Pith. sign in

REVIEW 4 major objections 4 minor 101 references

Towards a two-scale model for morphogenesis -- How cellular processes influence tissue deformations

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A two-scale model proposes that the local number of a cell's neighbors sets the tissue's bending stiffness, so cellular rearrangements directly reshape the tissue.

desk verdict A genuinely new two-scale coupling, benchmarked in the passive limit, but the central scale-separation assumption is asserted rather than demonstrated, and the 92-cell statistics are too thin for strong claims. read the letter →

arxiv 2506.19475 v1 pith:7MXGV22G submitted 2025-06-24 cond-mat.soft math-phmath.MP

Lea Happel , Axel Voigt This is my paper · ORCID
classification cond-mat.softmath-phmath.MP
keywords two-scalemodelmorphogenesismultiphase-fieldbendingrigiditytopologicaldefectsinepitheliaactivedeformablecellsevolvingsurfacescellneighbornumber
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 aims to establish that tissue-scale deformations can be driven by the arrangement of individual cells: each cell is resolved as a deformable phase field on an evolving surface, and the surface's bending stiffness is lowered wherever a cell has other than six neighbors. Because cells with five or seven neighbors are the defects of epithelial packing, this turns neighbor-count differences into local curvature changes, and shape changes feed back into the cell arrangement. The numerical explorations show transitions, such as cells rotating on a fixed vesicle-like shape versus carrying the whole shape around with them, and show that activity broadens the neighbor-number distribution and shifts its mean toward five. If the model is right, it provides a concrete computational route from cell-scale behavior to quantitative morphogenesis predictions.

What carries the argument

The load-bearing object is the relation $\kappa_{\mathrm{bending}}(x,t) = k_{\mathrm{fac}}(\tanh(-|n_{\mathrm{neigh}}(x,t)-6|)+1)+k_{\mathrm{lower}}$, which assigns a reduced bending rigidity to a cell whose neighbor count $n_{\mathrm{neigh}}$ deviates from six, the hexagonal-packing ideal. This single function carries the two-scale coupling: through it the cellular network modifies the surface's bending-energy evolution, while the surface's motion feeds back by advecting and deforming the phase fields. The second piece of machinery is the explicit scale-separation assumption that the cell-interface and cell-cell interaction energies are confined to thin interfaces and can be omitted from the surface evolution equations; this keeps the scheme computable while leaving the neighbor-dependent rigidity as the only cellular channel to the tissue scale. Each phase field is solved on its own refinement of the evolving surface, with neighbor interactions communicated on a common mesh, so the numerical cost scales essentially with the number of cells.

What would settle it

Re-run the N=12 and N=32 computations with the cell-interface and interaction energies kept in the surface evolution equations; if the equilibrium shapes change appreciably, the scale-separation premise fails. A complementary experimental check is whether five-neighbor cells in a curved epithelial monolayer sit in regions of measurably lower local bending stiffness, as the rigidity relation prescribes.

Watch

Extended reading notes

Core claim

The central claim is that a two-scale coupling through a neighbor-dependent bending rigidity is sufficient to capture how cellular processes influence tissue deformations. The surface evolves by the standard bending energy of a thin elastic sheet, with constraints on constant area and enclosed volume; on it, each cell is a phase field with its own self-propulsion, and the bending rigidity at a point is reduced the further the nearest cell's neighbor count deviates from six. The paper's computations show that this minimal coupling already produces symmetry breaking: for 12 cells, a prolate shape deforms below a reduced volume of roughly $V_r \approx 0.96$; for 32 cells, the six five-neighbor cells shape a truncated-icosahedral surface at higher $V_r$; and active rotation transitions from collective motion on a stationary surface to rigid rotation of the whole shape. For 92 cells, the mean neighbor number shifts from six toward five with activity, cells with five neighbors sit in regions of stronger negative curvature, mean-curvature gradients grow, and relative shape change increases with activity while absolute deviation from the equilibrium prolate decreases. These results are presented as qualitative numerical evidence, not as quantitative biological predictions.

Load-bearing premise

The model assumes that the forces concentrated in cell boundaries are too small to move the tissue surface, leaving the neighbor-dependent bending stiffness as the only channel from cell behavior to tissue shape; if those boundary forces matter, the coupling misses its main load.

Editorial extensions

If this is right

  • Neighbor exchanges (cellular rearrangements) become shape-changing events, because changing a cell's neighbor count locally softens or stiffens the surface.
  • Topological defects localize curvature: cells with five neighbors concentrate strong negative curvature, so the defect network determines the symmetry axes of the deformed tissue.
  • Cellular activity alone can switch the tissue between two dynamical regimes: collective rotation of cells on a fixed shape and rigid-body rotation of the whole tissue.
  • The model predicts enhanced gradients of mean curvature at the cell scale, a proposed source of active geometric forces, so cellular-scale activity could initiate further large-scale morphogenesis.
  • Because the numerics scale with the number of cells, simulations with hundreds of cells become feasible, enabling statistical comparisons with continuous tissue models.

Reading between the lines

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

  • Editorial inference: the scale-separation assumption can be tested directly by re-running the same computations with the cell-interface and interaction energies kept in the surface evolution equations; if equilibrium shapes change materially, the neighbor-dependent rigidity is not the only coupling that matters.
  • Editorial inference: the predicted shift of the mean neighbor number toward five with increasing activity is a measurable signature; time-lapse imaging of curved epithelial monolayers or organoids could look for whether local curvature correlates with five-neighbor cells as prescribed by the rigidity relation.
  • Editorial inference: the same coupling mechanism could be extended to cell growth or division by letting the rigidity field respond to newly created neighbors, turning the constant-cell-size assumption into a testable extension.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a two-scale model for tissue morphogenesis in which each cell is resolved as a phase field on an evolving surface, while the surface itself evolves under a Canham-Helfrich bending energy whose rigidity depends locally on the number of neighbors of the underlying cell. Cell activity is introduced through a self-propulsion term with rotational noise. The authors validate the passive limit against the classical Helfrich phase diagram and against known optimal arrangements of cells on a sphere, then study active cases with N=12, N=32, and N=92 cells, reporting collective rotation, rigid-body rotation of the surface, activity-dependent shifts in neighbor-number statistics, and correlations between local mean curvature and neighbor number. The paper is explicitly framed as a qualitative step toward quantitative morphogenesis modeling, and it discusses several limitations in the final section.

Significance. If the central modeling assumptions are accepted, the paper offers a computationally tractable route from cellular-scale activity to tissue-scale shape change, with a numerical method that scales essentially with the number of cells. The consistency checks are genuine strengths: the passive limit reproduces the Helfrich phase diagram from [74], and the N=12 and N=32 configurations reproduce known Thomson/Tammes arrangements on a sphere. The authors are also transparent about the phenomenological nature of the neighbor-dependent bending rigidity and about the computational constraints. However, the central claim that cellular processes influence tissue deformations rests on a scale-separation premise that is asserted rather than quantified, and the coupling via Eq. (12) is chosen by hand without sensitivity analysis. These issues affect the load-bearing mechanism of the model, so the manuscript requires substantial revision before the central claim can be considered established.

major comments (4)
  1. [Section 2.3, Eqs. (2)-(5) and (8)] The central scale-separation assumption is asserted, not demonstrated. The text states, "Due to separation of scales we question the necessity of F_CH and F_Int to be considered in eqs. (2)-(4). Both are essentially only nonzero within the vicinity of the cell interfaces and thus on a small scale, which is not relevant for the large scale surface evolution," but spatial localization is not a sufficient criterion. In two-component vesicle models, interface line tension is localized at the interface and yet directly drives budding and tubulation. For the parameters in Table 1 (Ca=10, In=0.05, epsilon=0.01, and kappa_bending in [0.008, 0.03]), the normal forces from F_CH and F_Int are never compared with delta(F_Helf)/delta(X). Since the stated objective is to let cellular processes influence tissue deformations, omitting the direct mechanical action of the very energies that define cell deformability and cell-cell adhesion weakens the central claim unless an order-of-magnitude estimate or a direct inclusion of these terms is provided.
  2. [Section 2.3, Eq. (12)] The neighbor-dependent bending rigidity is introduced phenomenologically as kappa_bending = kfac(tanh(-|n_neigh-6|)+1) + klower, with no derivation from cell mechanics and no calibration against experimental or defect-mechanics data. Since Eq. (12) is the only mechanism by which cellular topology affects the tissue-scale curvature evolution, the arbitrariness of the chosen form and of the parameters kfac and klower is load-bearing. The authors acknowledge the phenomenological character, but the manuscript would need at least a sensitivity study over kfac and klower, or a comparison with an independently derived defect-based estimate, to support the general conclusions drawn from Figures 4-16.
  3. [Section 3.4, Figs. 11-16] The N=92 results appear to be based on single stochastic trajectories. Because the active force contains a Wiener process and neighbor rearrangements are stochastic, the reported averages, standard deviations, and shape-change measures are not statistically characterized. Multiple realizations, or at least time-block averaging with confidence intervals, are needed before the monotonic trends in v0 and Vr claimed in Figures 11, 12, and 16 can be considered robust.
  4. [Section 2.4] The numerical scheme is validated only by reference to previous convergence results for v0=0, a single phase field, and a stronger area-conservation constraint. The coupled system here involves an evolving surface with kappa_bending depending on the discrete neighbor number n_neigh, which changes discontinuously during T1-type rearrangements. No convergence or mesh-resolution study is provided for this coupled regime, so the quantitative accuracy of the averaged quantities in Section 3.4 is not established.
minor comments (4)
  1. [Throughout] The word "resamples" is used repeatedly where "resembles" is intended (e.g., Sections 3.1 and 3.2, including the caption of Figure 4).
  2. [Reference [36]] The reference to Wenzel and Voigt lists volume 184, which is not a valid Physical Review E volume; the correct volume appears to be 103 (article 054410, 2021).
  3. [Figure 15] The axis labels in Figure 15 contain LaTeX markup artifacts ("D ´ cell H´ cell 1") that make the figure difficult to read and should be typeset properly.
  4. [Section 3.3] The distinction between collective rotation on a fixed shape and rigid-body rotation of the surface is inferred from the behavior of q_l and |q_3^m|, but no quantitative threshold or statistical test is given; a precise criterion would strengthen this central phenomenological claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cell-tissue coupling is a declared phenomenological input, benchmarks are external, and none of the reported shape changes is a fitted quantity renamed as a prediction.

full rationale

The derivation chain is self-contained. The surface evolution in eqs. (1)-(5) is the standard Canham/Helfrich gradient flow, the cell-scale multiphase-field equations in eqs. (6)-(11) are standard phase-field equations with an added activity term, and the only cross-scale coupling is eq. (12), which the paper explicitly labels phenomenological: 'The dependency of κbending on nneigh is phenomenological and chosen similar to the discrete defect localization approach in [64].' No parameter is fitted to a quantity that is later presented as an independent prediction. The consistency test in Section 3.1 is checked against the external Helfrich phase diagram of Seifert [74], and the comparisons with the authors' earlier stationary-surface results [39, 40] are validation checks of the numerical setting, not inputs that force the new symmetry-breaking or rigid-body-rotation behaviors. The self-citations that appear ([39], [40], [64], [81]) serve as numerical benchmarks, tool provenance, or admitted modeling inspiration; none is an unverified load-bearing theorem or ansatz disguised as a derivation. The observed concentration of mean-curvature magnitude at five-neighbor cells is a direct mathematical consequence of lowering κbending at such cells through eq. (12), and the paper states this mechanism explicitly ('Local variations in the bending rigidity lead to changes in mean curvature') rather than presenting it as an independent empirical discovery. The scale-separation neglect of F_CH and F_Int in eqs. (2)-(4) is a substantive modeling assumption whose quantitative accuracy could be questioned, but that is a correctness or validation concern, not circularity: the assumption is stated, not derived from the results it is used to produce.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The model's predictive content is carried by many hand-picked parameters and explicit assumptions. The most important free ingredients are the bending rigidity parameters kfac and klower, the cell interaction and deformability constants, and the activity values. The neighbor-dependent bending law is a postulate, and the scale-separation step that removes cell energies from the surface equation is a domain assumption stated but not derived.

free parameters (10)
  • kfac = 0.022
    Sets the range of the neighbor-dependent bending rigidity in Eq. (12); chosen by hand, not derived.
  • klower = 0.008
    Minimum bending rigidity approached for nonzero |n_neigh - 6| in Eq. (12); chosen by hand.
  • Ca = 10.0
    Capillary number controlling cell deformability in F_CH; chosen, not fitted to data.
  • In = 0.05
    Interaction strength in F_Int; chosen ad hoc.
  • arep = 0.0625
    Repulsive interaction coefficient in Eq. (7); chosen ad hoc.
  • aatt = 0.48
    Attractive interaction coefficient in Eq. (7); chosen ad hoc.
  • epsilon = 0.01
    Diffuse interface width; a numerical resolution parameter.
  • Dr = 0.005
    Rotational diffusion coefficient for activity noise; chosen ad hoc.
  • v0 = 0.1 to 0.4
    Self-propulsion strength; varied across simulations, not fitted.
  • packing_fraction = 0.94
    Packing fraction of cells; chosen because 100% packing is not feasible with diffuse interfaces.
assumptions (6)
  • domain assumption Epithelial tissue is modeled as a closed 2D surface of constant thickness with constant area and enclosed volume.
    Section 2.1 states this as the starting geometry; no thickness, growth, or volume change is included.
  • domain assumption Cells have identical, constant area and no proliferation; they are modeled as deformable active Brownian particles without sub-cellular details.
    Section 2.1 describes cells as active deformable objects with constant area; Section 4 acknowledges the neglect of proliferation and sub-cellular processes.
  • domain assumption Surface evolution is an L2-gradient flow of the Canham-Helfrich energy; the surface only resists bending.
    Section 2.3 uses Eq. (2) with bending energy only, plus area and volume constraints.
  • ad hoc to paper Phase-field and interaction energies F_CH and F_Int do not contribute to the surface evolution equations because of scale separation.
    Section 2.3 explicitly states: 'Due to separation of scales we question the necessity of F_CH and F_Int to be considered in eqs. (2)-(4)'. This is load-bearing for the two-scale coupling.
  • ad hoc to paper Bending rigidity depends on neighbor number only through Eq. (12), with a chosen tanh form and parameters kfac and klower.
    Eq. (12) is introduced as a phenomenological relation; the paper does not derive it from cell mechanics.
  • domain assumption Two cells are neighbors if their diffuse interfaces overlap, defined as {phi_i | phi_i > -0.5} intersecting {phi_j | phi_j > -0.5}.
    Section 2.3 defines neighbor detection based on the phase-field threshold, following [63].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Towards a two-scale model for morphogenesis -- How cellular processes influence tissue deformations." pith.science (2026). https://pith.science/paper/7MXGV22G

@misc{pith2026250619475,
  author       = {Pith},
  title        = {Pith review of: Towards a two-scale model for morphogenesis -- How cellular processes influence tissue deformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7MXGV22G}},
  note         = {Machine review of arXiv:2506.19475}
}
read the original abstract

We propose a two-scale model to resolve essential features of developmental tissue deformations. The model couples individual cellular behavior to the mechanics at tissue scale. This is realized by a multiphase-field model addressing the motility, deformability and interaction of cells on an evolving surface. The surface evolution is due to bending elasticity, with bending properties influenced by the topology of the cellular network, which forms the surface. We discuss and motivate model assumptions, propose a numerical scheme, which essentially scales with the number of cells, and explore computationally the effect of the two-scale coupling on the global shape evolution. The approach provides a step towards more quantitative modeling of morphogenetic processes.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

101 extracted references · 78 canonical work pages

  1. [74]

    Advances in Physics 46, 13–137 (1997)

    Seifert, U.: Configurations of fluid membranes and vesicles. Advances in Physics 46, 13–137 (1997)

    work page 1997

  2. [1]

    Cambridge University Press, Cam- bridge (1917)

    Thompson, D.W.: On Growth and Form. Cambridge University Press, Cam- bridge (1917)

    1917

  3. [2]

    Cell 153, 948–962 (2013)

    Heisenberg, C.-P., Bella¨ ıche, Y.: Forces in tissue morphogenesis and patterning. Cell 153, 948–962 (2013)

    2013

  4. [3]

    Trends in Cell Biology 22, 536–545 (2012)

    Salbreux, G., Charras, G., Paluch, E.: Actin cortex mechanics and cellular morphogenesis. Trends in Cell Biology 22, 536–545 (2012)

    2012

  5. [4]

    Journal of the Mechanics and Physics of Solids 164, 104876 (2022) 23

    Da Rocha, H.B., Bleyer, J., Turlier, H.: A viscous active shell theory of the cell cortex. Journal of the Mechanics and Physics of Solids 164, 104876 (2022) 23

    2022

  6. [5]

    Cell 186, 3049–3061 (2023)

    De Belly, H., Yan, S., Rocha, H.B., Ichbiah, S., Town, J.P., Zager, P.J., Estrada, D.C., Meyer, K., Turlier, H., Bustamante, C., Weiner, O.D.: Cell protrusions and contractions generate long-range membrane tension propagation. Cell 186, 3049–3061 (2023)

    2023

  7. [6]

    eLife 12, 75878 (2023)

    Khoromskaia, D., Salbreux, G.: Active morphogenesis of patterned epithelial shells. eLife 12, 75878 (2023)

    2023

  8. [7]

    Reports on Progress in Physics 81, 076601 (2018)

    J¨ ulicher, F., Grill, S.W., Salbreux, G.: Hydrodynamic theory of active matter. Reports on Progress in Physics 81, 076601 (2018)

    2018

Show all 101 references
  1. [8]

    Physical Review E 100, 022413 (2019)

    Morris, R.G., Rao, M.: Active morphogenesis of epithelial monolayers. Physical Review E 100, 022413 (2019)

    2019

  2. [9]

    Physical Review Research 4, 033158 (2022)

    Salbreux, G., J¨ ulicher, F., Prost, J., Callan-Jones, A.: Theory of nematic and polar active fluid surfaces. Physical Review Research 4, 033158 (2022)

    2022

  3. [10]

    Journal of Fluid Mechanics 957, 4 (2023)

    Al-Izzi, S.C., Morris, R.G.: Morphodynamics of active nematic fluid surfaces. Journal of Fluid Mechanics 957, 4 (2023)

    2023

  4. [11]

    Proceed- ings of the Royal Society A 481(2310), 20240380 (2025)

    Nitschke, I., Voigt, A.: Active nematodynamics on deformable surfaces. Proceed- ings of the Royal Society A 481(2310), 20240380 (2025)

    2025

  5. [12]

    Journal of Fluid Mechanics 872, 218– 271 (2019)

    Torres-S´ anchez, A., Mill´ an, D., Arroyo, M.: Modelling fluid deformable surfaces with an emphasis on biological interfaces. Journal of Fluid Mechanics 872, 218– 271 (2019)

    2019

  6. [13]

    Journal of Fluid Mechanics 900, 8 (2020)

    Reuther, S., Nitschke, I., Voigt, A.: A numerical approach for fluid deformable surfaces. Journal of Fluid Mechanics 900, 8 (2020)

    2020

  7. [14]

    Journal of Computational Physics 486, 112097 (2023)

    Krause, V., Voigt, A.: A numerical approach for fluid deformable surfaces with conserved enclosed volume. Journal of Computational Physics 486, 112097 (2023)

    2023

  8. [15]

    Physics of Fluids 36, 102120 (2024)

    Porrmann, M., Voigt, A.: Shape evolution of fluid deformable surfaces under active geometric forces. Physics of Fluids 36, 102120 (2024)

    2024

  9. [16]

    Advances in Differential Equations 30, 335–420 (2025)

    Nitschke, I., Voigt, A.: Beris-Edwards models on evolving surfaces: A Lagrange- D’Alembert approach. Advances in Differential Equations 30, 335–420 (2025)

    2025

  10. [17]

    Journal of Computational Physics 405, 109168 (2020)

    Torres-S´ anchez, A., Santos-Oliv´ an, D., Arroyo, M.: Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations. Journal of Computational Physics 405, 109168 (2020)

    2020

  11. [18]

    Journal of Computational Physics 389, 48–61 (2019) 24

    Nestler, M., Nitschke, I., Voigt, A.: A finite element approach for vector-and tensor-valued surface PDEs. Journal of Computational Physics 389, 48–61 (2019) 24

    2019

  12. [19]

    Nature Physics 17, 859–866 (2021)

    Kim, S., Pochitaloff, M., Stooke-Vaughan, G.A., Camp` as, O.: Embryonic tissues as active foams. Nature Physics 17, 859–866 (2021)

    2021

  13. [20]

    Science Advances 10, 0860 (2024)

    Fuhrmann, J.F., Krishna, A., Paijmans, J., Duclut, C., Cwikla, G., Eaton, S., Popovi´ c, M., J¨ ulicher, F., Modes, C.D., Dye, N.A.: Active shape programming drives drosophila wing disc eversion. Science Advances 10, 0860 (2024)

    2024

  14. [21]

    Nature Computational Science 4, 299– 309 (2024)

    Runser, S., Vetter, R., Iber, D.: SimuCell3D: three-dimensional simulation of tissue mechanics with cell polarization. Nature Computational Science 4, 299– 309 (2024)

    2024

  15. [22]

    bioRxiv (2025) https://doi.org/10.1101/2025.03

    Ouzeri, A., Kale, S., Chahare, N., Torres-Sanchez, A., Santos-Olivan, D., Trepat, X., Arroyo, M.: Theory of multiscale epithelial mechanics under stretch: from active gels to vertex models. bioRxiv (2025) https://doi.org/10.1101/2025.03. 23.644792

    2025 doi

  16. [23]

    Physical Review A 38, 1005–1018 (1988)

    Seung, H.S., Nelson, D.R.: Defects in flexible membranes with crystalline order. Physical Review A 38, 1005–1018 (1988)

    1988

  17. [24]

    Nature Communications 4(1), 2098 (2013)

    Lehtinen, O., Kurasch, S., Krasheninnikov, A., Kaiser, U.: Atomic scale study of the life cycle of a dislocation in graphene from birth to annihilation. Nature Communications 4(1), 2098 (2013)

    2013

  18. [25]

    Mechanics of Materials 198, 105114 (2024)

    Benoit–Mar´ echal, L., Nitschke, I., Voigt, A., Salvalaglio, M.: Mesoscale modeling of deformations and defects in thin crystalline sheets. Mechanics of Materials 198, 105114 (2024)

    2024

  19. [26]

    Nature 544, 212–216 (2017)

    Saw, T.B., Doostmohammadi, A., Nier, V., Kocgozlu, L., Thampi, S., Toyama, Y., Marcq, P., Lim, C.T., Yeomans, J.M., Ladoux, B.: Topological defects in epithelia govern cell death and extrusion. Nature 544, 212–216 (2017)

    2017

  20. [27]

    Proceedings of the Royal Society A 476, 20200313 (2020)

    Nitschke, I., Reuther, S., Voigt, A.: Liquid crystals on deformable surfaces. Proceedings of the Royal Society A 476, 20200313 (2020)

    2020

  21. [28]

    Nature Materials 21, 588–597 (2022)

    Guillamat, P., Blanch-Mercader, C., Pernollet, G., Kruse, K., Roux, A.: Inte- ger topological defects organize stresses driving tissue morphogenesis. Nature Materials 21, 588–597 (2022)

    2022

  22. [29]

    Science Advances 8, 2712 (2022)

    Hoffmann, L.A., Carenza, L.N., Eckert, J., Giomi, L.: Theory of defect-mediated morphogenesis. Science Advances 8, 2712 (2022)

    2022

  23. [30]

    Nature Physics 17, 251–259 (2021)

    Maroudas-Sacks, Y., Garion, L., Shani-Zerbib, L., Livshits, A., Braun, E., Keren, K.: Topological defects in the nematic order of actin fibres as organization centres of hydra morphogenesis. Nature Physics 17, 251–259 (2021)

    2021

  24. [31]

    Development 152, 204514 (2025)

    Maroudas-Sacks, Y., Suganthan, S., Garion, L., Ascoli-Abbina, Y., Westfried, A., Dori, N., Pasvinter, I., Popovi´ c, M., Keren, K.: Mechanical strain focusing at 25 topological defect sites in regenerating hydra. Development 152, 204514 (2025)

    2025

  25. [32]

    Physical Review X 6, 021011 (2016)

    Bi, D., Yang, X., Marchetti, M.C., Manning, M.L.: Motility-driven glass and jamming transitions in biological tissues. Physical Review X 6, 021011 (2016)

    2016

  26. [33]

    Proceedings of the National Academy of Science (USA) 111, 14770–14775 (2014)

    Camley, B.A., Zhang, Y., Zhao, Y., Li, B., Ben-Jacob, E., Levine, H., Rappel, W.-J.: Polarity mechanisms such as contact inhibition of locomotion regulate persistent rotational motion of mammalian cells on micropatterns. Proceedings of the National Academy of Science (USA) 111...

    2014

  27. [34]

    Physical Review Letters 122, 048004 (2019)

    Mueller, R., Yeomans, J.M., Doostmohammadi, A.: Emergence of active nematic behavior in monolayers of isotropic cells. Physical Review Letters 122, 048004 (2019)

    2019

  28. [35]

    Physical Review Letters 125, 038003 (2020)

    Loewe, B., Chiang, M., Marenduzzo, D., Marchetti, M.C.: Solid-liquid transition of deformable and overlapping active particles. Physical Review Letters 125, 038003 (2020)

    2020

  29. [36]

    Physical Review E 184, 054410 (2021)

    Wenzel, D., Voigt, A.: Multiphase field models for collective cell migration. Physical Review E 184, 054410 (2021)

    2021

  30. [37]

    Current Biology 17, 2095–2104 (2007)

    Farhadifar, R., Roeper, J.-C., Algouy, B., Eaton, S., J¨ ulicher, F.: The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing. Current Biology 17, 2095–2104 (2007)

    2007

  31. [38]

    Nature Communications 16, 530 (2025)

    Rozman, J., Chaithanya, K., Yeomans, J.M., Sknepnek, R.: Vertex model with internal dissipation enables sustained flows. Nature Communications 16, 530 (2025)

    2025

  32. [39]

    EPL 138, 67002 (2022)

    Happel, L., Wenzel, D., Voigt, A.: Effects of curvature on epithelial tissue - coordinated rotational movement and other spatiotemporal arrangements. EPL 138, 67002 (2022)

    2022

  33. [40]

    Physical Review Letters 132, 078401 (2024)

    Happel, L., Voigt, A.: Coordinated motion of epithelial layers on curved surfaces. Physical Review Letters 132, 078401 (2024)

    2024

  34. [41]

    Journal of Theoretical Biology 26, 61 (1970)

    Canham, P.B.: The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology 26, 61 (1970)

    1970

  35. [42]

    Zeitschrift f¨ ur Naturforschung C28, 693–703 (1973)

    Helfrich, W.: Elastic properties of lipid bilayers: Theory and possible experi- ments. Zeitschrift f¨ ur Naturforschung C28, 693–703 (1973)

    1973

  36. [43]

    Physical Review E 79, 031915 (2009)

    Arroyo, M., DeSimone, A.: Relaxation dynamics of fluid membranes. Physical Review E 79, 031915 (2009)

    2009

  37. [44]

    Multiscale Modeling & Simulation 13, 632–643 (2015) 26

    Reuther, S., Voigt, A.: The interplay of curvature and vortices in flow on curved surfaces. Multiscale Modeling & Simulation 13, 632–643 (2015) 26

    2015

  38. [45]

    Multiscale Modeling & Simulation 16, 1448–1453 (2018)

    Reuther, S., Voigt, A.: Erratum: The interplay of curvature and vortices in flow on curved surfaces. Multiscale Modeling & Simulation 16, 1448–1453 (2018)

    2018

  39. [46]

    Physical Review Letters 129, 048102 (2022)

    Grossman, D., Joanny, J.-F.: Instabilities and geometry of growing tissues. Physical Review Letters 129, 048102 (2022)

    2022

  40. [47]

    PRX Life 3(2), 023002 (2025)

    Claussen, N.H., Brauns, F.: Mean-field model for active plastic flow of epithelial tissue. PRX Life 3(2), 023002 (2025)

    2025

  41. [48]

    eLife 13 (2024)

    Armengol-Collado, J.-M., Livio, N.C., Giomi, L.: Hydrodynamics and multiscale order in confluent epithelia. eLife 13 (2024)

    2024

  42. [49]

    arXiv:2501.07280 (2025)

    Nejad, M.R., Yeomans, J.M.: Coarse-graining dense, deformable active particles. arXiv:2501.07280 (2025)

    2025 arXiv

  43. [50]

    arXiv:2503.05053 (2025)

    Monfared, S., Ardaˇ seva, A., Doostmohammadi, A.: Multi-phase-field models of biological tissues. arXiv:2503.05053 (2025)

    2025 arXiv

  44. [51]

    Scientific Reports 13, 10096 (2023)

    Jain, H.P., Voigt, A., Angheluta, L.: Robust statistical properties of T1 tran- sitions in a multi-phase field model of cell monolayers. Scientific Reports 13, 10096 (2023)

    2023

  45. [52]

    Journal of Fluid Mechanics 977, 41 (2023)

    Bachini, E., Krause, V., Nitschke, I., Voigt, A.: Derivation and simulation of a two-phase fluid deformable surface model. Journal of Fluid Mechanics 977, 41 (2023)

    2023

  46. [53]

    Journal of The Royal Society Interface 21, 20240056 (2024)

    Krause, V., Voigt, A.: Wrinkling of fluid deformable surfaces. Journal of The Royal Society Interface 21, 20240056 (2024)

    2024

  47. [54]

    Mathematical Methods in Applied Science 44, 5385–5405 (2021)

    Salvalaglio, M., Voigt, A., Wise, S.M.: Doubly degenerate diffuse interface mod- els of surface diffusion. Mathematical Methods in Applied Science 44, 5385–5405 (2021)

    2021

  48. [55]

    Journal of Computational Physics 275, 626–641 (2014)

    Gu, R., Wang, X., Gunzburger, M.: Simulating vesicle-substrate adhesion using two phase field functions. Journal of Computational Physics 275, 626–641 (2014)

    2014

  49. [56]

    Interface Focus 6, 20160037 (2016)

    Marth, W., Voigt, A.: Collective migration under hydrodynamic interactions: a computational approach. Interface Focus 6, 20160037 (2016)

    2016

  50. [57]

    Journal of Chemical Physics 150, 164108 (2019)

    Wenzel, D., Praetorius, S., Voigt, A.: Topological and geometrical quantities in active cellular structures. Journal of Chemical Physics 150, 164108 (2019)

    2019

  51. [58]

    Acta Numerica 22, 289–396 (2013)

    Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numerica 22, 289–396 (2013)

    2013

  52. [59]

    Journal of Geometry and Physics 173, 104428 (2022) 27

    Nitschke, I., Voigt, A.: Observer-invariant time derivatives on moving surfaces. Journal of Geometry and Physics 173, 104428 (2022) 27

    2022

  53. [60]

    SIAM Journal on Mathematical Analysis 55, 6625–6675 (2023)

    Caetano, D., Elliott, C.M., Grasselli, M., Poiatti, A.: Regularization and separation for evolving surface Cahn–Hilliard equations. SIAM Journal on Mathematical Analysis 55, 6625–6675 (2023)

    2023

  54. [61]

    Computer Methods in Applied Mechanics and Engineering to appear (2025)

    Sischka, J.M., Voigt, A.: Two-phase fluid deformable surfaces with constant enclosed volume and phase-dependent bending and gaussian rigidity. Computer Methods in Applied Mechanics and Engineering to appear (2025)

    2025

  55. [62]

    Physical Review Research 6, 033176 (2024)

    Jain, H.P., Voigt, A., Angheluta, L.: From cell intercalation to flow, the importance of T1 transitions. Physical Review Research 6, 033176 (2024)

    2024

  56. [63]

    eLife 12, 82435 (2023)

    Monfared, S., Ravichandran, G., Andrade, J., Doostmohammadi, A.: Mechanical basis and topological routes to cell elimination. eLife 12, 82435 (2023)

    2023

  57. [64]

    Multiscale Modeling & Simulation 10, 82–110 (2012)

    Aland, S., R¨ atz, A., R¨ oger, M., Voigt, A.: Buckling instability of viral capsids—a continuum approach. Multiscale Modeling & Simulation 10, 82–110 (2012)

    2012

  58. [65]

    Journal of Computational Physics 227, 4281–4307 (2008)

    Barrett, J.W., Garcke, H., N¨ urnberg, R.: On the parametric finite element approximation of evolving hypersurfaces in R3. Journal of Computational Physics 227, 4281–4307 (2008)

    2008

  59. [66]

    Archive of Numerical Software 6, 1–27 (2022)

    Praetorius, S., Stenger, F.: Dune-CurvedGrid - A Dune module for surface parametrization. Archive of Numerical Software 6, 1–27 (2022)

    2022

  60. [67]

    In: Binder, K., M¨ uller, M., Trautmann, A

    Praetorius, S., Voigt, A.: Collective cell behavior - a cell-based paralleliza- tion approach for a phase field active polar gel model. In: Binder, K., M¨ uller, M., Trautmann, A. (eds.) NIC Symposium 2018, vol. 49, pp. 369–376. Forschungszentrum J¨ ulich GmbH, Zentralbiblioth...

    2018

  61. [68]

    https://gitlab.com/amdis/amdis

    Adaptive Multi-Dimensional Simulations. https://gitlab.com/amdis/amdis

  62. [69]

    Computation and Visualization in Science 10, 57–67 (2007)

    Vey, S., Voigt, A.: AMDiS: Adaptive multidimensional simulations. Computation and Visualization in Science 10, 57–67 (2007)

    2007

  63. [70]

    Advances in Computational Mathematics 41, 1145–1177 (2015)

    Witkowski, T., Ling, S., Praetorius, S., Voigt, A.: Software concepts and numer- ical algorithms for a scalable adaptive parallel finite element method. Advances in Computational Mathematics 41, 1145–1177 (2015)

    2015

  64. [71]

    Computers & Mathematics with Applications 81, 75–112 (2021)

    Bastian, P., Blatt, M., Dedner, A., Dreier, N.-A., Engwer, C., Fritze, R., Gr¨ aser, C., Gr¨ uninger, C., Kempf, D., Kl¨ ofkorn, R., Ohlberger, M., Sander, O.: The Dune framework: Basic concepts and recent developments. Computers & Mathematics with Applications 81, 75–112 (2021)

    2021

  65. [72]

    Springer, Berlin (2020)

    Sander, O.: DUNE — The Distributed and Unified Numerics Environment. Springer, Berlin (2020)

    2020

  66. [73]

    Archive of Numerical Software 4, 1–28 (2016) 28

    Alk¨ amper, M., Dedner, A., Kl¨ ofkorn, R., Nolte, M.: The dune-alugrid module. Archive of Numerical Software 4, 1–28 (2016) 28

    2016

  67. [75]

    The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 7, 237–265 (1904)

    Thomson, J.J.: On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure. The London, ...

    1904

  68. [76]

    Acta Crystallographica Section A 43, 612–616 (1987)

    Tarnai, T., G´ asp´ ar, Z.: Multi-symmetric close packings of equal spheres on the spherical surface. Acta Crystallographica Section A 43, 612–616 (1987)

    1987

  69. [77]

    Physical Review E 81, 025701 (2010)

    Backofen, R., Voigt, A., Witkowski, T.: Particles on curved surfaces: A dynamic approach by a phase-field-crystal model. Physical Review E 81, 025701 (2010)

    2010

  70. [78]

    Multiscale Modeling & Simulation 9, 314–334 (2011)

    Backofen, R., Gr¨ af, M., Potts, D., Praetorius, S., Voigt, A., Witkowski, T.: A con- tinuous approach to discrete ordering on S2. Multiscale Modeling & Simulation 9, 314–334 (2011)

    2011

  71. [79]

    Physics of Fluids 35, 062111 (2023)

    Olshanskii, M.: On equilibrium states of fluid membranes. Physics of Fluids 35, 062111 (2023)

    2023

  72. [80]

    Proceedings of Applied Mathematics and Mechanics 2023, 202300044 (2023)

    Nestler, M., Voigt, A.: Stability of rotating equilibrium states of fluid deformable surfaces. Proceedings of Applied Mathematics and Mechanics 2023, 202300044 (2023)

    2023

  73. [81]

    arXiv:2506.13880 (2025)

    Happel, L., Hardering, H., Praetorius, S., Voigt, A.: Surface Minkowski tensors to characterize shapes on curved surfaces. arXiv:2506.13880 (2025)

    2025

  74. [82]

    Advanced Materials 23, 2535–2553 (2011)

    Schr¨ oder-Turk, G.E., Mickel, W., Kapfer, S.C., Klatt, M.A., Schaller, F.M., Hoffmann, M.J.F., Kleppmann, N., Armstrong, P., Inayat, A., Hug, D., Reichels- dorfer, M., Peukert, W., Schwieger, W., Mecke, K.: Minkowski tensor shape analysis of cellular, granular and porous stru...

    2011

  75. [83]

    Journal of Rock Mechanics and Geotechnical Engineering 14, 232–239 (2022)

    Wei, D., Zhao, B., Gan, Y.: Surface reconstruction with spherical harmonics and its application for single particle crushing simulations. Journal of Rock Mechanics and Geotechnical Engineering 14, 232–239 (2022)

    2022

  76. [84]

    International Journal for Numerical and Analytical Methods in Geomechanics 41, 93–109 (2017)

    Zhou, B., Wang, J.: Generation of a realistic 3D sand assembly using X-ray micro-computed tomography and spherical harmonic-based principal compo- nent analysis. International Journal for Numerical and Analytical Methods in Geomechanics 41, 93–109 (2017)

    2017

  77. [85]

    In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing

    Kazhdan, M., Funkhouser, T., Rusinkiewicz, S.: Rotation invariant spherical harmonic representation of 3D shape descriptors. In: Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing. SGP ’03, 29 pp. 156–164. Eurographics Association, Goslar, Germa...

    2003

  78. [86]

    Engineering Geology 184, 126–137 (2015)

    Zhou, B., Wang, J., Zhao, B.: Micromorphology characterization and reconstruc- tion of sand particles using micro x-ray tomography and spherical harmonics. Engineering Geology 184, 126–137 (2015)

    2015

  79. [87]

    Journal of Optical Society of America A 5, 1127–1135 (1988)

    Horn, B.K.P., Hilden, H.M., Negahdaripour, S.: Closed-form solution of absolute orientation using orthonormal matrices. Journal of Optical Society of America A 5, 1127–1135 (1988)

    1988

  80. [88]

    Physical Review Research 5, 043227 (2023)

    Al-Izzi, S.C., Alexander, G.P.: Chiral active membranes: Odd mechanics, spon- taneous flows, and shape instabilities. Physical Review Research 5, 043227 (2023)

    2023

  81. [89]

    Proceedings of the National Academy of Science (USA) 109, 1973–1978 (2012)

    Tanner, K., Mori, H., Mroue, R., Bruni-Cardoso, A., Bissell, M.J.: Coherent angular motion in the establishment of multicellular architecture of glandular tissues. Proceedings of the National Academy of Science (USA) 109, 1973–1978 (2012)

    2012

  82. [90]

    Proceedings of the National Academy of Science (USA) 110, 163–168 (2013)

    Wang, H., Lacoche, S., Huang, L., Xue, B., Muthuswamy, S.K.: Rotational motion during three-dimensional morphogenesis of mammary epithelial acini relates to laminin matrix assembly. Proceedings of the National Academy of Science (USA) 110, 163–168 (2013)

    2013

  83. [91]

    Nature 425, 821–824 (2003)

    Baumgart, T., Hess, S., Webb, W.: Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension. Nature 425, 821–824 (2003)

    2003

  84. [92]

    Communications in Computational Physics 13, 325–360 (2013)

    Elliott, C.M., Stinner, B.: Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements. Communications in Computational Physics 13, 325–360 (2013)

    2013

  85. [93]

    ESAIM: Mathematical Modelling and Numerical Analysis 51, 2319–2366 (2017)

    Barrett, J.W., Garcke, H., N¨ urnberg, R.: Finite element approximation for the dynamics of fluidic two-phase biomembranes. ESAIM: Mathematical Modelling and Numerical Analysis 51, 2319–2366 (2017)

    2017

  86. [94]

    Physical Review Letters 129, 118001 (2022)

    Bell, S., Lin, S.-Z., Rupprecht, J.-F., Prost, J.: Active nematic flows over curved surfaces. Physical Review Letters 129, 118001 (2022)

    2022

  87. [95]

    Nature 568, 395–399 (2019)

    M¨ unster, S., Jain, A., Mietke, A., Pavlopoulos, A., Grill, S.W., Tomancak, P.: Attachment of the blastoderm to the vitelline envelope affects gastrulation of insects. Nature 568, 395–399 (2019)

    2019

  88. [96]

    IMA Journal of Numerical Analysis 43, 1543–1585 (2023)

    Hardering, H., Praetorius, S.: Tangential errors of tensor surface finite elements. IMA Journal of Numerical Analysis 43, 1543–1585 (2023)

    2023

  89. [97]

    Journal of Numerical Mathematics 32(1), 55–75 (2024)

    Bachini, E., Brandner, P., Jankuhn, T., Nestler, M., Praetorius, S., Reusken, A., 30 Voigt, A.: Diffusion of tangential tensor fields: numerical issues and influence of geometric properties. Journal of Numerical Mathematics 32(1), 55–75 (2024)

    2024

  90. [98]

    Computers & Mathematics with Applications 146, 253–270 (2023)

    Sass, H., Reusken, A.: An accurate and robust eulerian finite element method for partial differential equations on evolving surfaces. Computers & Mathematics with Applications 146, 253–270 (2023)

    2023

  91. [99]

    Numerische Mathematik 157, 663–715 (2025)

    Elliott, C.M., Sales, T.: A fully discrete evolving surface finite element method for the cahn–hilliard equation with a regular potential. Numerische Mathematik 157, 663–715 (2025)

    2025

  92. [100]

    ACM Transactions of Mathematical Software 44 (2018)

    Engwer, C., N¨ ußing, A.: Geometric reconstruction of implicitly defined surfaces and domains with topological guarantees. ACM Transactions of Mathematical Software 44 (2018)

    2018

  93. [101]

    https://gitlab.mn.tu-dresden.de/spraetor/ SurfaceMinkowski.jl 31

    Praetorius, S.: SurfaceMinkowski.jl: a Julia package for computing sur- face Minkowski functionals. https://gitlab.mn.tu-dresden.de/spraetor/ SurfaceMinkowski.jl 31

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.