L4 · Chapter 7

Reading “Neither pulled nor pushed” and checking its drift claim

Sandbox Physics · 2026-10-30

Birzu, Hallatschek and Korolev claim that waves of growth and dispersal come in three classes, not two, and that the middle class keeps far less genetic diversity than a pushed wave should. This chapter is a guide to reading the paper with the Lab beside you. It shows where each claim sits, which quantity carries it, and which numbers we could reproduce, which we could not, and why.

The claim in one paragraph

Fluctuations in a pulled wave fall only logarithmically with the number of individuals per site, $N$ , because a few individuals at the tip control the wave. A naive expectation is that pushed waves average fluctuations like any large population, so their fluctuations should fall as $1/N$ . The paper finds that this holds only for fully pushed waves. In semi-pushed waves, with $1<v/v_{\mathrm F}<3/(2\sqrt2)$ , the front wandering and the rate of diversity loss both fall as a power of $N$ with an exponent between $0$ and $-1$ .

The evidence comes in two parts: simulations of a stepping-stone model across cooperativity and population density, and a perturbation theory that explains where the $1/N$ scaling breaks down. The claim of universality, that the boundary depends only on the speed ratio, rests on three different growth laws collapsing onto one curve.

Source: G. Birzu

Where to look first

Read Eq. (3) and Fig. 1 together: the growth law with cooperativity $B$ , and the speed rising above the Fisher speed beyond $B=2$ . The Methods section gives the simulation, Eq. (11), as dispersal followed by growth and multinomial resampling in each deme. That is the model the Lab runs.

Fig. 5 carries the claim about diversity. Panel A shows the heterozygosity $H$ decaying exponentially, panel B its rate $\Lambda$ against $N$ on log-log axes, and panel C the fitted exponent $\alpha_h$ against $B$ . Eqs. (6) and (7) give the boundary between the two pushed classes. The supplementary information holds the derivations and the closed forms.

H(t)=\bigl\langle 2f(1-f)\bigr\rangle\sim e^{-\Lambda t},\qquad \Lambda\propto N^{\alpha_h}

The measured quantity: average heterozygosity at the front, its decay rate, and the exponent that classifies the wave.

Source: G. Birzu

How the rate is measured

The supplementary section on the duration of simulations is short and worth reading closely. Heterozygosity is averaged over the demes of the simulation box and then over $1000$ independent runs. The fit of $\ln H$ against time ends at the last time when at least $50$ runs still hold both alleles. Its start is chosen to maximise the quality of the fit, with a window of at least $1000$ generations. The exponents are fitted only for $N>10^4$ , because the power laws are asymptotic.

Two consequences matter for a reader. The early, non-exponential part of the decay is excluded by design. And a fitted exponent over a finite range of $N$ need not equal its asymptotic value: Fig. 5B shows pulled waves with an effective exponent near $-0.33$ although the theory gives a logarithmic law, which corresponds to an exponent of zero. For front wandering, van Saarloos’s review records the same kind of law, a front diffusion constant falling as $1/\ln^3N$ , and notes that Brunet and Derrida needed a special method to reach large enough $N$ to see it.

Source: G. Birzu · W. van Saarloos

Reading Eq. (5)

Eq. (5) is the perturbative rate of diversity loss. The prefactor $1/N$ is the central-limit scaling. The ratio of integrals weighs each position in the front: the denominator counts ancestry, with the same weight $\rho^2e^{v\zeta/D}$ as in chapter 6, and the numerator counts where drift acts.

At the tip $\rho\sim e^{-k\zeta}$ , so the numerator integrand behaves as $e^{(2v/D-3k)\zeta}$ . It converges only if $2v/D<3k$ , which holds when $v/v_{\mathrm F}$ exceeds $3/(2\sqrt2)$ . Below it the integral diverges, a cutoff at low density is needed, and the cutoff brings in a power of $N$ other than $-1$ . For the cooperative growth law with $B>4$ , SI Eq. (S122) gives the integral in closed form.

\Lambda=\frac{1}{N}\,\frac{\displaystyle\int_{-\infty}^{\infty}\gamma_f\,\rho^3(\zeta)\,e^{2v\zeta/D}\,d\zeta}{\Bigl[\displaystyle\int_{-\infty}^{\infty}\rho^2(\zeta)\,e^{v\zeta/D}\,d\zeta\Bigr]^2}

Eq. (5) of the paper for the rate of diversity loss, with

\rho

the normalised density profile in the frame moving with the front.

\Lambda=\frac{\gamma_f}{4\pi N}\sqrt{\frac{g_0B}{2D}}\,(B+4)\tan\frac{2\pi}{B}\qquad(B>4)

SI Eq. (S122): the same rate for the cooperative growth law, valid for fully pushed waves.

Source: G. Birzu

Worked example: reproducing one rate

Problem. Take $B=10$ , $g_0=0.01$ , $D=0.125$ , $\gamma_f=1$ and $N=300$ . Predict $\Lambda$ and compare with the Lab model.

Closed form. $\sqrt{0.01\times10/0.25}=0.632$ , $B+4=14$ and $\tan(2\pi/10)=0.727$ . The product is $6.43$ , and dividing by $4\pi\times300=3770$ gives $\Lambda=1.71\times10^{-3}$ per generation. Evaluating Eq. (5) with the lattice front profile instead of the continuum one gives $1.70\times10^{-3}$ .

Simulation. $320$ seeded runs were warmed up for $1500$ generations, the two alleles were then remixed $50$ : $50$ , and the heterozygosity was averaged over the demes from ten behind to three ahead of the front. A straight-line fit of $\ln H$ between generations $1000$ and $2100$ gives $\Lambda=(1.68\pm0.20)\times10^{-3}$ , with a jackknife error over runs. The ratio to the perturbative value is $0.99$ .

Second point. At $N=1000$ the measurement is $(4.3\pm0.7)\times10^{-4}$ against $5.1\times10^{-4}$ , $1.1$ standard errors low. Between the two sizes the rate falls by $3.9\pm0.8$ , against $3.33$ for an exact $1/N$ law.

Answer. For a fully pushed wave the Lab model reproduces both the size of $\Lambda$ and its $1/N$ scaling within the errors of a few hundred runs.

Source: G. Birzu

What this reproduction does not show

For pulled waves the Lab finds $\Lambda=(1.96\pm0.21)\times10^{-3}$ at $N=100$ and $(0.92\pm0.11)\times10^{-3}$ at $N=1000$ . The ratio $2.1\pm0.3$ is far from the factor of $10$ that a $1/N$ law would give, but also below the $3.4$ of the asymptotic $\ln^3N$ law. As an effective exponent it is $-0.33\pm0.07$ , close to the value in Fig. 5B, though our range of $N$ is lower than the paper’s, so the agreement may be partly coincidental.

The Lab run differs from the paper in several ways: a few hundred runs instead of a thousand, a window ending at $30$ live runs instead of $50$ , heterozygosity over fourteen demes near the front rather than the whole box, and $N$ well below $10^4$ . We did not measure the semi-pushed exponents, the front diffusion constant or the collapse of three growth laws in Fig. 6. Those are the paper’s central new results, and checking them would need much larger runs. A follow-up by Birzu, Matin, Hallatschek and Korolev reports that dispersal which increases with density suppresses drift at the edge, so the classes depend on how individuals move as well as how they grow. The Lab keeps dispersal fixed.

Source: G. Birzu · G. Birzu

Try it in the Lab

See where the central-limit scaling holds, and where it fails

Choose Fully pushed: the bulk shares ancestry and watch the diversity readout fall. Predict how much more slowly it falls if the carrying capacity rises from one thousand to about three thousand.
Make the change and compare. Then choose Pulled: the tip wins and make the same change. Predict whether the slowdown is as large.
Set the cooperativity to three. Explain, with the tip of the front, why the paper needs a cutoff to estimate drift in this wave.

Exercises with explained answers

Use SI Eq. (S122) to compute $\Lambda$ for $B=10$ and $N=1000$ with the Lab parameters.
Show the answer
The rate scales as $1/N$ , so $1.706\times10^{-3}\times300/1000=5.12\times10^{-4}$ per generation. The Lab measures $(4.3\pm0.7)\times10^{-4}$ .
The supplement quotes the pulled-front transient as $v(t)=v_{\mathrm F}\bigl(1-3/(4r_0t)\bigr)$ . Show that this is the same as $v^*-3/(2\lambda^*t)$ from chapter 6 for the continuum equation.
Show the answer
For the continuum, $v^*=v_{\mathrm F}=2\sqrt{Dr_0}$ and $\lambda^*=\sqrt{r_0/D}$ . Then $v_{\mathrm F}\cdot3/(4r_0t)=3\sqrt{D}/(2\sqrt{r_0}\,t)=3/(2\lambda^*t)$ . The two formulas agree.
A rate falls by a factor of $2.12$ when $N$ grows tenfold. What effective exponent $\alpha_h$ does that imply, and what would a $\ln^3N$ law give between $N=100$ and $1000$ ?
Show the answer
$\alpha_h=-\log_{10}2.12=-0.33$ . A $\ln^3N$ law gives a ratio $(\ln1000/\ln100)^3=1.5^3=3.4$ , an effective exponent of $-0.53$ , even though its asymptotic exponent is zero. Effective exponents over one decade say little about asymptotic laws.

Sources and what we recomputed

G. Birzu, O. Hallatschek and K. S. Korolev, Neither pulled nor pushed: genetic drift and front wandering uncover a new class of reaction-diffusion waves, arXiv:1709.01601v2 (2018)
Version 2, full text. Eq. (2): the Fisher speed. Eq. (3): the cooperative growth law. Eq. (5): perturbative rate of diversity loss. Eqs. (6)-(7): the leading-edge decay and the speed ratio $3/(2\sqrt2)$ that separates semi-pushed from fully pushed waves. Fig. 5: diversity loss in the three classes. Methods, Eq. (11): the simulation model. SI Eq. (S28): ancestry weight; Eq. (S122): closed form of the loss rate; section Duration of simulations and data collection, Eqs. (S149)-(S150).
G. Birzu, S. Matin, O. Hallatschek and K. S. Korolev, Genetic drift in range expansions is very sensitive to density feedback in dispersal and growth, arXiv:1903.11627v1 (2019)
Version 1, full text. Abstract: dispersal that increases with population density suppresses genetic drift at the edge, and one ratio, the expansion speed over the geometric mean of dispersal and growth at the edge, controls the regimes of diversity loss.
W. van Saarloos, Front propagation into unstable states, arXiv:cond-mat/0308540v2 (2003), Physics Reports
Version 2, full text. Section 2.2, Eqs. (24)-(27): the linear spreading speed $v^*$ and steepness $\lambda^*$ from the dispersion relation. Section 2.9.1, Eq. (78): the universal slow approach $v(t)=v^*-3/(2\lambda^*t)$ of a pulled front. Section 7.1, Eqs. (241)-(246): the cutoff correction to the pulled speed in particle models, proportional to $1/\ln^2N$ ; Eq. (248): the front diffusion constant falling as $1/\ln^3N$ .