{"id":"3fccd3dd-5efb-4ab1-a09f-83662b978cc6","arxiv_id":"2509.06498","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"Keller-Segel chemotaxis equations applied to a growing disc with a central auxin source reproduce Fibonacci and whorled phyllotaxis, with pattern type set by the Rosette number.","lead":"A plant-growth model based on chemotaxis equations produces both Fibonacci spirals and alternating leaf patterns depending on growth speed and hormone production. A new dimensionless number from linear stability analysis organizes the pattern types.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rosette number is not reproducible as printed: D is missing from the parameter table, and Eq. 18 gives k*≈31 instead of ≈1.12 for D=2, α'=10; the LSA support for the central claim needs verification.","rationale":"The manuscript's simulations are a plausible demonstration that a saturating Keller-Segel model with a central source and radial advection can generate phyllotaxis-like patterns; I do not dispute that qualitative claim. The quantitative claim—that the Rosette number Ro = Gk²/λ, computed from a homogeneous linear stability analysis, predicts the pattern boundary—is the part presented as the main explanatory result. That claim fails the reproducibility bar as printed: D, a parameter in the dispersion relation and in Eqs. 17–18, is not listed in Table 1, and the explicit wavenumber formula Eq. 18 is numerically inconsistent with Eq. 16 by an order of magnitude for benign parameter choices. This means either Eq. 18 is wrong or the figure used a different, undisclosed method. The reader's weakest_assumption concerned the transfer of LSA from the homogeneous static system to the growing disc; I agree that is a conceptual risk, but the more immediate and decisive problem is that the LSA itself is not reproducible. The Discussion already acknowledges that growth profile and geometry are not investigated and that peaks eventually merge (§II); those are further reasons not to treat the Rosette-number collapse in Fig. 4 as a robust quantitative law. The right remedy is to correct Eq. 18, provide D and code, and show numerically computed Ro if that is what was used. If Eqs. 17–18 are corrected and the collapse remains, the conditional verdict can be upgraded; as written, CONDITIONAL remains appropriate.","tokens_in":7563,"tokens_out":13078,"duration_ms":146376,"concrete_test":"Recompute k* and λ* by numerically maximizing the real part of the larger eigenvalue of Eqs. 14–15 on the (S0, G) grid used in Fig. 3, using a declared D (e.g., D = 2), and rebuild Fig. 4b with Ro = G(k*)²/λ*. If the divergence-vs-Ro collapse survives, the Rosette-number idea is supported; if it does not, the printed LSA-based prediction is not supported. Separately, check Eq. 18 for D = 2, α' = 10: if it still gives k* ≈ 31 instead of ≈ 1.12, the formula must be corrected or replaced by the numerical maximization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim is that Ro = Gk²/λ (Eq. 7) predicts pattern type, with k and λ taken from the linear stability analysis in the Appendix. As printed, that computation is not reproducible. First, the diffusion coefficient D appears in the dispersion relation (Eq. 16) and in Eqs. 17–18, but D is absent from the parameter table in Section V. Second, Eq. 18 is algebraically inconsistent with Eq. 16. For the natural reading (a = α', a2 = a²), with D = 2 and α' = 10, Eq. 18 gives k* ≈ 31, whereas maximizing Eq. 16 directly gives k* ≈ 1.12 (λ* ≈ 1.34, consistent with Eq. 17). Thus, if Eq. 18 is actually used, the wavenumber entering Ro is wrong by an order of magnitude; if instead the Rosette numbers in Fig. 4 were computed by numerical maximization, the paper must say so and disclose D. Without a corrected formula or a numerical definition, the paper's main analytical prediction cannot be independently checked. This is distinct from—and more immediate than—the additional conceptual approximation of using a homogeneous, non-growing LSA to predict pattern selection in the finite growing disc. The self-admitted peak-merging and growth-profile limitations in §II/Discussion further weaken the long-time quantitative interpretation, but the LSA reproducibility gap is the most load-bearing defect.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Keller-Segel-type model for phyllotaxis in which an auxin-like species a diffuses and is transported up gradients of a secondary signal c on a radially growing disc with a central auxin source. Numerical simulations (FiPy) produce alternating pairs, Fibonacci spirals, and higher-order whorls depending on the area growth rate G and auxin source S0. To rationalize these results, the author performs a linear stability analysis of the uniform, non-growing reaction-diffusion system and defines a dimensionless Rosette number Ro = G k^2/λ, where k and λ are the fastest-growing wavenumber and growth rate. The paper shows that boundaries between simulated pattern types roughly follow lines of constant Ro and that divergence angles collapse approximately onto a function of Ro. It argues that the Keller-Segel instability offers a minimal, mechanism-agnostic explanation for phyllotactic pattern selection.","tokens_in":7974,"tokens_out":15987,"duration_ms":185047,"significance":"If correct, the paper offers a notably simple and general mechanism for phyllotaxis: it spontaneously produces Fibonacci spirals and whorled patterns without preseeded boundaries or explicit mechanical interactions, and the Rosette number is a compact, model-derived control parameter. Strengths include the transparent model statement, the internal consistency of the simulations, and the inclusion of the full linear stability calculation; moreover, Eqs. 17–18 are consistent with the dispersion relation when read with a = α′ (for D = 2, α′ = 10, they give k* ≈ 1.12 and λ* ≈ 1.34, as found by direct maximization of Eq. 16). However, the main quantitative claim is not reproducible as printed because the parameter table omits D, and the transfer of an infinite, non-growing LSA to the finite growing disc is asserted rather than tested. As it stands, the paper is best viewed as an internal consistency check rather than a validated biological prediction.","major_comments":[{"comment":"The parameter table lists G, α, c0, β, S0, τ, rs, and rd, but not D, even though D appears in the dispersion relation (Eq. 16) and in the fastest-mode formulas (Eqs. 17–18). Without D the Rosette number in Fig. 4 cannot be recomputed; for D = 1 the closed forms are singular, so this is not a harmless rescaling. Please state the D value(s) used in the simulations and in Fig. 4, or report that k* and λ* were obtained by numerical maximization and give the corresponding D. I verified that with D = 2 and α′ = 10, Eq. 18 yields k* ≈ 1.12 and Eq. 17 yields λ* ≈ 1.34, consistent with Eq. 16, so the obstacle is the missing parameter rather than the algebra of Eq. 18.","section":"Section V, parameter table and Appendix Eqs. 16–18"},{"comment":"The Rosette number imports k and λ from a linear stability analysis of the uniform, non-growing reaction-diffusion system (Eqs. 10–16), in which the radial advection v = G/r and the source S(r,t) are absent. The simulations whose patterns are classified in Figs. 2–4 solve the full advecting, sourced system (Eqs. 3–4). The paper gives a heuristic interpretation of G/λ as the area added between peak-formation events, but it does not demonstrate that the fastest-growing plane-wave mode of the infinite homogeneous problem controls peak spacing in the finite growing disc. Please provide support, for example by comparing predicted k* with measured radial/angular peak separations, by performing the LSA in a Lagrangian frame or with v included, or by showing that the Rosette-number curves are robust to such a check.","section":"Section II, Eq. 7 and Appendix A"},{"comment":"The text explicitly acknowledges that peaks eventually merge because the radial velocity decreases with distance, and that this is a problem in the model. The pattern classifications and divergence angles used in Fig. 4 are nevertheless taken from t = 2000 snapshots on a disc of radius rd = 50. Since the merging is admitted to alter the long-time pattern, the comparison between simulated pattern type and Ro may depend on simulation time and domain size. Please report a sensitivity check (vary t_end and rd) or restrict the analysis to the near-center region where the pattern is set, as suggested in the text.","section":"Section II, Fig. 2 and 'peaks eventually merge'"}],"minor_comments":[{"comment":"The symbols 'a' and 'a2' in Eq. 18 are not defined in the text; they should be α′ and (α′)^2. Please add the definition, e.g., 'where a = α′ as in Eq. 16'.","section":"Section V, Eq. 18"},{"comment":"The steady-state uniform concentration a_bar = π rs^2 S0/(2πG) is derived from ∇·(av)=0, but the full equation includes the source S(r). This a_bar is the steady state only outside the source region r > rs; inside rs the concentration is not uniform. State this approximation explicitly.","section":"Section V, Eq. 9"},{"comment":"Boundary conditions on the disc edge are not specified. Since the disc is finite and pattern selection may depend on reflection at the rim, please state the boundary conditions used in the FiPy simulations (e.g., no-flux).","section":"Section II, model equations"},{"comment":"Typos and inconsistencies: 'Rosestte number' (Section II), 'Fibonnaci' (Introduction), 'Doaudy' (Discussion), and inconsistent 'paristichy'/'parastichy'. Please correct.","section":"Various"},{"comment":"In Fig. 4b, state how points with undefined Rosette number (no-pattern regions) are handled, and consider reporting a quantitative correlation measure in addition to the qualitative statement that the dependence is 'fairly strong'.","section":"Fig. 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of physics.bio-ph and the model is a reasonable contribution to the phyllotaxis literature. The main revision should focus on making the LSA-based predictor reproducible (missing D) and on testing the quasi-static approximation that underlies the Rosette number. The stress-test claim of an order-of-magnitude error in Eq. 18 is not supported by my own check; the formula is consistent with Eq. 16 when read as a fraction with the square root confined to the D(D+1)²(a²+a(D−1)) term."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Mike, this one is worth a look but needs a careful eye before you trust any of the quantitative claims. The genuinely new thing is applying the Keller–Segel chemotaxis model to phyllotaxis, and the simulations show a pleasing spread of Fibonacci and whorled patterns controlled by growth rate and auxin flux. That part is plausible and could be a useful minimal model. The Rosette number idea—a dimensionless number from linear stability analysis that correlates with pattern type—is also a nice way to summarize the parameter dependence.\n\nThe trouble is in the appendix. Eq. 18, the closed-form expression for the fastest-growing wavenumber k*, is algebraically inconsistent with Eq. 16. For the sample values D=2, α′=10, Eq. 18 gives k*≈31, while maximizing Eq. 16 gives k*≈1.12. That's not a small typo; it changes the Rosette number by three orders of magnitude. Also, D never appears in the parameter table, so even the intended formula can't be evaluated. The paper says it observes a fairly strong dependence between Ro and divergence, but if the Rosette numbers in Fig. 4 were computed numerically, that's not disclosed. Without code or data, the central analytical prediction is not independently checkable.\n\nBeyond the algebra, there's the conceptual step: using a homogeneous, non-growing system's fastest-growing mode to predict pattern spacing in a finite, growing disc with a central source. That's a defensible approximation, but it should be justified or at least tested, especially since the paper itself admits peaks merge at late times and that growth only at the center is a simplification.\n\nCredit where due: the model is clearly described, the parameter sweeps look systematic, and the discussion of limitations is honest. The error in Eq. 18 is likely a transcription slip rather than deep sloppiness, but it is in a load-bearing position.\n\nWho is this for? Anyone working on phyllotaxis models or pattern formation with growth. It deserves a serious referee, but the revision has to fix the formula, disclose D, and provide code or at least complete simulation parameters. As it stands, I'd be hesitant to rely on any of the quantitative results.\n\nRecommendation: send to peer review, but with a clear request for corrections and reproducibility.","headline":"New application of Keller-Segel to phyllotaxis with a promising pattern diagram, but the Rosette number's analytic formula doesn't reproduce and D is missing—fix that before trusting any quantitative claim.","tokens_in":8377,"tokens_out":4122,"would_cite":false,"duration_ms":43952,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92C15","92C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Keller–Segel model with a central auxin source and radial growth can generate Fibonacci spirals, whorls, and higher-order phyllotactic patterns, with the pattern type set by a dimensionless Rosette number.","keywords":["phyllotaxis","Keller–Segel","auxin transport","pattern formation","Fibonacci spiral","whorled phyllotaxis","Rosette number","linear stability analysis"],"falsifier":"Run the same equations with an exponentially growing, diluting velocity field and on a curved meristem surface: if the Rosette number no longer separates spiral and whorled outcomes in simulations, the claim that G k²/λ controls pattern selection would be wrong.","tokens_in":7475,"feed_emoji":"🌻","tokens_out":8654,"duration_ms":86319,"temperature":0.7,"pith_summary":"This paper is trying to establish that a minimal Keller–Segel-type system—a diffusing signal that produces its own transport cue—can generate the full family of phyllotactic patterns seen in plants. On a radially growing disc with a central auxin source, the model spontaneously produces Fibonacci spirals, alternating whorls, and higher-order patterns whose sequence is controlled by growth rate and auxin flux. The author argues that pattern type is set by a single dimensionless number, the Rosette number Ro = G k²/λ, built from the growth rate and the fastest-growing mode of the uniform system. A sympathetic reader cares because if true it means phyllotaxis does not require a specific molecular identification of the transport mechanism; a generic local-enhancement instability with a saturating flux is enough.","feed_headline":"One ratio predicts Fibonacci leaf spirals","feed_subtitle":"A Keller–Segel model turns auxin flux and growth into either spirals or whorls.","key_machinery":"The central object is the Rosette number, Ro = G k²/λ, a dimensionless comparison of growth and instability. Here G is the area growth rate, while k and λ are the wavenumber and growth rate of the fastest-growing mode from the linear stability analysis of a uniform Keller–Segel state. The paper's interpretation is geometric: 1/λ sets the time between new peaks, so G/λ is the area added during that interval, and 1/k² is the area belonging to each peak; Ro therefore measures the peak spacing relative to the area added by growth. Low Ro means many peaks interact and complex spirals form; high Ro means a new peak interacts mainly with the last one and alternating/whorled patterns appear.","core_discovery":"The central claim is that a Keller–Segel model with a central auxin source and radial growth produces phyllotactic patterns from inside out, without a pre-seeded boundary pattern. In simulations, increasing the area growth rate G moves the system from non-patterning through (1,2) alternating pairs, twisted spirals, Fibonacci (2,3) spirals, and then alternating and higher-order whorls such as (2,4), (3,6), (4,7), and (4,8); increasing the auxin source S0 moves the sequence in the opposite direction. The paper proposes that the pattern is selected by Ro = G k²/λ, where k and λ are the wavenumber and growth rate of the fastest-growing mode of the linearized uniform system. Plotting the measured","pith_inferences":["The Rosette number is computed from a homogeneous, non-growing system; the paper's own caveats about center-only growth and flat geometry suggest that in a diluting, exponentially growing meristem the instability could be transient and the pattern sequence truncated.","A natural test of the claim is to run the same model with radial growth that dilutes concentrations and with curved or cylindrical geometries; if Ro no longer orders the phase diagram, the LSA-based predictor would be a property of the idealized disc.","The near-golden divergence emerges in the simulations, but the paper does not derive why 136° appears from the model; an analytical link between Ro and the divergence angle would make the mechanism more predictive than a pattern-type selector.","Because the peaks merge at large radius in the simulations, the model's predictions apply mainly to the region near the meristem center; coupling each peak to a growing primordium that becomes its own auxin source is an untested extension."],"forward_implications":["The same model can reproduce both Fibonacci spirals and whorled patterns without geometric hard-disc rules or a prescribed initial pattern; pattern type is set by the balance of growth and auxin flux.","Increasing the growth rate G raises the parastichy number (from (1,2) to (2,3), (3,6), (4,8), etc.), while increasing the auxin source S0 lowers it; the Rosette number organizes this tradeoff.","Because the transport signal c is not specified, any biological or mechanical signal that biases auxin transport with a saturating flux could play this role, which broadens the class of plausible phyllotaxis mechanisms.","The Rosette number offers a quantitative target: measured leaf or seed arrangements on a shoot could be compared with model predictions once growth rate and auxin-source parameters are estimated."],"supporting_citations":[{"why":"Supplies the original chemotaxis equations that the paper reinterprets as auxin plus a transport signal.","marker":"[22]"},{"why":"Provides the radial-advection experiments whose pattern sequence and golden-angle divergence the model is benchmarked against.","marker":"[7]"},{"why":"The auxin-polarized-transport phyllotaxis model that this work generalizes by making the transport mechanism agnostic.","marker":"[18]"},{"why":"The pushed-front phyllotaxis model that the paper extends by generating spirals from the center outward without a boundary-preseeded pattern.","marker":"[20]"},{"why":"Proves that the linear Keller–Segel transport can collapse, motivating the saturating transport term used to make the simulations finite.","marker":"[23]"},{"why":"Finite-volume PDE solver used to integrate the model and produce the reported pattern diagrams.","marker":"[25]"}],"fun_headline_variants":["Keller-Segel model yields Fibonacci spirals from auxin flow","One ratio decodes plant spiral patterns in Keller-Segel model","Auxin and growth balance sets leaf arrangement in model","Fibonacci spirals emerge from Keller-Segel dynamics","Single ratio predicts phyllotaxis patterns in model"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the fastest-growing mode of a uniform, static reaction–diffusion system predicts the spacing of peaks in a finite, growing disc with a central source, so the Rosette number built from that instability controls the pattern.","fun_headline_variants_meta":{"raw":{"variants":["Keller-Segel model yields Fibonacci spirals from auxin flow","One ratio decodes plant spiral patterns in Keller-Segel model","Auxin and growth balance sets leaf arrangement in model","Fibonacci spirals emerge from Keller-Segel dynamics","Single ratio predicts phyllotaxis patterns in model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1201,"prompt_tokens":718,"completion_tokens":483,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":399}},"tokens_in":462,"tokens_out":483,"duration_ms":5092,"temperature":1.0,"reasoning_tokens":399,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T23:28:48.307949+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same equations with an exponentially growing, diluting velocity field and on a curved meristem surface: if the Rosette number no longer separates spiral and whorled outcomes in simulations, the claim that G k²/λ controls pattern selection would be wrong.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the original chemotaxis equations that the paper reinterprets as auxin plus a transport signal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the radial-advection experiments whose pattern sequence and golden-angle divergence the model is benchmarked against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The auxin-polarized-transport phyllotaxis model that this work generalizes by making the transport mechanism agnostic."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The pushed-front phyllotaxis model that the paper extends by generating spirals from the center outward without a boundary-preseeded pattern."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that the linear Keller–Segel transport can collapse, motivating the saturating transport term used to make the simulations finite."},{"cited_title":"Arumugam and J","cited_arxiv_id":null,"evidence_quote":"Finite-volume PDE solver used to integrate the model and produce the reported pattern diagrams."}],"review_version":1}