Free Growth under Tension

Chenyun Yao; Jens Elgeti

REVIEW 3 major objections 4 minor 44 references

Free Growth under Tension

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A cell colony can develop tensile stress with no motility force at all, driven by growth plus substrate friction.

desk verdict A genuinely new mechanism for tensile stress in growing colonies without motility, with a clean analytic model and honest calibration; the expansion-speed prediction is partly self-consistency but the existence claim holds. read the letter →

arxiv 2502.03927 v1 pith:WVORXACY submitted 2025-02-06 physics.bio-ph cond-mat.softq-bio.CB

Chenyun Yao , Jens Elgeti This is my paper · ORCID

classification physics.bio-phcond-mat.softq-bio.CB

keywords tissuemechanicshomeostaticpressuretensilestresscellcolonygrowthsubstratefrictionretrogradeflowtwo-particlemodelactivematter

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 sets out to show that the tensile stress observed in expanding cell colonies does not require cells to push outward. In its model, cells proliferate faster in a thin rim at the free boundary; those extra cells flow inward, and substrate friction resists that flow, stretching the tissue into tension. The authors derive analytic pressure and velocity profiles, a constant expansion speed, and a phase diagram in which colonies can grow indefinitely while under tension even when the homeostatic pressure is negative. Adding random motility strengthens the tension and reproduces the stress profile reported in [7]. If correct, the work gives a purely mechanical origin for tissue tension and makes a testable prediction of retrograde cell flow.

What carries the argument

The load-bearing object is the growth law $k = \kappa(P_H - P) + \Delta k\Delta x\,\delta(x - x_0)$, which couples proliferation to mechanical stress through a bulk response coefficient $\kappa$ and a surface term $\Delta k\Delta x$: cells near the free boundary proliferate faster because growth there is mechanically cheaper. From that law, the continuity equation $\partial_x v = k$, and force balance with linear substrate friction, the argument reduces the stress field to the screened Poisson equation $\partial_x^2 P_x = (P_x - P_H)/\lambda^2$, whose solution is the hyperbolic-cosine pressure profile of Eq. (7). The same growth law, with a motility traction term added to the force balance, yields the motile pressure profile and the generalized expansion speed. The crucial entity is therefore the combination of negative homeostatic pressure and rim growth: the rim produces an inward flux, friction converts that flux into tension, and the tension suppresses growth in the bulk.

What would settle it

Measure the velocity field inside a non-motile expanding colony that is under tension: the model predicts a retrograde inward flow of cells throughout the interior, so the absence of such a flow would falsify the mechanism.

Watch

Extended reading notes

Core claim

The central discovery is that tension arises from the interplay of a negative homeostatic pressure and boundary-enhanced growth. The growth rate is written as $k = \kappa(P_H - P) + \Delta k \Delta x\,\delta(x - x_0)$, with the surface term representing faster growth in a thin layer at the free edge. Combining this with force balance, where substrate friction is the only external force, $\partial_x\sigma_{xx} = 2\rho\gamma v$, gives a screened Poisson equation for the pressure, $P_x = P_H\bigl(1 - \cosh(x/\lambda)/\cosh(x_0/\lambda)\bigr)$ with $\lambda^2 = (\rho\kappa\gamma)^{-1}$. For homeostatic pressures between a critical negative value $P_H^c = -\Delta k\Delta x\sqrt{4\rho\gamma/\kappa}$ and zero, colonies expand indefinitely at constant speed $v(x_0) = P_H\sqrt{\kappa/(4\rho\gamma)} + \Delta k\Delta x$ while their interior is tensile; the tension is balanced by friction against an inward retrograde flow of cells. Motility adds a boundary-localized traction that generates additional tension and extends the tensile-growth phase, and the two-dimensional circular case reduces to the same physics with growth proportional to the perimeter.

Load-bearing premise

The whole mechanism depends on the assumption that cells in a thin rim at the colony boundary proliferate faster than cells in the bulk; if that surface growth advantage disappeared, no inward flux would form and tension would not arise.

Editorial extensions

If this is right

Non-motile tensile colonies should show a retrograde flow: cells move inward from the proliferating rim toward the center, except in a thin layer at the boundary; this is directly measurable by particle image velocimetry.
The expansion speed of an indefinitely growing colony is constant and independent of colony size; it is linear in the growth force for non-motile colonies and quadratic in the motility speed for motile ones.
Motility generates its own boundary tension $T$ that is quadratic in $v_0$ and linear in $G$, and it widens the range of negative homeostatic pressures for which a colony can grow indefinitely under tension.
In two dimensions, growth is perimeter-controlled: cell number grows as $t^2$, radius grows linearly, and fingers at the frontier do not lead to fractal boundaries.
The analytical profiles allow experimental stress profiles to be decomposed into friction and motility contributions, providing a procedure for inferring the underlying mechanics from traction measurements.

Reading between the lines

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

The same mechanism should apply to other proliferating tissues or colonies on frictional substrates, wherever growth is enhanced at free edges and friction resists inward flow; this is an extension the paper does not develop.
Because the surface growth term $\Delta k\Delta x$ is partly calibrated from the expansion speed it later predicts, a full test of the theory would require measuring $\Delta k$ and $\Delta x$ independently; absent that, the quantitative match is less decisive.
A simultaneous measurement of stress and velocity fields in an expanding monolayer would separate this growth-friction mechanism from motility-alignment mechanisms: the former predicts inward flow correlated with tension, while alignment models predict outward-oriented traction.
The predicted exponential buildup length $\lambda$ could be extracted experimentally from stress and velocity profiles and compared with the bulk response coefficient $\kappa$ measured by independent deformation experiments.

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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

Colleague, what you should know: this paper shows that non-motile growing colonies develop tensile stress, purely from boundary-enhanced growth plus substrate friction. That is a new result relative to the cited literature, which ties tension to outward-oriented motility. The authors build a simple continuum model that reproduces the simulation pressure and velocity profiles with mostly independently measured parameters, and they make a sharp, testable prediction: tensile non-motile colonies should show retrograde cell flow. The two-particle growth simulation is standard, and the stress measurement technique is reasonable. Credit is due: the central existence claim is direct simulation observation, and the analytic derivation is clear and internally consistent. The paper honestly flags its own limitations, including the absence of motility alignment and the reliance on linear friction. Now the soft spot, confirmed on reading: the quantitative expansion-speed prediction in Eq. (11) is partly calibrated in-sample. The paper states, after Eq. (11), "We use this constant expansion speed to obtain ΔkΔx." So the surface-growth amplitude, which is the key input for the tensile phase, is fitted to the very front speed the model then claims to predict. The agreement is a consistency check, not a fully independent validation. That said, the pressure and velocity profile comparisons (Fig. 3) are less circular—PH, κ, ρ, γ are measured separately, and Δx is estimated—so the mechanism itself is not in doubt. The matching to Trepat et al. is also parameter-choice dependent, without robustness analysis. What would settle it: an independent measurement or derivation of ΔkΔx, plus release of simulation code and data. The paper is worth engaging seriously; the physics is plausible, the writing is honest, and the retrograde-flow prediction is falsifiable in experiments. I would accept it for peer review and ask for the calibration issue to be addressed, not because the result is wrong, but because the headline quantitative claim currently overstates its predictive status. For a reading group it would spark good discussion. I would not cite it for the expansion-speed formula without caveat, but I would cite it for the mechanism.

Referee Report

3 major / 4 minor

Summary. The manuscript combines 2PG-type agent-based simulations and a one-dimensional continuum theory to argue that cell colonies expanding on a substrate can be under tensile stress even in the absence of motility forces. The proposed mechanism is that cells near the free boundary grow faster than the bulk, producing an inward cell flux; substrate friction opposes this flux and generates tension when the homeostatic pressure is negative. The authors derive exponential pressure profiles, a constant expansion speed v(x0) = PH sqrt(kappa/(4 rho gamma)) + Delta k Delta x, and a phase boundary between finite and indefinitely growing colonies. They extend the model to motile colonies, where boundary polarization adds a motility tension term, and compare the resulting profiles and speeds with simulations.

Significance. If the central claim holds, the paper offers a mechanistically distinct route to tensile stress in freely expanding monolayers, complementing the established motility-alignment and confinement-sorting explanations. The analytical model is transparent, the simulation method is standard, and the predicted retrograde flow in non-motile tensile colonies is a concrete, falsifiable experimental target. The comparison between analytic pressure profiles and simulations in Fig. 3 is largely convincing for the shape of the profiles. However, the quantitative predictive claim for the expansion speed is weakened by the in-sample calibration of the surface-growth parameter Delta k Delta x from the very speed the theory is said to predict, as discussed below. The existence of tension in non-motile colonies is a direct simulation observation and is not undermined by this calibration issue.

major comments (3)

[Section I, after Eq. (11) and SI Fig. S7] The central quantitative claim v(x0) = PH sqrt(kappa/(4 rho gamma)) + Delta k Delta x is presented as a prediction, but the text states 'We use this constant expansion speed to obtain Delta k Delta x.' SI Fig. S7(b) confirms that the surface-growth contribution is obtained as the difference between the total measured front speed and the independently computed bulk term. Therefore the agreement between Eq. (11) and the simulated front speed used for calibration is a consistency check, not an independent prediction. The linearity in G of the simulated speed is also not an independent test, because Delta k Delta x is derived from that speed and could absorb the same linear dependence. Please either measure Delta k locally in the simulations, provide a microscopic derivation of its value, or explicitly re-frame Eq. (11) as a parameterized relation rather than a parameter-free prediction.
[Main text Fig. 3 caption vs. Section I, after Eq. (11)] The Fig. 3 caption claims that the theory curves are 'calculated from the piecewise solution with all parameters measured independently (i.e. not fitted)' and that only Delta x is estimated. This conflicts with the main-text statement that Delta k Delta x is obtained from the constant expansion speed. Since Delta k enters the piecewise pressure profile in Eq. (S6), the pressure and velocity comparisons in Fig. 3 are partially constrained by the fitted quantity, not fully independent. The manuscript should clearly list, for every parameter entering Eqs. (S6)-(S7), whether it is measured in a separate simulation, estimated from a structural assumption, or calibrated from the expansion speed.
[Section I, Eq. (1) and phase-II mechanism] The existence of indefinite tensile growth for PH < 0 rests entirely on the surface-growth term Delta k Delta x in Eq. (1): without it, Eq. (11) gives a negative expansion speed and no phase II. The physical motivation for boundary-enhanced growth is plausible, but the manuscript does not provide a direct measurement of the local growth rate k(x) in the simulation rim, nor a derivation of Delta k from the 2PG microscopic rules. Given that this term is load-bearing, please report the measured k(x) profile in the boundary layer and compare it with the assumed step/delta form, or state clearly that the surface growth amplitude is an inferred effective parameter.

minor comments (4)

[Throughout] The manuscript contains many typographical errors, including 'optianed', 'boudnary', 'therory', 'shortcommings', 'enabeled', 'reaveals', 'arrise', 'mechanims', 'accrding', 'consitituting', and 'partcile'. A thorough proofreading pass is needed.
[SI Section S2] The text refers to 'Main text Fig.7(a)' when comparing quasi-1d and 2d expansion speeds, but the main text has Fig. 6(a) for this comparison; please correct the cross-reference.
[SI Fig. S10 caption] The caption uses 'B^* = 11.4' where the context indicates this should be 'G^* = 11.4'.
[Main text Fig. 3(b) and Section I, paragraph after Eq. (11)] The description of the pressure peak for expanding fronts with PH < 0 is qualitatively clear, but the figure inset is too small to see the predicted boundary pressure peak; a larger inset or an explicit zoom would help the reader verify this nontrivial prediction.

Circularity Check

2 steps flagged · score 6.0 of 10

Expansion-speed prediction is calibrated in-sample: Eq. (11)'s ΔkΔx is obtained from the measured front speed it then claims to reproduce; the motile Eq. (14) uses the same subtraction.

fitted input called prediction [Section I, after Eq. (11)]
"Eq.(11) predicts a constant expansion speed for all colonies in phase II or III and is linear in PH− PC H . Indeed, the simulations display a constant expansion speed linear in G (see Fig.S7). We use this constant expansion speed to obtain ∆k∆x. Obtaining the homeostatic pressure and other bulk tissue properties from bulk simulations (i.e. without a fit, see SI), and estimating ∆x∗ = 0.7 reproduces the simulation data remarkably well (Fig.3)."

The claimed prediction v(x0)=PH sqrt(κ/(4ργ))+ΔkΔx is not out-of-sample: ΔkΔx is obtained from the very measured expansion speed by subtracting the independently computed bulk term PH sqrt(κ/(4ργ)) (SI Fig. S7 shows the surface contribution as the difference between total speed and bulk contribution). The subsequent agreement with the expansion speeds used for the subtraction is therefore a consistency check. Because ΔkΔx also enters the piecewise pressure and velocity profiles (Eq. S6), the profile comparisons in Fig. 3 are partly constrained by this calibration, although PH, κ, ρ, γ, and Δx are obtained separately.
fitted input called prediction [SI Fig. S10 caption]
"The surface growth contribution (green) is obtained from the difference between the total expansion speed (blue) and the bulk contribution (orange)."

For motile colonies, the same surface-growth parameter ΔkΔx is calibrated from the measured total expansion speed and then inserted into Eq. (14), v(x0) = (PH + T/(1+λm/λ)) sqrt(κ/(4ργ)) + ΔkΔx. The agreement between Eq. (14) and the expansion-speed data is therefore not an independent test of the theory; it is guaranteed by construction for the surface-growth part. The motility contributions T and λm are independently measured from polarization and traction profiles, so the motile tension mechanism retains independent content, but the quantitative speed prediction is in-sample.

full rationale

The paper's central existence claim—that a non-motile expanding colony can be under tension—is a direct simulation observation and does not reduce to a fit; the analytic mechanism (friction balancing the inward flux produced by boundary-enhanced growth) is a coherent derivation from stated mechanical assumptions. Homeostatic pressure PH, friction γ, compressibility κ, and density ρ are measured independently, and Δx is estimated rather than fitted. However, the load-bearing quantitative prediction of the expansion speed is circular in an explicit, quotable way: Eq. (11) contains the parameter ΔkΔx, and the paper states 'We use this constant expansion speed to obtain ΔkΔx.' SI Fig. S7 confirms that ΔkΔx is the difference between the measured front speed and the computed bulk term, so the agreement in Fig. 3 and Fig. S7 is a consistency check rather than a prediction. The same calibration is reused in the motile expansion speed Eq. (14) via the caption of Fig. S10. The phase boundary between finite and infinite tensile colonies also inherits this calibration through PC_H = −ΔkΔx sqrt(4ργ/κ), so the predicted transition is not fully out-of-sample. An independent measurement of the surface-growth contribution (e.g., from local division-rate profiles in the rim) would settle the quantitative claim; without it, the central speed prediction is forced by construction. Self-citations to Refs. [2] and [25] motivate the surface-growth ansatz but are not the main circular step; the circularity is the in-sample calibration of ΔkΔx. Overall score 6: partial circularity of the central quantitative prediction, while the qualitative tensile mechanism remains independently supported by simulation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on measured but not independently predicted tissue parameters (PH, κ), a calibrated boundary-growth parameter (ΔkΔx), an estimated boundary width (Δx), and an assumed exponential polarization profile (λm). No new entities are invented.

free parameters (3)

ΔkΔx (surface growth amplitude) = calibrated from expansion speed
Obtained from the measured constant expansion speed via Eq. (11) and then used in the analytic expansion-speed and pressure predictions. See Section I: 'We use this constant expansion speed to obtain ΔkΔx.'
Δx* (surface growth layer width) = 0.7 in units of σ
Estimated by hand, not measured directly; enters the piecewise solution and the comparison in Fig. 3.
λm (motility force decay length) = estimated from polarization/force density profile (Fig. S9)
Defines the exponential profile of the motility force in Eqs. (12) and (13); obtained from the same simulations, not independently predicted.

assumptions (6)

domain assumption Growth rate is linear in pressure: k = κ(PH - P).
Invoked in Eq. (1) and throughout; standard tissue-growth feedback from prior literature.
domain assumption The colony interior has constant density ρ.
Used to simplify the continuity equation from Eq. (2) to Eq. (3); stated in Section I.
domain assumption In the quasi-1D colony the y-direction is homeostatically balanced: σyy = -PH.
Reduces the 2D stress state to 1D pressure Px; stated at Eq. (4) in the main text.
domain assumption Substrate friction is linear: f_ext = -2ργv.
Essential to convert the retrograde cell flux into tension; the authors acknowledge in the conclusions that dry or active friction would change the mechanism.
ad hoc to paper Surface growth is confined to a layer of width Δx, or to a delta function.
The term ΔkΔx δ(x-x0) in Eq. (1) is introduced as the engine of the inward flux; its amplitude and width are fit or estimated rather than derived.
ad hoc to paper The average cell polarization decays exponentially from the boundary.
Motile theory assumes 2ργv0⟨cosθ⟩ = F exp((x-x0)/λm) before Eq. (13); the simulations show this is approximate, as density is not a step function.

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Cite this review

Pith. "Pith review of Free Growth under Tension." pith.science (2026). https://pith.science/paper/WVORXACY

@misc{pith2026250203927,
  author       = {Pith},
  title        = {Pith review of: Free Growth under Tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVORXACY}},
  note         = {Machine review of arXiv:2502.03927}
}

read the original abstract

Ever since the ground breaking work of Trepat et al. in 2009, we know that cell colonies growing on a substrate can be under tensile mechanical stress. The origin of tension has so far been attributed to cellular motility forces being oriented outward of the colony. Works in the field mainly revolve around how this orientation of the forces can be explained, ranging from velocity alignment, self-sorting due to self-propulsion, to kenotaxis. In this work, we demonstrate that tension in growing colonies can also be explained without cellular motility forces! Using a combination of well established tissue growth simulation technique and analytical modelling, we show how tension can arise as a consequence of simple mechanics of growing tissues. Combining these models with a minimalistic motility model shows how colonies can expand while under even larger tension. Furthermore, our results and analytical models provide novel analysis procedures to identify the underlying mechanics.

Figures

Figures reproduced from arXiv: 2502.03927 by the authors.

**Figure 1.** FIG. 1. Growth of tensile colonies. (a) Superimposed snap [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 3.** FIG. 3. Pressure and velocity profiles of growing non-motile [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 6.** Figure 6: FIG. 6. The growth of 2d colonies. (a) The comparison of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 5.** Figure 5: FIG. 5. Effects of motility on the growth of motile quasi-1d [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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x∈ [0,x 0− ∆x] P =PH +D1ex/λ +D2e−x/λ
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The boundary conditions read:

x∈ (x0− ∆x,x 0] P =PH + 2∆k κ +C1ex/λ +C2e−x/λ As in the main text, we assume mirror symmetry around x = 0 and thus only treat the positive half space here. The boundary conditions read:
[40]

Zero pressure at the boundary P (x0) = 0
[41]

Pressure and its derivative are continuous, in particular at x0− ∆x
[42]

Mirror symmetry at x = 0 implies dP dx = 0 at x = 0 Using the boundary conditions we fix the constants of integration: P = { PH(1−e x−x0 λ ) + 2∆k κ (cosh ∆x λ − 1)e x−x0 λ for x∈ [0,x 0− ∆x] (PH + 2∆k κ )(1−e x−x0 λ ) + 2∆k κ e− ∆x λ sinh (x−x0 λ ) for x∈ (x0− ∆x,x 0] (S6) ∗ ...

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Montel, F. et al. Stress Clamp Experiments on Multi- cellular Tumor Spheroids. Phys. Rev. Lett. 107, 188102 (2011)

work page 2011

[2] [2]

Delarue, M. et al. Mechanical Control of Cell flow in Multicellular Spheroids. Phys. Rev. Lett. 110, 138103 (2013)

work page 2013

[3] [3]

Hallatschek, O. et al. Proliferating active matter. Nat. Rev. Phys. 1–13 (2023)

work page 2023

[4] [4]

Shraiman, B. I. Mechanical feedback as a possible regu- lator of tissue growth. Proc. Natl. Acad. Sci. 102, 3318– 3323 (2005)

work page 2005

[5] [5]

Belteton, S. A. et al. Real-time conversion of tissue-scale mechanical forces into an interdigitated growth pattern. Nat. Plants 7, 826–841 (2021)

work page 2021

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