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Sandbox Physics
Evolution as statistical physics

FR24 · An interactive course

Evolution as statistical physics: drift, ancestry and expanding fronts

Sandbox Physics ·

Whether a new gene spreads depends partly on how good it is and partly on luck. In a finite population, luck alone fixes some genes and loses others, and at the edge of a growing colony it carves the population into sectors of single ancestry. This course treats that luck the way statistical physics treats noise: with sampling, diffusion and travelling waves, each checked against simulations you can run.

Start with drift in a small population Open an experiment
1010010100distance grown, cellssector boundaries
Mean number of sector boundaries across a two-colour Eden strip of width 10001000, averaged over 4040 seeded colonies, against the distance grown (both axes logarithmic). The solid guide falls as distance to the power −2/3-2/3, the dashed one as −1/2-1/2, which ordinary random walks would give.

From a coin-flip population to a published wave theory

The first chapter builds the simplest model of inheritance by random sampling and shows how diversity drains away. The second adds selection and computes exactly how often a favoured mutant still loses. The third turns time around and follows ancestors back until they meet. The fourth moves the population onto a growing edge, where sectors appear and their boundaries wander faster than a random walk. The fifth asks whether a large sector proves that its owner was fitter. The sixth explains why some expanding fronts are pulled by their tip and others pushed from behind, and the last chapter reads the paper that found a third class between them.

You need probability at the level of coin flips and averages, exponentials and logarithms, and a first taste of derivatives. Diffusion equations and travelling waves are introduced when they are first used. FR24

  1. 01 · L1

    Genetic drift lets a neutral gene take over a finite population

    Build the Wright–Fisher model from random sampling, see why a neutral gene wins as often as its starting share, and watch diversity drain away.

  2. 02 · L2

    Fixation probability and time, and why a better mutant often loses

    Add selection to random sampling, solve the Moran model exactly, derive the diffusion formula and test both on thousands of seeded populations.

  3. 03 · L2

    The coalescent traces a sample back to its common ancestor

    Trace a sample of a neutral population back in time, derive the mean age of its common ancestor and see why the last two lines take half the wait.

  4. 04 · L3

    Sectors at an expanding edge split a growing colony into colours

    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.

  5. 05 · L3

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

    Learn how a fitter mutant opens a wedge at a colony edge, how often it survives, and why a neutral colour can still take a large share by luck.

  6. 06 · L3

    Pulled and pushed waves differ in who leads an expanding population

    Derive the Fisher speed, see how cooperative growth makes a front pushed, find where the ancestors sit, and test speeds against lattice and continuum solutions.

  7. 07 · L4

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

    A guide to the 2018 paper on semi-pushed waves: where its claims sit, how it measures diversity loss, and which numbers the Lab reproduces.

Experiments along the way

01Watch chance fix a geneRun hundreds of small populations at once, see neutral and favoured mutants win or vanish, and trace a sample back to its common ancestor.02Grow a colony and count its sectorsLet a colony spread from many coloured founders, watch a few of them take the whole edge, and compare a lucky sector with a favoured one.03Turn a pulled wave into a pushed oneRaise cooperativity in a spreading population and see where its ancestors live move from the tip into the bulk, and how diversity survives.

Every estimate in these experiments comes from seeded simulations and carries its standard error. Exact solutions, diffusion formulas and published exponents are drawn as separate reference lines.

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)

    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. A. Traulsen and C. Hauert, Stochastic evolutionary game dynamics, arXiv:0811.3538v1 (2008), in Reviews of Nonlinear Dynamics and Complexity, Vol. II

    Version 1, full text. Section 1.3.1: the Moran process as a birth-death chain with transition probabilities. Section 1.3.2, Eqs. (1.7)-(1.20): fixation probabilities from a recursion, with the closed form of Eq. (1.20), which reduces to the starting fraction for neutral selection. Eqs. (1.26)-(1.35): mean and conditional fixation times.

  3. N. Berestycki, Recent progress in coalescent theory, arXiv:0909.3985v1 (2009), Ensaios Matemáticos, Sociedade Brasileira de Matemática

    Version 1, full text. Section 2.1.1: the Kingman coalescent, in which each pair of lineages merges at rate one. Theorem 2.3: the Moran genealogy becomes this coalescent after rescaling time. Theorem 2.4: the Wright-Fisher genealogy does too when time is measured in units of NN generations. Theorem 2.8, Eqs. (36)-(38): a neutral allele fixes with probability equal to its frequency, with the diffusion formula for the mean absorption time; the formula after it gives the time conditioned on fixation.

  4. O. Hallatschek and D. R. Nelson, Life at the front of an expanding population, arXiv:0810.0053v2 (2008)

    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⁡(Φ/2)\tan(\Phi/2) for opening angle Φ\Phi. Eq. (46): cos⁡(Φ/2)\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.

  5. 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)

    Preprint, version 1, full text. Fig. 2B and the text around it: the mean-square wandering of 92 domain boundaries grows as L2ζL^{2\zeta} with ζ=0.65±0.05\zeta=0.65\pm0.05. The scaling argument after it gives ζ=1+(χ−1)/z\zeta=1+(\chi-1)/z, which is 2/32/3 for the Eden model.

  6. 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/(22)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).

  7. 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.

  8. 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∗v^* and steepness λ∗\lambda^* from the dispersion relation. Section 2.9.1, Eq. (78): the universal slow approach v(t)=v∗−3/(2λ∗t)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⁡2N1/\ln^2N; Eq. (248): the front diffusion constant falling as 1/ln⁡3N1/\ln^3N.

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