System size and boundaries determine the patterning dynamics of attracting active particles

Alexander Aulehla; Anna Erzberger; Jan Rombouts; Michael L Zhao

REVIEW 4 major objections 5 minor 55 references

System size and boundaries determine the patterning dynamics of attracting active particles

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper shows that confinement size controls not just where patterns form, but how long they take: beyond a critical length, slow multi-peaked transient states cause an exponential delay, and mouse-cell experiments confirm the predicted

desk verdict Solid analytical core on bounded-domain nonlocal advection-diffusion; the experimental confirmation is suggestive but rests on a neutral-boundary assumption that is not directly measured. read the letter →

arxiv 2509.08533 v1 pith:6C6SHRU5 submitted 2025-09-10 nlin.PS cond-mat.softnlin.AOphysics.bio-phq-bio.CB

Jan Rombouts , Michael L Zhao , Alexander Aulehla , Anna Erzberger This is my paper · ORCID

classification nlin.PScond-mat.softnlin.AOphysics.bio-phq-bio.CB MSC 35B3635K5792C15

keywords nonlocaladvection-diffusionpatternformationconfinementactiveparticlesbifurcationanalysisslowtransientsmulticellularaggregationeigenvalueveering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the size and boundaries of a confined space change how a collective of self-attracting particles forms patterns—not just where the patterns sit, but how long they take to form. It studies a minimal nonlocal advection-diffusion model in which particles attract each other and also feel the walls, and shows that increasing the domain size triggers successive bifurcations that add unstable multi-peaked states. Trajectories pass near these saddles and leave slowly, so the time needed to reach the final polarized pattern, and its shot-to-shot variability, grow exponentially with system size. The boundary interaction determines which final pattern appears: repelling walls give a central peak, strongly attracting walls give boundary peaks, and weakly attracting walls give asymmetric polarized states. Experiments with mouse embryo cells on microprinted bars show the predicted shift: larger bars display more symmetric and multi-peaked transient patterns at early times, while bars below roughly 160 µm suppress multi-peak formation.

What carries the argument

The central object is the exact eigenvalue reduction for exponential interaction kernels: an eigenfunction u of the nonlocal problem (∂xx−ᾱ∂xA)u=ωu with no-flux boundary conditions also solves the constant-coefficient ODE u''''+(2ᾱ−1−ω)u''+ωu=0. This reduction turns the nonlocal stability problem into algebra and yields the instability locus e^{ℓz}=(−1)^n(z−1)/(z+1) with z=√(1−2ᾱ), giving the size thresholds at which each spatial mode destabilizes. The second element is the sequence of pitchfork bifurcations that adds unstable saddle steady states; their unstable manifolds carry the slow transients that delay polarization.

What would settle it

Measure the time to reach a polarized profile in the same cell assay across a continuous range of bar lengths spanning the predicted first instability; if the mean and variance of that time do not grow exponentially once the bar exceeds the threshold, or if no multi-peaked transients appear, the saddle-controlled slowing mechanism is falsified. A second check is to vary boundary adhesiveness: the model predicts neutral walls polarize fastest, so making walls more attractive or repulsive should both slow polarization, and a flat timing curve would rule out the proposed boundary mechanism.

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Extended reading notes

Core claim

For a generic class of nonlocal advection-diffusion equations describing self-attracting active particles in a confined one-dimensional domain, the paper derives that boundaries and system size are active controls of the dynamics. Assuming interaction kernels that are equal exponentials for particle-particle and particle-boundary forces, every eigenfunction of the linearized stability problem also satisfies a constant-coefficient fourth-order ODE, which yields algebraic equations for the eigenvalues and exact instability curves e^{ℓz}=(-1)^n(z-1)/(z+1), z=√(1−2ᾱ). These curves show that successive spatial modes become unstable as the domain length increases, each transition adding an unstab

Load-bearing premise

The quantitative predictions assume the particle-particle and particle-boundary forces have the same exponential range, and the experimental match assumes the wall is neutral (β=αρ0) with α fitted from the same data, so any departure from this interaction balance shifts the predicted thresholds and transient fractions.

Editorial extensions

If this is right

Systems confined below the first critical length polarize quickly and robustly; above it, polarization slows because multi-peaked saddle states trap trajectories.
The mean and standard deviation of the polarization time increase exponentially with system size, so larger systems are both slower and less reproducible in their patterning time.
Wall interaction type selects the final pattern: repelling boundaries yield a central peak, strongly attracting boundaries yield boundary peaks, and weakly attracting boundaries yield asymmetric polarized states.
The number of unstable multi-peak states is controlled by system size and attraction strength, so the sequence of transient patterns can be predicted from the analytical instability thresholds.
The eigenvalue-to-ODE reduction and bifurcation approach extend to other nonlocal kernels, applying to neural fields, ecological dispersal, and kernel-based reaction-diffusion models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the saddle-induced slowing is generic, tissues that grow across the critical size should suddenly display long, variable patterning times—an effect that could matter for interpreting developmental timing.
The exact reduction for exponential kernels likely extends to finite sums of the form s^m e^{-s}, yielding closed-form thresholds for more complex interaction kernels without further approximation.
Varying wall adhesiveness at fixed domain size would isolate the boundary's role: the model predicts neutral walls polarize fastest, and repulsive or strongly attractive walls both slow the approach to the final state.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take. The paper has a genuinely useful analytical core: exact eigenvalue reduction for exponential kernels on a bounded domain, closed-form instability curves (Eq. 12), and a clean demonstration that confinement generates a sequence of pitchfork bifurcations whose saddle branches control transient dynamics. That part is new and sound. The experimental part is suggestive but not conclusive: the match between simulation and cell patterns in Fig. 5b hinges on the neutral-boundary assumption beta = alpha*rho0, which is imposed rather than measured, and alpha is fit to the same data. I'd send this to review rather than desk reject, but I'd ask for robustness checks.

What the paper does well: the analytical machinery is real. The reduction of the nonlocal eigenproblem to a fourth-order ODE for exponential kernels is elegant, and the instability curves (Eq. 12) are exact within the stated assumptions. The numerical bifurcation analysis is careful, and the phase-plane picture of saddle-guided transients (Fig. 4) is a nice conceptual contribution that could generalize beyond this specific model. The exponential growth of polarization time is presented as a simulation result over a limited range of system sizes; that is a minor weakness since no analytical scaling argument is given, but it is clearly labeled as numerical.

The soft spots are real but addressable. First, the entire theoretical structure—uniform steady state, eigenmodes, saddle branches, and the transient pattern classification in Fig. 5b—is derived for the neutral boundary beta = alpha*rho0. This is not a small technical restriction: Fig. 3 shows that different beta gives different final states (center peak, boundary peaks, polarized), so the predicted distribution of transient patterns is tied to that neutral choice. The stress-test note is right that beta is not measured; if the real boundary interaction deviates, the experimental agreement could be coincidence or due to beta compensation while fitting alpha. Second, alpha is estimated from the same experiments used for validation, which introduces circularity, and the experimental sample sizes are small (N ranges from 2 to 26 per bar length) with no error bars. Third, the assumption that particle-particle and particle-boundary kernels have equal shape and range is stated but not tested.

These are not fatal flaws. The analytical results stand on their own, independent of the biology. The experimental claim is conditional, but the authors are appropriately cautious elsewhere—they call the patterns "transients" and the comparison "signatures." I would like to see the authors test the sensitivity to beta, report error bars, and either release the experimental data or point to a public version. The patterns themselves are from a companion paper (Zhao et al., in prep.), so some details are legitimately deferred.

Who should read this: anyone working on nonlocal aggregation models, active matter on finite domains, or biological self-organization on micropatterns. It deserves a serious referee—the theoretical payoff is enough, and the experimental claim, while not airtight, is worth scrutiny. I would accept it for peer review and push for a revision that makes the beta dependence explicit.

Referee Report

4 major / 5 minor

Summary. This Letter studies pattern formation in a one-dimensional nonlocal advection-diffusion model of self-attracting active particles confined by boundaries. For exponential interaction kernels of equal range for particle-particle and particle-boundary coupling, the authors derive an exact fourth-order ODE reduction of the linear-stability eigenproblem under the neutral boundary choice β=αρ0, yielding analytic instability curves in the (ℓ,α) plane and accurate perturbative thresholds. Numerical bifurcation analysis shows that stable polarized states coexist with a hierarchy of unstable saddle branches that appear as the system size grows; these saddles guide slow transients, producing exponentially growing polarization times. Experiments on primary mouse cells patterned on rectangular bars are used to fit α from early-time data and then compared with model simulations for the distribution of transient pattern types across system sizes. The authors claim the experimental data confirm the predicted size-controlled patterning dynamics.

Significance. If the theoretical results stand, the paper makes a valuable conceptual contribution: confinement size controls not only whether patterns form but also the dynamical pathway and timescale of patterning, via saddle-induced slow transients. The exact eigenvalue reduction for exponential kernels is elegant and provides a rare analytical window into nonlocal problems on bounded domains; the instability curves of Eq. (12) and the perturbative thresholds appear internally consistent and are checked against numerics. The availability of code and the explicit comparison with experiments are also strengths. However, the experimental confirmation is currently much weaker than the theoretical core: it relies on an unmeasured boundary-interaction strength, on a calibration/validation using the same dataset, and on a small number of replicates at the sizes that carry the main claim. The theoretical part is likely to be of lasting value; the experimental claim needs substantial reframing or additional evidence.

major comments (4)

[Experimentally observed patterns... / Fig. 5] The experimental confirmation is built on the neutral-boundary assumption β=αρ0, which is imposed rather than measured. The analytical saddle-branch analysis (Eqs. (7)-(12), Fig. 4) is derived only for this case, and the simulations in Fig. 5b set β=αρ0. Figure 3 shows that the stable pattern class depends strongly on β: β<0 gives a center peak, β>α gives boundary peaks, and only 0<β<α gives polarized states. A non-neutral β would therefore change the predicted fractions of 'polarized', 'middle peak', and 'two peaks' categories. The fit of α from the early experimental timepoints does not constrain β, so the agreement in Fig. 5b does not provide a stand-alone test of the saddle-induced slow-mode mechanism. Please estimate β independently, or demonstrate that the Fig. 5b predictions are robust to β over an experimentally plausible range, or explicitly reframe the experimental claim as con
[Fig. 4a (lower panel) and Conclusions] The exponential dependence of the polarization time τ_pol on system size is one of the two headline results (abstract and Conclusions), but it is asserted from simulations over a narrow range of ℓ without quantitative support. The lower panel of Fig. 4a appears to cover roughly ℓ∈[2,5] (up to the second saddle-branch threshold), and no fitted exponent, confidence interval, or comparison with a predicted scaling is reported. Over such a short range, an apparent exponential law can be a generic finite-size effect. Please report the fitted exponential rates for the mean and standard deviation, the number of ℓ values used, and a check that the rate is consistent with the number of saddle branches or the associated eigenvalue gaps.
[Fig. 5b-c] The statistical support for the size-controlled suppression of multi-peak patterns is weak. In Fig. 5b, the two smallest domain sizes, which underpin the claim that confinement below 160 µm suppresses multiple peaks, appear to have only N=3 and N=2 replicates (assuming the N labels are in the same order as the bar lengths). In addition, the comparison time is set by mapping 16 h to 2.4τ, but the fitted value of τ is not reported; the predicted pattern fractions are time-dependent, so the comparison is sensitive to this mapping. Please provide the fitted τ and its uncertainty, and show how the fractions in Fig. 5b change if the classification time is varied within the uncertainty.
[General: scope of the experimental claim] Figure 5 tests only the spatial pattern-type distribution at a single timepoint. It does not directly test the predicted exponential growth of the polarization time or the slow transient dynamics, which are the mechanism-specific predictions of the paper. The title and abstract claim that 'experimental measurements confirm our predictions' about patterning dynamics. If the experimental evidence is to support the dynamical claim, a time-resolved observable (e.g., the time at which the dominant mode emerges, or a comparison of τ_pol across sizes) is needed. Otherwise the conclusions should be limited to 'the observed spatial pattern statistics are consistent with the model under the neutral-boundary assumption'.

minor comments (5)

[Equation (9)] The perturbative expansion in Eq. (9) does not explicitly identify the small parameter. Please state that ar α is assumed small and define the regime of validity of the approximation beyond the statement that it works for ω_n<0.
[Figure 2] The inset in Fig. 2a, which illustrates the eigenvalue veering/cross-over, is too small to read. Please enlarge it or provide a separate panel.
[Order parameter (Fig. 3)] The definition of μ(ρ)=A(ρ)||ρ-ρ0||^2 is confusing because A(ρ) is already an integral functional of ρ. Rewriting A as a separate functional and then defining μ as its product with the L2 norm would improve clarity.
[References] Reference [34] is cited as 'in preparation' for the experimental system, and the full imaging/classification methods appear only in the supplementary material. Since Fig. 5 depends on these methods, please provide more details in the main text or update the reference if a preprint is available.
[Figure 3c caption] The dashed lines in Fig. 3c are described as 'multiple unstable multi-peak states' but are not defined as saddle-node or pitchfork branches. Please label the bifurcation types or refer explicitly to the supplement.

Circularity Check

1 steps flagged · score 4.0 of 10

The theoretical derivation is self-contained, but the experimental 'confirmation' is an in-sample calibration using a fitted interaction strength and an unmeasured neutral boundary, so the confirmation is partially circular.

fitted input called prediction [Experimental section, Fig. 5a-b and main text before 'In particular, we find...']
"By fitting the linearized model to the initial timepoints of these experiments, we estimated an interaction strength of roughly α≈20 (Fig. 5a, Fig. S6). Next, we examined whether using this parameter value, the theoretical distribution of transient patterns corresponds to experimentally observed intermediate patterns."

The model's key interaction strength α is estimated from the early-time portion of the same experimental time series. The 'theoretical distribution of transient patterns' is then generated by simulating the model with that fitted α and compared against the later-time portion of the same experiments. This is an in-sample consistency check, not an out-of-sample prediction: the later pattern distribution is a forward evolution of a model already calibrated to the same system's early dynamics. The abstract's claim that experimental measurements 'confirm our predictions' overstates the independence of this comparison. The qualitative size-dependent increase in multi-peaked patterns is not directly encoded in the single fitted value of α, so the central theoretical claim retains some independent

full rationale

The analytical core of the paper—the eigenvalue reduction (Eqs. 7–12), the exact ODE for exponential kernels (Eq. 11), the instability curves (Eq. 12), and the saddle-branch bifurcation sequence (Fig. 4)—is derived from the stated model (Eq. 5) without using experimental data. The references to the authors' own supplemental material [22] provide technical algebra for these derivations, but they are parameter-free and do not incorporate the target results, so they are not load-bearing circularity. The theoretical prediction of size-controlled slow modes and exponentially increasing polarization time is an independent mathematical consequence of the model under the neutral-boundary assumption β=αρ0. The circularity concern lies in the experimental confirmation: α is fitted from the same dataset, and β=αρ0 is imposed rather than measured, while Fig. 3 shows that different β values produce qualitatively different pattern classes. Thus the match in Fig. 5b is a calibrated, in-sample comparison, not an independent verification. This weakens the 'confirm' claim but does not make the central derivation circular. Score 4 reflects this partial circularity in the experimental confirmation while acknowledging that the theoretical result stands on its own.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central theoretical result rests on a specific nonlocal model with equal-range exponential kernels and on the neutral boundary choice for analytic tractability; the experimental confirmation uses one fitted interaction strength alpha, calibrated on the same data. No new physical entities are postulated.

free parameters (3)

Interaction strength alpha = alpha approx 20
Estimated by fitting the linearized model to early experimental density evolution (Fig. 5a, S6), then used in simulations for the experimental comparison.
Mean density rho0 = 0.4
Chosen for simulations, presumably from experimental average density; not independently fitted or varied in the reported comparison.
Boundary interaction beta = beta = alpha*rho0 (neutral)
Set to the neutral value in the analytical linear stability and in the experimental simulations; other values are explored numerically but the experimental comparison uses this choice.

assumptions (5)

domain assumption The dynamics follow the nonlocal advection-diffusion equation (5) with volume-filling mobility rho(1-rho) and exponentially decaying interaction kernels.
The model is assumed to capture cell/particle aggregation; it is not derived from first principles for the specific mouse cell system.
ad hoc to paper Particle-particle and particle-boundary interactions have kernels of equal shape and range sigma.
Stated before Eq. (5); this equal-shape assumption enables the exact ODE reduction but is a simplifying modeling choice.
ad hoc to paper The boundary interaction is neutral, beta = alpha*rho0, for the linear stability analysis and for the experimental simulations.
The neutral choice allows linearization around a uniform steady state (Eq. 7); for other beta values the paper gives only numerical results.
domain assumption The external material defining the boundary is constant in time (phi independent of t).
Stated in the text after Eq. (5); the boundary is treated as static, which is reasonable for the experiments but remains an assumption.
standard math The first-order perturbative approximation (Eqs. 9-10) is valid for small effective interaction strength.
Used to estimate the instability thresholds; the paper checks the approximation against exact numerics in the omega_n < 0 regime, finding good agreement for thresholds.

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Pith. "Pith review of System size and boundaries determine the patterning dynamics of attracting active particles." pith.science (2026). https://pith.science/paper/6C6SHRU5

@misc{pith2026250908533,
  author       = {Pith},
  title        = {Pith review of: System size and boundaries determine the patterning dynamics of attracting active particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6C6SHRU5}},
  note         = {Machine review of arXiv:2509.08533}
}

read the original abstract

Pattern formation often occurs in confined systems, yet how boundaries shape patterning dynamics is unclear. We develop techniques to analyze confinement effects in nonlocal advection-diffusion equations, which generically capture the collective dynamics of active self-attracting particles. We identify a sequence of size-controlled transitions that generate characteristic slow modes, leading to exponential increase of patterning timescales. Experimental measurements of multicellular dynamics confirm our predictions.

Figures

Figures reproduced from arXiv: 2509.08533 by the authors.

**Figure 1.** c shows simulations of Eq. (5) with no-flux boundary conditions, illustrating how spontaneous patterns form in this system. Conditions for patterning On periodic or infinite domains, patterns form spontaneously from a homogeneous particle distribution if [22, 23] αρ0(1 − ρ0) > 1 2 . (6) Thus, since α represents the relative strength of attraction compared to diffusion, patterning is possible if attraction overco… view at source ↗

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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