INTERACTIVE COMPANION · POPULATION GENETICS · FOUNDER EFFECTS

The Founder Bench

A live model accompanying “Everything a Small Founding Group Loses on the Way”

Two benches sit below. The first draws a founding group out of a source population and lets you watch heterozygosity and allele count come apart, then grows the new population back. The second asks a narrower question: given a founding size and a number of generations since, would a geneticist landing on the island today be able to tell that anything had happened?

Both run the same source population the paper used. Its twenty loci and all 231 of its allele frequencies are printed in the analysis output and pasted into this page unchanged, so the numbers on the bench are the numbers in the paper rather than a fresh draw that resembles them. The random number generator here is a small seeded one written into the page, not the PCG64 the analysis used, so Monte Carlo figures will land near the paper's rather than on top of them.

source gene diversity: –
alleles per locus: –
distinct alleles, 20 loci: –
allele states per locus: 20, K-allele mutation at 4e-4
page seed: –

Model 1. The draw

Each small square is one gene copy at one locus, coloured by which allele it carries. The top band is the source pool. The middle band is what the founding group actually carries, which is 2k squares and never more than 2k colours. The bottom band is the population after however many generations you have let it grow.

The two numbers to watch are on the readout. Heterozygosity is the probability that two gene copies drawn at random differ, and it falls by exactly a factor of 1 − 1/2k at the moment of founding. Allele count is how many distinct colours are left, and it falls much further.

Founding size k 2 individuals Starts at 2, the smallest founding group that can breed. Push it to 50, the number conservation plans tend to quote, and watch the two readouts disagree: heterozygosity goes to 99 per cent while the allele count is still near two thirds. At 500 the heterozygosity is 99.9 per cent and roughly one allele in twelve is still gone. Growth rate r 0.40 per generation How fast the new population climbs to a carrying capacity of 1,000. Slow growth is expensive, because the population keeps drifting while it is small. Set k to 2 and compare r = 0.05 with r = 1.00 at 120 generations. The founding draw was identical in both. Generations grown 0 Zero is the founding group itself. Scrub right and the bottom band refills with colour, but look at which colours. The rare ones do not come back; new mutations arrive in their place. Locus on show 1 of 20 The squares show one locus at a time; the readout always averages all twenty. Locus 12 is the least diverse in the source and locus 9 the most, which is a good pair to compare at k = 2.

heterozygosity: – closed form: – alleles per locus: – alleles lost of 231: – binomial prediction: – population size: –

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The two prediction fields are not fitted to anything. The closed form is H × (1 − 1/2k) and the binomial prediction is Σ(1 − p)^2k summed over all 231 source alleles. Both are computed from the printed spectrum before any random number is drawn. The Monte Carlo figures are means over 300 independent founding draws at the current k; the coloured squares show one of those draws.

Model 2. The detection window

Colonists land, the population grows, centuries pass. Somebody arrives with a sequencer and thirty animals' worth of tissue. The question this bench answers is whether the founder event is still visible in what they can measure.

The test is the heterozygosity excess of Cornuet and Luikart. After a bottleneck, rare alleles are gone but heterozygosity is not, so the population carries more diversity than its own allele count would predict. The bench builds that prediction from control populations founded by the full island capacity, counts how many of your loci sit above it, and takes the critical value from the controls so that the false positive rate is five per cent whatever else you change.

Founding size k 5 individuals The island holds 400 and grows at 0.25 a generation regardless, so this slider changes only how many colonists arrived. Recomputing takes a second or two. Generations since founding 10 Free to scrub once the run is computed. Find the generation at which power crosses one half, then keep going and watch the gap between the two populations stay open long after the test has stopped seeing it. Individuals genotyped 30 Sixty gene copies at thirty individuals is a modest study. Doubling it buys you real power early and almost nothing after generation 100. Loci scored 15 The sign test runs across loci, so this is the number that sets its resolution. Below about eight loci the test cannot reach five per cent significance at all and the critical value pins to the top.

power: – false positives: – critical value: – alleles, founded: – alleles, control: – H shortfall: –

Computing the run.

What the defaults reproduce

Model 1 opens at k = 2, r = 0.40, generation 0, which is the founding event the paper's abstract describes. Three of its numbers can be checked with a calculator and nothing else. The closed form reads 75.000 per cent, because 1 − 1/(2×2) is three quarters exactly. The binomial prediction reads 183.19 alleles lost of 231, which is the sum of (1 − p)⁴ over the printed spectrum. The source line at the top of the page reads 0.604432 and 11.550, the gene diversity and mean allele count of that spectrum. All three appear in Section 2 and Section 3 of the raw output with the same digits.

The Monte Carlo figures beside them will not be identical to the paper's, and should not be. This page averages 300 founding draws from a seeded generator written into the page; the paper averaged 20,000 from numpy's PCG64. At k = 2 the paper measured 74.973 per cent with a standard error of 0.048 per cent, and 300 draws here carry a standard error nearer 0.4 per cent.

Model 2 is a smaller version of the paper's detection run rather than a copy of it. It follows 100 independent locus trajectories per arm instead of 5,000 whole replicates, and assembles pseudo-replicates from that pool, which is why its power curve is rougher than Figure 5 in the article. The shape and the crossing points are the same. At the defaults the paper reports 67.08 per cent power at generation 10 for a five-founder event, and this bench should land within a few points of that.

Things worth trying

Set Model 1 to k = 2 and scrub the generation slider to 120 with r = 1.00, then drop r to 0.05 and scrub again. Same founders, same draw, and the ending is nowhere near the same. That single comparison is the second result in the paper and the one that surprised the club most.

Then put k at 50 and look only at the colours. Heterozygosity says 99 per cent and the picture says something else. Both are describing the same three hundred squares.

In Model 2, set the founding size to 100 and ask whether anything is detectable at all. A hundred colonists is a real founder event with a real cost, and it is almost invisible to a test of this kind within a decade of the landing. That is the case where the arithmetic and the evidence part company earliest.