FR24 · Evolution and ecology

Genetic Drift and Fixation: Who Takes Over a Small Population?

How often does a new mutation take over a population of fifty, and how long does that take? A neutral mutation wins as often as its starting share. One newcomer among fifty wins once in fifty tries, and a win takes about a hundred generations. A 5 percent advantage lifts that newcomer's chance only to about 9 percent, so most beneficial mutations are still lost by chance.

Watch one Wright–Fisher population pass labels down the generations, then compare 240 replicate runs with the exact fixation probability and the diffusion formula.

Mutant fixed in replicate runs $\hat P_{\mathrm{fix}}=0.508\pm0.023$

Exact fixation probability $P_{\mathrm{fix}}=0.5$

Mean time to fixation $\bar t_{\mathrm{fix}}=64.6\ \text{gen}$

Sample traced to one ancestor $T_{\mathrm{MRCA}}=48\ \text{gen}$

Change the population and the mutant

Population size

50

N

Selection coefficient

0

s

Starting mutant frequency

0.50

p_0

Drag the marker along the newest generation to trace another individual back to its ancestors.

Sampling and playback

Individuals traced back

6

n

Generations shown per second

8

\dot g

Model, method and sources

$k_{t+1}\sim\mathrm{Bin}\!\left(N,\ \frac{p_t(1+s)}{1+s\,p_t}\right),\qquad u(p_0)=\frac{1-e^{-2Nsp_0}}{1-e^{-2Ns}}$

Each generation the mutant count is a binomial draw from the selection-weighted frequency of the parents. The exact fixation probability and times solve the absorption equations of this finite chain. The diffusion formula is the continuum approximation; for a neutral mutant it reduces to the starting frequency. Replicates are scored per finished batch so slow runs are not left out.

Hallatschek & Nelson (2008), arXiv:0810.0053v2 · Berestycki, Recent progress in coalescent theory (2009), arXiv:0909.3985v1 · Traulsen & Hauert (2008), arXiv:0811.3538v1

Derive the fixation probability

Model assumptions

Each generation of a fixed number of haploid individuals is drawn from the previous one: every child picks a parent at random, and mutant parents are favoured in proportion to their selection advantage. The scene shows one population with every parent link; the white lines trace a sample of the newest generation back to its common ancestor. The fixation estimate comes from 240 replicate populations started together, scored only when a whole batch has finished, with a binomial standard error. The reference values are the exact absorption probabilities and times of the finite Wright–Fisher chain, solved as a linear system, and the diffusion formula in the form given by Hallatschek and Nelson (2008). Everything is simulated or computed; no measured data are shown.

Mutant frequency in replicate populations

Fixation probability: simulated, exact and diffusion

More numbers and how they are computed

Diffusion approximation of the fixation probability: $0$
Exact mean time to fixation, given fixation: $0$
Simulated mean time to fixation: $0$
Neutral diffusion fixation time: $0$
Kingman mean time to the sample ancestor: $0$
Replicate rounds scored: $0$
Mutants at the start: $0$
Round shown in the scene: $0$

Exact values solve the absorption equations of the finite Wright–Fisher chain. Replicates are scored only when a whole batch has finished, so slow rounds are not dropped. The exported file lists every round with its seed, outcome and duration.

Start with the essentials

Focus question: How often does a new mutation take over a population of fifty, and how long does that take?
One-sentence intuition: A neutral mutant fixes with probability equal to its starting frequency. With an advantage, the chance rises, but a single new mutant is still lost most of the time unless $Ns$ is large.

Core mathematical model

Wright–Fisher update

k_{t+1}\sim\mathrm{Bin}\!\left(N,\ \frac{p_t(1+s)}{1+s\,p_t}\right),\qquad p_t=\frac{k_t}{N}

Each child picks its parent independently, so the number of mutants in the next generation is binomial. Its mean is the selected frequency; its variance $Np(1-p)$ is drift.

Diffusion formula for fixation

u(p_0)=\frac{1-e^{-2Nsp_0}}{1-e^{-2Ns}}\ \xrightarrow{\,s\to0\,}\ p_0

Valid when $N$ is large and $s$ small. The Lab also shows the exact probability of the finite chain, which differs slightly.

Neutral time to fixation

\bar t_{\mathrm{fix}}(p_0)\approx-\frac{2N(1-p_0)\ln(1-p_0)}{p_0}

For $p_0=1/2$ this is about $1.39N$ generations; for a single new mutant it approaches $2N$ .

Common difficulties

Better does not mean safe

Typical misconceptionA beneficial mutation always spreads.

Better mental modelA single new mutant with $s=0.05$ in a population of fifty fixes only about one time in ten. Early on there are too few copies for the advantage to beat luck.

Run the experiment

01
Watch one population
Leave the half-and-half preset running.
What to observe: Amber and teal stripes drift until one colour fills the row; then the population is relabelled and the race starts again.
02
Count many populations
Look at the fixation chart while the replicate batch finishes.
What to observe: About half the replicate runs fix the mutant, and the interval covers the exact value of one half.
03
Give the mutant an advantage
Choose Better, yet usually lost.
What to observe: Most runs still lose the single mutant, yet the fixed fraction sits near the exact value of about 0.094, almost five times the neutral one in fifty.

Genetic Drift and Fixation: Who Takes Over a Small Population?

Drift decides who wins in a small population

Core mathematical model

Wright–Fisher update

Diffusion formula for fixation

Neutral time to fixation

Common difficulties

Better does not mean safe

Run the experiment

Watch one population

Count many populations

Give the mutant an advantage

Related experiments and articles

From random sampling to pulled and pushed waves