Empirical validation of the adaptive cost explanation of the golden angle in phyllotaxis: the physiological cost mechanism, quantitative genetics, and variance structure of the divergence angle

Okabe (2015) proposed that the golden angle of about 137.5 degrees, the divergence angle at which successive leaf primordia arise at the shoot apical meristem (the growing tip of the shoot, where new organ primordia are initiated) of most vascular plants, is an evolutionary optimum: a cost function summing the squared angular shifts between the innate divergence angle and the rational phyllotaxis fractions realized on the mature stem is globally minimized when the innate angle equals the golden angle, and local minima at other Fibonacci-like limit angles explain the anomalous patterns seen in nature. The model is theoretical, and its own discussion lists the empirical gaps that must be closed to confirm the evolutionary scenario: the physiological mechanism and actual magnitude of the posited cost are unidentified and unmeasured; the genetic architecture and heritability of the divergence angle and of its variance are unknown; systematic measurements of divergence-angle variance within and between species, and of its predicted effect on the frequency of anomalous patterns, are lacking; and the quantitative relationship between leaf-trace length and growth assumed in the model has not been determined. A decade of later work has refined the theory (a surface-area-to-volume fitness proxy for primary vascular tissue, phylogenetic scenarios supported by new fossil data), identified genes whose perturbation shifts divergence angles, and weakened rival light-capture optimality accounts, but none of these gaps has been closed. The problem is to supply the missing empirical content: measure the cost mechanism, establish the quantitative genetics of the divergence angle, test the predicted variance structure across species, and determine the assumed relationship between leaf-trace length and growth, so that the adaptive explanation of the golden angle is confirmed or refuted rather than merely proposed.

Background

Phyllotaxis is the regular arrangement of leaves, flowers, or other lateral organs around a plant stem. It is characterized by the divergence angle, the angle between successive organs as seen along the stem axis. On the mature stem the arrangement usually conforms to a rational fraction: in a 3/8 phyllotaxis, every eighth leaf sits directly above one below it after three turns of the generative spiral, producing eight vertical ranks (orthostichies) of leaves. As a plant grows, these fractions typically follow a Fibonacci rule — each fraction is obtained from the previous two by adding numerators and denominators (1/2, 1/3 give 2/5, then 3/8, 5/13, ...), a change called phyllotaxis transition. In sharp contrast, the nascent organs (leaf primordia) at the shoot apical meristem, the growing tip of the shoot that successively produces organ primordia, are placed at an essentially constant divergence angle close to the golden angle of 137.5 degrees, an irrational value related to the golden ratio. Anatomically, each leaf is connected to the stem's vascular system (the conduits for water and sugars) by a strand called a leaf trace; when the phyllotaxis fraction transitions during growth, these vascular connections must be readjusted. Whether phyllotaxis has adaptive value at all had been debated for two centuries without resolution. Okabe (2015) proposed an adaptive answer: the innate divergence angle is selected to minimize the cost of phyllotaxis transition. In the model, divergence angles are expressed as fractions of the total circumference restricted to the range 0 to 1/2 (180 degrees), so that the golden angle corresponds to 0.382; each plant carries an inherited innate divergence angle $\alpha_0$, the realized fraction on the stem is a rational value $\alpha_n=\alpha(\alpha_0,l_n)$ that depends on $\alpha_0$ and on the leaf-trace length parameter $l_n$ (assumed linear in the leaf index $n$, as $l_n=2+n/3$ for $1\leq n\leq 30$), and the transition cost is the summed squared angular shift $u(\alpha_0)=\sum_{n\geq 1}(\alpha_n-\alpha_0)^2$, realized in the paper's cost landscape over the bounded range $1\leq n\leq 30$ set by the leaf-trace assumption. Averaging over a population in which $\alpha_0$ is normally distributed with mean $\bar{\alpha}$ and standard deviation $\delta\alpha$ yields a population cost $U(\bar{\alpha},\delta\alpha)$ with an absolute global minimum at the golden angle, deeper for smaller $\delta\alpha$, and local minima at the limit divergence angles of other Fibonacci-like sequences — matching the rarity pattern of anomalous phyllotaxis in conifers and sunflowers. The model is consistent with classical observations (e.g., anomalous patterns in a few percent of Norway spruce cones and sunflower capituli (composite flower heads), with frequencies varying by species and locality) and with the extreme diversity of high-order fractions in the fossil lycopsid Lepidodendron (an extinct tree-sized relative of clubmosses), but every load-bearing empirical input is an assumption. The cost itself is a geometric proxy: the physiological or anatomical mechanism (presumably the construction and rearrangement of vascular connections) that would make an angular mismatch energetically expensive has never been identified or measured. The divergence angle is treated as a heritable quantitative trait, yet nothing is known about its genetic architecture or heritability, and the variance $\delta\alpha$ — which the model predicts should differ between species and should control the frequency of anomalous patterns — has never been systematically measured within or across species. Later work has reshaped the theoretical landscape without closing these gaps. Okabe and Yoshimura (2021) and Okabe (2024) recast the fitness proxy as the surface-area-to-volume ratio of primary vascular tissue, reproducing the canonical sequence of optimal fractions as internode length decreases and the approach to a goal angle near 137 degrees, and Okabe (2024) added a phylogenetic scenario informed by the fossil record, including the discovery of rare non-Fibonacci spirals in the Early Devonian lycopod Asteroxylon mackiei (Turner et al. 2023). Strauss et al. (2020) showed by simulation that many divergence angles, not just the golden angle, give comparable light capture, weakening the older light-capture optimality account and arguing that selection favors the organ-positioning mechanism rather than a fine-tuned angle — a challenge any cost-based selection story must meet. Godin, Golé and Douady (2020) frame the regularity as geometric canalization of a dynamical inhibitory mechanism, a proximate (developmental) explanation whose relation to ultimate (evolutionary) explanations remains actively debated (Reinhardt and Gola 2022). On the genetics side, perturbation of specific genes can shift divergence angles in particular organs (the ETT–RPL–TEC3 module, a set of interacting auxin-response and transcription-factor genes, in Arabidopsis flowers, Simonini et al. 2017; the receptor-like cytoplasmic kinase gene RLCKVI_A2 in siliques, the elongated fruits of Arabidopsis, Kenesi et al. 2024), and moss spiral phyllotaxis can be controlled by the rotation angle of stem-cell division planes (Kamamoto et al. 2021), but no mapping or heritability study of the divergence angle as a quantitative trait exists. Field data showing that the frequency of Fibonacci-category spiral counts in sunflower depends on genotype, location, and year (Dozet et al. 2026) is consistent with the model's variance prediction but does not measure divergence angles directly.

Problem Statement

The adaptive cost-minimization explanation of the golden angle — the divergence angle of about 137.5 degrees between successive leaf primordia at the shoot apical meristem — proposed in Okabe (2015) rests entirely on unmeasured quantities. The problem is to supply the empirical content that would confirm or refute it. Concretely, with the common objective of testing whether the divergence angle is shaped by selection minimizing the cost of phyllotaxis transition: (1) identify and measure the physiological or anatomical mechanism that makes an angular mismatch between consecutive phyllotaxis fractions costly — presumably the construction and rearrangement of vascular (leaf-trace) connections — so that the model's summed squared angular shift (or its successor, the vascular surface-area-to-volume proxy) corresponds to a real, quantifiable energetic or fitness cost; (2) determine the genetic architecture of the innate divergence angle as a quantitative trait, including its heritability and whether it is polygenic, together with the genetics of its variance; (3) measure, systematically within and across species, the mean and variance of the innate divergence angle at the shoot apical meristem, and test the model's prediction that this variance differs between species and controls the phenotypic frequency of anomalous phyllotactic patterns; and (4) determine empirically the quantitative relationship between leaf-trace length and plant growth assumed in the cost landscape (as the linear function $l_n=2+n/3$ for $1\leq n\leq 30$), since the relative depths of the local minima depend on the lower limit of $l_n$ and its fine structure on the upper limit. These four deliverables are all required components of the problem, not interchangeable alternatives; but because the measurements they call for are absent from the literature and cannot be produced within the task itself, a resolution consists of an evidence-grounded determination, for each component, of how far the existing record already closes it and what decisive measurement or experiment would close the remainder, together with the strongest confirm-or-refute verdict on the adaptive explanation that the record as a whole supports — a bare literature summary, or a research plan that stops short of such a per-component determination and verdict, does not resolve the problem. Deliverable (4) fixes the model's internal parameterization, and the paleobotanical comparison opened by Lepidodendron's high-order fraction diversity and by the non-Fibonacci Asteroxylon finds is a supporting line of evidence on the historical selective landscape rather than a separate requirement.

References

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