Sectors at an expanding edge split a growing colony into colours

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.

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.

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.

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