---
title: "Sectors at an expanding edge split a growing colony into colours"
description: "See why only the edge of a colony passes on its genes, why sector boundaries wander faster than random walks and how sectors die off as the colony grows."
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last_modified: "2026-10-30"
---

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

L3 · Chapter 4

# Sectors at an expanding edge split a growing colony into colours

Sandbox Physics · 2026-10-30

Mix red and green bacteria that differ only in a fluorescent label, put a drop on agar, and come back two days later. The colony is not a uniform brown. It is a pinwheel of red and green sectors, each founded by a few cells at the start. Drift made this pattern, and it is far stronger than the size of the colony suggests, because only a thin band of cells at the edge has any descendants.

## Before you begin

Chapters 1 and 3: drift as random sampling and the coalescence of lineages. Power laws on logarithmic axes. No knowledge of surface growth is assumed; the two exponents needed are introduced where they are used.

- Explain why genetic drift is concentrated at the edge of an expanding colony, and why that produces sectors.
- Derive the wandering exponent of sector boundaries from the roughness of the front, and use it to predict how fast sectors die off at a straight front.

## Only the edge has descendants

A colony on agar grows outwards because cells at the edge divide into empty space. Cells further in are surrounded by neighbours and nutrients run low there, so they stop contributing. In the E. coli colonies of Hallatschek and co-workers this growing band was about seventy micrometres wide, a small fraction of the whole population.

Every cell at the frontier of tomorrow is a daughter of a cell at the frontier of today. The frontier behaves like a small population that is resampled at every step, so drift is strong there even when the colony holds billions of cells. Neighbouring cells soon share an ancestor, and the front breaks into single-colour domains. Behind the front, those domains are frozen in place as sectors.

The Lab grows an Eden-type colony on a square lattice. Every occupied cell with an empty neighbour divides at rate $1$, and its daughter fills one of the empty neighbours chosen at random and inherits the colour. No cell is fitter than another in this chapter.

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

## Boundaries wander faster than random walks

A sector disappears when the two boundaries on its sides meet at the front. If each boundary performed an ordinary random walk around a straight line, its mean square sideways displacement $\langle y^2\rangle$ would grow in proportion to the distance grown $L$. In the paper by Hallatschek and co-workers, $92$ boundaries from E. coli colonies wandered faster: $\langle y^2\rangle$ grew as $L^{2\zeta}$ with $\zeta=0.65\pm0.05$, above the random-walk value $1/2$.

Their explanation is that a boundary runs locally perpendicular to the edge of the colony, and the edge is rough. A bump of width $\lambda$ has height of order $\lambda^{\chi}$, so it tilts the edge by $\lambda^{\chi-1}$, and it lasts for a time of order $\lambda^{z}$. The tilt pushes the boundary sideways. Over a time $t$ the bumps that matter have width $t^{1/z}$, so the boundary drifts at a speed of order $t^{(\chi-1)/z}$ and moves sideways by $t^{\,1+(\chi-1)/z}$.

For the Eden model the roughness exponent is $\chi=1/2$ and the dynamic exponent is $z=3/2$, so $\zeta=2/3$. The Lab measures $\zeta=0.656\pm0.045$ from $80$ seeded strips of width $400$. It agrees with both the prediction and the experiment, but this is a check of the lattice model, not a new measurement of bacteria.

$$
\langle y^2\rangle\sim L^{2\zeta},\qquad \zeta=1+\frac{\chi-1}{z}=1+\frac{1/2-1}{3/2}=\frac23
$$

Wandering of a sector boundary over a distance $L$, and the exponent predicted from the front roughness for the Eden model.

Source: O. Hallatschek

## Worked example: counting sectors at a straight front

Problem. A straight front $1000$ cells wide starts with two colours placed at random. After $32$ rows it carries $43.2$ sector boundaries on average. How many are left after $400$ rows?

Scaling. If boundaries wander a distance of order $L^{\zeta}$, then sectors narrower than that have had time to be squeezed out, and the typical surviving sector is about that wide. On a front of fixed width the number of boundaries falls as $L^{-\zeta}$.

Prediction. $43.2\times(400/32)^{-2/3}=43.2/5.39=8.0$. Ordinary random walks, with exponent $1/2$, would leave $43.2\times12.5^{-1/2}=12.2$.

Measurement. The average over $40$ seeded strips is $8.0\pm0.3$ boundaries. A straight-line fit of the logarithm of the mean against the logarithm of distance, from $32$ to $400$ rows, has slope $-0.669$. Fitting each strip on its own gives $-0.682\pm0.020$, which is about $9$ standard errors away from $-1/2$.

Answer. About eight boundaries, so about eight sectors on the periodic strip. The figure below shows the whole curve.

$$
\ell(L)\sim L^{\zeta}\ \Longrightarrow\ N_{\mathrm{b}}(L)\propto L^{-\zeta}=L^{-2/3}
$$

Typical sector width and number of boundaries on a straight front after it has grown a distance $L$.

Mean number of sector boundaries across a two-colour Eden strip of width $1000$, averaged over $40$ seeded colonies, against the distance grown (both axes logarithmic). The solid guide falls as distance to the power $-2/3$, the dashed one as $-1/2$, which ordinary random walks would give.

Source: O. Hallatschek · K. S. Korolev

## Round colonies keep some sectors

On a straight front the coarsening never stops: given enough distance, one colour takes the whole front. A round colony is different. Its circumference grows in proportion to its radius, while boundaries wander only as $L^{\zeta}$ with $\zeta<1$. After a crossover the inflation wins, neighbouring boundaries drift apart faster than they wander together, and the surviving sectors become permanent.

Balancing the two effects gives the scaling on the right for the number of surviving sectors $N_{\mathrm{s}}$ and the radius $r_0$ of the initial drop. With diffusive boundaries, $\zeta=1/2$, this is the square-root law that Hallatschek and Nelson derived for their model. With $\zeta=2/3$ it becomes the cube root. The Lab has not been benchmarked against this relation.

$$
N_{\mathrm{s}}\sim r_0^{\,1-\zeta}
$$

Number of sectors that survive in a round colony grown from a homeland of radius $r_0$.

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

## Try it in the Lab

### [Watch sectors form and die off](https://sandboxphysics.com/labs/range-expansion-sectors/)

1. Choose Two colours split into sectors. Predict whether the number of sectors keeps falling as the round colony grows, then watch the sector count.
2. Choose A straight front loses colours. Note the sector count when the front has grown about fifty cells, and predict it at two hundred cells from the two-thirds law before reading it.
3. Switch the colony shape between round and straight with the same colours. Explain the difference in the final sector count with the growth of the circumference.

## Exercises with explained answers

1. A straight front has $100$ sectors after growing $20$ rows. Predict the number after $160$ rows with $\zeta=2/3$ and with ordinary random walks.

   Show the answer

   The distance grows by a factor of $8$. With $\zeta=2/3$, $100\times8^{-2/3}=25$. With $\zeta=1/2$, $100\times8^{-1/2}=35$. Superdiffusive boundaries remove sectors faster.

2. The experiment fits $2\zeta=1.31\pm0.1$. Convert this to $\zeta$ with its uncertainty. How many uncertainties away is the random-walk value?

   Show the answer

   $\zeta=0.655\pm0.05$. The random-walk value $0.5$ lies about $3$ uncertainties below it, while $2/3$ is well inside.

3. By what factor does the number of surviving sectors in a round colony change when the radius of the starting drop is doubled, for $\zeta=2/3$ and for $\zeta=1/2$?

   Show the answer

   The number scales as $r_0^{\,1-\zeta}$. Doubling $r_0$ multiplies it by $2^{1/3}=1.26$ for $\zeta=2/3$ and by $2^{1/2}=1.41$ for $\zeta=1/2$. A wider homeland helps, but much less than in proportion.

[Previous: The coalescent traces a sample back to its common ancestor](https://sandboxphysics.com/stories/evolution-statistical-physics/ancestry-and-the-coalescent/) [Next: Does a big sector mean selection? Opening angles and luck in colonies](https://sandboxphysics.com/stories/evolution-statistical-physics/does-a-big-sector-mean-selection/)

## Sources and what we recomputed

1. [K. S. Korolev, M. Avlund, O. Hallatschek and D. R. Nelson, Genetic demixing and evolutionary forces in the one-dimensional stepping stone model, arXiv:0904.4625v2 (2011)](https://arxiv.org/abs/0904.4625v2)

   Version 2, full text. Section II, Population genetics in well-mixed populations: genetic drift as random sampling of a finite population, the heterozygosity as the chance that two sampled individuals differ, and its loss with time.

2. [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.

3. [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.