FR24 · Evolution and ecology

Range Expansion Sectors: Why a Mixed Colony Grows in Stripes

Why does a colony started from well-mixed colours grow into a few single-colour sectors? Only cells at the edge divide, so the edge drifts. It behaves like a small population: neighbouring lineages crowd each other out, colour boundaries wander until they meet, and the surviving colours form sectors. A faster founder opens a wedge whose angle is set by its speed ratio, but a big sector alone does not prove an advantage.

Grow an Eden-model colony from a mixed homeland. Neutral founders split into sectors at the edge, a faster founder opens a wedge and a rock leaves a shadow.

Sectors at the front $N_{\mathrm{sec}}=6$

Distance the front has grown $\Delta r=50\ \text{cells}$

Largest colour share at the front $f_{\max}=0.617$

Gold founder share at the front $f_{\mathrm{gold}}=0$

Seed a colony and let it grow

Colony shape

Growth advantage of the gold founder

0

s

Neutral founder colours

2

K

Rock radius

0

r_{\mathrm{rock}}

Drag the ring along the colony edge to trace a cell to its founder, or drag the rock and the colony regrows around it.

Sampling and playback

Homeland radius (round colony)

12

r_0

Divisions per frame

50

\dot n

Model, method and sources

$\langle\Delta x^2\rangle\propto\Delta r^{2\zeta},\quad \zeta=\tfrac23;\qquad \cos\frac{\Phi}{2}=\frac{v}{v_{\mathrm{gold}}}=\frac{1}{1+s}$

An occupied cell with an empty neighbour divides at rate one, or one plus the advantage for the gold lineage; the daughter takes a random empty neighbour. Sector counts come from the colours of the outermost cell in each column or ray. The wedge angle uses the speed ratio of pure colonies, which in this model equals one plus the advantage. Boundary wandering on a rough Eden front is expected to be superdiffusive with exponent two thirds.

Hallatschek & Nelson (2008), arXiv:0810.0053v2 · Hallatschek, Hersen, Ramanathan & Nelson (2007), arXiv:0812.2345v1

Read why sectors form

Model assumptions

Cells on a square lattice divide into an empty neighbour; descendants of the gold founder divide faster by the chosen advantage, and daughters keep their colour. Only cells at the edge can still divide, so a few founders end up owning the whole front. Sector boundaries wander; on this rough front they wander faster than a random walk. The dashed wedge is the Huygens construction of Hallatschek and Nelson (2008). Over the distances shown here, the simulated wedge opens more slowly than that construction predicts; the benchmark records quantify the gap. The colony is a lattice model, not a measured bacterial colony.

Sectors left as the front advances

Share of the front held by each founder

More numbers and how they are computed

Wedge opening angle from front speeds: $0$
Gold share expected from the wedge: $0$
Colours still at the front: $0$
Cells in the colony: $0$
Colony shown: $0$

Sectors are counted from the colour of the outermost cell along each ray or column. The wedge prediction uses the speed ratio of pure colonies and ignores the rough edge, so in a colony this size the measured wedge is narrower. The exported file lists the seed, the settings and every recorded front.

Physics tutorial

Only the edge reproduces, so the edge forgets

BackgroundIn a colony growing on agar, cells deep inside run out of room and food; only a thin rim keeps dividing. That rim is a small population moving into empty space, so drift acts on it strongly.

Why it mattersExperiments with fluorescently labelled bacteria and yeast show colonies started from a fine mixture growing into a few large single-colour sectors. The pattern is a record of drift at the front, and it changes how we should read spatial genetic data.

Start with the essentials

Focus question: Why does a colony started from well-mixed colours grow into a few single-colour sectors?
One-sentence intuition: Neighbouring founders compete only along the front, so each boundary between colours wanders as the front advances. When two boundaries meet, a colour is lost. On a straight front the number of sectors falls as $\Delta r^{-\zeta}$ ; on a round colony the growing circumference stops the loss and freezes a finite number of sectors.

Core mathematical model

Boundary wandering

\langle\Delta x^2\rangle\propto\Delta r^{2\zeta},\qquad \zeta=\tfrac12\ \text{(smooth)},\quad \zeta=\tfrac23\ \text{(rough, Eden)}

A random walk has $\zeta=1/2$ . Rough fronts carry boundaries along their bumps, which gives the larger exponent $\zeta=2/3$ predicted for Eden growth and measured as $0.65\pm0.05$ in E. coli colonies.

Huygens wedge of a faster founder

\cos\frac{\Phi}{2}=\frac{v}{v_{\mathrm{gold}}}=\frac{1}{1+s}

If the gold lineage alone would spread faster by the factor $1+s$ , geometric optics gives a wedge of opening angle $\Phi$ . In this lattice model the simulated wedge approaches that angle only slowly.

Common difficulties

A large sector is not proof of an advantage

Typical misconceptionThe biggest sector must belong to the fittest founder.

Better mental modelWith twelve equally fit colours on the straight front of this Lab, the luckiest colour typically ends up holding about two fifths of the edge. Only comparing many colonies, or the shape of the sector boundaries, can separate luck from selection.

Run the experiment

01
Watch two colours split
Let the round colony grow from the start.
What to observe: The fine mixture at the homeland gives way to a handful of sectors whose number stops changing once the colony is large.
02
Straighten the front
Choose A straight front loses colours and watch the sector chart.
What to observe: Sectors keep merging, and the curve follows the rough-front guide more closely than the random-walk guide.
03
Add an advantage
Choose A faster founder opens a wedge, then drag the ring onto the gold sector.
What to observe: The gold sector widens linearly with distance, and every gold cell traces back to the single gold founder.

Range Expansion Sectors: Why a Mixed Colony Grows in Stripes

Only the edge reproduces, so the edge forgets

Core mathematical model

Boundary wandering

Huygens wedge of a faster founder

Common difficulties

A large sector is not proof of an advantage

Run the experiment

Watch two colours split

Straighten the front

Add an advantage

Related experiments and articles

From random sampling to pulled and pushed waves