Prediction and control of geometry-induced nematic order in growing multicellular systems

Jonas Isensee; Lukas Hupe; Philip Bittihn; Ramin Golestanian

REVIEW 3 major objections 5 minor 47 references

Prediction and control of geometry-induced nematic order in growing multicellular systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In dense growing colonies of rod-shaped cells, the container geometry alone sets the steady-state nematic orientation pattern.

desk verdict Defect-charge law for polygons is a clean, useful result, but the paper doesn't directly verify that the isotropic-growth shear tensor is the mechanism driving it. read the letter →

arxiv 2506.10867 v1 pith:BEDB235M submitted 2025-06-12 cond-mat.soft physics.bio-ph

Lukas Hupe , Jonas Isensee , Ramin Golestanian , Philip Bittihn This is my paper · ORCID

classification cond-mat.softphysics.bio-ph

keywords growingactivematternematicorderisotropicgrowthshearalignmenttopologicaldefectchargepolygonalconfinementagent-basedsimulationbacterialcolonies

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

In a dense two-dimensional colony of growing rod-shaped cells, this paper argues, the steady-state pattern of nematic orientation is set by the geometry of the colony boundary rather than by active-stress feedback. The core assumption is that the expansion flow can be approximated as the gradient of a scalar pressure, so geometry and boundary conditions alone determine the velocity field; the traceless shear rate tensor of that field then gives the local preferred orientation. This reproduces previously reported alignment in channels, radial expansion, inward growth, and curved-strain geometries, and it predicts a new, simple law for polygonal domains: in an $n$-sided polygon the director rotates by the interior angle at each corner, giving a net topological defect charge of $s = 1 - n/2$. Agent-based simulations confirm this law for polygons from triangles to 17-gons. A minimal extension that adds advection and a linear decay of order turns the directional prediction into quantitative alignment-strength profiles, including tangential alignment when the inner growth rate is raised.

What carries the argument

The load-bearing object is the isotropic-growth approximation combined with the traceless shear rate tensor $\mathbf{u}^{ST}$. The approximation replaces the full active-nematic feedback loop with $\mathbf{v} = -\zeta\nabla p$ and $\nabla^2 p = -\alpha/\zeta$, so the velocity field is a gradient field with no vorticity and is determined entirely by the growth rate $\alpha$ and the boundary conditions. The preferred orientation is then read from the principal axis of $\mathbf{u}^{ST}$, with $\mathbf{Q}\approx\beta\mathbf{u}^{ST}$ at low alignment; in radial geometries this gives $\mathbf{u}^{ST} = \alpha (r/R_0)^{-2}(\hat{\mathbf{r}}\otimes\hat{\mathbf{r}} - \mathbf{I}/2)$. For polygonal domains the same Poisson problem is solved analytically for the equilateral triangle and numerically otherwise, producing the corner-rotation rule and the charge law $s=1-n/2$. The quantitative extension replaces instantaneous alignment with the advection-decay equation $(\mathbf{v}\cdot\nabla)\mathbf{Q} = \beta\mathbf{u}^{ST} - \mu\mathbf{Q}$, whose radial solution is governed by the ratio $\mu/\alpha$ and explains why the measured alignment decay is slower than the $r^{-2}$ shear profile.

What would settle it

Measure the steady velocity field in a growing colony inside a triangular microfluidic chamber and compare it with the curl-free solution of $\nabla^2 p = -\alpha/\zeta$; a significant $\nabla\times\mathbf{v}$, or a time-averaged director that does not rotate by the interior angle at each corner (charge $-1/2$), would refute the central claim. A cheaper test is to simulate division aspect ratio 3 or higher, where the paper already finds the charge law breaking at $n=12$, and check whether the breakpoint tracks the emergence of vorticity in the flow.

Watch

Extended reading notes

Core claim

The central claim is that geometry-induced nematic order in growing rods is slaved to the traceless shear rate tensor $\mathbf{u}^{ST}$ of an isotropic-growth expansion flow. Writing the flow as $\mathbf{v} = -\zeta \nabla p$ with $\nabla \cdot \mathbf{v} = \alpha$ makes the velocity field curl-free and computable from the domain shape and boundary conditions alone; at low alignment strength the nematic tensor is taken proportional to $\mathbf{u}^{ST}$. For regular polygons with absorbing boundaries, solving the Poisson problem yields a director that rotates by the interior angle at each corner, so the total topological charge is $s = 1 - n/2$. The paper verifies this charge law in agent-based simulations of dividing rods for $n = 3$ through $17$, and shows that the same framework, extended to the steady advection-decay equation $(\mathbf{v}\cdot\nabla)\mathbf{Q} = \beta \mathbf{u}^{ST} - \mu \mathbf{Q}$, quantitatively reproduces the measured radial alignment profiles in ring and wedge geometries, with deviations only near the outer boundary.

Load-bearing premise

The flow field is assumed to come from isotropic growth alone, with no feedback from local cell orientation to the velocity, and the director is assumed to follow the shear rate tensor of that flow; if active-stress feedback or vorticity is significant, the predicted patterns need not hold.

Editorial extensions

If this is right

Polygonal chambers become a design tool: since the net defect charge is $s = 1 - n/2$, choosing the number of sides prescribes the topological charge of the growing monolayer.
Alignment patterns reported for channels, free radial expansion, inward growth, and curvature-induced strain follow from a single mechanism, the anisotropic part of the growth flow, without invoking active-stress feedback.
The alignment-strength profile decays as a power law whose exponent is controlled by $\mu/\alpha$, and since $\mu$ is found to scale with the growth rate $\alpha$, the profile is independent of growth rate in the incompressible limit.
Local modifications of channel width can flip the sign of the shear rate tensor and rotate the nematic director by $90^\circ$ in the expanded region, turning static alignment patterns into prescribed rotational dynamics.
At division aspect ratio 2, the charge law holds on time-averaged fields over the full tested range $n=3$ to $17$; at aspect ratio 3 it starts to deviate at $n=12$, marking the limit of the isotropic-growth picture.

Reading between the lines

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

Editorial inference: if the corner-rotation law holds in experiments, sequences of polygon corners could be used like optical elements for the director field, writing arbitrary rotations and defect charges into a growing tissue.
Editorial inference: the curl-free flow assumption is the least protected part of the argument; a direct test is to measure vorticity in a growing colony and ask whether the director still tracks $\mathbf{u}^{ST}$ wherever $\nabla\times\mathbf{v}\neq 0$.
Editorial inference: the framework suggests that in dense short-rod colonies, orientation is a passive recorder of the expansion geometry; if so, the same Poisson calculation should predict order in smoothly curved or irregular tissue domains without any new physics.
Editorial inference: because the model suppresses feedback, it predicts that average orientation patterns should be identical for colonies with the same geometry but different noise levels, while fluctuations around the average grow with noise; varying division-rate randomization in simulations could test this.

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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,

The genuinely new and testable result here is the defect-charge law s = 1 - n/2 for polygonal domains, confirmed in agent-based simulations up to n = 17 at aspect ratio 2. That is a clean, falsifiable design rule. The isotropic-growth shear-alignment mechanism itself is not new—Langeslay and Juarez already used the shear component of the expansion flow—but the polygon charge law and the advection-decay extension are new. The paper is clearly written, the derivations from the Poisson problem are transparent, and the authors are candid about their model's limitations.

The soft spots, in proportion. First, the stress-test note is right: the central mechanism is not directly verified where it matters most. For the triangle, they show a qualitative match between simulated and theoretical velocity fields, but for the polygons n = 4 to 17 that establish Eq. (7), they never measure the actual velocity gradient, its vorticity, or the simulated uST. The predicted orientation fields (or at least their defect charge) match, but whether they arise from the isotropic-growth shear tensor is not demonstrated. This is a missing link, not a fatal flaw. Eq. (7) stands as an empirical law, but the "explained by uST" part is under-supported. The remedy is straightforward: compute the time-averaged velocity gradient in the simulations and compare with the theoretical curl-free field for all polygons, not just the triangle.

Second, the advection-decay model is a two-parameter fit with parameters that vary with ring curvature and growth excess. The authors admit this and rescue cross-prediction via the qc ratio, which does tie to the channel measurement. Still, the fit range is restricted after observing mismatches, and the model's predictive content is modest. This is a moderate weakness, largely acknowledged.

Third, the code and data for the specific results are not yet released, so independent reproducibility is hard to assess. The link to the simulation library is there, but the example code is promised "alongside final publication." For a preprint, acceptable, but worth noting.

Who gets value: experimentalists working with microfluidic chambers or growing colonies, and modelers looking for a simple design rule for topological defect charge. The paper deserves a serious referee. My recommendation would be "major revision," with the main request being the velocity-field verification across polygon geometries, plus release of the simulation analysis code.

Best.

Referee Report

3 major / 5 minor

Summary. The paper proposes that in dense colonies of growing rod-shaped cells, steady-state nematic orientation patterns can be predicted from boundary geometry alone. The key assumption is that the expansion flow is generated by isotropic growth, v = -ζ∇p, so the flow is curl-free and its traceless shear rate tensor uST can be computed from geometry and boundary conditions; the nematic director is then assumed to align with the principal axis of uST. The framework is first applied to radially symmetric geometries, recovering known channel, radial, and inward-growth results, and then to regular polygonal domains where it predicts a total topological defect charge s = 1 - n/2 (Eq. 7). This prediction is compared with agent-based simulations for polygons with n = 3 to n = 17. The paper then extends the model to a quantitative advection-decay equation, v·∇Q = βuST - μQ, whose solution is fitted to radial alignment profiles in ring and wedge geometries and to a channel-like geometry with excess growth in the center. The central claim is that orientation patterns, and hence defect charge, can be forward-engineered through domain geometry without invoking active-stress feedback.

Significance. If the geometry-only mechanism is correct, this is a valuable unifying framework: it links a wide class of previously studied growing-colony geometries to a single scalar Poisson problem and produces an explicit, falsifiable prediction for defect charge in polygons. The paper gives credit to several concrete strengths: Eq. (7) is tested against independent agent-based simulations for a wide range of polygon side counts at aspect ratio 2; the qualitative match between simulated and predicted velocity fields is shown for the triangle (Fig. 2e-f); the advection-decay model reproduces the radial alignment profiles in ring and excess-growth geometries; and the authors openly acknowledge the limitations of their assumptions, including approximate incompressibility and the geometry dependence of fitted parameters. The main significance hinges, however, on the unverified intermediate step for the polygon systems: the actual simulated velocity field is never compared with the isotropic-growth solution for n ≥ 4, so the observed defect-charge agreement is not yet causally tied to the proposed uST mechanism.

major comments (3)

[§IIC, Figs. 2e-f and 3] The central claim that geometry determines orientation through the isotropic-growth shear tensor requires that the simulated velocity field be, at least approximately, curl-free and equal to the predicted gradient-pressure solution. This is verified only for the equilateral triangle (Fig. 2e-f). For all other polygons used to establish Eq. (7), no simulated velocity field, no vorticity measurement, and no comparison between simulated and predicted uST is reported. The defect-charge agreement could therefore in principle arise from a different shear mechanism with the same boundary-driven topology. Please compute, for at least a representative subset of polygons (e.g., n = 4, 6, 8, 12), the time-averaged agent-based velocity field, its vorticity, and the symmetric shear-rate tensor uST, and compare them with the numerical solutions of Eq. (3) and the resulting uST. This is a load-bearing check for the mechanistic claim.
[§IIIB, Eqs. (10)-(11), Fig. 4d-g, Supp. Fig. S4] The quantitative advection-decay model is fitted with two free parameters β and μ, but the authors report that these parameters vary with ring curvature and with growth excess, and that the fit landscape is degenerate (Supp. Fig. S4). Consequently, the statement that the framework can be extended to 'quantitatively capture alignment strength' and to enable cross-prediction is stronger than the evidence supports: the only transferable parameter combination is qc = αβ/μ, and Fig. 4f shows that this quantity, computed from fits, is lower than the measured channel value at low curvature, with no reported uncertainty. To substantiate the quantitative claim, please either provide a microscopic or independent determination of β and μ, or perform an explicit out-of-sample transfer test (e.g., predict the channel alignment from ring fits and compare with simulation) with confidence intervals. Without this, the model remains a two-parameter interpolation of the radial profiles.
[§IV and Supp. Fig. S5] The authors correctly note that incompressibility is only approximate in the simulations and that density profiles are parabolic, and they restrict their fits to the inner region where density gradients are smaller. However, the polygon defect-charge comparison in Fig. 3 is not accompanied by any estimate of how much the density variation or finite cell hardness affects the validity of the isotropic-growth solution over the region used to compute the charge. Since Eq. (7) is a topological prediction derived from an ideal incompressible flow, please quantify the sensitivity of the computed charge to the observed density profile, for example by repeating the charge computation on a subset of polygons with a stiffness or growth-rate variation, or by showing that the measured velocity deviations remain small over the charge-integration path.

minor comments (5)

[Introduction] The phrase 'strongly effected by the system geometry' should read 'strongly affected'.
[§IV Discussion] The sentence 'any anisotropic redistribution induced by the boundary conditions results in nonzero uST = 0' contains a typo; the intended statement should be 'uST ≠ 0'.
[§IIIB] The text 'with decay exponent varying with decreasing with increasing R0' is ambiguous; please restate the monotonic trend of the fitted exponent with R0.
[§IIIC heading] The heading 'TANGENTIAL ALIGNMENT: EXCESS GROWTH' contains a typo; it should read 'TANGENTIAL'.
[Code availability] Example code is stated to be made available 'alongside the final publication'; please clarify the current availability of the simulation and analysis code used to produce the figures, since the model library link is given but the exact analysis scripts are not.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geometry-based orientation and defect-charge predictions are tested against independent agent-based simulations, and the fitted advection-decay parameters are used as fits with cross-geometry checks rather than as derived predictions.

full rationale

The central derivation chain is self-contained. The isotropic-growth approximation is introduced explicitly (Eqs. 2-3) as an assumption, not as a consequence of the target prediction; the velocity field and uST are computed from the geometry and boundary conditions, and the resulting director and defect-charge predictions (Eq. 7 and Figs. 2-3) are compared with agent-based simulations that do not implement the isotropic-growth ansatz. This is a genuine independent test, so the main prediction does not reduce to its input. The only direct velocity comparison is for the triangle (Figs. 2e-f); the absence of measured uST and vorticity checks for n>3 is a validation gap, not circularity. The advection-decay model (Eqs. 9-11) uses beta and mu as free parameters fitted to simulated q(r); the paper presents these as fits, not as parameter-free predictions of the same curves, and the channel comparison via qc = alpha beta / mu uses an independently measured channel alignment strength, making it a cross-check rather than a tautology. The Discussion explicitly notes that beta and mu vary with geometry and that incompressibility is approximate; these are acknowledged limitations that affect predictive strength but do not make the derivation circular. Self-citations to prior particle models and software are methodological rather than load-bearing, and no uniqueness theorem or ansatz is imported solely from the authors' prior work. Overall, no circular step can be exhibited.

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

The qualitative predictions rest on the isotropic-growth, Darcy-flow, and low-alignment assumptions; the quantitative extension adds two fitted parameters and a boundary condition. No new particles, forces, or conserved quantities are introduced. The linear decay term is phenomenological, not a new entity.

free parameters (3)

beta (flow-alignment strength) = fitted per geometry; about 0.5 to 1.5 in Fig 4d
Introduced in Eqs. (9)-(10) as the coefficient of the shear source term. Best-fit values are obtained by fitting Eq. (11) to simulated q(r) profiles (Fig 4c) and vary with ring curvature and growth excess (Fig 5e), so beta is not a fixed cell property.
mu (nematic order decay rate) = fitted per geometry; scales with growth rate alpha (Fig 4g)
Introduced as the linear decay term in Eq. (9) and fitted simultaneously with beta. The fit landscape is degenerate (Supp Fig S4), so separate values are poorly constrained; mu also changes with geometry and growth excess.
inner boundary alignment q(R0) = 0
In the excess-growth cases (Section III C), the authors impose q(R0)=0 based on the observation that the inner sector behaves like free expansion, not from the governing equations. This boundary condition is needed to fix the homogeneous solution for nonzero injection velocity.

assumptions (6)

domain assumption Colony is incompressible with constant density and isotropic growth source: div v = alpha.
Section II A, Eq. (2). Used to derive the pressure Poisson equation and velocity profiles. The paper later acknowledges density is only approximately constant (Section IV).
domain assumption Flow is overdamped Darcy flow from a scalar pressure, v = -zeta grad p, so vorticity vanishes.
Section II A, Eq. (3) and surrounding text. This makes the flow curl-free and removes orientation feedback; it is load-bearing for the geometry-only predictions.
domain assumption The nematic director follows the principal axis of the traceless shear tensor uST at low alignment strength.
Section II A: 'assuming that Q is approximately proportional to uST'. This is the basis for reading orientation directly from the shear; it is later relaxed by the advection-decay model.
domain assumption The colony is in steady state, with open boundaries removing material to balance cell growth.
Section II A: 'we restrict our considerations to systems in which particle density, flow and orientation have reached a steady state'. This determines the open and closed boundary conditions for the Poisson problem.
ad hoc to paper A linear decay term -mu Q is the correct minimal balance to the constant shear source.
Eq. (9): 'to compensate for the constant source term, we introduce a simple linear decay with rate mu'. No microscopic derivation is given; the rate is fitted to data.
standard math Q stays bounded at the inner radius, so the homogeneous integration constant C is set to zero.
Section III A, following Eq. (11): 'Assuming that Q does not diverge at R0, we set the integration parameter C to zero'. This is a regularity condition for the ODE solution.

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Cite this review

Pith. "Pith review of Prediction and control of geometry-induced nematic order in growing multicellular systems." pith.science (2026). https://pith.science/paper/BEDB235M

@misc{pith2026250610867,
  author       = {Pith},
  title        = {Pith review of: Prediction and control of geometry-induced nematic order in growing multicellular systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BEDB235M}},
  note         = {Machine review of arXiv:2506.10867}
}

read the original abstract

In densely-packed two-dimensional systems of growing cells, such as rod-shaped bacteria, a number of experimental and numerical studies report distinct patterns of nematic orientational order in the presence of confinement. So far, these effects have been explained using variations of growing active nematic continuum theories, which incorporate feedback between growth-induced active stresses, the resulting material flow and nematic orientation, and were adapted to the specific geometry under investigation. Here, we first show that a direct, analytical prediction of orientation patterns based on a simple isotropic-growth assumption and the shear rate tensor of the expansion flow already covers previously observed cases. We use this method to tune orientation patterns and net topological defect charge in a systematic way using domain geometry, confirmed by agent-based simulations. We then show how this framework can be extended to quantitatively capture alignment strength, and explore its potential for cross-prediction across different geometries. Our simplified and unifying theoretical framework highlights the role of domain geometry in shaping nematic order of growing systems, and thereby provides a way to forward-engineer desired orientation patterns.

Figures

Figures reproduced from arXiv: 2506.10867 by the authors.

**Figure 2.** (a) Simulation snapshot of a circular colony of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a) Defect charge in polygonal domains as a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (a) Snapshot of a colony in a ring-shaped domain with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: (a) Time-averaged orientation fields for wedge-shaped domains with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Snapshot of a channel-like system engineered to [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

Discussion (0). Continue with ORCID to comment.

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