Everything a Small Founding Group Loses on the Way

Two people get into a boat

Two people get into a boat. Whatever waits on the far shore waits for the genes those two carry. Nothing else made the crossing. Ernst Mayr named the thing in 1942, and he named it for this exact picture: a few individuals arriving somewhere with only a sample of what they left behind [1].

The sampling does the work. A population hands its colonists no representative parcel of itself, only \(2k\) gene copies drawn at random, and \(2k\) gene copies hold at most \(2k\) alleles, however rich the source. Four copies from a locus carrying twelve alleles will carry four at most, and usually fewer, because some of the four are the same allele twice.

The real cases are easy to find. A typhoon crossed Pingelap, in the Caroline Islands, in about 1775, and left a few dozen survivors. One of them carried a variant of the CNGB3 gene. Total colourblindness now runs at roughly one person in twelve on that atoll, against about one in thirty thousand elsewhere [19]. Silvereyes island-hopping across the southwest Pacific lost measurable diversity at each hop, and Clegg and colleagues read the order of the hops back out of the losses [14]. Human genetic diversity falls almost linearly with distance walked from East Africa [18], which is what a long chain of small founder events looks like from the far end. Across 202 island and mainland comparisons, islands usually held less variation [15].

We wanted numbers rather than a picture. How much heterozygosity survives a founding of size \(k\), how many alleles survive it, how much of either comes back as the population grows, and how long a geneticist arriving later could still tell that any of it had happened. Nei, Maruyama and Chakraborty asked the first three questions in 1975. They answered them in closed form, for the cases that have closed forms [2]. We contribute no theory. We contribute a careful measurement of the theory's own quantities. The simulation lets us watch every generation. One result those papers did not set out to give sits under all this. The evidence stops being visible long before the founder effect does.

The first answer has sat in textbooks for fifty years. Nobody disputes it. We measured it anyway. A simulation that cannot reproduce a known closed form is not to be trusted on the parts where no closed form exists.

What we built, and what we did not

The island does not exist. The club has no field site, no permit, no sequencing machine and no organisms. Every number in this paper came out of a program on a laptop, and the program sits in the repository. Where we write that something was measured, we mean one thing: we estimated it from our own pseudo-random replicates, and from nothing else.

Here is the model. A source population holds twenty unlinked neutral loci. Each locus has up to twenty possible allele states and a fixed frequency spectrum, drawn once by Poisson-Dirichlet stick breaking at \(\theta = 1.6\), which is the neutral equilibrium spectrum Ewens derived in 1972 [7], and then never changed. That spectrum is printed in full in Section 1 of the raw output. It averages 11.55 alleles per locus and a gene diversity of \(H = 0.604432\), with a long tail of rare alleles: 4.93 alleles per locus sit below 1% frequency.

A founding group of \(k\) diploids forms by the random union of \(2k\) gametes drawn from that source. Allele counts in the founders are Multinomial\((2k, p)\) at every locus independently. The new population then grows toward a carrying capacity of \(K = 1000\). Growth is capped geometric, \(N_{t+1} = \min(K, \text{round}(N_t(1+r)))\). It reproduces as a Wright-Fisher population, the chain Wright set out in 1931 [3]. Gametes mutate under the K-allele model at \(\mu = 4 \times 10^{-4}\), and the next generation's counts are Multinomial\((2N_{t+1}, q)\).

The model throws away a long list of things. The list is worth reading. Nothing is under selection, so the study says nothing about adaptation on islands. Generations do not overlap. Mating is random across the whole new population, so no spatial structure exists inside it. Growth is deterministic once \(r\) is fixed, which means our founding group of two never dies, and in the real world a founding group of two usually dies. The club's companion study on extinction by bad luck covers that case, and should be read as the other half of this one. We treat the census size as the effective size, which is generous: real effective sizes run well below census sizes, so a real founder event of \(k\) individuals loses more than ours does.

One assumption deserves its own sentence. Through most of what follows the source pool is infinite, and the closed form is exact only because of that, so Section 4 of the output puts a finite source in its place and shows what changes.

One line of algebra

Let the source have allele frequencies \(p_i\) at a locus, and let \(H_{\text{old}} = 1 - \sum_i p_i^2\) be its gene diversity. Draw \(2k\) gametes. The counts \(X_i\) are multinomial, so \(E[X_i] = 2k\,p_i\) and \(\mathrm{Var}(X_i) = 2k\,p_i(1-p_i)\). Writing \(x_i = X_i/2k\) for the founders' frequencies,

Subtract from one and the source's diversity factors straight out:

We made no approximation anywhere in those two lines. Nothing here is a large-\(k\) limit or a diffusion result. The line states an exact fact about the second moment of a multinomial draw, and that is why we could use it as a hard check rather than a soft one. If our simulator had disagreed at any founding size, the simulator was wrong.

Allele loss has an equally clean statement. Allele \(i\) survives the draw unless all \(2k\) gametes miss it, so it is absent with probability exactly \((1-p_i)^{2k}\), and the expected number of source alleles lost at founding is \(\sum_i (1-p_i)^{2k}\). Notice how differently those two expressions behave. Heterozygosity loses a term in \(1/2k\). That term is small once \(k\) passes a handful. Allele counts lose a term that depends on every \(p_i\) separately, and for a rare allele \((1-p_i)^{2k}\) stays close to one until \(2k\) is of order \(1/p_i\). Allendorf made this point in 1986. Nothing else about bottlenecks is as useful to know [4]. Maruyama and Fuerst worked out the full distribution of allele number after a bottleneck a year earlier, and the asymmetry is the whole content of their result too [5].

The arithmetic

Fifteen founding sizes, from 2 to 500. Twenty thousand replicates each. Twenty loci per replicate. The source holds 231 alleles across those twenty loci. Its gene diversity is 0.604432.

Two founders retained 74.973% of the source heterozygosity, standard error 0.048%. The closed form says 75.000%. Ten founders retained 95.039%, standard error 0.023%, against 95.000%. Fifty founders retained 99.002% against 99.000%. Five hundred founders retained 99.899% against 99.900%. The largest standardised deviation across the fifteen sizes was 2.13, at \(k = 3\), and the mean was +0.239 with a realised standard deviation of 0.904.

The allele counts run differently. Two founders held 2.39 alleles per locus of the source's 11.55, which is 20.69%. Ten founders held 4.76, or 41.24%. Fifty founders held 7.38, or 63.87%. Five hundred founders held 10.62, or 91.91%. Total alleles lost at \(k = 2\): 183.21 of 231. The binomial prediction was 183.19. Largest standardised deviation on allele loss across the fifteen sizes: 1.54.

Put the two measurements on one axis and the gap between them is the whole result of this section. Figure 1 does that.

Fifty founders is a number conservation biologists use. At fifty, this model keeps 99.0% of the heterozygosity and 63.9% of the alleles. Both numbers are correct. They describe the same founding group. A report quoting only the first would not be lying.

Which alleles go first

Rarity is the whole story. A 40% allele survives a founding of two: \(1 - 0.6^4 = 87\%\). A 0.1% allele survives with probability \(1 - 0.999^4\), which is four tenths of one per cent. The source spectrum we used carries 4.93 alleles per locus below 1% frequency, and a founding group of five kept none of them at all, to three decimal places, across two thousand replicates.

Figure 2 shows this allele by allele. Every one of the 231 source alleles is plotted at its source frequency against the fraction of 20,000 replicates in which two founders failed to carry it, with the curve \((1-p)^{4}\) drawn underneath. The points sit on the curve. They should. No fitting was involved, and the curve has no parameters.

Frequency classes make the loss concrete. Below 1%, the source holds 4.93 alleles per locus and five founders hold 0.000. Between 1% and 2%, the source holds 1.27 and five founders hold 0.000. Between 10% and 20%, the source holds 1.13 and five founders hold 1.65. More than the source had, because the founders' frequencies are coarse. With ten gene copies the only available frequencies are tenths, so alleles that were commoner get pushed down into that band while alleles that were rarer vanish entirely.

Fifty generations later the population is back at carrying capacity. It holds 3.69 alleles per locus below 1% frequency again. Almost none of them are the ones it lost. They are new mutations, which is a different thing wearing the same clothes. The spectrum recovers its shape. The population recovers nothing that was in it.

Notes from the club table

Figure 3 plots the running estimate as a standardised deviation rather than as a raw ratio, because the raw ratios for \(k = 2\) and \(k = 50\) differ by a quarter and would need different axes. On this axis all three sizes are asking the same question, which is whether the Monte Carlo error we quote is the error we are actually getting.

Growing back is not coming back

A founding group that survives grows. While it grows it is small. While it is small it loses diversity every generation, at a rate of \(1/2N_t\). The total loss over the growth phase depends on how long the population spends at each size, which means it depends on the growth rate, which is the second axis of our sweep.

Thirty-two cells, eight founding sizes by four growth rates, 5,000 replicates each. Against the exact recursion the mean standardised deviation was −0.004 with a standard deviation of 0.998, and the largest single deviation across all 32 cells was 2.52. The recursion and the simulation measure the same thing, and the recursion costs nothing to run, which matters in a moment.

Read the first four rows slowly. Two founders growing at 5% a generation took 147 generations to reach capacity, and arrived with 12.35% of the source's gene diversity. The same two founders growing at 100% a generation arrived with 60.15%. Identical founding event, identical starting sample. A factor of nearly five between the outcomes. The founder effect proper cost them 25%. The growth phase cost the slow ones another 62 percentage points.

Figure 4 shows that generation by generation. The exact recursion is drawn underneath each simulated curve.

The right panel of Figure 4 answers the question in the title. The recursion is exact and cheap, so we ran it out to twenty thousand generations, far past anything simulation could ever have afforded. A population founded by two reaches its minimum at generation 14. That minimum is 40.4% of the source. It then climbs, slowly, as mutation replaces what drift removed. Halfway back at generation 532. Within one per cent of its eventual level at generation 3,017.

Three thousand generations is not a recovery in any human sense. For an annual plant it is three thousand years. For a bird it might be ten thousand. Nothing in the model prevents the population from being wiped out, replaced, hybridised or simply outcompeted long before it arrives, and the model has no mechanism for any of those because we did not put one in. The number measures how long the arithmetic alone takes, with nothing else going wrong.

The evidence goes before the loss does

Somebody lands on the island four hundred years later. They carry a sequencer. Can they tell?

Cornuet and Luikart built the standard way to ask, and they call it the heterozygosity excess test [8]. Its logic falls straight out of Section 5: a bottleneck removes rare alleles faster than it removes heterozygosity, so for a while afterwards the population carries more diversity than its own allele count would lead you to expect. We built our own version. An island of carrying capacity 400, growth rate 0.25, 5,000 replicates, fifteen loci. At each checkpoint, a sample of 30 individuals. The control is a population founded by the full 400, which lost essentially nothing at the founding but lives in the same small place and drifts at the same rate afterwards. Calibrating against that control rather than against the source makes this a strictly harder test, because it asks whether the founding group was small and not merely whether the island is.

Critical values come from the control's own distribution, so the false positive rate is 5% by construction and not by assumption, and measured rates across the thirteen checkpoints ran from 1.1% to 4.9%. Anyone running this test on real data would use BOTTLENECK, which Piry, Luikart and Cornuet released in 1999 and which has been applied to several thousand populations since [10].

Two real studies bracket what follows. Hoelzel and colleagues ran simulations against northern elephant seal data, a species reduced to perhaps twenty or a hundred animals in the 1890s, and found the genetic signature still readable a century later [17]. Hundertmark and Van Daele looked at elk introduced to Afognak Island in 1929 from eight founders and found the founder signature clear but already partly obscured [16]. Eight founders, eighty years, and the answer was already becoming ambiguous.

At generation 10 a five-founder event was detected 67.08% of the time. At generation 17, 48.92%. At generation 60, 6.48%. At generation 200, 0.04%, which is below the false positive rate. A twenty-five-founder event did slightly better early and was gone by the same point.

Now look at the right panel. The same two populations, described rather than tested. At generation 200 the five-founder sample held a mean of 3.25 alleles against the control's 4.42. Its gene diversity was 0.3884 against the control's 0.5514. A shortfall of 29.6%, in a population where the test had no power whatsoever to say that anything had happened. The signature decayed. The loss did not.

Our club's methods paper on detection in this same issue makes the parallel point from the other direction: a survey that finds nothing has not proved anything, and a test that returns nothing is not evidence that nothing occurred. Here the failure is quantified. Between generation 60 and generation 200 the honest report from this test is "no detectable bottleneck", and the honest report from the population itself is "one third of the diversity is missing and is not coming back".

The strongest objection we can make to this

Every locus in this study is neutral, and neutrality is the assumption that does the most work in everything above. We would attack it first.

Consider what neutrality buys us. Under neutrality, sampling alone decides an allele's fate, which is the only reason the closed forms are exact and the only reason we could validate anything at all. Drop it and almost nothing survives. An allele under selection has a fate that depends on its own effect and on the population size. The club's companion paper on drift versus selection measured that dependence. At small sizes it is weak. A 1% beneficial mutation fixes about 2% of the time in a thousand, and is lost the rest of the time. In a founding group of two it barely matters what an allele does, because it will be kept or dropped by the draw before selection gets a vote. That argument cuts in our favour at small \(k\) and against us at large \(k\), where selection has generations to act and our model gives it nothing to act on.

We calibrated our detection test on the same model that generated the data, which is the second objection and the one we cannot fully answer. Real genetic data comes from a mutation process nobody has fully characterised, at loci whose mutation rates vary by an order of magnitude, in populations whose history includes things no model contains. Peery and colleagues looked at exactly this in 2012. Bottleneck tests on real data turned out to be considerably less reliable than their simulated power curves suggest, with results sensitive to the assumed mutation model and to the number of loci [11]. Our power figures are therefore upper bounds. The real detection window is shorter than the one in Figure 5, not longer.

A third: twenty allele states per locus is a small universe, where real microsatellites have more and sequence data has effectively unlimited states. A larger allele space would put more rare alleles in the source. More rare alleles means a larger fraction lost at founding. Our 20.69% for two founders would go down. We chose twenty because it keeps the arrays a fixed width and because it gives an exact recursion, and both of those are conveniences for us rather than features of biology.

The one thing we will defend without qualification is the founding arithmetic itself. It does not depend on the mutation model, the growth model, the selection regime or the number of loci. Two founders carry four gene copies. Four gene copies cannot hold five alleles. No amount of subsequent biology changes what was in the boat.

Where a different choice would have changed the answer

We modelled growth as capped geometric, the simplest thing that reaches carrying capacity in a predictable number of generations. Discrete logistic growth is the obvious alternative. It holds the population below capacity for longer. Run both for exactly 120 generations from five founders at \(r = 0.15\): the geometric path accumulated 89,715 individual-generations and ended at \(H = 0.343596 \pm 0.001238\), the logistic path accumulated 83,553 and ended at \(0.339325 \pm 0.001247\). The difference is −1.24%. In absolute terms, \(-0.004270\) against a standard error of \(0.001757\). Small, and in the direction you would guess.

Mutation matters far more. Switch it off over the same 120 generations and final gene diversity falls by 16.44% and the final allele count by 72.34%, from 7.19 alleles per locus to 1.99. Our recovery numbers are entirely a statement about \(\mu = 4\times10^{-4}\). Halve that rate and every recovery time in Section 7 roughly doubles.

The finite-source correction changes the answer wherever \(k\) is an appreciable fraction of \(N\). With a source of \(N = 60\), a founding group of 30 has a predicted diversity of 0.598690 by the hypergeometric formula and 0.593701 by the infinite-source one. The simulation gave \(0.598670 \pm 0.000057\). Anyone applying \(1 - 1/2k\) to a founding group taken from a small remnant population will overstate the loss, and the club made that mistake for a full meeting.

Colonisation happens once in our model. A real range expansion is a sequence of founder events, each one drawing from a population that was itself founded. Excoffier, Foll and Petit showed what repeated sampling does at an advancing front, and our single event cannot do it: sampling can carry a rare allele up to high frequency across the new range [20]. A serial version of this study would lose more diversity overall. It would also throw up common alleles that were rare at the origin. We did not build that. The silvereye data [14] is where it would matter most.

Sample size and locus count change only the detection result, and they change it a lot: thirty individuals at fifteen loci is a modest study by current standards. A hundred individuals at fifty loci would push the power curves of Figure 5 right. It would not push them far enough to matter at generation 200, because by then the two populations differ in mean sampled allele count by 1.18 alleles against a between-replicate spread of similar size.

Reproducing this

The analysis is one file. No arguments, no configuration:

python analysis/founder-effect.py > analysis/founder-effect-output.txt

Python 3.12.3 and numpy 2.4.2. Wall clock on the laptop we used was 276.4 seconds. Most of that went to the 32-cell recovery grid and the detection run. Master seed 20241108, spawned into twelve independent PCG64 streams through numpy's SeedSequence. One stream per experiment, so the whole output is deterministic. Your numbers should match ours to within the printed standard errors, and if they do not, something is wrong and we would like to hear about it. The interactive model runs a smaller version of the same draw in your browser. On its default settings it reproduces the headline numbers.

One number stayed with us longer than the others. Not the one in the abstract. Take a population founded by two individuals, growing at 40% a generation. Mutation runs the whole time, and nothing else goes wrong. Its expected heterozygosity settles within one per cent of where it would have been had the boat never left, and it does so at generation three thousand and seventeen. Three thousand and seventeen.

References