The problem of being rare
Imagine a new allele — a variant spelling of a gene — that appears in a population of 100 diploid individuals. Because the population is diploid, there are 200 gene copies in total. Our new allele starts as just 2 of those 200 copies, giving it a starting frequency of 1%.
Nothing is wrong with this allele. It does not make its carrier sick. It does not make them faster or smarter. It is, in the language of genetics, selectively neutral. And yet, if we watch what happens over the next hundred generations, we will find that in the vast majority of parallel universes — parallel populations — it is simply gone.
Why? The answer is genetic drift: the random fluctuation in allele frequencies that arises from the sampling process of reproduction itself.
The Wright–Fisher model: how the simulation works
The simulation you will run below implements the classical Wright–Fisher model. Each generation, the next generation's allele pool is formed by randomly sampling — with replacement — from the current gene pool. If a diploid population has N individuals, each new generation draws 2N allele copies from a binomial distribution.
Each generation, the number of copies of the rare allele in the next generation is drawn as:
copies ~ Binomial(2N, p)
where 2N is the total number of allele copies and p is the current allele frequency. The new frequency is simply copies / (2N). If it hits zero, the allele is permanently lost — extinction is an absorbing state.
Each replicate is an independent population following its own random walk. Running 50 replicates in parallel lets you see the distribution of outcomes rather than a single lucky or unlucky trajectory.
Interact with the simulation
Adjust the parameters below and run the simulation. The experiments in §4 will tell you exactly what to look for.
Survived Extinct (allele lost)
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What each parameter actually does
Each of the four parameters controls a fundamentally different aspect of the evolutionary process. Understanding their effects — separately and in combination — is the core lesson of this simulation.
| Parameter | Name | Effect on extinction |
|---|---|---|
| N | Population size |
Strong The single most powerful control. Drift strength scales as 1/(2N) — halving N quadruples the variance per generation. At N = 10, nearly all rare alleles vanish within 20 generations. At N = 5,000 with the same p₀ = 1%, the allele often survives for centuries. |
| p₀ | Initial frequency |
Strong A rare allele's fate is closely approximated by its initial frequency — the probability of fixation (reaching 100%) for a neutral allele is exactly p₀, and the probability of extinction is 1 − p₀. Start at 1% and you expect ~99% extinction across replicates. Start at 50% and the allele is nearly as likely to fix as to be lost. |
| G | Generations |
Moderate Increasing generations gives drift more time to act. Alleles that survived to generation 50 have already partially escaped the most dangerous early phase. Very long runs show that every neutral allele eventually fixes or goes extinct — drift always wins in finite populations. |
| R | Replicates |
Precision only Replicates do not change the biology — they change how precisely we estimate the extinction rate. 10 replicates gives a noisy estimate; 100 gives a reliable one. In a real experiment you only have one population, so replicates represent the space of possible histories your population could have had. |
Guided experiments: what to look for
Run these four experiments in sequence. They are designed to isolate the effect of each parameter.
Set p₀ = 0.01, G = 100, R = 50. Run first with N = 10, then with N = 1,000. At N = 10 you should see near-total extinction within 20–30 generations; at N = 1,000, many lines survive and the trajectories are much smoother. This contrast shows why conservation biologists are so concerned about minimum viable population size.
Keep N = 100, G = 200, R = 50. Run with p₀ = 0.01 and then p₀ = 0.50. At 1%, expect ~99% extinction. At 50%, expect roughly half the replicates to go extinct and half to rise toward fixation. This is the neutral allele fixation probability theorem due to Kimura.
Set N = 500, p₀ = 0.01, R = 50. Run with G = 50, then G = 500. At 50 generations, many alleles appear stable. By 500 generations, most have resolved — either driven to zero or climbed substantially.
Set N = 100, p₀ = 0.05, G = 100. Run with R = 10 then R = 100. The extinction rate estimate will vary substantially across runs of R = 10; at R = 100 it converges. This illustrates why published simulation studies typically use hundreds or thousands of replicates.
Neutrality is not protection
The results of this simulation carry a counterintuitive message. In everyday thinking, we might assume that a gene variant that causes no harm will persist indefinitely — why would it disappear if nothing is selecting against it? The Wright–Fisher model shows this intuition is wrong.
In small populations, drift overwhelms selection. Even mildly beneficial mutations are regularly lost to drift when population sizes are low. This is the theoretical basis for Kimura's neutral theory of molecular evolution: the majority of genetic variants fixed in natural populations are fixed not because they are beneficial, but because they survived the lottery of drift.
The practical implications extend well beyond theory. In conservation genetics, small isolated populations — island species, fragmented habitats, captive breeding programs — lose genetic diversity not through any environmental pressure but through the relentless grind of drift across generations.
The simulation makes this visceral. Watch the trajectories at N = 50 over 200 generations: one by one, the lines collapse to zero. The ones that survive are not fitter — they were luckier. And in small populations, the lucky ones are rare.
How the simulation runs
The simulation runs entirely in your browser using JavaScript. The binomial sampling at each generation step uses a loop of independent Bernoulli trials — equivalent to R's rbinom(1, size=2N, prob=p). Once an allele's frequency reaches exactly zero, extinction is permanent, matching the absorbing-barrier condition of the theoretical model.
The model assumes: (1) discrete non-overlapping generations, (2) constant population size N, (3) random mating with no selection, mutation, migration, or recombination. Real populations violate all of these assumptions to varying degrees, which is what makes population genetics an active area of research.
For a rigorous treatment, see Hartl & Clark's Principles of Population Genetics or Ewens' Mathematical Population Genetics.