{"id":"94e49681-7167-4bd8-b05a-10c18fecb7ed","arxiv_id":"2602.13049","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A rotationally invariant viscous vertex model with junctional and bulk dissipation, regularized at zero substrate friction by Lagrange multipliers, predicts tissue viscosity from cell-level viscosities.","lead":"This paper builds a rotationally invariant \"viscous vertex model\" of epithelial cell sheets, adding friction both at cell junctions and inside cells, and making the equations work even when cells do not touch a substrate. It also gives a way to measure the tissue's effective viscosity and shows how viscosity changes cell alignment and flow in active tissues.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Zero-friction well-posedness is conditional on self-balanced activity: polar traction F_J^active=T0 p_J requires Lagrange multipliers to supply an external force, so the free-floating claim is not general; Eq. 65 prefactor is internally consistent.","rationale":"The reader's weakest assumption correctly identifies the zero-friction polar-traction issue as the most load-bearing concern. The paper's well-posed-at-zero-friction claim is central to its free-floating-tissue motivation, and the polar active traction model is not self-balanced by construction. The Lagrange multipliers can mathematically restore solvability, but they then act as external forces, which is not the physics of an isolated tissue. This warrants a conditional verdict: the model is well-posed at zero friction for self-balanced active stresses and for passive rheology, but not for the polar traction model unless a substrate or another external momentum sink is retained. The reader's separate claim that Eq. (65) is off by a factor of two appears to be a misreading of the parenthesization: the displayed Eq. (63) should be read as 1/(4√3)η_s + 1/(2√3)η_b, not (1/4)√3 η_s + (1/2)√3 η_b; the latter would contradict Eq. (65), while the former reproduces it exactly. Recomputing the honeycomb sums with shared interfaces counted once confirms the factor 1/(4√3) for the interfacial term. Thus the central quantitative prefactor and rotational-invariance claims hold up; the zero-friction polar-activity limitation is the genuine qualifier.","tokens_in":24030,"tokens_out":17549,"duration_ms":165516,"concrete_test":"At γ=0, compute the total active force and torque from the polar traction model, P=Σ_i F_i^active=T0 Σ_J p_J and T=Σ_i r_i×F_i^active, for a generic configuration under the polarization dynamics Eq. (74) in a periodic domain. If P or T is nonzero, then F^(t) is not orthogonal to ker(C) and the unconstrained zero-friction problem has no solution; only the saddle-point system Eq. (47) restores solvability, with the Lagrange multipliers providing an external balancing force. Repeating the same computation for the nematic stress σ=−βQ_J should give P=T=0, confirming that the zero-friction regularization is physical only for self-balanced activity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is the zero-friction regularization in Sec. II C 3 when combined with the polar active traction model of Sec. IV B. At γ=0, C is singular with nullspace containing global translations and rotations; Eqs. (44)-(45) impose Σv_i=0 and Σr_i×v_i=0. This is physically justified only if the non-dissipative force vector F^(t) is orthogonal to that nullspace, i.e., if the active forces are net force-free and torque-free. For the polar traction model, F_J^active=T0 p_J is an imposed body force: generically Σ_J F_J^active = T0 Σ_J p_J ≠ 0 and Σ_i r_i×F_i^active ≠ 0. These unbalanced components lie in ker(C), so the unconstrained Eq. (1) has no finite-velocity solution; the Lagrange multipliers in Eq. (47) restore solvability only by supplying a compensating external force −D^T λ. That is a hidden substrate, not a free-floating tissue. Thus the abstract and conclusion claims of applicability to free-floating epithelia/organoids are conditional on replacing polar traction by self-balanced active stresses such as the nematic stress σ_J^active=−βQ_J, which is force- and torque-free by construction. The reader's additional concern about Eq. (65) does not survive recalculation: counting each shared cell-cell interface once, the interface contribution is (1/4√3)η_s γ and the bulk contribution is (1/2√3)η_b γ, giving Eq. (64) and hence Eq. (65) η_tissue^ST=(√3/4)η when η_s=η_b=η.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a viscous vertex model for epithelial tissues in which dissipation arises from two microscopic sources: a cell–cell interfacial viscosity proportional to the rate of change of edge length, and a cell-bulk viscosity associated with the rate of change of virtual vertex-to-cell-center links. The dissipative operator is assembled into a symmetric positive semi-definite matrix C, and rotational invariance is shown explicitly. For vanishing substrate friction, global translation and rotation constraints are imposed through Lagrange multipliers, yielding a saddle-point system; the paper analyzes its condition number and identifies floppy modes when bulk viscosity vanishes. A slab-shear protocol is used to define short- and long-time tissue viscosities; an analytical calculation for a regular hexagonal monolayer gives η_tissue^(ST) = (√3/4)η when η_s=η_b=η. The model is then applied to polar and nematic active tissues, where viscosity is shown to increase cell alignment, reduce defect density, and promote coherent flows. The paper claims that the model remains well-posed at zero substrate friction and is therefore suited to free-floating epithelia and organoids.","tokens_in":24462,"tokens_out":17853,"duration_ms":151849,"significance":"The paper's main contribution is a rigorously formulated internal-dissipation vertex model that avoids the usual substrate-friction crutch and provides a well-conditioned numerical scheme at γ=0. The matrix formulation, symmetry arguments, positive semi-definiteness proofs, and condition-number scaling analysis are careful and reproducible. The analytical benchmark Eq. (65) is a useful closed-form result that can be used to test numerical implementations. If the zero-friction claim is qualified appropriately, the framework genuinely extends the reach of vertex models to free-floating epithelia and organoids and provides a concrete bridge to continuum active gel descriptions. However, as written, the free-floating claim is too broad for polar active traction.","major_comments":[{"comment":"The zero-friction regularization via the constraints (44)–(45) is physically valid only when the non-dissipative forces are net force-free and torque-free. For the polar active traction F_J^(active)=T0 p_J (Sec. IV B), the unconstrained force vector generically has nonzero total force and torque, i.e., components in ker(C). The Lagrange multipliers in Eq. (47) then supply an external reaction force −D^T λ, which is a hidden substrate rather than a free-floating boundary condition. As a result, the abstract's claim that the model is 'naturally suited to describing free-floating epithelia and organoids' is not supported for polar activity; it holds only for self-balanced active stresses such as the nematic stress σ_J^(active)=−βQ_J, or if a substrate is retained. Please state this condition in Sec. II C 3 and soften the corresponding claims.","section":"Sec. II C 3; Sec. IV B"},{"comment":"The derivation of the interfacial contribution to Eq. (63) is internally consistent only if one accounts for the fact that each cell–cell interface is shared by two cells and the global sum in Eq. (57) counts each interface once. Recomputing from Eqs. (60)–(62), the sum over the six edges of one hexagonal cell gives an interfacial stress of 1/(2√3) η_s γdot; multiplying by the shared-interface factor 1/2 yields the quoted 1/(4√3) η_s γdot. This factor is not explained in the text, so a careful reader cannot reproduce Eq. (63). Please add a sentence making the shared-interface convention explicit.","section":"Sec. III B1, Eq. (63)"}],"minor_comments":[{"comment":"In the single-hexagon calculation, the affine velocity profile Eq. (60) assigns v1=v4=(√3/2)Rγdot even though these vertices have y=0. This follows from taking the bottom edge (vertices 5 and 6) as the fixed origin of the shear profile; please state this explicitly to avoid confusion.","section":"Sec. III B1"},{"comment":"The summation notation ⟨i,j⟩ over cell–cell interfaces should be defined as an unordered-pair sum over unique interfaces, to distinguish it from directed edge sums used elsewhere.","section":"Eq. (57)"},{"comment":"The sentence 'Either substrate friction or bulk viscosity is needed' is imprecise when combined with the floppy-mode discussion; it would be clearer as: 'With interfacial viscosity alone (γ=0 and η_b=0), the extended matrix remains singular; a finite bulk viscosity or substrate friction is required to lift the floppy modes.'","section":"Sec. II C3"},{"comment":"Reference [39] and reference [68] are the same arXiv entry (arXiv:2603.04170). Please merge the duplicate and renumber the citations accordingly.","section":"References"},{"comment":"The notation 'X_{i,j∈cellJ}' in the off-diagonal bulk term is nonstandard; please define it (e.g., sum over cells J that contain both i and j) or rewrite with an explicit summation.","section":"Eq. (19)"}],"recommendation":"major_revision","confidential_remarks":"The core formalism is sound and the paper is well-executed, but the free-floating claim in the abstract and conclusion overreaches for polar active traction. The analytical viscosity is correct, but the shared-interface convention in Eq. (63) needs to be stated explicitly. These are fixable with revision, hence major_revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. It builds a rotationally invariant vertex model with both junctional and bulk dissipation, handles the zero-friction limit with a saddle-point Lagrange-multiplier system, and gives a clean slab-shear protocol to extract coarse-grained tissue viscosity. The dissipation matrices are symmetric and positive semi-definite, the rotational invariance argument is straightforward and correct, and the floppy-mode analysis is honest about when the extended system becomes singular. The short-time viscosity result for a hexagonal cell, Eq. (65), checks out: counting each shared interface once gives (1/(4√3) + 1/(2√3))η = √3/4 η, so the reader's worry about a factor of two does not survive a careful read. The linear scaling with microscopic viscosities is also confirmed numerically in disordered packings, and the active nematic simulations show viscosity-dependent defect organization that is consistent with continuum expectations. That is a genuinely useful contribution for people working on free-floating epithelia, organoids, or low-friction monolayers.\n\nThe real soft spot is the zero-friction claim. The constraints Σv_i=0 and Σr_i×v_i=0 remove the rigid-body nullspace of C, but they are physically justified only if the non-dissipative forces are net force-free and torque-free. The polar traction model F_J^active = T0 p_J is not self-balanced; generically it has unbalanced net force and torque. At γ=0, those components lie in the nullspace, so the Lagrange multipliers end up supplying a compensating external force. That is a hidden substrate, not a free-floating tissue. The nematic stress σ=-βQ is fine because it is force- and torque-free by construction, but the paper should state this limitation explicitly rather than claiming blanket applicability to free-floating systems. This is a qualifier, not a fatal flaw: the formalism is correct for self-balanced active stresses and for any activity when friction is present.\n\nMinor issues: no code or data are provided, which limits reproducibility, and the long-time viscosity discussion is more phenomenological than the short-time derivation. Neither undermines the core results.\n\nThis paper deserves a serious referee. I would send it out, and with a modest revision that clarifies the zero-friction applicability, I would accept it.","headline":"A solid, useful viscous vertex model with a correct analytical prefactor; the main caveat is that the zero-friction limit is only physical for force- and torque-free active forces, so the free-floating claim needs qualification.","tokens_in":24901,"tokens_out":2334,"would_cite":true,"duration_ms":25438,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A rotationally invariant vertex model with junctional and bulk cell viscosities predicts tissue-scale viscosity and stays well-posed at zero substrate friction.","keywords":["vertex model","epithelial tissue","cell viscosity","tissue rheology","active nematic","topological defects","Lagrange multipliers","zero friction"],"falsifier":"Shear a modeled hexagonal tissue with η_s=η_b=η at zero friction and measure the viscous shear stress; if it deviates from (√3/4)η times the imposed strain rate within numerical precision, the central scaling is wrong. Equivalently, at γ=0 and η_b=0, invertibility of the extended matrix is required for the well-posed claim; the paper's own floppy-mode analysis predicts singularity in that limit.","tokens_in":23878,"feed_emoji":"🧫","tokens_out":4832,"duration_ms":44617,"temperature":0.7,"pith_summary":"The paper tries to establish that epithelial tissue dynamics can be captured at cell scale by two dissipation coefficients — one at cell-cell junctions, one between vertices and cell centers — and that this cell-based model remains rotationally invariant and solvable even when the tissue has no substrate friction. It further claims that the coarse-grained tissue shear viscosity scales linearly with these microscopic viscosities, with the exact ratio √3/4 for equal viscosities in a hexagonal tissue. If true, this gives a practical bridge between cell-resolved simulations and continuum active-nematic descriptions, and opens free-floating epithelia and organoids to quantitative rheological modeling. The paper also shows that increasing cell viscosity reorganizes active flows, from defect-rich turbulence to coherent rotation in confinement.","feed_headline":"Tissue viscosity follows a √3/4 rule from cell viscosity","feed_subtitle":"Rotational invariant vertex model shears free-floating epithelia; viscosity drives coherent tissue flows.","key_machinery":"The central objects are two dissipation quadratic forms: S_interface penalizes edge elongation rates projected along each cell-cell interface, and S_bulk penalizes stretching of virtual vertex-to-cell-center links. These define viscous line tensions proportional to projected relative velocities, assembled into a symmetric positive semi-definite coefficient matrix C. At zero substrate friction, global translation and rotation constraints (Σv_i=0 and Σr_i×v_i=0) are appended via Lagrange multipliers, forming a saddle-point matrix C_ext that regularizes the otherwise singular system. The slab-shear protocol uses this constrained system to impose a fixed strain rate and measure the resulting vis","core_discovery":"The paper claims that dissipation in an epithelial sheet can be resolved into two microscopic viscous coefficients: a junctional viscosity η_s acting along each cell-cell interface and a bulk viscosity η_b acting along virtual vertex-to-cell-center links. Both are defined through relative velocities projected along those segments, making the model rotationally invariant—unlike earlier formulations that penalize absolute velocity differences and thus spuriously dissipate under rigid rotation. The resulting coefficient matrix is symmetric positive semi-definite, and in the zero-friction limit the paper enforces global translation and angular-momentum conservation through Lagrange multipliers,","pith_inferences":["Because the dissipation law projects velocities along edge directions, the model is objective: pure rigid rotation produces no dissipation. Continuum viscosities extracted from this vertex model should therefore be frame-invariant, a property not guaranteed by earlier viscous vertex models.","The equal-viscosity prediction η_tissue = (√3/4)η is directly testable in micropatterned hexagonal epithelia by measuring shear force at fixed strain rate — an experiment the paper does not propose but its framework makes concrete.","The polar-traction activity is not globally force- and torque-free, so at strictly zero friction the Lagrange multipliers would have to supply unphysical body forces; applying this activity to free-floating tissues requires either a substrate or a self-balanced stress, which may change the interpretation of the polar-activity results.","The linear-scaling law suggests that tissue viscosity measurements could be used to infer local variations in cell-cell adhesion strength or cortical contractility, since both enter through η_s and η_b."],"forward_implications":["Short-time tissue viscosity of a regular hexagonal tissue is exactly (√3/4)η_s + (√3/2)η_b, so a single tissue-level rheology measurement constrains the two cell-scale viscous coefficients.","The same linear scaling survives in disordered packings and in long-time steady shear with cell rearrangements, making it a robust coarse-graining relation.","At zero friction, global momentum and angular-momentum constraints regularize the dynamics, enabling simulations of free-floating epithelia and organoids without a substrate.","In polar-active tissues, increasing viscosity enlarges swirl size and reduces the number of topological defects, eventually crossing over to a global rotation with two +1/2 defects in confinement.","In nematic-active tissues, higher viscosity produces longer-ranged cell-shape correlations and well-defined topological defects whose stress and flow fields match active gel theory."],"fun_headline_variants":["Rotational invariance fixes viscous vertex model for epithelia","Two viscosities drive shape textures in active epithelial sheets","Zero-friction vertex model reveals tissue viscosity from cell-level friction","Slab-shear rheology extracts cell-scale viscosities in epithelia"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Well-posedness at zero friction assumes the tissue is an isolated system whose only zero-dissipation motions are global translation and rotation; that physical picture requires all non-dissipative forces to be force- and torque-free (the nematic stress is, the polar traction is not) and requires a nonzero bulk viscosity, since with only junctional viscosity the network still has area- and length-preserving floppy modes.","fun_headline_variants_meta":{"raw":{"variants":["Rotational invariance fixes viscous vertex model for epithelia","Two viscosities drive shape textures in active epithelial sheets","Zero-friction vertex model reveals tissue viscosity from cell-level friction","Slab-shear rheology extracts cell-scale viscosities in epithelia"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1042,"prompt_tokens":600,"completion_tokens":442,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":344,"completion_tokens_details":{"reasoning_tokens":385}},"tokens_in":344,"tokens_out":442,"duration_ms":4639,"temperature":1.0,"reasoning_tokens":385,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T23:39:12.160208+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Shear a modeled hexagonal tissue with η_s=η_b=η at zero friction and measure the viscous shear stress; if it deviates from (√3/4)η times the imposed strain rate within numerical precision, the central scaling is wrong. Equivalently, at γ=0 and η_b=0, invertibility of the extended matrix is required for the well-posed claim; the paper's own floppy-mode analysis predicts singularity in that limit.","supporting_citations":[],"review_version":1}