Actin driven morphogenesis in hydra

Anirban Sain; Sabyasachi Mukherjee

REVIEW 4 major objections 5 minor 29 references

Actin driven morphogenesis in hydra

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

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Pith's one-line read Tentacle growth in hydra is driven by active actin bundles bending the membrane into tubes, and the head's +1 topological defect plays no positive role in where the tubes emerge.

desk verdict The paper's central claim about head-defect insensitivity is built into the model's rigid inactive head patch; the fluidity and threshold results are more solid, but the manuscript needs major rework. read the letter →

arxiv 2509.08655 v1 pith:EZKKYOYH submitted 2025-09-10 cond-mat.soft physics.bio-ph

Sabyasachi Mukherjee , Anirban Sain This is my paper · ORCID

classification cond-mat.softphysics.bio-ph MSC 92C1592C3782B80 PACS 87.17.Aa

keywords hydramorphogenesisactinnematicordertopologicaldefectsmembranetubulationactiveMonteCarlosimulationWnt/beta-cateninsignalingtissuefluiditymechanics

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

This paper tries to establish that hydra's tentacles are not placed by the +1 topological defect in the actin nematic field at the head, as earlier work suggested, but by local signaling hotspots that drive actin bundles to polymerize and mechanically bend the epithelial membrane into tubes. In a non-equilibrium simulation of a fluid membrane bearing a nematic actin field, tubular tentacles form with an aster-like +1 defect at the head, with a ring-like +1 defect, or with no defect at all — the defect is, at most, a passive placeholder. Two ingredients carry the growth: a Hamiltonian term giving the growing actin bundle a preferred curvature perpendicular to its axis (c⊥), and membrane fluidity delivered by link-flip moves — without link flips, tubes do not grow at all. The result matters because it turns tentacle formation from a chemical-morphogen or defect-driven story into a mechano-chemical one where active actin polymerization does the morphogenetic work, and it explains why beta-catenin over-expression makes tentacles grow all over the body.

What carries the argument

The central mechanism is the c⊥ term in the modified Helfrich–Canham Hamiltonian: a preferred curvature perpendicular to the local nematic (actin) director, which models a growing actin bundle pushing the membrane into a tube of radius about 1/c⊥ with the nematic aligned axially. Around it the model layers non-equilibrium conversion rates between 'active' and 'inactive' vertices — active vertices stand for Wnt3/β-catenin signaling hotspots, and the rates break detailed balance so that active patches self-organize autocatalytically — plus link-flip moves on the triangulated mesh, which give the membrane its fluidity. The claim is that all three, not the head defect, carry tentacle morphogenes

What would settle it

The paper's own suggested experiment is the clean test: decouple growth from defect formation, for example by suppressing actin bundling proteins. If tentacles still emerge and elongate while the head keeps its +1 defect, the bundle-driven c⊥ mechanism is not the cause. Conversely, a direct look at a growing tentacle base — measuring membrane curvature and actin bundle orientation as the tube forms — would show whether tubes adopt a radius set by bundle-induced curvature and whether the bundle grows before the tube or merely reorganizes on an already formed tube.

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Extended reading notes

Core claim

This paper proposes that hydra's tentacles are tubes that grow where local signaling hotspots drive actin to polymerize into bundles, and the growing bundles mechanically bend the epithelial membrane outward. The head's +1 nematic defect is not the driver. In the authors' non-equilibrium Monte Carlo simulation, tubular tentacles emerge with an aster-like +1 defect at the head, with a ring-like +1 defect, or with no imposed defect at all; the +1 defect merely marks a place where tubes do not form. Each growing tube locally creates a charge-neutral triplet of defects (−1/2, +1, −1/2), which the authors propose arises from splitting and reorganization of the long contiguous actin fibers seen in

Load-bearing premise

The load-bearing assumption is that a growing actin bundle mechanically bends the membrane into a tube (the c⊥ term); this coupling is put into the Hamiltonian by hand and is not derived from actin dynamics or measured in hydra, and the paper itself notes there is no direct experimental evidence yet linking beta-catenin signaling to actin polymerization.

Editorial extensions

If this is right

Tentacle position is set by signaling-driven actin bundling, not by the head's +1 defect: the defect type can be changed or removed without changing where tubes emerge.
A minimum density of active (signaling) sites is required: below an initial active fraction of about 0.5, no tubes form, so tentacle number is a threshold-controlled readout of signaling density.
Tissue fluidity is essential for growth: if the epithelial layer cannot rearrange its connectivity (no link flips), the tubes do not grow even with active actin bundles present.
The model reproduces the beta-catenin over-expression phenotype, where making every vertex active makes tubes grow all over the body, matching experiments.
Tube nucleation creates a local, net-neutral defect triplet, so the hairy-ball constraint on total charge is preserved and local deformations can occur anywhere.

Reading between the lines

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

A direct test the paper leaves implicit: in a live hydra, pharmacologically suppressing actin-bundle assembly should stop tentacle elongation even though the +1 defect at the head remains — growth would decouple from the defect.
The mechanism could extend beyond hydra: any epithelial sheet under aligned filamentous actin with chemical hotspots is a candidate for bundle-driven tubulation, so the same model could be pointed at other cnidarian structures or at branching epithelia.
The sigmoidal active-fraction curve gives a quantitative handle: measuring the density of nuclear beta-catenin-expressing cells near the head should predict the number and spacing of tentacles, with a sharp onset at the threshold.
If the c⊥ coupling is real, tube radius should be set (about 1/c⊥) by the bundle's preferred curvature, giving a measurable relation between actin bundle diameter and tentacle radius.

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Signed reviews

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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

Quick take: the main biological claim—that tentacle growth is insensitive to the +1 defect at the head—is not actually supported by the simulations, because the head region is artificially rigid and inactive. That conclusion is close to an input. But there is a real model here, and two findings stand on their own: tubules require link-flip fluidity, and there is a threshold active fraction around 0.5.

What the paper does well: it extends the authors' earlier active-membrane machinery to hydra geometry, with a clean treatment of activity via non-equilibrium switching rates. The Kolmogorov loop argument showing broken detailed balance is coherent. The sigmoidal dependence of active fraction on input fraction, with a threshold, is a genuine simulation result, as is the absence of tube growth when link flips are turned off. These are worth reporting.

The soft spots are significant. The stress-test note is correct. The head is defined as the top vertex plus two rings where positions are fixed (for defect cases) and where the authors 'maintain high bending rigidity' and set c_perp=0. So in all three cases, the head patch cannot deform, cannot tubulate, and has no active vertices. The three cases differ only in the director field on that rigid patch. Claiming that the defect type doesn't matter is then near-tautological. The biologically relevant question—whether a deformable, active head with a +1 defect organizes tube formation in surrounding tissue—is not tested. The authors even phrase it as 'place holder where tubes do not form,' which is exactly what they constructed.

Related to this, the model has the mechanism injected by hand: c_perp is defined as the preferred curvature induced by a growing actin bundle, so the proposal that actin polymerization drives tubes is an assumption, not an emergent result. The paper admits there is no direct evidence for β-catenin to actin polymerization. That doesn't invalidate the model, but it weakens the biological extrapolation. Also, defect charges are identified visually from snapshots, not computed; no code or data are provided. Those are minor-to-moderate issues.

Who should read it: people working on active nematic membranes and hydra morphogenesis. The fluidity and threshold results are of some interest. But I would not put the head-defect claim in a paper without redoing the setup with an active, deformable head. As it stands, the manuscript needs major revision before it should be published. For a journal, I would send it to review despite the flaw, because the model and question are important enough that a referee could help the authors reframe the claim. For your own work, don't cite the defect-insensitivity conclusion.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a Monte Carlo model of hydra epithelial morphogenesis. A triangulated fluid membrane carries a nematic director field representing actin; a subset of 'active' vertices, interpreted as Wnt3/beta-catenin signaling hot spots, have a finite spontaneous curvature c_perp perpendicular to the director. Non-equilibrium conversion rates between active and inactive states break detailed balance, as shown by a Kolmogorov-loop argument. Simulations produce tubular tentacles around the upper hemisphere, with the tube axis aligned with the nematic director. The authors report three main results: (i) tube formation is insensitive to the +1 defect at the head, whether aster-like, ring-like, or absent; (ii) each tube contains a +1/-1/2/-1/2 defect triplet of zero net charge; (iii) link-flip moves conferring membrane fluidity are necessary for tube growth. They propose that active actin polymerization, not the head defect, drives tentacle emergence.

Significance. The paper addresses a topical question in active matter and developmental biophysics: whether topological defects in the actin nematic field actively organize morphogenesis or merely accompany it. The non-equilibrium construction with activity-dependent rates and the Kolmogorov-loop breaking is a useful formal feature, and the predicted threshold in the active-site fraction N+/N is falsifiable. The head-defect-insensitivity claim, if established, would directly challenge the mechanical-cue hypothesis of Maroudas-Sacks et al. The link-flip requirement is also a concrete, testable prediction about tissue fluidity. However, the force of these claims is currently weakened by the fact that the tubulation mechanism is inserted via a preferred-curvature term, by the absence of quantitative defect characterization, and by the fixed-total-vertex implementation of 'growth.'

major comments (4)

[Model, Eq. (1)] The central biological mechanism is assumed, not tested. Eq. (1) introduces c_perp > 0 as the preferred curvature 'induced by growing actin bundles,' and active vertices are defined by c_perp != 0. This is the sole tubulation drive in the model. Thus the paper's proposal that actin polymerization drives tentacle emergence is placed directly into the Hamiltonian; no derivation connects actin bundle growth or polymerization to this curvature, and the text admits there is no direct experimental evidence for the beta-catenin-to-actin link. The simulations therefore demonstrate tubulation of a membrane with an imposed anisotropic spontaneous curvature rather than that actin polymerization drives morphogenesis. To support the biological claim, the authors need either an independent derivation or measurement of c_perp, or a clear re-framing of the model as phenomenological.
[Results, Fig. 3] The central negative claim about head defects is confounded by the rigid, inactive head patch. The head (central vertex plus two surrounding rings) is excluded from the active population, assigned high kappa, c_perp=0, and its vertex positions, directors, and bond connectivity are held fixed. Tubes are therefore structurally prohibited on the head, and the three compared cases differ only in the director boundary condition on a patch that cannot participate in tubulation. The statement that the +1 defect 'merely acts like a place holder where tubes do not form' is an input, not an outcome. A fair test would allow the head region to deform and become active while varying the defect type, or at least quantitatively show that tube positions and orientations in the surrounding tissue are independent of the head boundary condition.
[Results, second paragraph] The paper states that each tube carries a +1 defect at its tip and two -1/2 defects at its base, produced by defect splitting, but no winding-number calculation or local topological-charge measurement is reported. This is inferred from snapshots (Fig. 3 and movies). Since the defect triplet is central to the proposed mechanism and to the claim that tubes have zero net topological charge, the authors should compute the local charge along loops around the tube tip and base and report the values, with error estimates from the discrete director field.
[Results, 'growth' in fixed-N simulation] The model is a closed triangulated network with fixed total vertex number N; no vertices are inserted and no area is created. Yet the abstract lists 'membrane growth and polymerization of actin' as part of the simulation, and the Results state that 'new vertices are continuously drawn from the cylindrical zone to the head zone.' At most, vertices are redistributed from the body to the tube; this is elongation by material transport, not growth. The authors should either implement actual area growth (e.g., vertex insertion/removal) or revise the language so that 'growth' means 'tubule elongation via redistribution of existing tissue.' This distinction matters for the claimed connection to actin polymerization and cell proliferation.

minor comments (5)

[Abstract] Typo: 'e p t i t h e l i a l' should be 'epithelial.'
[Results, last paragraph] Typo: 's u f f i c i e n t' appears as 'su fficient' with a ligature or encoding error; correct to 'sufficient.'
[Results, Fig. 3C] The 'no defect' case is not fully defined: if no defect is imposed, what director boundary condition is applied on the two fixed rings of the head? Specify the initial and boundary conditions for this control.
[Fig. 2 and Eqs. (4)-(5)] The transition labels p,q,r,s and p',q',r',s' are introduced in the text, but the figure alone is hard to follow. Define each symbol explicitly in the caption or near the equations, including which transitions correspond to energy-based MC moves and which to active-inactive conversion.
[Model, parameters] The total number of vertices N, physical units, and the values of mu and A0 are not stated. Please provide N and the full parameter set, since the threshold and tube-count results depend on system size.

Circularity Check

2 steps flagged · score 6.0 of 10

Two headline results reduce to model inputs: the c⊥ term is the claimed actin polymerization, and head-defect irrelevance is enforced by a rigid, inactive head.

self definitional [Section I, Eq. (1) and Results ("This axial arrangement...")]
"we set c⊥ > 0 to promote development of narrow membrane tubules induced by growing actin bundles. ... Once defects are created tubes can grow by taking advantage of the c⊥ term in the Hamiltonian which lowers energy costs by arranging the nematics parallel to the tube axis, provided the adjoining vertices are in active state. This axial arrangement of nematics can be interpreted as active polymerization."

The proposal that active polymerization of actin bundles is an important player in tentacle growth is not a simulation output; it is the meaning assigned to the c⊥ term inserted in Eq. (1). The term is defined as a preferred curvature 'induced by growing actin bundles,' and 'active' vertices are exactly those with c⊥ ≠ 0. Tube emergence then occurs because the Hamiltonian is lowest for a tube of radius 1/c⊥. Calling this 'active polymerization' renames the input mechanism as a finding: the model demonstrates that a curvature-coupling term can make tubes, not that actin polymerization independently drives hydra tentacles.
self definitional [Section I setup and Section II summary ("In summary...")]
"The simulation was started with N+0 active vertices randomly distributed on the upper hemispherical cap (excluding the head). ... Neither MC moves (which alter positions of vertices) nor bond flip moves (responsible for implementing membrane fluidity) were permitted for these set of vertices in order to keep their connectivity and the +1 defect intact. ... In summary, ... the positioning of the +1 defect at the center of the head merely acts like a place holder where tubes do not form. We implemented this by assigning a high value of κ and setting c⊥=0 on two rings of vertices around the centr"

The conclusion that the +1 defect is 'a place holder where tubes do not form' is exactly what the simulation enforces. The head is excluded from the active-vertex population, its vertices are held fixed (no MC or link-flip moves), and c⊥=0 there, so tubes cannot form on the head in any run. Comparing aster, ring, and no-defect director patterns on this rigid, inactive patch can only test boundary-condition effects on surrounding tissue; it cannot test whether the head defect 'plays a positive role' in tube emergence. The negative claim is therefore largely fixed by construction.

full rationale

The paper is a self-contained simulation with several genuinely emergent results, such as the threshold fraction of active vertices, the sigmoidal N+/N response, the dependence of tube number on c⊥ and ϵll, and the necessity of link-flip moves for tube growth. The nonequilibrium character is demonstrated by an explicit Kolmogorov-loop calculation. However, the two headline conclusions are partially circular. First, the central mechanistic proposal—that active actin-bundle polymerization drives tentacle growth—is encoded directly in the Hamiltonian as the c⊥ term, and the text then interprets axial nematic arrangement on tubes as 'active polymerization.' That is an input renamed as a finding, not a derivation. Second, the negative result about the head defect is weakened by construction: the head is deliberately made rigid, inactive, and c⊥=0, so the statement that the defect is 'a place holder where tubes do not form' is the implementation itself, not an emergent discovery. The citation of the authors' own prior work (Ref. [20]) for the c⊥ strategy is not load-bearing in a circular way, since the term is presented as an explicit modeling assumption rather than as an externally justified theorem. Overall, the model is internally consistent and explores a plausible mechanism, but the two central claims reduce in part to the model's own definitions; hence a partial circularity score of 6.

Assumptions & free parameters 7 free parameters · 5 assumptions · 2 invented entities

The model rests on a small set of hand-chosen parameters and two ad hoc constructs (active vertices and the c_perp curvature field). The central negative result about defect insensitivity is a simulation outcome rather than a fitted prediction, so the ledger is moderate rather than extreme.

free parameters (7)

c_perp (intrinsic curvature perpendicular to nematic) = 1.0 (chosen, not measured)
Sets the preferred tube radius; the mechanism of tubulation is introduced through this term.
kappa (isotropic bending rigidity) = 10
Chosen to keep the body stiff except the head.
kappa_parallel = 20
Anisotropic bending rigidity along the nematic.
kappa_perp = 30
Anisotropic bending rigidity perpendicular to the nematic; controls tube energy cost.
epsilon_ll (Lebwohl-Lasher alignment strength) = 6
Nematic alignment strength; higher values reduce the number of tubes.
N0_plus/N (initial fraction of active vertices) = 0.9
Tunes the number of tubes; the emergent threshold near 0.5 is a result.
mu and A0 = not stated numerically
Control the rates in Eq (2)-(3) toward the target population difference; values not given in the text.

assumptions (5)

ad hoc to paper c_perp > 0 represents the mechanical effect of a growing actin bundle pushing the membrane into a tube
Model section after Eq (1): 'we set c_perp > 0 to promote development of narrow membrane tubules induced by growing actin bundles.' This presumes the mechanism the paper proposes.
ad hoc to paper Active vertices are sites of high Wnt3/beta-catenin signaling that induce actin polymerization
Model section: 'we speculate such a possibility' and 'We assume active vertices are the ones where Wnt3/beta-catenin signaling is high.' The paper notes there is no direct experimental evidence.
domain assumption Epithelial tissue behaves as a fluid visco-elastic membrane with link-flip fluidity
Model section: triangulated fluid membrane; fluidity models T1 transitions, cell sorting, division/apoptosis. A standard but strong simplification.
standard math Helfrich-Canham bending and Lebwohl-Lasher nematic energy are valid coarse-grained free energies
Used in Eq (1); standard in membrane and liquid-crystal physics.
standard math Poincare-Hopf theorem constrains total defect charge on the closed vesicle
Invoked in the Introduction and Discussion to justify defect charge balance.

invented entities (2)

Active vertex states (+/-)
purpose: Carry the c_perp curvature source and switch via autocatalytic non-equilibrium rates, generating localized tubulation
The paper links these sites to Wnt3/beta-catenin without direct evidence ('we speculate'), and provides no falsifiable handle outside the simulation.
c_perp intrinsic curvature field
purpose: Represents growing actin bundles pushing the membrane outward into tubes
A modeling device; the paper gives no independent measurement of this preferred curvature in hydra tissue.

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Cite this review

Pith. "Pith review of Actin driven morphogenesis in hydra." pith.science (2026). https://pith.science/paper/EZKKYOYH

@misc{pith2026250908655,
  author       = {Pith},
  title        = {Pith review of: Actin driven morphogenesis in hydra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EZKKYOYH}},
  note         = {Machine review of arXiv:2509.08655}
}

abstract

Hydra, a centimeter long cylindrical-shaped freshwater organism, has emerged as an interesting model system for studying morphogenesis in animals. Recently, fluorescent imaging of cytoskeletal actin filaments on the outer surface of hydra has revealed nematic-type arrangement of actin filaments. {Several topological defects in the nematic field have also been detected. In particular, aster-like +1 defects appear at the curved head of hydra and at the tip of its tentacles, while -1/2 defects are seen at the base of the tentacles. However, functional role of these defects in tissue development is not clear. Motivated by these observations, we here model hydra's epthelial tissue as a visco-elastic membrane and the tentacles as growing membrane tubes driven by a nematic interaction among actin. We consider the epithelial layer of hydra as a fluid membrane and carry out a non-equilibrium simulation which also includes membrane growth and polymerization of actin. We show that specific kind of defect at the head does not play any positive role in emergence of the tentacles. The reorganization of actin at the base and the tip of growing tentacles are consistent with other possible defect structures at the head as well. While it is known that regions of tentacle growth are hot spots of chemical signaling, involving Wnt3/$\beta$-catenin pathway, we propose that active polymerization of actin bundles could also be an important player in the growth of tubular tentacles. In addition to polymerization, fluidity of our model membrane, capturing effective fluidity of the epithelial tissue, turns out to be essential for enabling such growth.

Figures

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**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Thompson, On Growth and Form (Cambridge Uni- versity Press, 1917)

D. Thompson, On Growth and Form (Cambridge Uni- versity Press, 1917)

1917

[2] [2]

A. D. Bershadsky and J. M. Vasiliev, Cytoskeleton (Springer Science & Business Media, 2012)

2012

[3] [3]

Maroudas-Sacks, L

Y. Maroudas-Sacks, L. Garion, L. Shani-Zerbib, A. Livshits, E. Braun, and K. Keren, Topological de- fects in the nematic order of actin fibres as organization centres of hydra morphogenesis, Nature Physics 17, 251 (2021)

2021

[4] [4]

Livshits, L

A. Livshits, L. Garion, Y. Maroudas-Sacks, L. Shani- Zerbib, K. Keren, and E. Braun, Plasticity of body axis polarity in hydra regeneration under constraints, Scien- tific Reports 12, 13368 (2022)

2022

[5] [5]

Z. Wang, M. C. Marchetti, and F. Brauns, Patterning of morphogenetic anisotropy fields, Proceedings of the National Academy of Sciences 120, e2220167120 (2023)

2023

[6] [6]

Wolpert, A

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