# Response to Will Duffy | PART I - Population Genetics

**URL:** <https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094>\
**Category:** Conversation\
**Tags:** Science, Communication\
**Created:** [October 5, 2026, 10:09pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094 "2026-10-05T22:09:25Z")\
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**Author:** ![Nesslig20](https://sea2.discourse-cdn.com/flex016/user_avatar/discourse.peacefulscience.org/nesslig20/32/11875_2.png) [@Nesslig20](https://discourse.peacefulscience.org/u/Nesslig20)\
**Post date:** [October 5, 2026, 10:09pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/1 "2026-10-05T22:09:26Z")

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I first wrote this response to Will Duffy with a total word count of 60529 words, exceeding the 32000 limit. Here is PART I. A separate topic will contain [PART II](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-ii-clades-and-chromsomes-and-ervs-oh-my/18095).

# PART I

## 1. PROLOGUE

### 1.1 Lotteries & Probabilities

John won a lottery, and the chance of winning was 1 in 100 million. Dale claims he can prove John cheated with the formula: [P = 1 - (1 - p)^n]. This calculates the probability of winning the lottery at least once with a given number of tickets [n]. Since [p = \frac{1}{100,000,000}]. Dale calculates how many tickets one needs to have a ≥50% probability of having at least one winning ticket:

1 - (0.99999999)^n ≥ 0.5

Which simplifies to:

(0.99999999)^n ≥ 0.5

Which can be expressed as:

\log(0.99999999) \times n ≥ \log(0.5)

We can rearrange it as:

n ≥ \frac{\log(0.5)}{\log(0.99999999)}

Which solves:

n ≥ 69,314,717

So having 69.3 million tickets gives you a 50% probability of winning at least once. However, this calculates the **median** number required to win. The **average** is simply the inverse of [p], so 100 million tickets. The median is lower than the average is because it’s possible to lose, even with 200 million tickets [P=0.86] or 400 million tickets [P=0.98]. This produces a heavy right-skewed probability distribution.

 ![Probabilities calculated with the dbinom() function and image generated with the plot() function in R.](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/d/8/d8a3f87b2978a7f9fee665ad9421940033691def.png)  
Dale notes that John only got one ticket… at least… John says he did. So Dale accuses John of cheating, because it’s so unlikely that he won fairly. Of course, people do win lotteries despite very similar odds. **So, where did Dale go wrong?** His flawed reasoning commits the **[Texas sharpshooter fallacy](https://en.wikipedia.org/wiki/Texas_sharpshooter_fallacy)**.  
 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/7/d/7d840b1a2cfa5556650a7014ea9b21efcbb33d28.jpeg)  
The name of the fallacy is in reference to an analogy. The proverbial sharpshooter professes to be an excellent marksman, claiming he can hit his targets accurately at a distance. His evidence is a wall filled with bullet holes, each are directly in the center of a bullseye. Except, the marksman omits to mention that he painted those bullseyes AFTER the bullets were fired.

The fallacy does not have to involve intentional deception. The Texas sharpshooter fallacy applies whenever someone is mistakenly inferring a cause from mundane or random event(s) after the fact. The key is that the erroneous inference is made in a _post hoc_ manner. In the case of Dale, he accuses John of cheating only after the fact of John winning the lottery. Dale is ignoring the other 70 million participated who lost. The probability that at least 1 out of 70 million wins is [P =0.51]. So, the fact that one person won is not remarkable. If someone else won in place of John, Dale could’ve just as easily accused that other person of cheating. Dale is drawing the proverbial bullseye around any winner, regardless who it ends up being.

### 1.2 The waiting-time problem

This fallacy applies to a variety of arguments commonly made Cdesign proponentsists [**[not misspelled](https://en.wikipedia.org/wiki/Of_Pandas_and_People#From_creationism_to_intelligent_design:_Pandas_and_%22cdesign_proponentsists%22)**]. These are often referred to as **“the waiting-time problem”** which claims that there is not enough time for a specific evolutionary change to have occurred. If you calculate the probability (or average time required) for a set of specific mutations to arrive and become fixed in a population by drift and/or selection, you often get very low odds - just like the low odds of John winning the lottery. However, these arguments omit the fact that there could have been numerous other ways of getting the same mutations fixed and/or all the other potential mutations that could have been fixed in their place, but weren’t - just like how John was not the only possible lottery winner.

The Texas sharpshooter fallacy is not the only issue with the waiting-time problem arguments. These have been more thoroughly covered by @dsterncardinale [[here](https://www.youtube.com/watch?v=F748itCI_es)] and [[here](https://www.youtube.com/watch?v=uXtrH1aAhLA&t=206s&pp=ygUQaGFsZGFuZXMgZGlsZW1tYQ%3D%3D)], @talkpopgen [[here](https://www.youtube.com/watch?v=KLPZxAU9nAM)]. I also encountered this when writing a script for a response video on @JacksonWheat1’s channel many years ago [[here](https://www.youtube.com/watch?v=Giylyd9zBN8)]. That brings me to Will Duffy’s recap presentation during the latest livestream on @GutsickGibbon’s channel. His presentation lasted from [4:28 - 43:51] minutes.

[![](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/5/d/5d93cfe8c01a13844ee53e0fdbd5e2afa8f0820b.jpeg "Teaching Famous Creationist Will Duffy Evolutionary Theory (LIVE) Human Evolution #2 Australopiths +") ](https://www.youtube.com/watch?v=6jXiwrcC5PQ)

The main argument was a type of ‘waiting-time problem’, which I will address here. The other minor arguments are covered in [PART II](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-ii-clades-and-chromsomes-and-ervs-oh-my/18095).

## 2. The MAIN argument

Will Duffy claims the rate at which mutations are fixed is far to slow to account for the fixed differences we observe between humans and chimps if they diverged from each other ~6.3 million years ago. Will gets the calculations very wrong, but he is just borrowing erroneous equations from someone else [see section 2.1 in [PART II](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-ii-clades-and-chromsomes-and-ervs-oh-my/18095)]. This would not be the first instance where Will Duffy comes to a wrong conclusion simply because he is repeating an argument from someone else (see my previous response to Will [here](https://discourse.peacefulscience.org/t/response-to-will-duffy-phylogenetic-systematics/17985)).

### 2.1 Neutral Theory

**J.B.S. Haldane** in his 1957 seminal paper _[“The Cost of Natural Selection”](https://www.blackwellpublishing.com/ridley/classictexts/haldane2.pdf)_ noted that selection has a limit, since it imposes a reproductive cost. He argues that, for a single beneficial allele, the cost of fixation via selection is [30 N\_e]. Meaning, 30 times the effective population need to “die” (i.e. are not permitted to reproduce). Of course, a population cannot die 30 times over in a single generation, so that number needs to be reached after many generations. Haldane argued that a reasonable 10% of the population can be culled per generation. So, it takes [\frac{30 N\_e}{0.1 N\_e} = \frac{30}{0.1} = 300] generations per beneficial mutation to be fixed via selection. The problem is that mutations are being fixed in populations at greater rates than this limit. This has been called _‘Haldane’s dilemma’_. Many waiting-time problem arguments make references to it.

**Motoo Kimura** proposes **Neutral theory** as a solution in his landmark 1968 paper _[“Evolutionary rate at the molecular level”](https://www.nature.com/articles/217624a0)_. There are other solutions that allow selection to exceed the limit argued by Haldane (see [this recent paper](https://academic.oup.com/genetics/article/229/4/iyaf011/7979206)), but I will stick with neutral theory for now. That’s not to say I am “abandoning selection” by any stretch of the imagination. Covering neutral evolution for now will be enough. According to neutral theory, the majority of mutations are fixed (or lost) by drift, and these mutations are also mostly neutral. Haldane’s reproductive cost limit does not apply to drift, but can it account for most of the genomic differences between humans and chimps?

The probability that a neutral allele will be fixed by drift alone [P\_{fix}] is exactly proportional to its frequency [p] in the population, which can be calculated from [n] number of times the neutral allele occurs in the population, and 2 times the effective population size [2 \times N\_e]. It is multiplied by 2 for diploid organisms:

P\_{fix}=p=\frac{n}{2 N\_e}

For example, if the frequency of the allele is 50% [p=0.5], it is intuitive that it has a 50% coin-flip probability of being fixed (or lost). What about neutral alleles that are freshly generated by mutations? Since each new mutation appears as 1 new allele in the whole population [n = 1], thus the probability that each new neutral mutation will eventually be fixed is:

P\_{fix}=\frac{1}{2 N\_e}

For even small effective population sizes [N\_e ≥ 1,000], ≤ 0.05% of all neutral mutations will be fixed and ≥ 99.95% will be lost due to drift. Are such small odds enough? The total number of mutations that arise in the entire population per generation can be calculated from the neutral mutation rate per haploid genome per generation [μ\_G] and [2N\_e]. We again multiply by 2 since each diploid individual is carrying 2 whole genomes:

2N\_e \times µ\_G

Next we can calculate [k], the number of neutral mutations that will fix per generation:

k = (2N\_e \times µ\_G) \times P\_{fix} = (2N\_e \times µ\_G) \times \frac{1}{2 N\_e} = µ\_G \times \frac{2 N\_e}{2 N\_e}= µ\_G

By doubling the effective population size you may cut the probability of fixation [P\_{fix}] in half, but the population-wide mutation rate [2N\_e \times µ\_G] also doubles. These effects cancel each other out. The fixation rate of neutral mutations per generation is the same as the genome-wide neutral mutation rate per generation [k=µ\_G]. This is often expressed as [k = µ\_S \times L] if you use the per-nucleotide neutral mutation rate [μ\_S] and the genome size [L], but it comes to the same value.

### 2.2 Fixation of neutral mutations

What is the genome-wide mutation rate in humans? Estimates vary from 100 to 200 per generation. Let’s assume a conservative neutral mutation rate of [μ\_G = 75]. That means [k = 75] mutations will fix per generation. Let’s also assume for the sake of simplicity that chimps experience a similar neutral mutation rate. How many generations are humans separated from chimps? Let’s take the value that Will Duffy used for the generation time in years [G\_T = 25], and for time to last common ancestor in years [T\_{LCA} = 6,300,000]. We need to multiply the whole figure by 2, since both humans and chimps fixing neutral mutations at the same time.

2 \times k \times \frac{T\_{LCA}}{G\_T}= 2 \times 75 \times \frac{6,300,000}{25} = 37,800,000

So, the expected number of fixed neutral mutations that separates humans and chimps is ~37.8 million based on this rough calculation. That is close to the estimated 1-2% difference between the human and chimp genome due to single-nucleotide differences, corresponding to 31-62 million single-nucleotide variants (SNVs). Such results were also obtained from the recent paper by [Yoo et al. 2025](https://www.nature.com/articles/s41586-025-08816-3). PLEASE NOTE: since they use one (or a few) reference genomes, not all of the differences they identified between genomes are actually fixed in either the human or chimp populations. Nevertheless, the estimate is close. Most differences between humans and chimps can be accounted for by the fixation of neutral mutations. Selection was still involved, but likely for a minor fraction of all the fixed mutations.

Yoo et al. 2025 also used the SNVs they identified to estimate the time of divergence of between humans and chimps. Basically, using the same principle of the formula above, but calculating the expected [T\_{LCA}] instead. Albeit, their calculations are far more rigorous and includes ranges of uncertainties ( **emphasis mine** ).

> We focused on segments that could be reliably aligned and then we estimated speciation times and modelled incomplete lineage sorting (ILS) across the ape species tree[19](https://www.nature.com/articles/s41586-025-08816-3#ref-CR19 "Rivas-González, I., Schierup, M. H., Wakeley, J. & Hobolth, A. TRAILS: tree reconstruction of ancestry using incomplete lineage sorting. PLoS Genet. 20, e1010836 (2024).") (Fig. [2b](https://www.nature.com/articles/s41586-025-08816-3#Fig2) and Supplementary Table [VI.26](https://www.nature.com/articles/s41586-025-08816-3#MOESM4)). **Our analyses dated the human–chimpanzee split between 5.5 and 6.3 million years ago** (Ma; minimum to maximum estimate of divergence), the African ape split at 10.6–10.9 Ma and the orangutan split at 18.2–19.6 Ma (Fig. [2a](https://www.nature.com/articles/s41586-025-08816-3#Fig2)). We inferred ILS for an average of 39.5% of the autosomal genome and 24% of the X chromosome. Compared with recent reports[20](https://www.nature.com/articles/s41586-025-08816-3#ref-CR20 "Rivas-González, I. et al. Pervasive incomplete lineage sorting illuminates speciation and selection in primates. Science 380, eabn4409 (2023)."), these values represent a 7.5% increase in autosomal ILS, results that are in part due to the inclusion of more repetitive DNA (Supplementary Fig. [VI.16](https://www.nature.com/articles/s41586-025-08816-3#MOESM1)). Consequently, we estimated that the human–chimpanzee–bonobo ancestral population size (average _N_e = 198,000) is larger than that of the human–chimpanzee–gorilla ancestor (_N_e = 132,000), a result consistent with an increase in the ancestral population 6–10 Ma (Supplementary Note [VI](https://www.nature.com/articles/s41586-025-08816-3#MOESM1)).

## 3. The many mistakes

### 3.1 Gap Divergence

In addition to the aforementioned 1-2% divergence between human and chimp genomes due to SNVs, [Yoo et al. 2025](https://www.nature.com/articles/s41586-025-08816-3) also identified gap divergences. These gaps corresponds to regions that could not be aligned base-by-base, which cover 12.5% (human v. chimp) or 13.3% (chimp v. human) of the genomes (see [supplementary figure III.12](https://media.springernature.com/original/springer-static/esm/art%3A10.1038%2Fs41586-025-08816-3/MediaObjects/41586_2025_8816_MOESM1_ESM.pdf)). These numbers have recently revived the creationist argument that humans and chimps are not genetically ≥ 98.0% identical, including the instance when [Casey Luskin LIED](https://discourse.peacefulscience.org/t/casey-luskin-deletes-inconvenient-evidence/17299/31) by doctoring supplementary figure III.12 in order to cover-up the within-species gap divergences, such as the gap divergence between two gorillas of 13.8%.

I am not going to reiterate that ~98% vs ~85% similarity argument here. Thorough breakdowns of this have been written by @GutsickGibbon ([here](https://discourse.peacefulscience.org/t/no-a-new-paper-did-not-discover-humans-and-chimps-are-only-85-similar/17309)) and Martin Neukamm on the [Panda’s Thumb blog](https://pandasthumb.org/archives/2025/07/human_and_chimpanzee.html). What is relevant here is the fact that gap divergences are due to large-scale mutations such as insertions/deletions (InDels). While one point mutation only affects one base-pair (bp), one InDel can affect many base-pairs, ranging from 10s to 10s of thousands or even 100s of thousands of bp. Below is a simple illustration where a single point mutation is contrasted with an InDel (courtesy of [Martin Neukamm](https://pandasthumb.org/archives/2025/07/human_and_chimpanzee.html)).

 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/d/3/d325fae7ed5a2683b859227c18ec147c555adb73.png)

If we calculated the divergence percentage base-by-base, then we would end up with a divergence of [1/60=] 1.7% in terms of SNVs and [9/60=] 15.0% in terms of gaps, but only if we used the human genome as the query. If we used the chimp as the query, the gap divergence would be 0%. That is why gap divergences change depending on the orientation of the comparison. One can sum up the values to 16.7%, which corresponds to 10 bp of the human genome. But these 10 bp differences do not correspond to the mutational divergence, for which there are only 2.

Remember that the gap divergence between two gorillas is 13.8%, in contrast to the 12.5% or 13.3% gap divergence between humans and chimps, but this does not mean gorillas are separated by more mutations from each other than humans are from chimps. The number of SNVs does correspond to the number of point mutations very neatly. Although, it is possible that the same nucleotide experienced multiple point mutations, which leads to an underestimation of point mutations from the number of SNVs, but this is minor. SNVs are also far more common and easier to identify with greater accuracy than InDels. That is why Yoo et al. 2025…

> …focused on segments that could be reliably aligned and then we estimated speciation times.

**DUFFY ERROR # 1** : As explained above, it would be an error to use the 16.7% figure and come to believe that [60 \times 0.167=] 10 mutations separate humans from chimps, while in reality there are only 2 mutations. This exact mistake produces **the erroneous 205 million mutations fixed in the human lineage** in Duffy’s slide at [28:18] minutes. This comes from using the SNV+gap divergence of ~14%, which covers ~410 Mbp of the 3 Gbp human genome. Split 410 Mbp in half (one half for humans and one for chimps), and you end up with the 205 million figure… which does NOT represent the number of mutations that is fixed in the human lineage.

### 3.2 The average is not the limit

One objection that Duffy makes against neutral theory at [36:11 to 38:14] is claiming the speed at which neutral mutation fix is too slow. Is it? For neutral mutations that are “destined” to be fixed, the average generations required to reach fixation [T\_{fix}] is also dependent on its initial frequency in the population [p]. Here is the formula:

T\_{fix}(p) = - 4N\_e \left(\frac{1-p}{p} \right) \ln(1-p)

For mutations that are “destined” to be lost, the average generations required is:

T\_{loss}(p) = - 4N\_e \left(\frac{p}{1-p} \right) \ln(p)

These formulas are based on approximating the process of random diffusion. When populations are sufficiently large, the initial frequency of a new mutation approaches zero [p \rightarrow 0]. Thus, for new mutations arriving at generation [T=0], the first formula approximates to:

T\_{fix} ≈ 4N\_e

**DUFFY ERROR # 2** : Will Duffy also shows this formula at [36:50] minutes. He uses a reasonable [N\_e = 10,000]. With this value, the average number of generations it takes to reach fixation is [4N\_e = 4 \times 10,000 = 40,000] generations. That is accurate. However, Duffy concludes that this is still not enough… arguing fixation only makes things worse since it is much slower than selection. **The mistake here is that he (knowingly or not) restricts drift such that it is only fixing one mutation at a time**… as if a bunch of mutations are… for some reason… stuck at the starting frequency [p = 1 / 2N\_e], or these mutations haven’t even occurred yet, and all of them have to wait until the first mutation is fixed after 40,000 generations, and… somehow… only then is the next mutation permitted to (appear and) increase in frequency, while all the others are still stuck and have to wait again until the second mutation is fixed. That just not how this works. Neutral mutations are changing their frequencies all at the same time, such that in the real world populations carry mutations at various frequencies. For neutral mutations, their expected ‘Site Frequency Spectrum’ (SFS) follows a power law [\xi \_{i} = 1/i] with [i = p \times 2N\_e]. See further ahead [section 3.4] for what an SFS looks like on a plot.

The mutations “destined” for fixation will increase their frequencies all at the same time, forming a parade of random-walks towards fixation. Some of them will arrive much earlier than [4N\_e] while other arrive much later. When we take all the values for the generation times of each individual mutation, and averaged them out, that gives us [4N\_e], but that does not mean only one mutation fixes every [4N\_e] interval. **The average is not the limit!**

**ANALOGY** : Let’s say we follow one hundred cars which drive from point A to B. They start driving at the same time, but they all take different roads and drive at different speeds. Most of them arrive within an hour, while a few took many hours. After 10 hours, all cars have arrived at B. If look at how long each car took to arrive and averaged out all these values, we get something like 2 hours. Obviously, that does not mean only one car was able to go from A to B within each 2 hour time frame. This average does not mean the number of cars that can go from A to B within 10 hours is limited to just 5 cars.

Will Duffy has a few slides arguing against parallel fixation, but these are about the supposed limits of selection. As I said before, I am sticking with neutral theory here, and his objection (whether correct or not) do not apply here.

### 3.3 How many will fix after how many generations?

We need to calculate what proportion of mutation are expected to be fixed by a specific generation. In other words, for a given number of mutations “destined” for fixation that appear at [t=0], what proportion will be fixed by the 100th generation [t=100], 1,000th [t=1,000], or 10,000th [t=10,000], etc. This proportion corresponds to the probability of fixation, which is given by this ‘cumulative distribution function’ (CDF):

F(t)= 1 + \sum\_{i=1}^{\infty} (-1)^i (2i+1) e^{- \frac{i(i+1)t}{4N\_e}}

It includes a sum of an infinite terms [i=1,2,3 ... , \infty] . But we don’t need all of these terms to come to an approximate estimate. The first 50 terms [i = 1, 2, ... 49, 50] is good enough. When we reach high generation times [t ≥ 4N\_e], the first term [i=1] alone is a decent estimator:

\hat{F}(t) = 1 - 3e^{- \frac{t}{2N\_e}}

In R, I calculated the first 50 terms with the same [N\_e = 10,000] value that Will Duffy used, and plotted the curve below. The second plot below is the corresponding non-cumulative probability curve from the ‘probability density function’ (PDF). I also notated the average and the median times to fixation.

 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/6/b/6b015d5b634d47a8c6b11474714605980b09d6dd.png)

 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/9/d/9dbbada2cc34303ed297f379fc54f8ad32df7830.png)

What these curves show is that 50.0% of the neutral mutations “destined” for fixation that appeared at [t=0] will be fixed by generation [t≈3.48N\_e ≈ 34,790]. Around 60.6% are fixed by generation [t≈4N\_e ≈ 40,000]. The latter is still the average, since there will be a few mutations that linger on before they are fixed, producing a right-skewed distribution as you can see. A similar effect was noted back in the lottery analogy [section 1.1]. 95.0% of all the mutations will be fixed by [t≈8.19N\_e ≈ 81,900], and 99.0% will be fixed by [t≈11.41N\_e ≈ 114,100], and 99.9% at [t≈16.01N\_e ≈ 160,100]. In other words, after 114-160 thousand generations, pretty much all of the mutations “destined” for fixation that appeared at [t=0] will be fixed. How many? I already explained the math before [section 2.1] that the answer is [k=µ\_G]. If we assume [µ\_G=75], then ~75 of all the neutral mutations that appeared at generation [t=0] will be fixed by generation [114,100 \< t \< 160,100].

### 3.4 How long until we reach a constant fixation rate?

**Someone might say** : _‘But wait! That is only 75 mutations fixed, and it took over a hundred thousand generations. That’s still not enough!’_

The plots above only show the fate of 75 mutations appearing at generation [t=0]. Another 75 neutral mutations “destined” for fixation will appear at the next generation [t=1], all of which will fix around [114,101 \< t \< 160,101]. And for those appearing at [t=2] they will fix at [114,102 \< t \< 160,102], and for [t=3] it is [114,103 \< t \< 160,103], etc. We will reach a constant neutral fixation rate per generation of [k=µ\_G] .

**Someone might say** : _‘But wait! It takes a very long time to reach the constant fixation rate. The lag is around 140,000 generations. This cuts down the number of generations since the last common ancestors in half!’_

That lag is only seen when you calculate the curve for the neutral mutations “destined” for fixation that appear at generation [t=0]. When we add up all the curves for all mutations that appear in each generation [t=0,1,2 ..., \infty] the lag is not severe. You will approach the constant rate rather quickly. In the plot below, the green curve is the sum of all the CDF curves starting at each generation. The summed up probabilities are multiplied by 75, thereby expressing the cumulative fixation of neutral mutations when [µ\_G=75]. I also added a black-dashed line notating a constant rate of fixation [k=µ\_G=75]. From this, we can see that the lag for reaching the constant rate is exactly [4N\_e]. What a coincidence! Well, not really. That’s just how the math works out.

 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/e/2/e21291a529571f42410fbf3bca8150fa91bd0f3c.png)

Furthermore, this “lag” does not even apply in the real world. In the plots above, the ancestral population at [t=-1] are all assumed to be identical clones (zero mutants in the population). Obviously, neither of the ancestral populations for humans and chimps consisted of purely identical clones. Not even right after these went their separate ways ~6.3 million years ago. Each started with a standing variation of neutral mutations at various frequencies.

As mentioned before, the expected ‘Site Frequency Spectrum’ (SFS) for neutral alleles, within a population at a **mutation-drift equilibrium** , follows a power law [\xi \_{i} = 1/i] with [i = p \times 2N\_e]. We can include the factor [\theta = 4N\_e \times µ\_G], which is the product of the rate at which new mutations enter the population [2N\_e \times µ\_G] and the average time mutations spend drifting until they are fixed or lost [2N\_e]. The SFS formula for the expected number of mutations for each allele count becomes [\mathbb{E}(\xi \_{i}) = \theta/i]. Below is the SFS plotted showing the expected number of mutations (y-axis) sharing the same allele counts (x-axis).

 ![image](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/3X/6/3/63f39ea52c957e934e84a1045afb8e0f255ae9ad.png)

Bear in mind that, while the individual mutations are jumping randomly from one allele count to another after each generation, these random fluctuations cancel each other out. The rate of fixation+loss is also balanced by the rate at which new mutations enter the population. Thus, the distribution of mutation number per allele count remains stable, in the case of neutral mutations and a constant effective population size. We observe similar distribution in real populations too (see figure 2 of [this paper](https://doi.org/10.1534/genetics.116.193821)).

### 3.5 Correcting bad math

At [30:06] minutes, Will Duffy does the following calculation to argue that humans could’ve only accumulated 180 mutations since they diverged from chimps 6.3 million years ago:

F\_{max} = \frac{t\_{div} \times d}{g\_{len} \times G\_f} = \frac{6,300,000 \times 1}{25 \times 1,400} = \frac{6,300,000}{35,000}=180

The time since LCA [t\_{div} = 6,300,000] and the generation time of [g\_{len} = 25] are fine. The most confusing aspect is factor [d]. It is supposedly the _“selective turnover coefficient”_ and _“corrects for overlapping generations”_, but Will does not explain what it actually means, nor what the correct value is supposed to be. I am guessing that it is similar to Haldane’s limit of 10%, which I mentioned previously [section 2.1]. Would that mean [d = 0.1] or [d = 0.9]? I have no idea. Will Duffy uses [d = 1] anyway, because (as he says) we are looking at the values from bacteria, and bacteria supposedly don’t need this adjustment because they don’t have overlapping generations. He does not say what it must be for humans. I will just ignore this factor for now on.

**DUFFY ERROR # 3** : That leaves us with the number of generations per fixation [G\_f = 1,400]. Duffy says this is _“the fastest rate of mutational fixation ever observed”_, which wrong for several reasons. First, Duffy cites [Barrick et al. 2019](https://www.nature.com/articles/nature08480) published by the team conducting the LTEE. **I cannot find where 1,400 value comes from in this paper**. Figure 2: _“Rates of genomic evolution and fitness improvement”_ provides a linear curve (colored blue) showing how many mutations accumulated over the generations. That curve starts from [y(0) = 0] to [y(20,000) = 45] So, after 20 thousand generations, you get 45 mutations which…

> …include 29 single-nucleotide polymorphisms (SNPs) and 16 deletions, insertions and other polymorphisms (DIPs)

That would give us [G\_f = \frac{20000}{45} ≈ 444] not 1,400. There was also a mutator phenotype that appeared around generation [t = 26,500] which had accumulated approximately [y(26,500) = 60] mutations at that point, and later accumulated to [y(40,000) = 653] mutations, which composed of…

> …627 SNP and 26 DIP mutations

That would give us [G\_f = \frac{40,000 - 26,500}{653 - 60} ≈ 23]. The rate of [G\_f = 1,400] is certainly NOT the fastest ever observed.

**DUFFY ERROR # 4** : Remember our formula [k=µ\_G] from neutral theory? [G\_f] is just the inverse of [k], such that [k = \frac{1}{G\_f}]. So we can use the [G\_f] to estimate the neutral mutation genome-wide mutation rate [µ\_G]. However, that is assuming all these mutations are neutral, and [Barrick et al. 2019](https://www.nature.com/articles/nature08480) argue that most of the accumulated mutations in _E. coli_ were beneficial. But let’s for the sake of the argument take these as neutral. That would mean [µ\_G = \frac{1}{444} \< 0.0023] mutations per genome per generation. This is actually close the the observed mutation rate in bacteria. Bacteria, such as _E. coli_, have very small genomes [L = 4,600,000] and the [mutation rate per nucleotide](https://pmc.ncbi.nlm.nih.gov/articles/PMC5289635/) is very small [µ\_S \< 10^{-9}]. This gives us an approximate genome-wide mutation rate of [µ\_G = µ\_S \times L \< 0.0046]. In contrast, the human genome is much larger [L = 3,200,000,000] and the per nucleotide mutation rate is also much higher [µ\_S \> 1 \times 10^{-8}], giving us a genome-wide mutation rate of [µ\_G = µ\_S \times L \> 32] So, we are not comparing apples to apples here. Heck, we are not even comparing eukaryotes to eukaryotes. The fixation rate of (neutral) mutation are very different between humans and _E. coli_. **Let’s use a realistic value of ‘generations per fixation’ [G\_f] for humans**, which is [\frac{1}{k} = \frac{1}{µ\_G} = 1/75 = 0.013]. We can also add a factor of 2 to account for the fact that we have 2 lineages (humans and chimps) which accumulate mutations after they diverged:

F\_{max} = \frac{2 \times t\_{div}}{g\_{len} \times G\_f} = \frac{2 \times 6,300,000}{25 \times 0.013} = \frac{12,600,000}{0.33}=37,800,000

**37.8 million fixed neutral mutational differences between humans and chimps**. The same number I calculated previously [section 2.2].

## 4. EPILOGUE

Before someone comments, yes, I have seen [the feature length](https://youtu.be/_Vu0ZVVjwHc) film with @GutsickGibbon and @talkpopgen doing a thorough breakdown of Vox’s nonsense and shenanigans. I was close to finishing writing when it dropped. Stealing my thunder, are we? 😉

_FIN_

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**Author:** ![Mercer](https://avatars.discourse-cdn.com/v4/letter/m/e274bd/32.png) [@Mercer](https://discourse.peacefulscience.org/u/Mercer)\
**Post date:** [October 6, 2026, 1:11am UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/2 "2026-10-06T01:11:03Z")

</div>

For 1.2, you’re missing a larger problem–selection and drift are acting far more on existing variation, which in humans (the only species IDcreationists care about) outnumbers new mutations a million to one.

IOW, the idea of a population waiting for mutations is simply silly.

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<div class="post-metadata">

**Author:** ![Nesslig20](https://sea2.discourse-cdn.com/flex016/user_avatar/discourse.peacefulscience.org/nesslig20/32/11875_2.png) [@Nesslig20](https://discourse.peacefulscience.org/u/Nesslig20)\
**Post date:** [October 6, 2026, 10:36am UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/3 "2026-10-06T10:36:09Z")

</div>

Already covered in section [3.4].

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<div class="post-metadata">

**Author:** ![Mercer](https://avatars.discourse-cdn.com/v4/letter/m/e274bd/32.png) [@Mercer](https://discourse.peacefulscience.org/u/Mercer)\
**Post date:** [October 6, 2026, 3:33pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/4 "2026-10-06T15:33:27Z")

</div>

Not explicitly. IDcreationist propaganda capitalizes (likely unwittingly) on the common lay misunderstanding that evolution doesn’t occur without a new mutation happening.

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**Author:** ![Rumraket](https://sea2.discourse-cdn.com/flex016/user_avatar/discourse.peacefulscience.org/rumraket/32/9328_2.png) [@Rumraket](https://discourse.peacefulscience.org/u/Rumraket)\
**Post date:** [October 6, 2026, 3:53pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/5 "2026-10-06T15:53:27Z")

</div>

Ahh but Will had an expert mathematician teach him population genetics _in a month_. Checkmate.

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**Author:** ![system](https://us1.discourse-cdn.com/flex016/uploads/peacefulscience/original/2X/6/6b7bf72453a5c60b0919b6b89a27dc18bc52382c.png) [@system](https://discourse.peacefulscience.org/u/system)\
**Post date:** [October 13, 2026, 3:53pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/6 "2026-10-13T15:53:30Z")

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This topic was automatically closed 7 days after the last reply. New replies are no longer allowed.

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<div class="post-metadata">

**Author:** ![Dan\_Eastwood](https://sea2.discourse-cdn.com/flex016/user_avatar/discourse.peacefulscience.org/dan_eastwood/32/2368_2.png) [@Dan\_Eastwood](https://discourse.peacefulscience.org/u/Dan_Eastwood)\
**Post date:** [October 15, 2026, 10:08pm UTC](https://discourse.peacefulscience.org/t/response-to-will-duffy-part-i-population-genetics/18094/7 "2026-10-15T22:08:20Z")

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