:31 - 35:34] Will begins the segment on Vox Day to complain that people don’t understand what factor d is.
[30:23] WILL DUFFY: When Zach was presenting, he said… so he Vox has this other term d… but we don’t know what it means. So he admitted that he doesn’t know what it means which makes sense. Unfortunately that entire section of the video was a category error and I didn’t find one person in the thousands of comments who caught this. I could be wrong here but I think it’s because people did not read probability zero.
Will should not be surprised to see people not understanding what factor d is supposed to be, since, in the previous live stream, he did not even bother to explain what it is. The only information he provided was the factor is called the “selective turnover coefficient” and that it “corrects for overlapping generations”, but did not provide any details. He used the value 1.0 for bacteria (no overlapping-generations), which means the factor can be ignored. What is the supposed value for humans or chimps? He did not provide any. He still does not in the latest live stream. So that was entirely unhelpful. Apparently, Will was expecting others to figure all of this out on their own, or by reading Vox’s book… which… NO WILL! If you present an argument (even if it is borrowed from someone else), it is your job to explain it thoroughly so everyone can understand what you are claiming. On top of that, I am definitely not going to waste my money (not even a penny) to buy AI generated slop.
Next, Will is responding to @talkpopgen who pointed out that by correcting the census population size into the effective population size N_e already corrects for overlapping generations.
[33:55] WILL DUFFY: Zach says here:
ZACH: “here are two classic models. There’s the Wright-Fisher model and the Moran model. And in its classic sense, the Wright-Fisher model does not have over overlapping generations. But you can equate the two by setting the population size the N value in the Wright-Fisher model to an effective population size N_e and you just scale that effective population size by the slowdown of drift due to overlapping generations. So even the Wright-Fisher model can account for overlapping generations.”
Zach is correct here in what he’s saying. But since he didn’t read probability zero and doesn’t understand day’s selective turnover coefficient d, he doesn’t know why it’s still necessary and why it captures something that N_e misses. Both models, Wright-Fisher and Moran are answering this question: How fast does genetic drift move neutral alleles through the population? Day selective turnover coefficient d is designed to answer an entirely different question: How fast does the entire gene pool turn over?
Changing the total population size N to the effective size N_e corrects for the rate of genetic drift per generation. The rate at which the gene pool turns over is definitionally the average number of generations per unit time. If the number of generations per unit time doubles, the gene pool turnover rate doubles, and so does the rate of genetic drift. If the rate of generations per unit time is cut in half, so is the gene pool turnover rate, and the rate of genetic drift. It scales linearly. Therefore, correcting for the former automatically corrects for the latter.
The error of using factor d stems form the thinking that overlapping generations somehow “slows down” the effective generation time. Their thinking: if the generation time is 30 years, the standard PopGen model assumes everyone reproduces at the age of 30 simultaneously and promptly die afterwards. But in real populations, older individuals of the first generations survives and reproduces alongside younger individuals of later generations. If this generational overlap is about 50%, then d=0.5. That would double the “effective” generation length to 60 years, thereby slowing down the rate of fixation. However, this is NOT how you account for the variation in ages of reproduction.
Let’s set-up a simple hypothetical of a population wherein 50% of babies are born to parents aged 20 and 50% to parents aged 40. This means the average generational overlap 50%, so d = 0.5. However, the ‘effective’ generation time still remains the average of 30 years. The average already balances out the “decelerating effect” of reproduction at older ages with the “accelerating effect” of reproduction at younger ages. Factor d = 0.5 pushes the generation time to 60 years, which is completely erroneous!! It implies that… for some reason… due to overlapping generations… there can effectively be only be 10 generations within 600 years. However, the average number of generations for all lineages would still be 20 generations; with an absolute minimum of 600/40= 15 and an absolute maximum of 600/20= 30 generations. Not just 10 generations!!
Overlapping generations actually does have an effect on the rate of drift. However, it does not change the “effective” generation time. It alters the sampling error of neutral alleles. It alters the degrees at which the allele frequencies are changing per generation. This is indeed accounted for when using the effective population size N_e instead of the census population size N.
The William Hill formula estimates N_e from N_{gen}, total number births per generation, and V_k the variance of reproductive success:
In an ideal Wright-Fisher population that remains constant in size, the average number of offspring each individual has is 2, and the number of offspring per individual is random (following a Poisson distribution). So, some individuals leave no offspring, a few individuals have a lot more, but the average remains 2 to keep the population size stable. In this case, V_k = 2, which means…
…the effective population size is approximately equal to the number of births per generation, which in turn also equals to the census population size of N (including reproducing adults only).
However, as noted before, in a Wright-Fisher model everyone has the exact same lifespan, everyone has the same reproductive odds, and everyone reproduces at the same age. In reality, this is not the case. For most of human history, almost 50% died in childhood before reproducing. The reproductive success of people aged 20-30 is also far greater than those reproducing at younger or older ages. Because there are distinct within-age and between-age variances of survivorship and reproductive success, that will increase the value of V_k far above 2. For most of human history, the variance ranged between 5 and 12, which means:
Overlapping generations increasing the sampling error, which reduces the effective population size down to 57% or 29% relative to the census (reproducing adults only) population size. Our N_e was close to 10,000 while our census population was much larger.
NOTE: by reducing N down to N_e to account for overlapping generations, this will actually increase the rate of genetic drift per generation. NOT slow it down. The complete opposite of what Will is arguing. Below notates the variance regarding the frequency change of one neutral allele after on generation, with a starting frequency of p. This means the maximum variance is created when the frequency is 50% p=0.5. This is intuitive since the frequency change of a neutral allele becomes most unpredictable if it is an exact 50/50 coin-flip. The variance (rate of drift) also increases with lower effective population sizes N_e.
Lastly, Will makes the following claim:
[35:01] WILL DUFFY: The d formula involves variables that simply have not been accounted for in standard population genetic models. And the d formula is this. It’s very well thought out.
Will makes NO attempt to substantiate this claim. All he does showing this formula for less than four seconds without explaining even a single factor, calling it “very well thought out”, and promptly moves on. Once again, this was completely unhelpful.
REQUEST TO WILL: If you plan to show this, or any other mathematical formula, in an upcoming slide show, please… PLEASE!!.. explain every single term that is included in the formula, and provide a simplified hypothetical example showing how the calculation is done step-by-step (like how I explained the formulas above). This is not an unreasonably difficult request. It would only take a few minutes of your presentation. Don’t flash a formula for less than 4 seconds and expect everyone else to instantly understand it.
2. Mutation Rate
Moving on to segment [35:34 - 37:40] Will covers the bit when @talkpopgen corrects himself when he used the diploid genome size of 6.4 Gbp to calculate that the genome wide mutation rate u_G of 76.8. He should have used the haploid genome size of 3.2 Gbp instead. Nevertheless, the rate of 76.8 can still be used, citing Porubsky et al. 2025.
[1:52:48] ZACH: Importantly, we can’t use 6.4. This is the diploid genome size for the fixation rate because you fix on haploid genome. So, it should be 3.2. Okay. So, I want to quickly clarify that value. But I also want to say that we can still use this value in the following derivations. And this is why. So I looked up this paper Porubsky et al. 2025 and they estimate the de novo human mutation rate for not just single nucleotide polymorphism but also for lots of other kinds of structural variation and they estimate between 98 and 206 de novo mutations every single generation. That’s 152 on average. If we divide that in half to take the haploid genome size that gives us 76. Okay, so just to to clarify that we can still use the 76.8 number that I derived just a moment ago uh in in the future derivations.
I concur that neutral mutation rates between 50 and 100 are reasonable. I used a haploid genome-wide neutral mutation rate of 75 in my own calculations in PART I. But Will has the following objection.
[36:37] WILL DUFFY: Zach referenced Porubsky et al 2025 and said they estimate an average of 152 de novo mutations every single generation and dividing that in half for the haploid genome size gives us 76.8 meaning the original number can still be used. But this is a pretty significant error. The Porubsky et al. number includes large structural variants, insertions, deletions, and duplications, not just single nucleotide polymorphisms, SNPs. So here’s the problem. Zach’s entire calculation that you see on the whiteboard is explicitly about single nucleotide differences, and he uses a single nucleotide mutation rate. And you can’t validate a snip only calculation with a number that includes large structural variation.
That’s not true at al. Zach’s calculation which he showed at [1:41:09 - 1:48:55] which explained why the neutral mutation rate of u_G is the same as the neutral substitution rate of k is not about SNPs exclusively. Neither is his calculation at [1:50:20 - 1:51:50] where Zach calculated the genome-wide mutation rate. I am guessing that Will is referring to the per-base mutation rate of 12e^{-8} that Zach used in the latter calculation. However, that mutation rate is not exclusively the mutation rate for point mutations (generating SNPs). It estimates how often a mutation (point mutation, or InDel, or whatever) occurs at each base pair. Zach never explicitly stated or implied that it was exclusively the point mutation rate.
3. The Texas Sharpshooter Fallacy
Zach calculated that, just by neutral theory alone, we expect that humans and chimps would diverge from another by around 38 million mutations. I personally calculated a similar number of 37.8 million in PART I. At [37:39 - 38:50] Will has three objections to this figure.
[37:49] WILL DUFFY: The 38 million number is incorrect for three reasons:
- First, the calculation as we just talked about is being run on the diploid genome size 76.8 instead of the haploid genome of 38.4.
Zach already explained (citing Porubsky et al. 2025) that the diploid mutation rate lies between 98 and 206 mutations per generation. Thus, the corresponding haploid rate would be between 49 and 103. So, a mutation rate of 76.8 is still not unreasonable, despite Zach’s error of using the diploid genome size.
- Number two, minor. He’s using 252,000 generations instead of the d corrected size of 201,600.
I covered why including this factor d is erroneous previously in section 1.
- Most importantly, neutral theory applies to the entire genome. Zach is using circular reasoning in assuming the neutral prediction of 35 million differences all fall perfectly into the divergent sites.
In reality, divergent sites are only 1.09% of the genome. So statistically only 1.09% of the neutral substitutions would end up in the divergent zone. So the proper formula would therefore be: 15,482,800 (35,000,000 / 3,200,000,000) = 169,344
So Zach’s original 38 million number is is in actuality just 169,344.
I don’t understand why Will uses 35 million instead of Zach’s 38 million. Secondly, the figure of 15,482,800 for the neutral theory predicted differences is based on Will’s haploid mutation rate of 38.4 and the “d corrected” number of generations of 201,600. I already noted previously why these are incorrect.
But… even putting those issues aside… this is ASTONISHINGLY wrong. I was completely gobsmacked when I saw this for the first time. It’s the weirdest instance of the Texas Sharpshooter fallacy I have ever seen. If you don’t see what is happening, here is what Will did. He takes all the locations of the human genome that are divergent from chimps, which represents around 1.09% of the human genome. He defines those locations as the ‘targets’. Will’s thinking goes like this: since neutral mutations randomly occur across the whole genome, only a small fraction (1.09%) of the 38 million fixed neutral mutations predicted by neutral theory will fall within the ‘targets’… and these tiny fraction are the only differences that count in the mind of Will. Of course, this complete asinine. It does NOT matter where in the genome each of the predicted 38 million fixations occur have fallen. Each and every site will… DEFINITIONALLY… be a divergence site. What else can such a site be? Every location wherein humans have fixed a neutral mutation which makes it different from chimps (and vice versa) is by definition a divergent site. This is not “circular reasoning”. It’s true by definition. It’s as circular as assuming that all bachelors are not married.
4. Latency vs. Rate
At [38:51 - 41:37] Will has a problem with Zach pointing out that Vox’s calculation assuming sequential fixation (only one mutation can go from the starting frequency to fixation at one time) instead of allowing many neutral fixations to wander in the population and reach fixation in parallel. Will asserts that Vox’s calculation is not a “latency” calculation but a “rate” calculation. To Will’s credit, he adequately explained the differences using intuitive between a latency-based calculation (how long it takes for each mutation to go from the starting point to fixation on average) and a rate-based calculation (how many mutations fix per generation) using an intuitive high-way analogy:
A rate-based calculation: 3,600 seconds / 360 cars = 10 seconds per car
However, Will DID use a latency calculation specifically to argue against fixation by drift during the previous livestream at 36:50. He used this formula to claim that, with N_e = 10,000, neutral fixation takes 40,000 generations or around 800,000 years (assuming 20 years per generation). Therefore, the rate of neutral fixation is too slow.
4N_e estimates the average number of generations it takes for a neutral mutation (destined for fixation) to go from the starting frequency to reach fixation (within a diploid population). This is a latency calculation. It does not mean the rate of neutral allele fixation is only 1 per 800,000 year interval. To borrow Will’s car analogy, it only calculates the average time it takes for one car to go from point A (the left-side of the slide) to point B (crossing the red line). It does not calculate how many cars cross the red line per unit of time. I already pointed this out this error in section 3.2 in PART I.
However, Will is likely specifically referring to this formula when he is claiming Vox is using a rate calculation:
Duffy simply (repeatedly) asserts that Vox’s calculation is a rate calculation, not a latency calculation. He actually doesn’t explain or demonstrate why this is the case. He only explains the car analogy. Will doesn’t even show the formula that is supposedly the rate calculation… again… I am assuming he is referring to this formula shown in the slide above. The formula shown here is neither a latency calculation nor a rate calculation. That entirely depends on the value you use for factor G_f. If you use a latency figure, the formula will be a latency calculation. Will used the figure of G_f = 1,400 citing one paper Barrick et al. 2019 on the LTEE with E. coli. However, as I explained in section 3.5 of PART I, the predicted fixation rate of neutral mutations in humans means the figure is G_f = \frac{1}{k} = \frac{1}{u_G} = \frac{1}{75} = 0.0133. Zach’s figure of u_G = 76.8 would give virtually the same value G_f = \frac{1}{76.8} = 0.0130.
SELF-CORRECTION: I previously said that the actual figure for E. coli from Barrick et al. 2019 is G_f = 444 for normal E. coli, and G_f = 23 for the mutator phenotype. However, these rates represent mutation “accumulation” rates, not how many mutations were fixed. So G_f = 1,400 can still be accurate for E. coli. However, this is NOT applicable to humans. Remember that the human genome-wide mutation rate is close u_G = 75, which means G_f = \frac{1}{75} = 0.0133. In contrast, the value for E. coli is closer to u_G = 0.000075 because its genome is almost 3 orders of magnitude smaller, and the per base-pair mutation rate is also much lower. This means their value is closer to G_f = \frac{1}{0.000075} = 13,333. It’s a vastly different situation.
EXTRA INFORMATION: I should also mention that the E. coli populations in the LTEE experiment started with clones at the first generation. This means there is a delay of about 2N_e generations until the E. coli population experiences a constant neutral fixation rate of G_f = \frac{1}{0.000075} = 13,333.(I explained this math in section 3.4 PART I). The N_e of E. coli during the LTEE experiments is over 30 million. That means we would have to wait 60 million generations until we reach a constant neutral fixation rate. The fact that we see a fixation rate of 1,400 very rapidly (after just 60,000 generations) means that fixation is mostly driven by selection (not drift), which is also concluded by the authors of the LTEE experiment.
5. MITTENS is a waiting-time problem argument
[42:01 - 42:40] WILL DUFFY: No, MITTENS is not a waiting time problem. It’s not accurate to describe MITTENS as the waiting time problem. The waiting time problem refers to the significant amount of time required for beneficial genetic mutations to occur in a population. The MITTENS argument isn’t concerned with mutation rates at all. It is an empirical audit of observable fixations. MITTENS would be better referred to as the fixation problem.
There are different types of waiting-time problem arguments floating around. All of them share the conclusion that the timelines (accepted by evolutionists themselves) are too short for evolution to work with, because evolutionary processes happen too slowly to account for the changes that must have happened during these timelines.
“MITTENS” attempts to argue that evolution requires more time; ~6 million years is not enough time for the rate of mutation fixation to generate the differences we see between chimps and humans. MITTENS is indeed a waiting-time problem argument. It is not “THE” waiting-time problem argument. There is not just one waiting-time problem argument. No one is saying MITTENS is identical to every other waiting-time problem argument, but it is indeed a waiting-time problem argument, because that’s what it is arguing for… that evolution supposedly requires more waiting-time.
Furthermore, the specific waiting-time problem argument that concerns ‘beneficial mutations’ (like those put forward by Michael Behe) concerns BOTH mutation and fixation rates. MITTENS is also concerned with both mutation and fixation rates. It has to since the fixation rate is dependent on the mutation rate. If the neutral mutation rate doubles, so does the neutral fixation rate.
6. Bacteria vs. Humans
[43:00 - 43:26] WILL DUFFY: I don’t know anyone who argues that humans fixate mutations faster than bacteria in terms of generations per fixation. I think it’s possible the audience walked away from that video thinking that is what was being argued here. So I’m hoping Erica and the others can clarify this for everyone whether or not you actually believe humans fixate mutations faster than bacteria.
Will should pay more attention. I previously explained this is indeed the case in PART I, and Zach also explained this in the video with Erika at [1:53:42 - 1:54:58].
There are two main reasons for this:
- Bacteria have much smaller genomes by almost 3-orders of magnitude. For the human genome L = 3.2 \text{ Billion}. For the E. coli genome L = 4.6 \text{ Million}.
- The per base-pair mutation rate is also much smaller, by almost 3-orders of magnitude. In other words, their DNA polymerases make far few mistakes for every base pair. The base pair mutation rate of humans is around u = 10^{-8} while that of E. coli is around u=10^{-11}.
- BONUS: The N_e sizes of humans is around 10,000 while those of bacteria can be many millions or even billions. That means new mutations will reach fixation by drift much faster in humans. The latency times is much shorter in humans. Although, does not affect the rate of neutral fixation, which is calculated as k=u \cdot L = u_G.
All of this means that the neutral fixation rate of k=75 for humans is almost 6-orders of magnitude greater compared to the rate of k=0.000075 for E. coli. Now this only counts for NEUTRAL evolution. More about selective evolution later.
[43:27 - 44:27] - WILL DUFFY: Zach said something good here. He said:
ZACH: “The issue with using even conceptually the Lensky long-term evolution experiment for a fixation rate is that the Lensky long-term experiment begins with a bacterial population that is clonal and has no diversity.”
I think Zach makes a really valid point here. This was also caught by Mike Griffiths, who brought it to my attention before this video was released. Now, here’s Vox Day’s response, which I’m not claiming to understand:
VOX: "The polymorphism objection sounds legitimate until you think about what standing variation is. It’s not a warehouse of pre-adapted beneficial mutations waiting for their moment. It’s mostly neutral variation, which is why it’s still segregating. If those variants were beneficial, they’d have already swept to fixation. If they were deleterious, purifying selection would have removed them. What mutation drift equilibrium gives you is a population full of genetic noise at low frequencies, not a head start on adaptation.
I don’t understand this enough yet to know who’s right. But here’s what both sides are saying.
First issue: If you don’t understand the argument, then there is no good reason for you to convey it.
Second: The standing neutral variation is important regarding the neutral rate of fixation. I also explained that here in previous section. If you start with clonal population without any standing variation, you need to wait for 2N_e generations for haploid populations, or 4N_e generations for diploid populations, until you reach the constant neutral fixation rate of k = u_G. If you already start with a standing variation of a population at drift-mutation equilibrium, you don’t need to wait.
Third: Standing variation often DOES contain mutations “waiting for their moment”. Yes, it’s mostly neutral, but some mutations change from being neutral (or deleterious) to becoming beneficial when environmental conditions change. Populations that are already rich in standing variation are able adapt to changing conditions - via evolution by natural selection - much faster relative to populations that start out as clones.
[44:30 - 45:10] WILL DUFFY: The main argument with the bacteria thing seems to be that Vox shouldn’t apply bacterial fixation rates to humans because bacteria are clonal and slow. Dr. Michael Lynch, who I’m told is one of the most respected population geneticists in the field, has stated that bacteria are actually faster and more efficient at molecular adaptation than eukaryotes, us because of their population size advantage. Here’s what he says.
MICHAEL LYNCH: Prokaryotes appear to be much more efficient than eukaryotes, at promoting simple to moderately complex molecular adaptations.
Michael Lynch is not saying evolution is faster. He is saying that prokaryotes are more efficient than eukaryotes at evolutionary adaptation. Those two are not the same. Lynch points out that the N_e sizes in prokaryotes are often many orders of magnitude greater than those in eukaryotes. The efficiency of natural selection is dependent on N_e because - within extremely large population sizes - the effect of selection far outweighs the random effect of drift. In other words, a beneficial mutation that appears in prokaryotes has a much higher likelihood to be fixed by selection (not lost by drift).
That does not mean that the total mutation fixation rates are greater in prokaryotes overall. It only means that the relative fixation rate for beneficial mutations (compared to neutral ones) is greater in prokaryotes. Eukaryotes still have a greater total fixation rate, mainly because their fixation rates are dominated by neutral mutations.