---
title: "Does a big sector mean selection? Opening angles and luck in colonies"
description: "Learn how a fitter mutant opens a wedge at a colony edge, how often it survives, and why a neutral colour can still take a large share by luck."
canonical_url: "https://sandboxphysics.com/stories/evolution-statistical-physics/does-a-big-sector-mean-selection/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/evolution-statistical-physics/does-a-big-sector-mean-selection/"
last_modified: "2026-10-30"
---

[Evolution as statistical physics](https://sandboxphysics.com/stories/evolution-statistical-physics/)

L3 · Chapter 5

# Does a big sector mean selection? Opening angles and luck in colonies

Sandbox Physics · 2026-10-30

A colony seeded with twelve colours ends with one of them holding forty per cent of the edge. Is that colour fitter? It might be. A faster-growing mutant opens a wedge whose angle depends on how much faster it grows. But the previous chapter showed that drift at the edge also produces big sectors with no selection at all. This chapter separates the two with the numbers from the Lab.

## Before you begin

Chapter 2: the diffusion formula for fixation. Chapter 4: sectors, sector boundaries and their wandering. Basic trigonometry: the cosine and tangent of a half angle.

- Relate the opening angle of a beneficial sector to the speed advantage of the mutant, and invert the relation to estimate the advantage from a measured angle.
- Compare the survival chance of one fitter founder with how large a neutral sector luck alone can make, and decide what a single big sector can and cannot show.

## Selection gives the boundaries a drift

For a neutral sector, each boundary wanders with no preferred side, so the expected change in sector width $X$ is zero. Hallatschek and Nelson add selection as a mean drift: each boundary of a beneficial sector moves outwards by $m_{\perp}$ per unit of front advance, while the random part keeps the diffusion constant $D_X$ of the neutral case.

Once the sector is wide enough that drift wins, its boundaries become straight lines and the sector becomes a wedge. The drift then equals the tangent of half the opening angle $\Phi$.

$$
\langle\Delta X\rangle=2m_{\perp}\,\Delta r,\qquad \langle\Delta X^2\rangle=4D_X\,\Delta r,\qquad m_{\perp}=\frac{v_{\perp}}{v}=\tan\frac{\Phi}{2}
$$

Mean and variance of the change in sector width over a front advance $\Delta r$, and the drift written as an angle.

Source: O. Hallatschek and D. R. Nelson

## A geometric rule for the angle

Suppose the only effect of the mutation is that mutant regions grow into empty space at speed $v^{\star}$ instead of $v$. Where the wedge meets the wild-type front, both populations must reach the kink at the same time. That gives the cosine rule on the right.

How $v^{\star}/v$ depends on the growth advantage $s$ is a property of the growth process. In the Lab every gold cell divides at rate $1+s$ instead of $1$. A colony made only of gold cells is the same random process run $1+s$ times faster, so $v^{\star}/v=1+s$ exactly. Hallatschek and Nelson give this linear rule for fronts where drift dominates, and a square-root rule for smooth Fisher waves.

$$
\cos\frac{\Phi}{2}=\frac{v}{v^{\star}}=\frac{1}{1+s}
$$

Opening angle of a beneficial sector when the mutant front is $1+s$ times faster.

Source: O. Hallatschek and D. R. Nelson

## How often a fitter founder survives

A new mutant starts as a sector one cell wide, and drift can close it before the drift of its boundaries opens it. With the biased walk above, the chance that a sector of width $x_0$ on a front of width $L$ eventually takes over has the same form as Kimura’s formula from chapter 2. Hallatschek and Nelson point this out in a footnote: the product of population size and selection coefficient is replaced by $m_{\perp}L/(2D_X)$.

Sectors much narrower than the establishment length $D_X/m_{\perp}$ survive with probability close to $m_{\perp}x_0/D_X$; wider ones almost always survive. In the Lab strip, one gold founder reached the far end of the strip in $3.3\pm0.9$ per cent of $400$ seeded runs when neutral, $13.5\pm1.7$ per cent at $s=0.1$ and $29.5\pm2.3$ per cent at $s=0.2$.

$$
u(x_0)=\frac{1-e^{-m_{\perp}x_0/D_X}}{1-e^{-m_{\perp}L/D_X}}
$$

Long-time survival probability of a beneficial sector of initial width $x_0$ at a straight front of width $L$.

Source: O. Hallatschek and D. R. Nelson

## Worked example: reading an advantage from an angle

Problem. A gold block $10$ cells wide, with $s=0.1$, starts on a straight front $700$ cells wide. What half-angle does the geometric rule predict? If you only had the measured angle, what advantage would you infer?

Prediction. $\cos(\Phi/2)=1/1.1$, so $\tan(\Phi/2)=\sqrt{1.1^2-1}=0.458$ and $\Phi=49^{\circ}$.

Measurement. In $20$ of $24$ seeded runs the wedge survived. Its half-angle tangent, measured from the change in wedge width between bands of rows, rose from $0.296\pm0.020$ between rows $50$ and $150$ to $0.392\pm0.014$ between rows $450$ and $600$. At $s=0.2$ it rose from $0.510$ to $0.603$ against $0.663$ predicted.

Inference. Reading $0.392$ as the final angle gives $1+s=\sqrt{1+0.392^2}=1.074$, so $s=0.074$, a quarter below the true value. The wedge is still opening at the end of the strip. The rule assumes a steady angle after transients, and in this lattice model the transient is long.

Answer. The geometric rule predicts $\tan(\Phi/2)=0.458$, and angles measured over a few hundred rows give only a lower bound on the advantage.

Source: O. Hallatschek and D. R. Nelson

## Luck alone makes big sectors

Now remove selection. On the Lab strip of width $200$ and height $240$, twelve neutral colours start in equal numbers, a fair share of $1/12=0.083$ each. Over $200$ seeded colonies the largest colour at the final front held a median share of $0.41$, between $0.31$ and $0.61$ in four colonies out of five. In $97.5$ per cent of them it held at least a quarter of the front.

So a single colour holding a large share of the edge is weak evidence of selection. Two kinds of evidence are stronger. One is the same opening angle in many independent sectors of the same mutant. The other is a survival rate of single founders well above the neutral rate. Hallatschek and Nelson propose combining the opening angle and the establishment rate of sectors to measure fitness and mutation rates in colonies.

Geometry can also reshape sectors without selection. A rock of radius $14$ cells bends the sectors around it, but over $40$ paired seeds the final front carried $9.0\pm0.2$ sectors with the rock and $9.1\pm0.2$ without, no measurable difference.

Source: O. Hallatschek and D. R. Nelson · O. Hallatschek

## Try it in the Lab

### [Tell a fitter founder from a lucky one](https://sandboxphysics.com/labs/range-expansion-sectors/)

1. Choose A faster founder opens a wedge. Predict the half-angle from the geometric rule, then compare it with the wedge you see after the front has grown across the strip.
2. Set the advantage to zero and replay. Predict how often the gold founder survives now, and explain why it usually disappears.
3. Choose A big sector by luck alone. Before reading the largest share, predict how far above the fair share of one in twelve it will be, then explain the result without selection.

## Exercises with explained answers

1. A mutant front is $20$ per cent faster than the wild type. What opening angle does the geometric rule predict?

   Show the answer

   $\cos(\Phi/2)=1/1.2=0.833$, so $\Phi/2=33.6^{\circ}$, $\Phi=67^{\circ}$ and $\tan(\Phi/2)=0.663$.

2. For small $s$, show that the geometric rule gives $\tan(\Phi/2)\approx\sqrt{2s}$ when $v^{\star}/v=1+s$. How good is this at $s=0.1$?

   Show the answer

   $\tan(\Phi/2)=\sqrt{(1+s)^2-1}=\sqrt{2s+s^2}\approx\sqrt{2s}$. At $s=0.1$, $\sqrt{0.2}=0.447$ against the exact $0.458$, two per cent low. Small advantages give relatively large angles, which is why angles are a sensitive measure.

3. In $400$ neutral runs a single founder survives $13$ times. At $s=0.1$ it survives $54$ times. Is the difference significant?

   Show the answer

   The fractions are $0.033\pm0.009$ and $0.135\pm0.017$. The difference is $0.102$ with standard error $\sqrt{0.009^2+0.017^2}=0.019$, more than five standard errors. The advantage shows up only when many founders are compared.

[Previous: Sectors at an expanding edge split a growing colony into colours](https://sandboxphysics.com/stories/evolution-statistical-physics/sectors-at-an-expanding-edge/) [Next: Pulled and pushed waves differ in who leads an expanding population](https://sandboxphysics.com/stories/evolution-statistical-physics/pulled-and-pushed-waves/)

## Sources and what we recomputed

1. [O. Hallatschek and D. R. Nelson, Life at the front of an expanding population, arXiv:0810.0053v2 (2008)](https://arxiv.org/abs/0810.0053v2)

   Version 2, full text. Eq. (41): a selective advantage gives sector boundaries a mean drift on top of their diffusion. Eq. (42): the drift is $\tan(\Phi/2)$ for opening angle $\Phi$. Eq. (46): $\cos(\Phi/2)$ is the ratio of wild-type to mutant front speed. Eq. (49): survival probability of a beneficial sector at a linear front; its footnote 4 notes the formal similarity to the Kimura fixation probability in a well-mixed population.

2. [O. Hallatschek, P. Hersen, S. Ramanathan and D. R. Nelson, Genetic drift at expanding frontiers promotes gene segregation, arXiv:0812.2345v1; PNAS 104, 19926 (2007)](https://arxiv.org/abs/0812.2345v1)

   Preprint, version 1, full text. Fig. 2B and the text around it: the mean-square wandering of 92 domain boundaries grows as $L^{2\zeta}$ with $\zeta=0.65\pm0.05$. The scaling argument after it gives $\zeta=1+(\chi-1)/z$, which is $2/3$ for the Eden model.