Starved first-stage (L1) larvae of the nematode C. elegans arrest development and, above a density threshold, aggregate into circular clusters of up to roughly 700 μm diameter with roughly uniform ~1 mm spacing that equilibrate within hours. A Keller–Segel-type chemotaxis model with only a self-secreted diffusible attractant fails to reproduce this: simulated aggregates have no maximum size, non-uniform spacing, and coarsen toward a single aggregate. Adding a second, roughly ten-times-longer-range repellent signal numerically reproduces the observed pattern, but the actual biological mechanism remains unidentified. The open problem is to determine what actually arrests runaway aggregation and selects the observed finite wavelength: identify or decisively discriminate among the candidate interactions (short-range attractant plus long-range repellent, a dual-valence single molecule, a locally destroyed ubiquitous repellent such as oxygen, physical forces, density-dependent slowing, or an extracellular scaffold) with perturbation evidence, and establish in the corresponding aggregation–diffusion system that the identified mechanism stabilizes finite aggregates and selects the observed scale, through validated model–data comparison and/or rigorous analysis.

Background

When Caenorhabditis elegans eggs hatch without food, the first-stage larvae (L1s) arrest their development and disperse in search of food; after two or more days of starvation they instead aggregate, forming circular clusters of several hundred worms (Artyukhin et al. 2015). The behavior requires a small amount of ethanol or acetate in the medium (ethanol must be metabolized, via alcohol dehydrogenase sodh-1, into acetate for de novo fatty-acid synthesis, and some newly made fatty acids are released as glycerophosphoethanolamides), appears only above a worm density of about 1500 worms per cm², and produces aggregates no larger than roughly 700 μm in diameter that equilibrate within hours and are fairly uniformly spaced at about 1 mm — a striking finite-wavelength pattern rather than unbounded clumping.

The classical Keller–Segel model (Keller and Segel 1970) describes aggregation driven by chemotaxis, i.e. directed movement of organisms up gradients of a diffusible chemoattractant that the organisms themselves secrete: a worm-density equation with a diffusive term and a chemotactic drift is coupled to a diffusion–decay–production equation for the signal. In its standard form the model is notorious for blow-up: in two spatial dimensions, aggregation above a critical mass collapses into point concentrations, and on finite domains solutions coarsen toward one aggregate. Extensive variants (volume filling, saturating sensitivity, nonlinear diffusion; see the review by Hillen and Painter 2009) were developed precisely to remove this runaway aggregation.

Avery, Ingalls, Dumur and Artyukhin (2021) formulated a Keller–Segel-type model of L1 aggregation. The worm density ρ\rho obeys a continuity equation with a conservative velocity field v=−∇Vv=-\nabla V plus random movement of strength σ\sigma, and each signal concentration UiU_i diffuses with constant DiD_i, decays at rate γi\gamma_i, and is secreted at a rate proportional to ρ\rho. The potential VV is the sum of a signal-dependent term — a saturating, Weber–Fechner-type (logarithmic-magnitude) function of each signal concentration, with a response threshold α\alpha below which worms do not respond — and a density-dependent close-packing term that caps the density inside aggregates at the physical packing limit (about 28 000 worms per cm²). With only a short-range attractant (range ≈ 100 μm, Da=10−6D_a = 10^{-6} cm² s⁻¹), numerical solutions reproduce sharp, circular aggregates but fail exactly where the classical model fails: aggregates have no maximum size, spacing is non-uniform, and the system keeps coarsening (the authors believe the only true equilibria place nearly all worms in a single aggregate). Adding a repellent with ten times the range (≈ 1 mm, Dr=10DaD_r = 10 D_a, γr=0.1γa\gamma_r = 0.1\gamma_a) produces aggregates of characteristic size and near-hexagonal spacing matching experiments, close to equilibrium, with the repellent range approximately equal to the observed inter-aggregate spacing. The need for a short-range activation combined with long-range inhibition echoes an older two-signal Keller–Segel model of Alzheimer's senile plaques (Luca et al. 2003).

The mathematics of the idealized attraction–repulsion Keller–Segel (ARKS) system, in which the chemotactic drift is a linear combination of attractant and repellent gradients, is partially understood: Liu, Shi and Wang (2013) proved existence of steady-state and time-periodic patterns via bifurcation theory and showed that competitive attraction–repulsion can generate periodic patterns impossible with a single signal, and Jin and Wang (2020) proved global stabilization (convergence to the spatially constant state) when the repulsive strength dominates in an explicit parameter sense. These results, however, concern linear chemotactic sensitivities; the Avery model's thresholded, saturating, density-dependent potential is not covered.

Adjacent collective phenomena in nematodes have competing mechanistic accounts: aggregation and swarming on food share density- and oxygen-dependent behavioral mechanisms (Ding et al. 2019; Demir et al. 2020, where bacteria-driven oxygen decrease slows worms locally so that they accumulate — a transition analogous to motility-induced phase separation, i.e. aggregation caused purely by reduced movement in dense regions); dauer colony formation — dauer is a non-feeding, stress-resistant alternative larval stage of C. elegans — can be explained as a condensation (phase separation) phenomenon with random movement plus slowdown-capture, exhibiting fusion and Ostwald-ripening-like coarsening rather than stable uniform spacing (Chen and Ferrell 2021); and starved worms can deposit a proteinaceous, chemoattractive extracellular material that scaffolds web-like aggregation (Li, Kerr and Kato 2025). Secreted soluble compounds also mediate density-dependent survival and recovery of L1-arrested larvae (Mata-Cabana et al. 2020). None of these identifies the signals or the arrest mechanism for the L1 finite-wavelength pattern.

Problem Statement

Determine the mechanism that arrests runaway aggregation and selects the stable, finite-sized, roughly uniformly spaced aggregates observed in starved C. elegans L1 populations, as posed by the Keller–Segel model of Avery, Ingalls, Dumur and Artyukhin (2021). Concretely: identify experimentally — or decisively discriminate among by elimination — the interactions responsible for the arrest and the ~1 mm wavelength, namely the postulated short-range attractant and roughly ten-times-longer-range repellent, or the alternative mechanisms noted in the source and in later work (a single molecule that repels at low and attracts at high concentration; a ubiquitous repellent locally destroyed by worms, as oxygen acts in social feeding; physical forces such as surface tension; density-dependent slowing as in condensation; an extracellular scaffold as in Li, Kerr and Kato 2025); and establish, in the corresponding Keller–Segel-type aggregation–diffusion system, that the identified mechanism stabilizes finite aggregates and selects the observed scale. Determining the mechanism therefore requires both naming the interaction (with perturbation evidence linking it to the observed aggregate sizes and spacing, e.g. genetic or chemical interventions that shift aggregate size/spacing in the predicted direction) and showing at the model level — via validated numerical model–data comparison (aggregate size distributions, nearest-neighbor spacing statistics, Fourier power spectra) and/or rigorous analysis of the nonlinear system (global boundedness, existence and asymptotic stability of nonconstant finite-aggregate steady states, absence of coarsening or blow-up, wavelength-selection criteria) — that this mechanism accounts for the finite-wavelength pattern. The two facets are distinct and jointly required: neither facet alone resolves the problem. The identification must narrow to a single mechanism — either a positive identification or an elimination that leaves no other candidate standing — and a partial discrimination that leaves two or more candidates open does not resolve it. This facet can rest on published genetic or chemical perturbation evidence, without new experimental data, provided the surviving mechanism is linked to the observed aggregate sizes and spacing by that evidence and every other enumerated candidate class is excluded, either by perturbation evidence or by being inconsistent with the published observations. For that single surviving mechanism, the model-level facet must then be shown — through the model–data comparison and/or analysis above, and beyond a bare restatement of the numerical agreement already reported in the cited source — to stabilize finite aggregates and reproduce the observed scale, i.e. the aggregate-size distribution and nearest-neighbor spacing measured by Artyukhin et al. (2015) (aggregates up to roughly 700 μm in diameter spaced roughly 1 mm apart); selecting some finite wavelength without accounting for that scale is not sufficient. The problem is not narrowed to any single signal chemistry, mechanism class, spatial dimension, or analysis method.

References

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