Population Genetics

"The effective population size of some population is no more than the size of a simple Wright-Fisher model population having some characteristic in common with the population of interest…  the implication that however it is calculated, the effective size bears any necessary similarity to an actual population size is without any foundation." — Warren Ewens, Mathematical Population Genetics, 2004¹

"they argued for a bottleneck size of 1,280 individuals, but we note that their estimate actually corresponds to an effective population size of N_e = 1,280.  Effective population sizes can be substantially smaller than the actual number of individuals, especially when there is a high variance in offspring numbers among individuals." — Deng, Nielsen and Song, correcting a 2023 Science paper²

"I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge, and I should have expected the very simple point which I wish to make to have been familiar to biologists." — G. H. Hardy, opening the letter that gave us the Hardy-Weinberg principle³

Summary

Arguments about evolution and the age of the earth almost always come down to six numbers:  how big a population is, how fast it mutates, how much a mutation helps or hurts, how likely that mutation is to spread to everybody, how long that takes, and how much damage a population can carry.  Creationists and evolutionary biologists both use these numbers constantly.  Both sides get them wrong, in documented and specific ways.

This page is not an argument.  It is the reference the other articles on this site stand on, written once so they do not each have to re-derive it.  For every quantity:  what it means in plain words, the formula, a worked number you can check, what the measured value actually is, and where the quantity stops working.

Every closed-form number here is reproducible in the Pop Gen Calculator, which runs in your browser and grades itself against 91 checks — including twelve deliberately wrong inputs it is required to reject.  Figures from simulation or from the literature name their own source instead.  If a number here looks wrong, re-run it rather than taking it on trust.

The twelve things worth knowing before reading anything else

1. Effective population size is not a headcount, and this is the single most abused number in the whole debate. N_e is the size of an imaginary, perfectly behaved population that would lose genetic variety at the same speed as the real one. A species with a million individuals can have an effective size of ten thousand. The confusion is not a layman's error: a 2023 Science paper described an effective size of 1,280 as "about 1280 breeding individuals," and the conflation was corrected in December 2024, in a critique published in GENETICS.^{4 2}
2. Over time, N_e is a harmonic mean, so one bad generation dominates all the others. A population that spends a single generation at N_e = 2 and then sits at a million for four thousand generations has a long-run effective size of 7,938 — not 999,750.⁵ And it does not wash out: it takes about 4.5 million further generations at a million before the running average climbs back to 90% of a million. That cuts against "the bottleneck was long ago, so it stopped mattering" — whoever says it.
3. "Human N_e ≈ 10,000" is a division, not a measurement. It is π/(4μ) with π ≈ 0.001 per base pair and μ = 2.5 × 10^-8. Put the modern measured pedigree rate μ = 1.25 × 10^-8 into the same data and you get 20,000. Nothing was measured differently; the divisor halved. This page therefore writes it as "≈10,000 — and ≈20,000 at the pedigree rate", always, and never mixes the two.
4. A new mutation almost always disappears, even a good one. A mutation that gives its carrier a 1% advantage is lost about 98 times in 100. Fisher published the exact figure — 0.0197 — in 1930, and said in his own words that such a mutation "has itself a chance of about 1 in 50 of establishing itself and sweeping over the entire species."^{6 6}
5. Whether selection can "see" a mutation depends on the population, not just the mutation. The only thing that matters is the product 4·N_e·s_het = 2·N_e·s_hom. The same mutation is invisible in a human population and strongly selected in a bacterium — because effective size across life spans 5.4 orders of magnitude, from about 10,900 in the vervet monkey to about 2.7 billion in Pseudomonas fluorescens.⁷
6. The "drift barrier" is a soft shoulder about a decade wide, not a wall. At the barrier itself a harmful mutation still fixes at 58% of the neutral rate. Anyone who draws it as a cliff — in either direction — is drawing it wrong.⁵
7. Fixation time is the number everybody forgets. A neutral mutation that does spread takes about 4·N_e generations, which for humans is 40,000 generations or 1.16 million years. A strongly favoured one with s_het = 0.01 takes 1,312 generations, about 38,000 years. And the mutations that fail — 99.995% of them — are gone in about 20 generations.^{8 8}
8. A bottleneck destroys allele counts instantly and heterozygosity slowly, and this asymmetry decides most bottleneck arguments. Two founders retain 75% of the source population's heterozygosity at generation two, and less every generation after, but at most four alleles at any autosomal locus — and only 36% of the alleles at a locus as diverse as the ones Nei measured (θ = 0.16), falling to 13% at θ = 1. Whichever number you reach for, a critic will reach for the other, so both belong on the page.⁹
9. Genetic load is a ratio against a reference genotype that usually does not exist. Ten different things are called "load." A sentence that says "the load is 0.9" means nothing until it says which load and against what reference — and Crow's own textbook says the segregational reference is an asexual population that a sexual species cannot produce.¹⁰
10. Mutation rates are one of the best-measured numbers in biology, and one of the worst-quoted. Seven independent whole-genome trio studies since 2011 put the human rate between 1.1 and 1.29 × 10^-8 per site per generation. But "per site" means per haploid site copy, so the expected count in a newborn is μ × 2 × G. Drop the 2 and every downstream date halves. Mix up per-site and per-genome and you are wrong by a factor of 6.2 billion.
11. Molecular clocks have an error floor that more data cannot lower. Fitting nearly nine million sites of primate genome, dos Reis and Yang found the credibility interval converges to w = 0.322·t — for every million years of divergence, 0.322 million years of irreducible uncertainty — and concluded that "adding more sequence data is unlikely to produce more precise estimates."¹¹
12. The same measured data, carried honestly to a date, moves by a factor of 4.3 on convention choices alone. That is worked all the way through in one measured number, carried to a date below. Every step is arithmetically correct at every setting. This is why this page states its conventions before it states any number.

What this page will not do

It will not settle whether the earth is young.  None of these quantities dates anything by itself;  they constrain, and the constraints run in both directions.  Where a result helps the young-earth case it is stated plainly, and where it hurts it — which is often — it is stated just as plainly, in the section where it belongs rather than in a footnote.

How to read a number here

Measured, model output, or assumed

Three kinds of number appear below and they carry very different weight.

Measured means somebody counted something.  The human mutation rate comes from sequencing parents and their children and counting what is new.  It needs no assumption about how long humans have existed or how big the population was.

Model output means a number that fell out of fitting a model, and the model contains a history.  "Human N_e ≈ 10,000" is model output.  So is any distribution of fitness effects inferred from polymorphism data, because demography and selection are fitted jointly and you cannot get one without assuming the other.

Assumed means somebody had to pick a value to proceed.  Twenty-nine years per generation is assumed.  So is the shape parameter of a gamma distribution of fitness effects.

Every table below carries this distinction where it matters.  It is not a hidden flaw — the people who do this work say so in their own titles — but it changes what an argument can carry.  Adopting an opponent's timescale for the sake of argument is legitimate.  Quietly importing a number whose value was produced by assuming that timescale is not.

Error bars

An error bar is a statement about a procedure, not about the number in front of you.  A 95% confidence interval is "an interval (L, U) generated by a procedure that in repeated sampling has an X% probability of containing the true value."¹²  A Bayesian HPD interval genuinely is a statement about the parameter, and the price of that is a prior.  A bootstrap interval is a statement about resampling, and is blind to anything every resample shares.

Two practical rules.  Overlapping intervals prove nothing — two 95% intervals can overlap while the underlying difference is significant at P = 0.03.¹³  And a two-significant-figure point estimate with no interval is not a measurement.  This page tries hard to obey its own rule here, and says so where it cannot.

The worked example is the headline human N_e itself.  Li and Durbin print "at 110kya, N_e^YRI=15313±559."  That ± is a block bootstrap, computed with μ held fixed at 2.5 × 10^-8.  Carry the trio-sequenced pedigree rate through instead and the same analysis gives 30,626.  Published interval 14,217–16,409;  honest interval 14,217–32,817, about 8.5 times wider.  Li and Durbin say as much in their own discussion;  the abstract-facing sentence does not.¹⁴

The conventions this page is bound to

Population genetics is full of factors of two that depend on bookkeeping rather than biology.  Nearly every one traces to three switches:  ploidy, where s is measured, and which allele h describes.  This page fixes them all, in the first place they could bite, and the full audit is on the conventions page.

Two of these are worth a sentence each, because they have already caused published errors.

Lethal equivalents.  Morton, Crow and Muller defined B per gamete in 1956 and said so in words:  "1.5-2.5 lethal equivalents per gamete, or 3-5 per zygote."¹⁶  Conservation genetics quotes the zygote figure.  Feed a diploid count into the per-gamete formula and you double the predicted death rate.  Frankham's own published arithmetic settles it three separate ways:  12 diploid lethal equivalents at F = 5% gives "a decline in total fitness of 26%", which reproduces as 25.9% only after halving, against 45.1% without.¹⁷

The drift barrier.  "1/(2N_e)" names two different thresholds, because there are two independent factors of two — the s convention and how strictly you read the inequality.  Kimura published both himself:  his fixation formula uses genic s, but his definition of the effectively neutral class integrates to 1/(2N_e) in that same genic s, which is 4N_es < 2, twice as wide as the strict reading.¹⁸  The way out is to stop printing a threshold and print the master parameter, which is unambiguous once you say which s built it.

Population size

N is a headcount. N_e is not.

The census size N is the number of breeding adults.  Wright restricted it explicitly:  "Obviously N applies only to the breeding population and not to the total number of individuals of all ages."¹⁹

The effective size N_e is something else entirely.  It is the size of an ideal population — random mating, constant size, equal sex ratio, Poisson offspring numbers — that would drift at the same rate as the real one.  It is a way of expressing how fast a population loses variety, converted into units of headcount because those are units people have intuitions about.  The intuitions are the problem.

There are at least five effective sizes, and they are different numbers for the same population.  Inbreeding N_e, variance N_e, eigenvalue N_e, coalescent N_e and linkage-disequilibrium N_e.  They coincide for an isolated population of constant size and can differ without limit otherwise.  Ewens gives the extreme case:  a single heterozygote leaving a huge number of offspring has variance and eigenvalue effective sizes that are "very large" while its inbreeding effective size is unity.¹  A gentler version:  a population that doubles every generation has a variance N_e exactly twice its inbreeding N_e.

Robin Waples, writing in a US-government paper that is therefore public domain, gives the statement worth memorising:  "N_e is not a physical entity (as is, say, allele frequencies).  Rather, N_e is a conceptual entity that we use to measure genetic drift," and "a commonly repeated misconception is that N_e is the number of parents that actually contribute genes to the next generation."²⁰

The three things that push N_e below N

Notice the second row.  Minimising family-size variance doubles the effective size — that is real biology, not a bookkeeping factor, and Crow and Kimura say so:  with V_k = 0, "the effective size is approximately twice as large as the actual size."¹⁰

Frankham's meta-analysis of 192 estimates from 102 species ranked five variables, and fluctuation in population size came first, ahead of variance in family size;  unequal sex ratio came fifth.²¹

The harmonic mean is the one the bottleneck arguments turn on

The harmonic mean is dominated by its smallest term.  This single fact does most of the work in every argument about population crashes, in both directions.

The third row is Wright's own example, worked in Crow and Kimura's textbook, and the answer really is 54.¹⁰

This cuts two ways and the article states both.  For a young-earth model it means N_e-based refutations of a brief bottleneck are largely beside the point:  Ne is an average over a long window, and a short crash barely moves it.  Stephen Schaffner puts it exactly:  N_e "represents a kind of average population size over extended times… this makes it rather insensitive to very short bottlenecks."²²  Against a young-earth model it means a crash never stops counting.  Four and a half million generations after that single generation at N_e = 2, the running harmonic mean is still under 90% of the census.

Measured N_e/N

Roughly one tenth, with a range spanning six orders of magnitude — and each ratio has a different denominator, which is where most of the confusion lives.

• N_e / adult census ≈ 0.10–0.11. Frankham's comprehensive estimates, the ones that include all three reducers.²¹ This is the figure conservation genetics uses, and it is why "N_e ≥ 100" translates to 500–1,000 adults on the ground.
• N_e / total census ≈ 1/3 for modern humans. Browning and Browning measured 0.34 (95% CI 0.27–0.51) six generations back from identity-by-descent segments.²³ This is the friendliest possible case — a large, stable, low-variance population — and should not be transferred to wildlife.
• 0.994 is a human N_e/N, but not one anybody should quote. Frankham calls it the top of his range of 192 heterogeneous published estimates — though his own Appendix carries a human row above it, at 1.18 — and Frankham's very next sentence explains it away: "A major reason for the large variation among estimates was that they were estimating different things."²¹ His human rows in the Appendix run from about 0.12 to 1.18, and Frankham says on the same page that "less than one third were comprehensive estimates that included the effects of all relevant variables." Quoted alone, as a species figure, it is meaningless.

Waples and colleagues modelled 63 iteroparous species from vital rates and found N_e/N running from 0.19 in hoop pine to 3.69 in a mosquito — nine animal species had N_e greater than N.²⁴  The ratio is not a constant of nature.

Human N_e: the six figures, reconciled

Six different numbers circulate as "the human effective population size."  None is wrong.  They differ in the mutation rate assumed, the estimator, and the epoch the estimator can actually see.

This page leads with "≈10,000, and ≈20,000 at the pedigree rate."  Not because it is the best number, but because it is the number every party to the debate uses, and this page's job is to explain it rather than replace it.  The rule that goes with it:  N_e = 10,000 is never combined with μ = 1.25 × 10^-8 in a formula that assumes mutation-drift equilibrium.  That mistake is internally inconsistent and a professional will see it immediately.

Two honest additions.  Tenesa and colleagues, whose linkage-disequilibrium estimates are the lowest in the table, wrote plainly that "the estimates of N_e appear to be much lower than the usually quoted value of 10,000."²⁶  And their own spread across 23 chromosomes — one sample, one method — is mean 2,772 with a standard deviation of 598 and a range from 1,459 to 3,613.  That is the cleanest available picture of how firm this quantity is, and it is not a confidence interval at all.

N_e across life

The only dataset with both a directly measured mutation rate and a measured natural-population diversity for the same species is Lynch's 2023 compilation, 144 usable species.  Rebuilt from its raw values:⁷

• Span: 5.4 orders of magnitude, from ≈1.09 × 10⁴ (vervet monkey) to ≈2.69 × 10⁹ (Pseudomonas fluorescens).
• Humans sit fifth from the bottom of 144, at N_e = 22,118. Only the vervet monkey, the walrus, the lesser rhea and the roseate spoonbill are lower.
• This is a lower bound on the real span, and the reason is sampling. The bottom is all vertebrates, because vertebrates are what get trio sequencing; the top is bacteria with mutation-accumulation lines. Endosymbionts, island endemics and most eukaryotes are simply absent.

Charlesworth's independent 2009 table, fifteen years earlier and a different method, agrees on the order of magnitude and not on the digit:  humans 10,400 against Lynch's 22,118 (the whole difference is the mutation rate), Drosophila melanogaster 1,150,000 against 627,824, E. coli 25,000,000 against 170,918,080.²⁷  That is the right size of error bar to carry.

The consequence for everything below:  because the master parameter is N_e·s, the same mutation is in completely different regimes in different species.  At the human N_e the effectively-neutral band is |s| < 2.3 × 10^-5.  At the median species in Lynch's table it is 1.3 × 10^-6.  In Pseudomonas fluorescens it is 1.9 × 10^-10.

Drift

Drift is sampling error, and it has a formula

Genetic drift is not vague randomness.  It is the sampling error you get when a finite number of gametes make the next generation, and its size is exactly what you would get from tossing 2N coins:

variance in allele frequency per generation = p(1-p) / (2*Ne)

Wright put the picture in words:  "The conception is that of two random samples of gametes, N sperms and N eggs, drawn from the total gametes produced by the generation in question."¹⁹

Because drift has a known variance, it makes quantitative predictions that can fail.  That is what makes it science rather than a shrug, and it is what Drift Lab exists to test.

Heterozygosity decays, and the textbook shortcut gets it wrong for a founding pair

H_t = H_0 * (1 - 1/(2*Ne))^t
F_t = 1 - H_t/H_0

At N_e = 2 this gives 0.75, 0.5625, 0.4219, 0.3164, 0.2373 …  But a male and a female cannot self-fertilise, and the exact recursion for a dioecious pair gives 1, 0.75, 0.625, 0.5, 0.4063, 0.3281 …¹⁰

The shortcut is already wrong at the first generation — it says 0.75 where the exact answer is 1 — and the gap widens, because the shortcut decays at a flat 25% per generation while the exact sib-mating series settles to about 19.1%.  By the fifth generation the shortcut is down to 0.2373 against the exact 0.4063 — it understates the surviving heterozygosity by 42%.  Use the exact recursion;  do not use the shortcut shifted by a generation, which looks right at the first step and is not right after it.

The error runs in the direction of overstating the damage, so a critic using the shortcut against a founding-pair model is being unfair — and this page says so even though the correction helps the young-earth side.

The inbreeding coefficient F has three definitions that agree:  the probability that both alleles at a locus are identical by descent, the correlation between uniting gametes, and the fraction of the genome sitting inside runs of homozygosity.  The first two are Wright's²⁸;  the third is what modern genomics actually measures.²⁹

Mutation

Get the units right or be wrong by a billion

Four units are in play and they differ by enormous factors.

"Per site" means per haploid site copy.  Kong's 2012 paper prints 63.2 mutations per trio and 1.20 × 10^-8 per nucleotide, and those are consistent only because 63.2/(2 × 2.6318 × 10⁹) = 1.20 × 10^-8.  Drop the 2 and every date downstream halves.

The human rate is one of biology's best-measured numbers

Seven independent whole-genome trio studies from 2011 to 2025, across three generations of sequencing technology, span 1.1 to 1.29 × 10^-8 per site per generation.

Most of the remaining disagreement is the callable fraction, and the papers say so.  Rahbari observed 64 de novo mutations per child and reported 76.9 after correcting for regions the pipeline could not reach — a 20% upward correction, which is the entire difference between their 1.28 × 10^-8 and Keightley's 1.1 × 10^-8.  Porubsky's 2025 study reached 91.9% of the autosomes using five sequencing technologies and near-complete assemblies, and 8% was still invisible.

Two things complicate the headline count.  Add indels, tandem-repeat length changes and structural variants and Porubsky gets 98 to 206 events per transmission, not 70.  And 16% of the de novo point mutations they see are post-zygotic — they arose in the embryo, not in a parent's germline, so strictly they are not part of μ at all.

The paternal age effect is large and well measured:  1.51 extra mutations per year of the father's age (95% CI 1.45–1.57), against 0.37 per year from the mother, in 1,548 trios.³⁰  It also varies strongly between families — measured per-family slopes run from 0.19 to 3.24 mutations per year.

Rates across life, and why they vary

Per generation, mutation rates vary about 1,000-fold across the tree of life, and over 5,000-fold among eukaryotes alone in the largest meta-analysis.  Bacteria and unicellular eukaryotes run 10^-11 to 10^-9 per site per generation;  multicellular eukaryotes 10^-9 to 10^-8.  Viruses, in their own unit, run 10^-8 to 10^-6 (DNA) and 10^-6 to 10^-4 (RNA).

Per cell division the picture inverts, and the human germline turns out to be the most accurate replication system measured:  0.06 × 10^-9 per site per division, "approximately one fifth that in unicellular organisms."³¹

Lynch's explanation is the drift barrier:  selection improves replication fidelity until further improvement is worth less than 1/N_e, at which point drift stops it.  Small-N_e species therefore stall at higher mutation rates.  The hypothesis predicts the pattern quantitatively and it is the best explanation on offer — but it is contested, and the dissent should be stated rather than characterised.  Charlesworth and Jensen's sharpest sentence:  within multicellular organisms the relation "is less apparent, especially if phylogenetic corrections are applied," and the per-bp rates of D. melanogaster and H. sapiens are 5 × 10^-9 and 7 × 10^-9 — a 1.4-fold difference across a 357-fold gap in generation time and a 7-fold gap in N_e.³²

U, the genomic deleterious rate

U is μ × the functional fraction × 2 (for a diploid genome).  It is the number that genetic-entropy arguments turn on, and it is genuinely disputed — not because the mutation rate is uncertain, but because the functional fraction is.

There are two routes and they do not agree.

• The sequence route — μ times a measured constrained fraction — gives 2.2 to 22 for humans.
• The fitness route — mutation-accumulation lines, measuring actual fitness decline — gives 0.01 to 1.0 for multicellular eukaryotes generally.

An order of magnitude apart, and the article must not blur them.  The full audit of what has actually been measured about mutational effects is on this site already, in what has actually been measured about mutations, and is not repeated here.

One correction to our own published page.  The genetic entropy article's middle estimate of U used Zoonomia's 10.7% constrained figure.  That is an element-level number — the fraction of the genome covered by constrained elements — not the fraction of individual bases under constraint.  Zoonomia's own single-base figure is 3.3%, which gives U = 2.25, i.e. Keightley's 2.2 rather than 7.3.  Using the element-level number inflates U by about 3.2×.  That correction was owed to the reader and is recorded here rather than made quietly;  the genetic entropy page now treats 3.26% as a floor (U ≈ 2.25) and 10.7% as a ceiling (U ≈ 7.3), as of 26 September 2026.

Do mutation rates change?

Yes, measurably, and a young-earth reader will want them to have been faster in the past.  The measured evidence is mixed and mostly points the other way for humans.

They do change.  A single mismatch-repair knockout raises the E. coli rate 138-fold.  A natural, segregating Mutyh allele raises the mouse C>A rate by 50%.  The European TCC→TTC rate rose about 50% and fell again in a burst dated to roughly 15,000–2,000 years ago.

But the human germline is remarkably hard to perturb.  Yeager and colleagues sequenced 130 children of Chernobyl cleanup workers with paternal preconception gonadal doses up to 4,080 mGy and found no elevation in de novo mutations:  −0.0007 mutations per mGy, 95% CI −0.003 to 0.002, P = 0.56.³³

And the comparative evidence runs against a faster past.  Chimpanzee, gorilla and orangutan run about 1.48 ± 0.08 times faster per year than humans do today, which is read as a recent human slowdown.  What is genuinely open is whether that slowdown is biochemical or just a change in generation time;  generation time alone can account for the whole gap.

Selection

What s is, and the six ways people define it

A selection coefficient is a difference between two fitnesses, per generation.  "s = 0.01" means one genotype leaves 1% more offspring than another.  It does not mean it leaves 1% of anything.

At least six conventions are in current use.  The most common puts s on the mutant homozygote with a dominance coefficient h on the heterozygote — that is what Hartl and Clark, Crow and Kimura, and Ewens use, and it is what this page uses.  But a large part of the empirical literature puts s on the allele, so the heterozygote carries s and the homozygote 2s.  A number quoted without its convention can be off by a factor of two.

The factor is not hypothetical.  Bersaglieri's famous lactase-persistence paper defines the model in its methods:  "We assumed a dominant model for lactase persistence—that is, w11 = w12 = 1 and w22 = 1 − s."³⁴  So s is the penalty borne by the non-persister.  Their results sentence glosses it as "the coefficient of selection associated with carrying at least one copy," which reads like a carrier bonus and is not one.  Re-based on the ancestral genotype, their 0.19 becomes 0.2346 — a 23% inflation from nothing but the convention.

The six ways s gets measured, and which three give an error bar

Two of these deserve emphasis because they get quoted as though they were selection coefficients.

Sweep statistics do not return s.  iHS, EHH and XP-EHH are standardised rank statistics with mean 0 and variance 1.  They tell you a locus looks unusual.  They do not tell you how unusual, in fitness units.

Selection gradients are not selection coefficients.  β is the slope of relative fitness on a quantitative trait, in units of the trait's standard deviation.  It is a property of a trait, not of an allele, and converting it needs the allele's effect size:  s_genic = β·a.  The bridge is written out on the conventions page.

What the measured values actually are

Human alleles that made it into the literature run roughly s = 0.01 to 0.2.  That range is censored at the bottom, not measured there.  The best current human method cannot resolve anything below about 0.02:  "we cannot reliably distinguish smaller selection coefficients from zero."³⁵

So the honest statement is that the distribution of measured s is dominated by small values with a long right tail, and that the part of the distribution most arguments depend on has never been measured in humans at all.

And the published values are systematically too big.  The field documents this itself, with numbers.  Kingsolver's meta-analysis found only 25% of linear gradients individually significant, a median sample size of 134, and an explicit file-drawer effect in the small-sample estimates.  Hereford and colleagues found a median mean-standardised |β| of 0.54 — 0.31 after correcting for the upward bias of taking absolute values — and said in print that this "cannot be representative of selection on all traits."³⁶  Morrissey and Hadfield found that about 90% of the apparent temporal variation in selection is sampling error.

The meta-analyses also disagree with each other by a factor of two to three, and both figures belong on the page rather than the convenient one.

The nearly-neutral zone is a gradient

The often-quoted threshold |s| < 1/(2N_e) marks where the bias stops being large, not where it starts.

At the barrier itself, a harmful mutation still fixes at 58% of the neutral rate and a beneficial one at 1.58 times it.  The two always differ by exactly the master parameter:  1.58 − 0.58 = 1.  Tomoko Ohta, whose name the nearly-neutral theory carries, says it herself:  "the fixation probability is a continuous monotone function of Ns.  Thus, when discussing molecular evolution by nearly neutral mutants, one has to consider all mutants around Ns = 0."³⁷

What a selection coefficient cannot tell you

Nothing about mechanism, and nothing about whether the change built or broke anything.

Sickle-cell HbS has one of the largest measured heterozygote advantages in humans, and it is a broken β-globin.  Lactase persistence has a comparable s, and it is a regulatory gain of function.  The series taxonomy in Measuring Biological Information scores its main variants as a modification, up, because an enhancer that already worked binds its activator more strongly;  see Lactase Persistence.  The number is identical in kind;  the biology is opposite.  Any argument that uses a fitness measurement as a proxy for information gain has to answer that pair.

Fixation

The probability

The whole subject is one formula, from Kimura in 1962 — written here with N_e where Kimura wrote N:³⁸

u(p) = (1 - e^(-4*Ne*sHet*p)) / (1 - e^(-4*Ne*sHet))

one new copy:              p = 1/(2N)
strong selection, Ne = N:  u -> 2*sHet          (Haldane 1927)
neutral limit:             u -> 1/(2N)

The neutral limit deserves its own sentence because the derivation is the best plain-language moment in population genetics.  A new mutation is one copy out of 2N.  Every copy alive today is equally likely to be the one all future copies descend from.  So its chance of taking over is 1/(2N).  No calculus needed.  Crow and Kimura state it that way and note that the N here is the census number, not N_e.¹⁰

Even a 1% advantage fails 98 times in 100.  Fisher tabulated the exact answer as 0.0197 in 1930 and wrote:  a selective advantage of 1% "has itself a chance of about 1 in 50 of establishing itself and sweeping over the entire species."⁶  He also gave the arithmetic that makes the point unforgettable:  of 10,000 such mutations, 3,642 leave no descendant at all — against 3,679 with no advantage whatever.⁶

The time

Two things fall out of that table.

The arithmetic of neutrality is brutal.  99.995% of neutral mutations vanish, on average within about twenty generations.  The surviving 0.005% take forty thousand.  The mean time to loss is the clock nobody quotes, and it is where nearly all the action is.

The textbook sweep-time shortcut is wrong by half.  (2/s)ln(2N) integrates the deterministic logistic curve from p = 1/(2N).  But a mutation conditioned on eventually fixing is by construction one that escaped the low-frequency danger zone unusually fast.  Print 1,312 and mention 1,981 as the shortcut, not the other way round.

Kimura's cancellation, in two lines

Under strict neutrality the substitution rate equals the mutation rate, and population size drops out entirely:

new neutral mutations per generation = 2*N*mu
each fixes with probability            1/(2N)
substitutions per generation           = 2*N*mu * 1/(2N) = mu

This is the reason a molecular clock should exist at all, and nothing in selectionist theory predicts it.  It holds only for strict neutrality, only per generation (not per year), and only if the neutral fraction of mutations is itself constant — which the nearly-neutral theory denies.

Two things that change the answer

Dominance.  Haldane's sieve:  a fully recessive beneficial mutation is nearly invisible to selection while rare, because it is almost never expressed.  At s_hom = 0.01 it fixes 12.5× less often than an additive one at N_e = 10⁴, and 125× less often at N_e = 10⁶ — the recessive probability goes as 1/√N, so the sieve gets worse in bigger populations.  Orr and Betancourt showed the sieve switches off entirely when adaptation draws on standing variation.

Population growth counterfeits selective advantage.  In a population growing at rate r per generation, the fixation probability is about 2(s + r).³⁹  Growth at 1% per generation doubles the fixation chance of an s = 0.01 mutation.  This matters directly for any post-bottleneck population — including a post-Flood one.

But the same paper closes the obvious follow-up:  "population bottlenecks do not change the fixation flux of beneficial alleles, at least for additively acting genes."  The per-mutation probability rises during recovery, and there were fewer mutations to begin with;  the two cancel.  Recessives are the exception.

Recombination and linkage

A Morgan is a count

A map length of one Morgan means one expected crossover on that stretch per gamete per meiosis.  So the total map length of a genome in Morgans is the average number of crossovers per meiosis.  "The human genome is about 34 Morgans" and "a human gamete carries about 34 crossovers" are the same sentence.

Summed from deCODE's own published map files:  male 2,602 cM, female 4,180 cM, sex-averaged 3,391 cM over 2,754 Mb — 1.23 cM/Mb.  Female meiosis makes 1.61 times as many crossovers as male.  Direct counting agrees:  22.8 ± 3.7 crossovers in 91 single sperm, and 41.6 ± 11.3 maternal crossovers in 51 oocytes and embryos.

Recombination is not spread evenly:  80% of sex-averaged recombination falls in 10.6% of the genome, and for males 80% falls in 6.1%.  Hotspot positions are set by PRDM9 and are almost completely different between humans and chimpanzees despite 99% sequence identity.

Why linkage matters

Selection acts on whole individuals, not on single sites.  With little shuffling, a good mutation is stuck beside whatever bad ones share its chromosome, and selection cannot separate them.

Hill-Robertson interference, simulated directly:  two beneficial mutations of s = 0.1 on different backgrounds in N = 1,000 haploids both fix 3.1% of the time under free recombination, 1.9% at c = 10^-4, 0.8% at c = 10^-5, and never at c = 0.  Those are 4,000-replicate Monte Carlo figures with a standard error of about 0.3 percentage points, so only the trend is real.

An attribution worth getting right:  Hill and Robertson's 1966 paper never uses the word "interference."  It is about artificial-selection limits, and states its conclusion as a reduced effective population size.⁴⁰  Felsenstein named it the Hill-Robertson effect in 1974.⁴¹

Muller's ratchet turns when the least-loaded class is small.  Haigh's formula is the whole story:

n0 = N * e^(-U/s)        [haploid:  U per genome, s per mutation]

At N = 10⁶, U = 0.1 and s = 0.01, n₀ = 45 individuals and the ratchet turns.  Raise U to 0.5 and n₀ = 2 × 10^-16 — there is no mutation-free class at all.  This is the one haploid formula on the page, and feeding it a diploid U understates the exponent by a factor of two.

But the ratchet is slow in DNA organisms.  Andersson and Hughes bottlenecked 444 Salmonella lineages for 1,700 generations and only 1% showed detectable fitness loss.  The dramatic ratchet results all come from RNA viruses, whose mutation rates are 10^-5 to 10^-3 per base per generation.

Background selection reduces to one ratio:  B = e^(−U/M), where M is map length in Morgans.  It does not depend on how strong the selection is.  Across the whole human autosome (U ≈ 1.5, M = 33.9) that is only a 4% reduction in N_e — but measured genome-average reductions are 19–26%, because deleterious mutations are concentrated near genes where the local U/M is far higher.

So N_e is not one number for a genome.  It is a function of position, set by local gene density and local recombination rate.  Any argument quoting a single N_e for a species is averaging over a quantity that varies several-fold across the genome.

The measured cost of linkage in humans is smaller than you would guess

Holding everything else fixed in Drift Lab and changing only the genetic map, deleterious fixations over 3,000 generations came out at 729 ± 38 under free recombination and 705 ± 38 at the human sex-averaged map — the same number within the seed-to-seed spread.  The female and male maps give 715 ± 46 and 710 ± 25.  One tenth of the map gives 994 ± 90, 1.1 Morgans about 1,650 across three fresh seeds (1,631–1,695), against the 1,403–1,591 recorded earlier and a single run of 1,484 — the one row here that does not reproduce, and no crossing over at all 3,712 ± 233 — about five times the free-recombination count.

A young-earth reader would want this section to say that linkage makes selection much weaker than the textbook formulas assume, so genetic-entropy-style arguments are conservative.  The measured answer is mixed and the honest version is worth more.  At the real human map length, linkage costs essentially nothing at the genome scale.  The cost is real and large in low-recombination regions — where an excess of damaging alleles has in fact been measured — and it is severe where recombination is absent altogether, which is the mitochondrion and the Y.

Diversity, and reading the past off it

The statistics

The neutral expectation for the spectrum is E[ξ_i] = θ/i, and its whole content is one fact about tree shape:  the coalescent devotes branch length 4N_e/i to branches with i descendants.  A 400,000-replicate coalescent simulation reproduces that to within 1.3%, and rejects 4N_e/i² on the same data.

One practical point:  plot i·ξ_i against i, not ξ_i against i.  Under the standard neutral model the first is flat;  on the raw plot every spectrum looks like the same decaying hyperbola and nothing is visible.

The ploidy trap

theta = 4*Ne*mu     autosome
      = 3*Ne*mu     X
      = 2*Ne*mu     mtDNA, Y, any haploid

Autosome : X : Y : mtDNA carry 4 : 3 : 1 : 1 gene copies at an even sex ratio, so expected diversity runs 1 : 0.75 : 0.25 : 0.25, and expected coalescent times scale the same way.

Humans break it in both directions, and the breaks are informative.  X/A runs 0.85–1.08 across six populations against the expected 0.75, implying more breeding females than males.  Y/A is about ten times below the 0.25 expectation, attributed to purifying selection on a chromosome that never recombines rather than to demography.  Tenesa measured the X effective size 30–50% larger than the autosomal one, wrote plainly "It is not clear why," and added that the difference "disappears when using HapMap Phase II data."²⁶

The coalescent bounds what diversity can see

E[TMRCA] = 4N_e(1 − 1/n) generations, so the expected TMRCA never exceeds 4N_e no matter how many genomes you sequence.  Any one realised genealogy is a single draw around that mean with a standard deviation of about 55% of it.  And most branch length sits near the tips:  the deepest interval is only 1/a_n of the tree — 35% at n = 10, 19% at n = 100, 13% at n = 1,000.

That is precisely why an old, brief bottleneck leaves so little trace, and it is the technical heart of the two-first-parents debate.

One measured number, carried all the way to a date

This is the most useful thing on the page, because a reader can follow every step and check it.  Start with a measured diversity and end with a date, carrying the conventions and error bars the whole way.⁵

Now change one convention at a time.  Every one of these answers is arithmetically correct.

Total spread on conventions alone:  a factor of 4.3.  The measurement in step 1 has a ±13.9% error bar — Yu's own ±0.016% on 0.115% — and combining it with the ±8.1% on the pedigree mutation rate gives the ±16.1% carried down the table.  The coalescent variance of a single realised genealogy adds ±55% on top of that.  And the convention choices swamp both.

This is not an argument that molecular dates are worthless.  It is the reason this page fixes its conventions in writing, and the reason any date quoted without them should be treated as a range rather than a number.

Structure, and the assumption underneath everything

Hardy-Weinberg

The one equation a high-school reader has already met, and the one every genotype statement above silently assumes.  For one locus with two alleles at frequencies p and q:

AA : Aa : aa  =  p^2 : 2pq : q^2

It arrives in one generation, not gradually — which was Hardy's actual point, and which Crow and Kimura flag as "of great importance":  "it is not necessary to inquire into the past history of a population;  if mating was at random during the preceding generation, the principle holds."¹⁰

A small point of credit, checked against both originals:  Hardy never writes p² : 2pq : q².  He works in counts, as p : 2q : r, and states the condition for stability as q² = pr.  The modern form is Weinberg's, on p. 378 of his 1908 paper.⁴²

The full assumption list is ten items:  diploid, sexual, non-overlapping generations, two alleles, equal allele frequencies in the sexes, random mating, very large size, no migration, no mutation, no selection.

A failed Hardy-Weinberg test names no cause.  The classic worked example is 18,121 Oslo schoolchildren whose colour-blindness data fail an X-linked test at P ≈ 0.03.  The violated assumption is not inbreeding and not structure — it is two alleles, because "colour blindness" is four conditions at two loci.

With inbreeding, the genotype frequencies become p²(1−F) + pF and so on.  That form is what the bottleneck sections below need.

F_ST and population structure

Wright's F-statistics measure how much of the total variation sits between groups rather than within them.  For humans the continental figure is about 0.10–0.15 — the famous result that most human genetic variation is within populations, not between them.

The island model converts F_ST into a migration rate:

Nm = (1/F_ST - 1) / 4

It is famously fragile, and should be quoted with that warning attached.

The finding that matters most here is that structure and size are formally confounded.  Mazet and colleagues proved that "any demographic model with structure will necessarily be interpreted as a series of changes in population size by inference methods ignoring structure."⁴³  Cousins, Scally and Durbin fitted an explicitly structured model to human data and recovered a deep split and re-merger instead of a size change.

This is not a technicality.  It means a population that was subdivided and a population that crashed can leave partly-confusable signatures, and any argument reading a bottleneck off an N_e curve has to say why structure is not the explanation.  It cuts both ways:  it weakens confident mainstream reconstructions of ancient population size, and it equally weakens any young-earth reading of the same curves.

Mazet and colleagues put a number on it.  For large M, a constant-size n-island model is read by a panmixia-assuming method as a decline by n + (n−1)²/(nM) — their eq. (19), p. 364.  Their worked human case is ten demes of 530 diploids, constant in size, with the migration rate changing three times.  Read exactly rather than by that large-M formula, the apparent size their model produces at M = 0.55, the value it holds in the recent past, is 13,600 — within a few per cent of the 13,500 peak Li and Durbin publish for the European and East Asian curve at 150 kya.  At the M = 0.85 that governs the ancient end of their scenario the same formula gives about 10,400.  Meanwhile the apparent current size is the deme size, 530, not the metapopulation's 5,300.⁴³

Genetic load

The definition, and the problem with it

L = (w_max - w_bar) / w_max

The argument is always about the numerator's reference, and w_max is a reference population, not an observed organism.  Crow's own textbook says so in the next sentence, and for the segregational load the reference is an asexual population fixed for the best heterozygote — something a sexual species cannot produce by construction.¹⁰

Ten different things are called load.  A sentence saying "the load is 0.9" is meaningless until it names which load and against what reference.

Mutational load does not depend on how harmful the mutation is

This is the most surprising exact result in the subject.  Haldane, 1937, in his own words:  the loss of fitness "depends entirely on the mutation rate and not at all on the effect of the gene upon the fitness of the individual carrying it, provided this is large enough to keep the gene rare."⁴⁴

Per locus the load is u for a fully recessive mutant and about 2u for anything with appreciable dominance.  Genome-wide with multiplicative fitness, L = 1 − e^-5  with u = 10^-5 and h = 0.5, the per-locus load is 2.000 × 10^-5 at s = 0.001, 0.01, 0.1 and 1 alike.  Four orders of magnitude in s, the same load.

Worth noting because it is routinely misreported:  Haldane's own worked number was about 4%, not 90%.  The catastrophic figures came later, from a much larger U.

Lethal equivalents, and the measured human cost

A lethal equivalent is a count:  "a group of mutant genes of such number that, if dispersed in different individuals, they would cause on the average one death."¹⁶  Survival = exp(−B·F), with B per gamete.

The measured human value has fallen by a factor of three since 1956.  Morton, Crow and Muller: 1.5–2.5 per gamete.  Bittles and Black: 0.56–0.7 per haploid genome.  Gao and colleagues, from Hutterite pedigrees: 0.29 recessive lethals per haploid set (95% CI 0.10–0.84).

And the measured cost of first-cousin marriage has two components that are usually reported as one:

• Mortality: 3.5% excess deaths from about six months' gestation to age ten, from a meta-analysis of 69 populations.
• Fertility: a 55% decrease (95% CI 44–66%) in the odds of having children, measured in 1.4 million people and confirmed within full-sibling pairs so that environment is controlled.⁴⁵

The mortality figure is the small one and it is the one everybody quotes.  A bottleneck argument that cites only mortality is cherry-picking, and this page says so because the fertility figure hurts the young-earth case rather than helping it.

Why the load paradox dissolves, and where load arguments survive

Under soft selection — selection that is density- and frequency-dependent — the reference genotype stops being an abstraction, and "those with the highest fitnesses (rather than an abstraction) become the real standards of comparison."⁴⁶  Made quantitative:  under relative fitness "a species could tolerate 10's or even 100's of new deleterious mutations per genome each generation."⁴⁷

The genetic entropy article concedes the load argument in full on exactly these grounds, and that concession is not revisited here.

Load arguments survive in three places, all of them cases where absolute fitness is what matters:  mutational meltdown in small populations, lethal mutagenesis in viruses, and conservation genomics.  All three use the same inequality — mean absolute fitness × maximum fecundity < 1 — which is Haldane's e^(−U) multiplied by a demographic ceiling.

Bottlenecks and small populations

A founding pair is not "N_e = 2 from generation one"

The children of one unrelated pair are outbred — each gets one allele from each parent, and those are two independent draws from the source population, so their expected heterozygosity equals the source's exactly.  F = 0 at generation 1, not 0.25.  The exact recursion for a dioecious pair is F(t) = ¼(1 + 2F(t−1) + F(t−2)).  The calculator reads N_e = 2 as one breeding pair and uses this recursion, so every row of the table above can be re-run there.⁵

Depth is cheap;  duration is expensive.  One generation at two individuals costs almost nothing.  Ten generations of continued sib mating costs 86% of the heterozygosity.

The asymmetry that decides most bottleneck arguments

Heterozygosity is nearly immune to a short bottleneck.  Allelic richness is not.  This is fifty years old and routinely mis-stated in both directions.

Two diploid founders retain 75% of the source population's heterozygosity at generation two, less every generation after that, and only a fraction of its alleles at any generation — and that fraction depends on how diverse the source was:

The first two rows are Nei, Maruyama and Chakraborty's own published values.⁹  The rest is the same Ewens sampling formula at modern diversity levels, against a source population of 8.1 × 10⁶ gene copies — the size implied by Nei's own E(K) = 3.599, and a number that has to be printed with the θ because the answer moves by a factor of two across plausible source sizes.  An allele-retention percentage without its θ is meaningless — and the high-θ end is exactly where a young-earth model is in most trouble, because those are the HLA-like loci.

There is a hard ceiling underneath all of it.  One couple can carry at most 4 autosomal alleles, 3 X alleles, 1 Y lineage and 1 mtDNA lineage at any one locus.  Note the X is three, not two.

And the eight people of Genesis 7:7 are not eight independent genomes.  The text names Noah, his wife, three sons and three sons' wives.  The three sons are full brothers, so every allele they carry came from Noah or his wife.  The real ceilings are:

Two caveats belong with those numbers.  "At most" is doing real work — the daughters-in-law could have been related, and the text is silent.  And 10 is the ceiling, not the expectation:  three sons draw six copies from their parents' four, so on average 9.5 of the ten survive to the next generation — which is the 4.75 "genome equivalents" figure Carter computes.

Recovery is glacial

The gap back to equilibrium closes at 2μ + 1/(2N_e) per generation.  For a human population restored to N_e = 10,000, the half-life is about 13,856 generations — 402,000 years.

Real populations founded from tiny numbers

Eight populations founded from twenty individuals or fewer are alive today:  the Chatham Island black robin (5, then a single breeding female), the Mauritius kestrel (4 known wild birds in 1974, of which one breeding pair founded the recovery), the pink pigeon (~10), the Mauritius parakeet (~20), the northern elephant seal (~20), the whooping crane (~20), the European bison (12, seven in one line) and Przewalski's horse (12).

All of them are recovered demographically and impaired genetically.  The two best-measured — elephant seal and whooping crane — each lost 70–88% of their heterozygosity permanently.  Northern elephant seals fell 8-fold in heterozygosity, from 0.00142 to 0.000176 per bp, and have not recovered it in 22 generations and 220,000 animals.

One counter-case is worth stating because it is the closest thing to a controlled experiment:  the Isle Royale wolves, founded by two to three animals in a closed system, went extinct in about seventy years, with 23–48% of the genome in runs of homozygosity and 33% vertebral malformation.

And one case cuts the other way.  Ashkenazi Jews passed through N_e ≈ 250–420 about 25–32 generations ago, and today have heterozygosity 2.4% higher than the European comparison group.

Purging is real and much weaker than advertised

The founding case — Speke's gazelle — was reanalysed and does not hold up.  Across 119 zoo populations and 88 species, purging was detectable but "the change in inbreeding depression due to purging averaged across the 119 populations is <1%."⁴⁸  Genomics has since found genuine purging of strongly deleterious recessives in island foxes, Alpine ibex and mountain gorillas — while mildly deleterious mutations accumulate at the same time.

50/500, and what it actually claims

The two famous numbers answer two different questions, and neither is "minimum viable population."

• N_e ≥ 50 (revised to 100) is about limiting inbreeding depression over about five generations.
• N_e ≥ 500 (revised to 1,000) is about retaining evolutionary potential in perpetuity.
• MVP is a census count for demographic persistence, median 4,169 adults (95% CI 3,577–5,129).

Frankham's own arithmetic behind the revision gives N_e = 142, rounded down to 100.  And with N_e/N ≈ 0.1–0.2, "N_e ≥ 100" means 500–1,000 adults on the ground.

Mutational meltdown sets a harder boundary:  "populations with effective sizes smaller than 100 (and actual sizes smaller than 1,000) are highly vulnerable to extinction via a mutational meltdown on timescales of approximately 100 generations."⁴⁹  For humans that is roughly 2,900 years at this page's 29 years per generation.  That is a real problem for a 4,500-year model, not a friendly one, and it belongs in the body of the page rather than a footnote.

How fast a population can respond to selection

Population genetics tracks alleles.  Breeding tracks traits.  The bridge is heritability, and getting it wrong is one of the documented errors on this site.

Broad-sense H² and narrow-sense h² are different numbers, and only h² predicts response.  H² = V_G/V_P counts everything genetic;  h² = V_A/V_P counts only the additive part that survives meiosis intact.

This distinction is load-bearing here.  Our own audit of Sanford's book shows him quoting Kimura's 0.004 as an h².  Kimura wrote H².  The underlying primary is a broad-sense heritability of egg-to-adult viability in one Drosophila population, extrapolated genome-wide by multiplying a second-chromosome estimate by three — and its value swings six-fold, from 0.0026 to 0.0157, on a gene-action parameter the authors could not measure.

The breeder's equation is R = h²S, and the standardised form is R = i·h·σ_A.

It decays, and there is a hundred-year experiment showing by how much.  The Illinois maize selection experiment has run since 1896.  Realised heritability for kernel oil fell 0.34 → 0.31 → 0.14 → 0.08 across four segments, and response per generation fell from 0.22 to 0.11 percentage points.  The trait kept moving;  the heritability behind it shrank four-fold.

Robertson's theory predicts both the ceiling and the clock:  total advance ≈ 2N_e × the first-generation gain, half-life ≈ 1.4N_e.  Six Drosophila lines give total response = 51.3 × R₁ and a half-life of 34.7 generations, implying N_e = 25.6 and N_e ≥ 24.8 by two independent routes that agree to 4%.

The Illinois high-oil line beats that ceiling, and mainstream quantitative genetics says why:  new mutation.  More than 100 generations at N_e under 30, "yet there is a continuing near linear increase in percentage of oil in the kernel that can be explained only by mutational input."⁵⁰  The low-oil line did hit a hard limit — because it has almost no oil left in its kernel.

The heritability of fitness itself is about 3%, measured directly across 19 wild populations:  "we found that h²(w) was generally small, with a meta-analytic average of 2.99%, 95% CI [0.80; 6.60%]."⁵¹

But low h²(w) does not mean slow adaptation, and the authors say so on the same page.  What sets the pace is V_A(w) = 0.185.  For a trait with h² = 0.3 the maximum response is 0.24 phenotypic standard deviations per generation, against observed phenotypic change that is "rarely above 0.03 standard deviations."  The measured ceiling on selection is about eight times the observed rate of change.  That cuts against genetic-entropy-style arguments, and the page states it here rather than burying it.

Molecular clocks

Why a clock should exist at all is Kimura's cancellation from the fixation section:  under neutrality the substitution rate is μ, independent of population size and selection.  Nothing in selectionist theory predicts rate constancy, which is why the clock was the neutral theory's best evidence.  How the clock performs against known dates and fossil ages is tested in Molecular Clocks.

The clock's founders hedged it in the same paragraph.  Zuckerkandl and Pauling wrote "There may thus exist a molecular evolutionary clock" and, two pages later, ended the argument with "The verdict of the facts will be looked forward to."⁵²  The famous phrase is a hypothesis offered for test.

And it was never a good clock.  The index of dispersion — variance over mean of substitution counts across lineages — is 1 for a Poisson clock.  Measured genome-wide means are 2.10 in yeast, 4.19 in Drosophila, 6.48 in mammals, with individual loci running from 0.13 to 43.82.  Substitution is roughly n-fold overdispersed, and n rises as effective population size falls.

Modern practice does not assume a clock;  it assumes a distribution of clocks.  Measured branch-rate coefficients of variation are 0.51 for influenza A, 0.39 for dengue-4, 0.32 for marsupials — and the covariance between parent and child branch rates has 95% intervals straddling zero, which undercuts the older autocorrelated models.

Fossil calibrations are minimum ages, and that has a signed consequence.  A fossil dates the oldest specimen you have, not the node.  Pin the node at the fossil and the inferred rate is too fast, so every other node comes out too young.  The mammal-bird node was fixed at 310 Ma "in some 500 or more publications since 1990" while the defensible range for the same fossils is 330–288 Ma.⁵³

And more data cannot fix it.  Fitting 8,708,584 sites of primate genome, dos Reis and Yang found the posterior interval width converges to w = 0.322·t and concluded that "nearly all the uncertainty in the posterior time estimate is due to uncertainties in the fossil calibrations, and adding more sequence data is unlikely to produce more precise estimates."¹¹

Where these quantities break

Every quantity above assumes things that are often false.  The honest list:

1. Non-equilibrium populations. θ = 4N_eμ, Tajima's D and most N_e estimators assume mutation-drift equilibrium. No real population that has recently grown, crashed or admixed is at equilibrium, and humans have done all three.
2. Structure read as size. Formally proven, not a caveat: a structured population is necessarily read as a series of size changes by any method that ignores structure.
3. Linked selection. N_e varies several-fold across a genome with local gene density and recombination rate, so a single species-level N_e is an average over a varying quantity.
4. Epistasis. Every fitness formula above assumes mutations combine additively or multiplicatively. Neither is generally true, and which one you pick changes a 1,000-locus calculation by a factor of 1,900.
5. The DFE below the detection floor. Human s below about 0.02 is unmeasured, not measured to be small. Arguments that depend on that region — and the most interesting ones do — are extrapolating.
6. Conventions. As the worked chain shows, convention choices alone move a date by a factor of 4.3.

How these numbers get misused

Both camps, with the passage quoted and a verdict on whether the error is load-bearing, are on the companion page:  how population-genetic quantities get misused.  Three examples to show the page is even-handed.

Mainstream, load-bearing.  Dennis Venema writes in Adam and the Genome that the human population "never dipped below about 10,000 individuals,"⁵⁴ and in a peer-reviewed paper that any bottlenecks "were a reduction only to a population of several thousand breeding individuals."⁵⁵  That converts a harmonic-mean average into a floor.  The correction came from Richard Buggs and Joshua Swamidass — two Christians who accept common descent — not from creationists.

Mainstream, in a top journal.  A 2023 Science paper described an effective size of 1,280 as "about 1280 breeding individuals."  Science published no correction;  the conflation was pointed out in print by a different group, in a different journal, in December 2024.  The confusion is not a layman's problem.

Creationist, load-bearing.  Sanford's Genetic Entropy renders Kimura's broad-sense H² = 0.004 as a narrow-sense h², treats it as additive, and generalises one Drosophila study to humans.  The full page-by-page audit is here.

Objections

1. "This is all theory. Real populations do not obey these equations."

They obey them well enough to be tested, and the tests are in the article.  A direct Wright-Fisher simulation of 60,000 replicates gives a fixation time of 1,327 ± 5 generations against Kimura and Ohta's analytic 1,312.  Fisher's 1930 table value of 0.0197 reproduces to 0.01981 in 200,000 runs.  Where the approximations fail, this page says where:  at s_het = 0.05 the diffusion formula over-predicts by 1.5%, and above about s = 0.02 the exact branching-process root should be used instead.

2. "Effective population size is unfalsifiable — you just adjust it to fit."

It is a fitted quantity, and that is a fair criticism of any single estimate.  What answers it is that independent estimators, seeing different epochs, are checkable against each other and against known history.  Where they disagree — LD giving 3,100–7,500 against a textbook 10,000 — the disagreement is published by the people who measured it, in their own words:  "the estimates of N_e appear to be much lower than the usually quoted value of 10,000."

3. "If most mutations are nearly neutral, how can there be a molecular clock at all?"

There cannot be a strict one, and there is not.  The measured index of dispersion is 2 to 6, not 1.  The clock works as a rough average over long spans and fails as a precision instrument, which is exactly what the nearly-neutral theory predicts.

4. "A bottleneck of two is genetically impossible, so Genesis is refuted."

As stated, that is wrong, and the correction is mainstream rather than creationist.  N_e-based methods cannot rule out a brief bottleneck even in principle, because N_e is a harmonic mean over a long window.  Swamidass states the corrected version:  "The genetic evidence only rules out a single-couple bottleneck (2, not 10,000) going back about 500,000 years (not 18 million)."⁵⁶

But the same literature is not friendly to a recent bottleneck.  Recent bottlenecks are the easy case to detect — "the more recent a bottleneck, the more likely it is to be detected" — and a 4,500-year-old bottleneck of eight sits in the visible part of the window, not the invisible part.  The load-bearing statistic against a recent single couple is TMR4A, with a median of about 495,000 years.

5. "Creationists just cherry-pick the numbers they like."

Sometimes, and this page documents specific cases.  So does the mainstream side, and this page documents those too.  The remedy is the one used throughout:  print the conventions, print the range, print who is at each end, and make every number re-runnable.

6. "Why should a reference article state which way each result cuts?"

Because the alternative is worse.  A reference that quietly omits the results that hurt its author's position is not a reference.  The results in this article that damage the young-earth case — mutational meltdown at N_e < 100 on a 2,500-year timescale, the 55% fertility cost of first-cousin inbreeding, the allele-count ceiling of ten for the eight of Genesis 7:7, TMR4A at 495,000 years — are stated in the sections where they belong, at the same weight as everything else.

Companion pages

• Conventions and notation — the full symbol table, the factor-of-2 and factor-of-4 audit with each formula in both conventions, and a nine-question checklist to run against a draft. This is the page other articles will cite most.
• Measured values — the big tables: N_e/N across species, measured s values, mutation rates across life, genetic maps, diversity, lethal equivalents, and every bottleneck case study, each row tagged measured / model output / assumed.
• How these quantities get misused — fourteen documented cases from both camps, each with the passage quoted verbatim and a verdict on whether the error is load-bearing.

Test it yourself

• The Pop Gen Calculator — every formula on this page, evaluated in your browser, with the citation and locator beside each answer. Twenty-six calculators in nine groups, and the Verify tab runs 91 checks in front of you, including twelve deliberately wrong inputs it is required to reject.
• Drift Lab — the WebGPU population simulator. Where this page gives closed forms, Drift Lab runs the actual populations.
• Genetic entropy, Haldane's dilemma and the waiting-time problem and mitochondrial Eve — the three articles this page was factored out of.
• Neutral Theory — whether most DNA differences between species spread by chance rather than by selection. Its verdict: most DNA differences between mammal species are neutral or nearly so, by the mainstream's own count, and that rests mainly on the measured finding that most of the genome is not under constraint; inside the proteins of species with large populations, half or more of the changes look adaptive and the original theory has failed. For the design argument the theory concedes that selection did not build most differences, but it counts against evolution only if a large, sequence-specific function is shown in the DNA that differs.

Sources