---
title: "Genetic Drift and Fixation: Who Takes Over a Small Population? Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/neutral-drift-fixation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/neutral-drift-fixation/"
last_modified: "2026-10-30"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Drift decides who wins in a small population

**Background** A population that stays the same size still changes its genes, because who has children is partly luck. In a population of fifty, that luck is large enough to fix or remove a mutation within a few dozen generations.

**Why it matters** Evolution is often told as selection alone. Counting outcomes in many identical populations shows how much of the result is chance, and how a small advantage changes the odds.

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 misconception** A beneficial mutation always spreads.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

Physical conclusion

In a finite population, fixation is a probability. Drift sets the baseline equal to the starting frequency, and selection tilts the odds by an amount set by the product of population size and advantage.

## Related experiments and articles

Evolution as statistical physics · Reading path

### From random sampling to pulled and pushed waves

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Genetic drift lets a neutral gene take over a finite population**](https://sandboxphysics.com/stories/evolution-statistical-physics/drift-in-a-finite-population/)
2. [Article **Fixation probability and time, and why a better mutant often loses**](https://sandboxphysics.com/stories/evolution-statistical-physics/fixation-probability-and-time/)
3. [Article **The coalescent traces a sample back to its common ancestor**](https://sandboxphysics.com/stories/evolution-statistical-physics/ancestry-and-the-coalescent/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)