How Plant Stress May Change Leaf Spacing Patterns

In the paper's words

Plants that develop spiral leaf or floret arrangements predominantly exhibit Fibonacci divergence angles near 137.5^∘, yet agronomic guidelines treat phyllotaxis as a static trait, so growers and breeders cannot anticipate the pattern changes observed under water or nutrient stress. Existing analytical models fix the golden angle and provide no criterion for when a stand will depart from it. We propose a stress-dependent threshold model in which a normalized stress parameter σ interpolates between a Fibonacci-weighted and a Lucas-weighted contribution to the primordium-packing problem, drawing on algebraic identities that link the two sequences.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Equating the weighted contributions at sequence index $n=5$ yields a critical stress $\sigma_c = F_5/(F_5+L_5) = 5/16 = 0.3125$; recomputing at $n=6$ shifts the threshold by only $1.54\%$.

   from Abstract

2. The model predicts an abrupt switch of the divergence angle from $\theta_F = 360^\circ/\varphi^2 \approx 137.508^\circ$ to a Lucas-related angle $\theta_L = 360^\circ/(\varphi+2) \approx 99.506^\circ$, a relative angular loss $R = (\theta_L-\theta_F)/\theta_F \approx -0.2764$, that is, a $27.6\%$ reduction in divergence angle.

   from Abstract

3. Explicit computation of a critical stress threshold $\sigma_c = 5/16 = 0.3125$ from the fifth terms of the two sequences, with a robustness check at $n=6$ showing a $1.54\%$ shift (Section 4).

   from 1. Introduction

4. An alternative definition based on the ratio $\psi = L_5/F_5 = 11/5$ was also considered and rejected: it yields $\theta = 360^\circ(1-1/\psi) = 360^\circ \cdot 6/11 \approx 196.36^\circ$, which lies outside the range of divergence angles observed in spiral phyllotaxis, whereas $\theta_L = 360^\circ/(\varphi+2) \approx 99.5^\circ$ does not.

   from 3.1 Model overview

5. The threshold is therefore nearly independent of the reference index over $n \in \{5,6\}$, because $F_n/(F_n+L_n)$ converges rapidly: as $n$ grows, $L_n/F_n \to \sqrt{5} \approx 2.236068$, so $\sigma_c(n) \to 1/(1+\sqrt{5}) \approx 0.309017$, between the two computed values.

   from 4.3 Critical stress at $n=6$ and robustness

6. We report $R \approx -0.2764$, a $27.6\%$ reduction in divergence angle on switching.

   from 4.5 Resilience index

How it is done

1. Agronomic guidelines treat phyllotaxis as a fixed trait, so they cannot prescribe irrigation or fertilisation strategies that prevent a detrimental pattern switch; whether such switches occur under stress is precisely the premise the proposed experiment of Section 7.3 would test.

   from 1. Introduction

2. We use the same continued-fraction condition, under which the codenominator function takes Lucas values, to motivate the Lucas-related angle $\theta_L$ adopted in Section 3.

   from 2. Background and Related Work

3. We model the divergence angle $\theta$ of successive primordia as a weighted average of two canonical angles: $$\theta(\sigma)= (1-\sigma)\,\theta_F + \sigma\,\theta_L, \tag{1}$$ where $\sigma \in [0,1]$ is a normalized stress variable ($\sigma = 0$ for optimal conditions, $\sigma = 1$ for extreme stress), $\theta_F = 360^\circ/\varphi^2$ is the golden angle with $\varphi = (1+\sqrt{5})/2$, and $\theta_L = 360^\circ/(\varphi+2)$ is a Lucas-related angle.

   from 3.1 Model overview

4. To connect $\sigma$ with the underlying integer sequences, we define a stress-weighted term $$W_n(\sigma)= (1-\sigma)F_n + \sigma L_n, \tag{2}$$ where $F_n$ and $L_n$ are the $n$-th Fibonacci and Lucas numbers.

   from 3.2 Stress-weighted sequence

What would falsify it

1. The main result, $\sigma_c \approx 0.31$ with an abrupt $38.0^\circ$ angular jump, is falsified by any of the following observations: Continuous response without a threshold.

   from 6.2 Falsification criteria

2. If severely stressed plants retain $\theta_F \approx 137.5^\circ$, the premise that stress drives the switch at all is refuted.

   from 6.2 Falsification criteria

3. The abrupt switch requires that the meristem commit to a locally optimal packing; without it the model predicts gradual drift.

   from 6.3 Limitations

Where the claim stops

1. The weighting is linear by assumption; a nonlinear stress response would shift $\sigma_c$ by an amount not computed here.

   from 6.3 Limitations

2. This is the assumption to which the results are most sensitive.

   from 6.3 Limitations

3. The model never connects $\sigma$ to water potential, nutrient concentration, or any agronomic measurement, so the numerical threshold cannot yet be converted into an irrigation prescription.

   from 6.3 Limitations

Where to read

1. Abstract195 words · 1 min
2. 1. Introduction412 words · 2 min
3. 2. Background and Related Work600 words · 3 min
4. 3. Methods518 words · 2 min
5. 4. Analysis612 words · 3 min
6. 5. Results222 words · 1 min
7. 6. Discussion568 words · 2 min
8. 7. Practical Implications and Engineering Pathways652 words · 3 min
9. 8. Conclusion182 words · 1 min
10. Provenance and AI Involvement240 words · 1 min