Phyllotaxis in a Keller-Segel model

Michael F. Staddon

REVIEW 3 major objections 5 minor 26 references

Phyllotaxis in a Keller-Segel model

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2509.06498 v1 pith:PUED3CRD submitted 2025-09-08 physics.bio-ph

Michael F. Staddon This is my paper · ORCID

classification physics.bio-ph MSC 92C1592C80

keywords phyllotaxisKeller–SegelauxintransportpatternformationFibonaccispiralwhorledRosettenumberlinearstabilityanalysis

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

Beyond 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.

Credit 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.

Who 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.

Recommendation: send to peer review, but with a clear request for corrections and reproducibility.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section V, parameter table and Appendix Eqs. 16–18] 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 II, Eq. 7 and Appendix A] 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 II, Fig. 2 and 'peaks eventually merge'] 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.

minor comments (5)

[Section V, Eq. 18] 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 V, Eq. 9] 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 II, model equations] 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).
[Various] Typos and inconsistencies: 'Rosestte number' (Section II), 'Fibonnaci' (Introduction), 'Doaudy' (Discussion), and inconsistent 'paristichy'/'parastichy'. Please correct.
[Fig. 4] 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'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Rosette number is derived from the model's own LSA and compared to simulations of the same model, but no simulation outcome enters its definition.

full rationale

The paper's central claim is that a Keller-Segel model with a central auxin source and radial growth produces phyllotactic patterns, and that the Rosette number Ro = Gk^2/λ, computed from the linear stability analysis in the Appendix, correlates with the pattern type seen in simulations. This is an internal consistency check, not a circular reduction: Ro is constructed from model parameters (G, k, λ) derived analytically from the model equations, and no simulation output or observed phyllotaxis data is used to define or fit Ro. The correlation between Ro and simulated divergence is a genuine prediction from the model's own equations, even though it does not independently validate the model against external data. There is no fitted parameter renamed as a prediction, no self-citation (the single author's prior work is not used as load-bearing evidence), and no uniqueness theorem imported from the authors. The LSA reproducibility concerns noted in the skeptical reading (D missing from the parameter table, Eq. 18 appearing algebraically inconsistent) are correctness/reproducibility issues, not circularity. The limitations stated in the Discussion about growth profile and geometry are honest caveats, not circular moves. Therefore, no circular step can be quoted and exhibited; the derivation chain is self-contained.

Assumptions & free parameters 7 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the Keller-Segel model with a saturating transport rate, a uniform steady state that ignores the source region, and the transfer of a homogeneous infinite-domain instability analysis to a growing finite disc. The transport signal is an invented, unidentified field. Several model parameters are hand-chosen, and the most important (D) is not specified.

free parameters (7)

D
Diffusion coefficient of the transport signal c. Used in LSA formulas (Eqs 16-18) but absent from the parameter table, preventing reproduction.
alpha = 10
Auxin transport rate, chosen by hand. Controls the instability condition; not fitted to data.
c0 = 5
Saturation constant for auxin transport. Introduced ad hoc to prevent Keller-Segel singularities.
beta = 2
Production rate of the transport signal by auxin. Hand-chosen; affects LSA and pattern stability.
tau = 100
Source onset timescale. Chosen to smooth pattern formation; not varied systematically.
r_s = 2
Source radius. Hand-chosen; sets the steady-state auxin level together with G and S0.
r_d = 50
Disc radius. Chosen large enough to observe patterns; affects finite-size effects.

assumptions (4)

domain assumption The plant surface is a flat, uniformly growing disc with radial velocity v = G/r * r_hat.
Introduced in Eq (6); ignores growth dilution and curved geometry, as acknowledged in the Discussion.
domain assumption Auxin transport follows the gradient of a saturating transport signal c with saturation parameter c0.
Eqs (3)-(4) use this Michaelis-Menten-type term to prevent singularities; no biological evidence is given for c0.
ad hoc to paper The LSA of the uniform, non-growing reaction-diffusion system determines the pattern wavelength in the growing disc.
The appendix computes stability for a uniform state without advection or source, then uses these k and λ to define the Rosette number for the full model.
ad hoc to paper The fastest-growing mode from the LSA is the pattern wavenumber on the finite disc.
Standard LSA selection rule, but not justified for the moving-boundary, source-driven problem studied here.

invented entities (1)

Transport signal c
purpose: A generic chemical or mechanical field that directs auxin transport, standing in for PIN1 polarisation or stress.
The paper explicitly labels c as unidentified (Discussion); no experimental signature or independent handle is proposed.

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Cite this review

Pith. "Pith review of Phyllotaxis in a Keller-Segel model." pith.science (2026). https://pith.science/paper/PUED3CRD

@misc{pith2026250906498,
  author       = {Pith},
  title        = {Pith review of: Phyllotaxis in a Keller-Segel model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUED3CRD}},
  note         = {Machine review of arXiv:2509.06498}
}

read the original abstract

Plants often exhibit regular arrangements of seeds or leaves, known as phyllotaxis, and can famously exhibit Fibonacci spirals, as in sunflower heads or on pine cones. While there are many models which can reproduce spiral formation, the actual mechanism of arrangement remains unclear. Here, we test a more general class of model, based on the Keller-Segel model, in which a diffusing chemical signal produces its own transport signal and can form local instabilities. By modelling the plant as a disc with a signal source at the center and radial growth, we show that we can reproduce both Fibonacci spirals and alternating patterns, depending on a balance between growth and source. Finally, using linear stability analysis, we show how the pattern arises due to a balance between the growth of instabilities and the growth of the plant. Overall, this work demonstrates how the Keller-Segel model can reproduce a wide range of observed patterns, and may act as a phenomenological description of a number of processes in plants.

Figures

Figures reproduced from arXiv: 2509.06498 by the authors.

**Figure 2.** FIG. 2. The pattern depends on the growth rate. (a) A (1, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Auxin flux and area growth rate control the pattern. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Linear stability analysis can predict pattern type. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Levitov, Fibonacci numbers in botany and physics: Phyllotaxis, JETP letters54, 546 (1991)

L. Levitov, Fibonacci numbers in botany and physics: Phyllotaxis, JETP letters54, 546 (1991)

work page 1991

[2] [2]

(b) A Fibonacci spiral (2, 3) pattern,S= 1, G= 1.2

pattern,S 0 = 1, G= 0.5. (b) A Fibonacci spiral (2, 3) pattern,S= 1, G= 1.2. (c) An alternating (2, 4) pattern, S0 = 1, G= 1.6. (d) A (3, 6) pattern,S 0 = 1, G= 2.5. (e) A (4, 7) pattern,S 0 = 1, G= 2.9. (f) A (4, 8) pattern, S0 = 1, G= 3.1. (g) Divergence against area growth rateG. The golden line shows the golden angle, Φ≈137 ◦. Pattern analysis a b FIG...

work page

[3] [3]

Adler, D

I. Adler, D. Barabe, and R. V. Jean, A history of the study of phyllotaxis, Annals of botany80, 231 (1997)

work page 1997

[4] [4]

M. F. Pennybacker, P. D. Shipman, and A. C. Newell, Phyllotaxis: Some progress, but a story far from over, Physica D: Nonlinear Phenomena306, 48 (2015)

work page 2015

[5] [5]

Hofmeister,Allgemeine morphologie der gew¨ achse, Vol

W. Hofmeister,Allgemeine morphologie der gew¨ achse, Vol. 1 (Wilhelm Engelmann, 1868)

work page

[6] [6]

Airy, On leaf-arrangement, Proceedings of the Royal Society of London21, 176 (1873)

H. Airy, On leaf-arrangement, Proceedings of the Royal Society of London21, 176 (1873)

work page

[7] [7]

G. Van Iterson, Mathematische und mikroskopisch- anatomische studien ¨ uber blattstellungen nebst betra- chtungen ¨ uber den schalenbau der miliolinen, Mathe- matische und mikroskopisch-anatomische Studien ¨ uber Blattstellungen nebst Betrachtungen ¨ uber den Schalen- bau der Miliolinen (1907)

work page 1907

[8] [8]

Douady and Y

S. Douady and Y. Couder, Phyllotaxis as a physical self-organized growth process, Physical review letters68, 2098 (1992)

work page