Tag: MITTENS

People who believe the earth was created 6000 years ago, when it’s actually 4.5 billion years old, should also believe the width of North America is 8 yards. That is the scale of the error.
—Richard Dawkins

And 8 yards has to be wrong, because an evolutionary biologist like Richard Dawkins believes that the width of North America is 8 and 1/4 inches. That is the scale of the error committed by someone who believes in the evolution of Man and thinks that there was time for the evolution of 205,000,000 base pairs in the time that was sufficient for, ironically, 8.

It’s more than a little amusing to see how evolutionists are observably worse at science than young-earth creationists.

Read the 2nd edition of Probability Zero, the number one bestseller in Biology, Evolution, and Genetic Science if you want to see how comprehensively and conclusively that statement is backed up. The hardcover and paperback editions will be available soon.

DISCUSS ON SG

It occurred to me that since the population genetics and evolutionary biology fields are obsessed with Kimura’s substitution formula to the point of literal unreason, instead of trying to show them how Kimura made an algebraic mistake and why the formula only applies to one specific case instead of everything, it would be much more useful to demonstrate how, with a few modifications, Kimura’s equation could serve as the foundation of a predictive calculator that is considerably more accurate and useful than the original equation.

Kimura’s Fixation Calculator: Providing Neutral Theory With Predictive Capacity

Neutral theory has stood for fifty-seven years on a simple result: the substitution rate k equals the per-site mutation rate μ. This identity, derived by Kimura in three lines, rests on canceling two quantities that share a letter but not a meaning: the census number of breeding adults N (which supplies mutations) and the variance effective population size Nₑ (which governs drift and fixation). The cancellation in the derivation is valid in the special case of asexual bacteria where N ≈ Nₑ. It does not hold in sexually reproducing species, where Nₑ/N is typically ~0.1 (Frankham 1995).

Rejecting the incorrect application of the derivation and treating the realized substitution rate as the minimum of three serial constraints—input flux, polymorphism throughput, and selection cost—yields Kimura’s Fixation Calculator. The selection-cost term is a simple expression in four independently measurable parameters (maximum reproductive differential s_max ≈ 1, Selective Turnover Coefficient d, genome length L, and effective population size Nₑ). The full calculator recovers k ≈ μ for bacteria while predicting the observed compression of rates across sexual eukaryotes, where the selection term sets a ceiling two to five orders of magnitude below textbook expectations based on the standard derivation.

Validated on fourteen sexual species pairs plus the E. coli LTEE (all calibrations independent of molecular clocks), the calculator provides forward prediction of k from organismal parameters, inverse inference of divergence time or Nₑ from observed substitutions, and joint constraint surfaces. Where the textbook supplies a single number, the calculator returns a mechanistically grounded range consistent with observable biological reality.

You can read the whole paper if you are a serious glutton for punishment or if you want to understand why no less than nine scientific fields will be seeing significant future adjustments. This paper will be one of the new appendices in the second edition of Probability Zero, since there really is no need for the Sakana study and the rejection of the MITTENS paper means that there is no reason to add it at the back as well.

DISCUSS ON SG

It was just remarkable, with this evolutionary distance, that we should see such coherence in gene expression patterns. I was surprised how well everything lined up.

—Dr. Robert Waterston, co-senior author, Science (2025)

If one wanted to design an experiment to give natural selection the best possible chance of demonstrating its creative power, it would be hard to improve on the nematode worm.

Caenorhabditis elegans is about a millimeter long and consists of roughly 550 cells. It has a generation time of approximately 3.5 days. It produces hundreds of offspring per individual. Its populations are enormous. Its genome is compact—about 20,000 genes, comparable in number to ours but without the vast regulatory architecture that slows everything down in mammals. The worms experience significant selective pressure: most offspring die before reproducing, which means natural selection has plenty of raw material to work with. And critically, worms have essentially no generation overlap. When a new generation hatches, the old generation is dead or dying. Every generation represents a complete turnover of the gene pool. There is no drag, no cohort coexistence, no grandparents competing with grandchildren for resources.

In the notation of the Bio-Cycle Fixation Model, the selective turnover coefficient for C. elegans is approximately d = 1.0. Compare that to humans, where we have shown d ≈ 0.45. The worm is running the evolutionary engine at full throttle. No brakes, no friction, no generational overlap gumming up the works.

Now consider the timescale. C. elegans and its sister species C. briggsae diverged from a common ancestor approximately 20 million years ago. At 3.5 days per generation, that is roughly two billion generations. To put that in perspective, the entire history of the human lineage since the putative chimp-human divergence—six to seven million years at 29 years per generation—amounts to something like 220,000 generations. The worms have had nearly ten thousand times as many generations to diverge. Ten thousand times.

Two billion generations, running the evolutionary engine at maximum speed, with enormous populations, high fecundity, complete generational turnover, and all the raw material that natural selection could ask for. If there were ever a case where the neo-Darwinian mechanism should produce spectacular results, this is it.

So what did it produce? Nothing.

In June 2025, a team led by Christopher Large and co-senior authors Robert Waterston, Junhyong Kim, and John Isaac Murray published a landmark study in Science comparing gene expression patterns in every cell type of C. elegans and C. briggsae throughout embryonic development. Using single-cell RNA sequencing, they tracked messenger RNA levels in individual cells from the 28-cell stage through to the formation of all major cell types—a process that takes about 12 hours in these organisms.

What they found is what Dr. Waterston described, with evident surprise, as “remarkable coherence.” Despite 20 million years and two billion generations of evolution, the two species retain nearly identical body plans with an almost one-to-one correspondence between cell types. The developmental program—when and where each gene turns on and off as the embryo develops—has been conserved to a degree that startled even the researchers.

Gene expression patterns in cells performing basic functions like muscle contraction and digestion were essentially unchanged between the two species. The regulatory choreography that builds a worm from a fertilized egg—which genes activate in which cells at which times—was so similar across 20 million years that the researchers could map one species’ cells directly onto the other’s.

Where divergence did occur, it was concentrated in specialized cell types involved in sensing and responding to the environment. Neuronal genes, the researchers noted, “seem to diverge more rapidly—perhaps because changes were needed to adapt to new environments.” But even this divergence was modest enough that Kim, one of the co-senior authors, noted the most surprising finding was not that some expression was conserved—the body plans are obviously similar, so that’s expected—but that “when there were changes, those changes appeared to have no effect on the body plan.”

Read that again. The changes that the mechanism did produce over two billion generations had no detectable effect on how the organism is built. The divergence was, as far as the researchers could determine, functionally trivial.

Murray, the study’s third senior author, offered the most revealing comment of all: “It’s hard to say whether any of the differences we observed were due to evolutionary adaptation or simply the result of genetic drift, where changes happen randomly.”

After two billion generations, the researchers cannot confidently identify a single adaptive change in gene expression. They cannot point to one cell type, one gene, one regulatory switch and say: natural selection did this. Everything they found is equally consistent with random noise.

Now, the standard response to findings like this is to invoke purifying selection, also known as stabilizing selection. The argument goes like this: most mutations are deleterious, so natural selection acts primarily to remove harmful changes rather than to accumulate beneficial ones. Gene expression patterns are conserved because any change to a broadly-expressed gene would disrupt too many downstream processes. The machinery is locked down precisely because it works, and selection fiercely punishes any attempt to modify it.

This is true. Purifying selection is real, well-documented, and no one disputes it. But invoking it as an explanation only deepens the problem for the neo-Darwinian account of speciation.

The theory of evolution by natural selection claims that the same mechanism, random mutation filtered by selection, both preserves existing adaptations and creates new ones. The worm data shows empirically what the constraint looks like. The vast majority of the genome is locked down. Expression patterns involving basic cellular functions are untouchable. The only genes free to diverge are those expressed in a few specialized cell types, and even those changes are so subtle that the researchers can’t distinguish them from genetic drift.

This is the genome’s evolvable fraction, and it is small. The regulatory architecture that controls development, the transcription factor binding sites, the enhancer networks, the chromatin structure that determines which genes are accessible in which cells, is so deeply entrenched that two billion generations of nematode reproduction cannot budge it.

And here’s the question no one asked: how did that regulatory architecture get there in the first place?

If the current architecture is so tightly constrained that it resists modification across two billion generations, then building it in the first place required an even more extraordinary series of changes. Every transcription factor binding site had to be fixed. Every enhancer had to be positioned. Every element of the chromatin landscape that determines which genes are expressed in which cell types had to be established through sequential substitutions. This is what we call the shadow accounting problem. The very architecture now being invoked to explain why the worm hasn’t changed is itself a product that requires explanation under the same model. The escape hatch invokes a mechanism whose existence demands an even larger prior expenditure of the same mechanism—an expenditure that the breeding reality principle tells us was itself problematic.

Let us be precise about the scale of the failure. The MITTENS analysis, as published in Probability Zero, establishes that the neo-Darwinian mechanism of natural selection faces multi-order-of-magnitude shortfalls when asked to account for the fixed genetic differences between closely related species. The worm study provides an independent empirical check on this conclusion from the opposite direction.

Instead of asking “can the mechanism produce the required divergence in the available time?” and discovering that it cannot, the worm study asks “what does the mechanism actually produce when given enormous amounts of time under ideal conditions?” and discovers that the answer is exactly what MITTENS proves: essentially nothing.

Two billion generations with every parameter set to maximize the rate of adaptive change, with short generation times, high fecundity, large populations, complete generational turnover, and a compact genome, nevertheless produced two organisms so similar that researchers can map their cells one-to-one. The divergence that did occur was concentrated in a few specialized cell types and could not be confidently attributed to adaptation.

Now scale this down to the conditions that supposedly produced speciation in large mammals. A large mammal has a generation time of 10 to 20 years. Its fecundity is low, with a few offspring per lifetime instead of hundreds. Its effective population size is small. Its generation overlap is substantial (d ≈ 0.45, meaning that less than half the gene pool turns over per generation). Its genome is vastly larger and more complex, with regulatory architecture orders of magnitude more elaborate than a nematode’s.

The number of generations available for speciation in large mammals is measured in the low hundreds of thousands. The worms had two billion and produced nothing visible. On what basis should we believe that a mechanism running at a fraction of the speed, with a fraction of the population size, a fraction of the fecundity, a fraction of the generational turnover, and orders of magnitude more regulatory complexity to navigate, can accomplish what the worms could not?

The question answers itself.

“The worms are under strong stabilizing selection. Other lineages face different selective pressures that drive divergence.”

No one disputes that stabilizing selection explains the stasis. The problem is what happens when you look at the fraction that isn’t stabilized. Two billion generations of mutation, selection, and drift operating on the unconstrained portion of the genome produced changes that (a) affected only specialized cell types, (b) didn’t alter the body plan, and (c) couldn’t be distinguished from drift. If the creative power of natural selection operating on the evolvable fraction of the genome is this feeble under ideal conditions, it does not become more powerful when you make conditions worse.

“Worms are simple organisms. Complex organisms have more regulatory flexibility.”

This gets the argument backward. Greater complexity means more regulatory interdependence, which means more constraint, not less. A change to a broadly-expressed gene in an organism with 200 cell types is more dangerous than a change to a broadly-expressed gene in an organism with 30 cell types, because there are more downstream processes to disrupt. The more complex the organism, the smaller the evolvable fraction of the genome becomes relative to the locked-down fraction.

“Twenty million years is a short time in evolutionary terms.”

It is 20 million years in clock time but two billion generations in evolutionary time. The relevant metric for evolution is not years but generations, because selection operates once per generation. Two billion generations for a nematode is equivalent, in terms of opportunities for selection to act, to 58 billion years of human evolution at 29 years per generation. That’s more than four times the age of the universe. If the mechanism can’t produce meaningful divergence in the equivalent of four universe-lifetimes, the mechanism obviously doesn’t function at all.

“The study only looked at gene expression, not genetic sequence. There could be extensive sequence divergence not reflected in expression.”

There is sequence divergence, and it’s well-documented. C. elegans and C. briggsae differ at roughly 60-80% of synonymous sites and show substantial divergence at non-synonymous sites as well. The point is that this sequence divergence has not produced meaningful functional divergence. The genes have changed, but what they do and when they do it has remained largely the same. Sequence divergence without functional divergence is exactly what you’d expect from neutral drift operating on a tightly constrained system—and it is exactly the opposite of what you’d expect if natural selection were the creative engine the theory claims it to be.

The Science study is good science. The researchers accomplished something genuinely unprecedented: a cell-by-cell comparison of gene expression between two species across the entire course of embryonic development. The technical accomplishment is significant, and the evidence it produced is highly valuable.

But the data is reaching a conclusion that the researchers are not eager to draw. Two billion generations of evolution, operating under conditions more favorable than any large animal will ever experience, failed to produce any meaningful or functional divergence between two species. The mechanism ran at full speed for an incomprehensible span of time, and the result was the same worm.

This is not a philosophical objection to evolution. It is not an argument from personal incredulity or religious conviction. It is the straightforward empirical observation that the proposed mechanism, given every possible advantage, does not produce the results attributed to it. The creative power of natural selection, when measured rather than assumed, turns out to be approximately zero.

Two billion generations of nothing. A worm frozen in time. That’s what the data shows. And that’s exactly what Probability Zero predicted.

DISCUSS ON SG

Dennis McCarthy noted an interesting statistical fact about the geneology of Charlemagne.

Every person of European descent is a direct descendant of Charlemagne. How can this possibly be true?

Well, remember you have 2 parents, 4 grandparents, 8 grandparents, etc. Go back 48 generations (~1200 years), and that would equate to 248 ancestors for that generation in the time of Charlemagne, which is roughly 281 trillion people.

The actual population of Europe in 800 AD was roughly 30 million. So what happened? After roughly 10 to 15 generations, your family tree experiences “pedigree collapse.” That is, it stops being a tree and turns into a densely interconnected lattice that turns back on itself thousands of times—with the same ancestors turning up multiple times in your family tree.

Which, of course, is true, but considerably less significant in the genetic sense than one might think.

Because the even more remarkable thing about population genetics is that despite every European being a descendant of Charlemagne, very, very few of them inherited any genes from him. Every European is genealogically descended from Charlemagne many thousands of times over due to pedigree collapse. That’s correct. But genealogical ancestry ≠ genetic ancestry. Recombination limits how many ancestors actually contribute DNA to you.

Which means approximately 99.987% of Europeans inherited zero gene pairs from Charlemagne.

And this got me thinking about my previous debate with Mr. McCarthy about Probability Zero, Kimura, and neutral theory, and led me to another critical insight: because Kimura’s equation was based on the fixation of individual mutations, it almost certainly didn’t account for the way in which gene pairs travel in segments, and that this aspect of mutational transmission was not accounted for in the generational overlap constraint independently identified by me in 2025, and prior to that, by Balloux and Lehmann in 2012.

Which, of course, necessitates a new constraint and a new paper: The Transmission Channel Capacity Constraint: A Cross-Taxa Survey of Meiotic Bandwidth in Sexual Populations. Here is the abstract:

The molecular clock treats each nucleotide site as an independent unit whose substitution trajectory is uncorrelated with neighboring sites. This independence assumption requires that meiotic recombination separates linked alleles faster than mutation creates new linkage associations—a condition we formalize as the transmission channel capacity constraint: μ ≤ r, where μ is the per-site per-generation mutation rate and r is the per-site per-generation recombination rate. We survey the μ/r ratio across six model organisms spanning mammals, birds, insects, nematodes, and plants. The results reveal a sharp taxonomic divide. Mammals (human, mouse) operate at or above channel saturation (μ/r ≈ 1.0–1.5), while non-mammalian taxa (Drosophila, zebra finch, C. elegans, Arabidopsis) maintain 70–90% spare capacity (μ/r ≈ 0.1–0.3). The independent-site assumption underlying neutral theory was developed and validated in Drosophila, where it approximately holds. It was then imported wholesale into mammalian population genetics, where the channel is oversubscribed and the assumption systematically fails. The constraint is not a one-time packaging artifact but a steady-state throughput condition: every generation, mutation creates new linkage associations at rate μ per site while recombination dissolves them at rate r per site. When μ > r, the pipeline is perpetually overloaded regardless of how many generations elapse. The channel capacity C = Lr is a physical constant of an organism’s meiotic machinery—independent of population size, drift, or selection. For species where μ > r, the genome does not transmit independent sites; it transmits linked blocks, and the number of blocks per generation is set by the crossover count, not the mutation count.

There are, of course, tremendous implications that result from the stacking of these independent constraints. But we’ll save that for tomorrow.

DISCUSS ON SG

May I suggest a gift of some light reading material that will surely bring joy to any evolutionist’s naturally selected heart?

Sadly, this Darwin Day, there is some unfortunate news awaiting this gentleman biologist who is attempting to explain how the molecular clock is not supported by the fossil record.

As I explain in my book the Tree of Life, the molecular clock relies on the idea that changes to genes accumulate steadily, like the regular ticks of a grandfather clock. If this idea holds true then simply counting the number of genetic differences between any two animals will let us calculate how distantly related they are – how old their shared ancestor is.

For example, humans and chimpanzees separated 6 million years ago. Let’s say that one chimpanzee gene shows six genetic differences from its human counterpart. As long as the ticks of the molecular clock are regular, this would tell us that one genetic difference between two species corresponds to one million years.

The molecular clock should allow us to place evolutionary events in geological time right across the tree of life.

When zoologists first used molecular clocks in this way, they came to the extraordinary conclusion that the ancestor of all complex animals lived as long as 1.2 billion years ago. Subsequent improvements now give much more sensible estimates for the age of the animal ancestor at around 570 million years old. But this is still roughly 30 million years older than the first fossils.

This 30-million-year-long gap is actually rather helpful to Darwin. It means that there was plenty of time for the ancestor of complex animals to evolve, unhurriedly splitting to make new species which natural selection could gradually transform into forms as distinct as fish, crabs, snails and starfish.

The problem is that this ancient date leaves us with the idea that a host of ancient animals must have swum, slithered and crawled through these ancient seas for 30 million years without leaving a single fossil.

I sent the author of the piece my papers on the real rate of molecular evolution as well as the one on the molecular clock itself. Because there is another reason that explains why the molecular clock isn’t finding anything where it should, and it’s a reason that has considerably more mathematical and empirical evidence support it.

DISCUSS ON SG

One of the fascinating things about the Probability Zero project is the way that the desperate attempts of the critics to respond to it have steadily led to the complete collapse of the entire evolutionary house of cards. MITTENS began with the simple observation that 9 million years wasn’t enough time for natural selection to produce 15 million fixations. Then it turned out that there were only 6-7 million years to produce 20 million fixations twice.

After the retreat to neutral theory led to the discovery of the twice-valued variable and the variant invariance, the distinction established between N and N_e led to the recalibration of the molecular clock. And the recalibration of the molecular clock led, inevitably, to the discovery that the evolutionists no longer have 6-7 million years for natural selection and neutral theory to work their magic.

And now they have as little as 200,000 years, with an absolute maximum of 580,000, with which to work. And they still need to account for the full 20 million fixations in the human lineage alone, while recognizing that zero new potential fixations have appeared in the ancient DNA pipeline for the last 7,000 years. Simply pulling on one anomalous string has caused the entire structure to systematically unravel. The whole system proved to be far more fragile than I had any reason to imagine when I first asked that fatal question: what is the average rate of evolution?

So if your minds weren’t blown before, The N/N_e Distinction and the Recalibration of the Human-Chimpanzee Divergence should suffice to do the trick.

Kimura’s (1968) derivation of the neutral substitution rate k = μ rests on the cancellation of population size N between mutation supply (2Nμ) and fixation probability (1/2N). This cancellation is invalid. The mutation supply term uses census N (every individual can mutate), while the fixation probability is governed by effective population size Ne (drift operates on Ne, not N). The corrected substitution rate is k = μ × (N/Ne). Using empirically derived Ne values—human Ne = 3,300 from ancient DNA drift variance (Day & Athos 2026a) and chimpanzee Ne = 33,000 from geographic drift variance across subspecies—we recalibrate the human-chimpanzee divergence date. The consensus molecular clock estimate of 6–7 Mya collapses to 200–580 kya, with the most plausible demographic parameters yielding 200–360 kya. Both Ne estimates are independent of k = μ and independent of the molecular clock. The recalibrated divergence date increases the MITTENS fixation shortfall from ~130,000× to 4–8 million×, rendering the standard model of human-chimpanzee divergence via natural selection mathematically impossible by an additional two orders of magnitude.

There are a number of fascinating implications here, of course. But in the short term, what this immediately demonstrates is that all the heroic efforts of the evolutionary enthusiasts to somehow defend the mathematical possibility of producing 20 million fixations in 6.5 million years were utterly in vain. Because, depending upon how generous you’re feeling, MITTENS just became from 10x to 45x more impossible.

Here is the correct equation to calculate the amount of time for evolution from the initial divergence for any two lineages.

t = D / {μ × [(N_A/N_eA) + (N_B/N_eB)]}

Where:

• D = observed pairwise sequence divergence
• μ = per-generation mutation rate (from pedigree data)
• N_A= census population size of lineage A
• N_B = census population size of lineage B
• N_eA = effective population sizes of lineage A (from historical census demographics)
• N_eB = effective population sizes of lineage B (from historical census demographics)

Which, by the way, finally gives us the answer to the question that I asked at the very start: what is the rate of evolution?

R = μ(N/N_e) / g

This is the number of fixations per site per year. It is the rate of evolution for any lineage from a specific divergence, given the pedigree mutation rate, the census-to-effective population size ratio estimated from historical census demographics, and the generation time in years.

And yes, that means exactly what you suspect it might.

DISCUSS ON SG