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

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 $\Phi$.

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.

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 $s=0.2$.

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