Elasticity and plasticity of epithelial gap closure

Maryam Setoudeh; Pierre A. Haas

arxiv: 2509.05110 · v1 · submitted 2025-09-05 · ❄️ cond-mat.soft · physics.bio-ph· q-bio.TO

Elasticity and plasticity of epithelial gap closure

Maryam Setoudeh , Pierre A. Haas This is my paper

Pith reviewed 2026-05-18 18:46 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.bio-phq-bio.TO

keywords epithelial gap closurecell intercalationactomyosin cabletissue fluidisationepibolywound healingcontinuum modelplasticity

0 comments

The pith

A minimal continuum model reveals that epithelial gap closure switches between cell intercalation into the gap and deintercalation from the boundary depending on the relative size of the intercalation energy barrier and released energy.

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

The paper builds a minimal continuum model of a circular epithelial gap bounded by a contractile actomyosin cable to examine how plastic cell intercalations interact with tissue elasticity during closure. When the energy barrier to intercalation greatly exceeds the energy released, cells move into the gap and close it. In fluidised tissues where the barrier is low, cells instead move from the boundary into the tissue bulk, with inhomogeneities in the actomyosin cable gaining an emergent mechanical function. This accounts for the observed role of tissue fluidisation in beetle serosa closure and related processes such as epiboly and wound healing.

Core claim

In the minimal continuum model of the closure of a circular gap bounded by a contractile actomyosin cable, the interplay between elasticity and plasticity is governed by the energy barrier Eb to cell intercalation and the energy ΔE released by it: if Eb ≫ ΔE, cells intercalate into the gap to close it, whereas for a fluidised tissue with Eb ≪ ΔE, cells deintercalate from the boundary into the bulk and inhomogeneities of the actomyosin cable acquire an emergent mechanical role.

What carries the argument

The single energy barrier Eb and released energy ΔE for cell intercalation in the minimal continuum description, which sets whether the tissue closes by adding cells at the edge or by ejecting them from the boundary.

If this is right

When Eb greatly exceeds ΔE, cells intercalate into the gap and close it through addition at the edge.
When Eb is much smaller than ΔE in a fluidised tissue, cells deintercalate from the boundary into the bulk.
In the fluidised regime, inhomogeneities in the actomyosin cable play an emergent mechanical role in closure.
The model explains the mechanical contribution of tissue fluidisation to serosa closure in Tribolium and to epiboly and wound healing more generally.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Modulating tissue fluidity might offer a way to steer gap closure outcomes in wound-healing contexts.
The same energy-barrier logic could apply to gap closure in other epithelial systems beyond the circular geometry studied here.
Testing the model on tissues with controlled actomyosin cable tension variations would check whether inhomogeneities indeed dominate only in the low-barrier regime.

Load-bearing premise

A minimal continuum model that uses only one energy barrier Eb and one released energy ΔE for intercalation is enough to capture the main mechanical competition between elasticity and plasticity in real epithelial tissues.

What would settle it

Live imaging of cell trajectories at the gap boundary in an epithelial tissue whose fluidity (and thus Eb relative to ΔE) can be tuned, checking whether cells move into the gap or outward into the bulk during closure.

Figures

Figures reproduced from arXiv: 2509.05110 by Maryam Setoudeh, Pierre A. Haas.

**Figure 2.** Figure 2: FIG. 2. Intercalation dynamics. (a) Discrete model of cell [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Intercalation energetics. (a) Sketch of the energy [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Dynamics of epithelial gap closure. Top row: dynamics for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Dynamics of epiboly for ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

read the original abstract

Epiboly, during which a tissue closes around the surface of the egg, pervades animal development. This epithelial gap closure involves cell intercalations at the edge of the gap. Here, inspired by serosa closure in the beetle Tribolium, we study the interplay between these plastic cell rearrangements and the elasticity of the tissue in a minimal continuum model of the closure of a circular gap bounded by a contractile actomyosin cable. We discover two different closure mechanisms at the tissue scale depending on the energy barrier $E_\text{b}$ to and the energy $\Delta E$ released by intercalation: If $E_\text{b}\gg\Delta E$, cells intercalate into the gap to close it. For a fluidised tissue in which $E_\text{b}\ll\Delta E$, however, cells deintercalate from the boundary into the bulk of the tissue, and we reveal an emergent mechanical role of inhomogeneities of the actomyosin cable. Our work thus explains the mechanical role of tissue fluidisation in Tribolium serosa closure and processes of epiboly and wound healing more generally.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a minimal continuum model that splits gap closure into two regimes set by the ratio of intercalation barrier to released energy, with an emergent role for cable inhomogeneities when the tissue fluidises.

read the letter

This paper builds a stripped-down continuum description of a circular epithelial gap closed by a contractile actomyosin cable. The central finding is that closure proceeds by two distinct routes depending on whether the energy barrier Eb to cell intercalation is much larger or much smaller than the energy ΔE released by the rearrangement. In the high-barrier case cells move into the gap; in the low-barrier fluidised case they move out of the boundary into the bulk, and inhomogeneities along the cable start to matter mechanically. The authors tie this split to observations in Tribolium serosa closure and suggest it organises similar processes in epiboly and wound healing. That mapping and the emergent cable effect are the genuinely new pieces inside the class of continuum epithelial models. The work is honest about its minimal assumptions and does not overclaim quantitative fits. The main limitation is that the model remains very coarse. Intercalation is encoded through a single barrier and energy release whose ratio controls the plastic flow direction, but the continuum limit erases the discrete cell-cable contacts that actually set local force balance at the free edge. It is not obvious that the predicted deintercalation survives once those length scales and steric constraints are restored, which is the point raised in the stress-test note. Without the explicit equations, numerical scheme, or side-by-side comparison to Tribolium data it is hard to judge how sensitive the result is to those choices. The paper is written for people who already work with continuum descriptions of tissue mechanics and want a simple organising principle for when fluidisation helps or hinders closure. A reader who cares about developmental biophysics or wound-healing models will find the two-regime distinction worth thinking about, even if they later add more cell-level detail. I would send it to peer review. The idea is clear, the parameter dependence is testable in principle, and the mechanical account of fluidisation is worth checking against experiments or finer-grained simulations.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a minimal continuum model of circular epithelial gap closure bounded by a contractile actomyosin cable, motivated by serosa closure in Tribolium. Plasticity is encoded via a single energy barrier Eb to cell intercalation and an energy release ΔE upon intercalation. The central result is a bifurcation in closure mechanism: when Eb ≫ ΔE cells intercalate into the gap, whereas when Eb ≪ ΔE (fluidised regime) cells deintercalate from the boundary into the tissue bulk, with inhomogeneities in the actomyosin cable acquiring an emergent mechanical role.

Significance. If the two-regime distinction and the emergent role of cable inhomogeneities survive scrutiny, the work supplies a compact mechanical explanation for how tissue fluidisation modulates gap closure in epiboly and wound healing, linking a simple energy-barrier model to observable cell behaviours at the tissue scale.

major comments (1)

[Model setup] Model setup / continuum limit: the claim that cells deintercalate from the gap boundary when Eb ≪ ΔE is generated inside a coarse-grained description that ties the plastic strain rate to an effective potential whose minimum shifts with the Eb/ΔE ratio. Because the continuum smoothing erases discrete cell–cell and cell–cable contacts that set the local force balance at the free edge, it is unclear whether the predicted direction of plastic flow is robust or an artefact of the coarse-graining. This issue is load-bearing for the fluidised-regime mechanism.

minor comments (1)

[Abstract] The abstract states the qualitative outcomes clearly but does not indicate the explicit form of the energy functional or the constitutive relation for the plastic strain rate; adding one sentence on these points would improve accessibility without lengthening the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading of our manuscript and for identifying a key point concerning the robustness of the continuum description in the fluidised regime. We address this comment below and have revised the manuscript accordingly to strengthen the justification of our approach.

read point-by-point responses

Referee: [Model setup] Model setup / continuum limit: the claim that cells deintercalate from the gap boundary when Eb ≪ ΔE is generated inside a coarse-grained description that ties the plastic strain rate to an effective potential whose minimum shifts with the Eb/ΔE ratio. Because the continuum smoothing erases discrete cell–cell and cell–cable contacts that set the local force balance at the free edge, it is unclear whether the predicted direction of plastic flow is robust or an artefact of the coarse-graining. This issue is load-bearing for the fluidised-regime mechanism.

Authors: We agree that the continuum limit requires careful justification, as averaging necessarily smooths discrete cell–cell and cell–cable interactions. In our model the plastic strain rate is obtained from the derivative of an effective potential whose minima are set by the ratio Eb/ΔE; this potential is constructed by coarse-graining the microscopic energy landscape of intercalation events. When Eb ≪ ΔE the minimum shifts to favour negative plastic strain (deintercalation), reflecting a thermodynamic preference for reducing the number of boundary cells. The contractile cable enters through the boundary stress condition, which remains well-defined after averaging. While local force fluctuations at individual contacts are lost, the net direction of plastic flow is preserved because it is dictated by the global energy balance rather than by any single contact geometry. To make this explicit we have added a paragraph in the Model section together with a supplementary derivation that starts from a discrete energy functional and shows that the sign of the coarse-grained flow is unchanged under spatial averaging. We have also included a brief discussion of the regime of validity of the continuum approximation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; outcomes follow from independent model parameters

full rationale

The paper presents a minimal continuum model in which Eb and ΔE are introduced as independent parameters whose ratio determines the direction of plastic flow (intercalation into the gap when Eb ≫ ΔE versus deintercalation when Eb ≪ ΔE). These regimes and the emergent role of actomyosin inhomogeneities are obtained by solving the model's equations rather than by fitting to data or by any self-referential definition. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior work are invoked to force the central claims; the derivation chain remains self-contained against the stated modeling assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model treats Eb and ΔE as independent control parameters whose ratio determines the closure mode; the circular gap geometry and the contractile cable are modelling choices taken from the biological system. No new particles or forces are postulated.

free parameters (2)

Eb
Energy barrier to intercalation, introduced as a tunable parameter that controls whether cells intercalate inward or deintercalate outward.
ΔE
Energy released upon intercalation, introduced as a second tunable parameter whose magnitude relative to Eb selects the closure mechanism.

axioms (1)

domain assumption The tissue can be described by a minimal continuum model that couples elasticity to discrete plastic rearrangements via an energy barrier.
Invoked in the model setup to justify the continuum approach to cell intercalations.

pith-pipeline@v0.9.0 · 5725 in / 1330 out tokens · 30986 ms · 2026-05-18T18:46:05.731565+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We discover two different closure mechanisms ... depending on the energy barrier Eb to and the energy Delta E released by intercalation: If Eb ≫ Delta E, cells intercalate into the gap ... For a fluidised tissue in which Eb ≪ Delta E, however, cells deintercalate ...
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the elastic energy density ... e(s,t)=C/2[Es^2 + Es Ephi + Ephi^2] f(s,t)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Code is available at zenodo:17062742. Elasticity and plasticity of epithelial gap closure • Supplemental Material • Maryam Setoudeh and Pierre A. Haas Max Planck Institute for the Physics of Complex Systems, N ¨othnitzer Straße 38, 01187 Dresden, Germany Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden, Ger...

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S2 and only diﬀer at a quantitative level: the boundary conditions thus have but a slight eﬀect on the dynamics of epiboly

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20.1, pp

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work page

[1] [1]

Solnica-Krezel, Conserved patterns of cell movements during vertebrate gastrulation, Curr

L. Solnica-Krezel, Conserved patterns of cell movements during vertebrate gastrulation, Curr. Biol. 15, R213 (2005)

work page 2005

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R. E. Keller and J. P. Trinkaus, Rearrangement of en- veloping layer cells without disruption of the epithelial permeability barrier as a factor in Fundulus epiboly, Dev. Biol. 120, 12 (1987)

work page 1987

[3] [3]

K¨ oppen, B

M. K¨ oppen, B. G. Fern´ andez, L. Carvalho, A. Jacinto, and C.-P. Heisenberg, Coordinated cell-shape changes control epithelial movement in zebraﬁsh and Drosophila, Development 133, 2671 (2006)

work page 2006

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S. E. Lepage and A. E. E. Bruce, Zebraﬁsh epiboly: mechanics and mechanisms, Int. J. Dev. Biol. 54, 1213 (2010)

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[5] [5]

Behrndt, G

M. Behrndt, G. Salbreux, P. Campinho, R. Hauschild, F. Oswald, J. Roensch, S. W. Grill, and C.-P. Heisenberg, Forces driving epithelial spreading in zebraﬁsh gastrula- tion, Science 338, 257 (2012)

work page 2012

[6] [6]

Campinho, M

P. Campinho, M. Behrndt, J. Ranft, T. Risler, N. Minc, and C.-P. Heisenberg, Tension-oriented cell divisions limit anisotropic tissue tension in epithelial spreading during zebraﬁsh epiboly, Nat. Cell Biol. 15, 1405 (2013)

work page 2013

[7] [7]

A. E. E. Bruce and C.-P. Heisenberg, Mechanisms of ze- braﬁsh epiboly: A current view, in Gastrulation: From Embryonic Pattern to Form , Current Topics in De- velopmental Biology, Vol. 136, edited by L. Solnica- Krezel (Academic Press, Elsevier, London, England,

work page

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Chap. 11, pp. 319–341

work page

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M. A. Benton, M. Akam, and A. Pavlopoulos, Cell and tissue dynamics during Tribolium embryogenesis revealed by versatile ﬂuorescence labeling approaches, Develop- ment 140, 3210 (2013)

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