A full-range computational analysis of plant shoot architectures showed that the golden angle of 137.5° is only one of many divergence angles — including the Lucas angle of 99.5° and other Fibonacci-like sequence angles — that near-maximize light capture in simplified projection models, leading to the hypothesis that evolution selects the organ-spacing mechanism at the shoot apex rather than any specific angle. The open problem is whether this equivalence, and the mechanism-not-angle selection hypothesis built on it, survives physiological realism: realistic radiative transfer (diffuse light, sunflecks, canopy dynamics, leaf optics) and photosynthetic acclimation, conversion of intercepted area into integrated carbon gain, extension beyond rosette plants to other growth forms with costs and trade-offs, and empirical tests linking variation in the organ-initiation mechanism and divergence-angle distributions to lifetime fitness in the field. The functional and evolutionary significance of the divergence angle remains open.

Background

Phyllotaxis is the arrangement of leaves and other lateral organs around a plant stem. In spiral phyllotaxis, each new organ primordium is initiated at a fixed divergence angle from its predecessor at the shoot apical meristem (the growing tip of the stem, a radially symmetric dome of dividing cells), and by far the most common divergence angle in vascular plants is the golden angle of approximately 137.5°, related to the golden ratio. Angles generated by other Fibonacci-like sequences — sequences in which each term is the sum of the two preceding ones but started from different integers, such as the Lucas sequence yielding about 99.5° and the first anomalous sequence yielding about 151.14° — also occur in nature, though less frequently, and some plants initiate organs in simultaneous pairs (bijugate phyllotaxis), halving the relevant angle.

A hypothesis dating to the nineteenth century holds that the golden angle is prevalent because it spaces leaves so as to minimize self-shading and thereby maximize light capture, photosynthesis, and ultimately fitness. Computational tests of this idea have conflicted. A detailed light model of the redwood-forest understory herb Adenocaulon bicolor (Pearcy & Yang, 1998) and a shadow-function analysis calibrated against it (King et al., 2004) found light capture to be maximal near the golden angle, whereas analyses of Mediterranean woody species (Valladares & Brites, 2004) and of tomato canopies with a functional-structural plant model — a simulation framework coupling 3-D plant architecture with physical light and physiological processes (Sarlikioti et al., 2011) — found that the divergence angle had little effect on light absorption compared with other architectural traits. A plant-competition model (Niklas & Hammond, 2013) showed that the interception advantage at the golden angle can be reduced or eliminated when petiole length, internode distance, or leaf dissection increase.

Strauss et al. (2020) swept the entire 0°–180° range of divergence angles at 0.25° resolution using simulated plants with idealized and scanned leaves of Arabidopsis thaliana and Cardamine hirsuta. Light capture was quantified with two projection-based, unitless measures: the light capture area (LCA) and light capture efficiency (LCE), the fraction of total leaf area exposed to the light source (normalized to the final or the current total leaf area, respectively), computed by orthographic projection of the leaves onto a plane perpendicular to a direct light source. They confirmed that the golden angle is a maximum, but found that many other Fibonacci-like angles are equally good; the essential requirement is only to avoid small-denominator rational angles (such as 90° or 120°) that superpose successive leaves. Varying leaf shape, petiole length, incident-light angle, leaf inclination, translucency, and age-dependent photosynthetic decline changed the curves quantitatively but not the equivalence of the optimal angles, whereas long internodes damped the angle dependence — so the conclusions hold mainly for rosette plants with short internodes. Developmental noise in the divergence angle degraded light capture (with dissected leaves more robust than simple leaves), and the heteroblastic progression of leaf shape (the programmed, age-dependent change in leaf form during development) improved early light capture by about 32% for the first five leaves in A. thaliana. Because several angles achieve equal light capture, and because different angles can even appear on the same plant, the authors proposed that the iterative organ-spacing mechanism that self-organizes primordia on the growing apex — not any particular divergence angle — is the more plausible target of natural selection. They explicitly acknowledged, however, that their measures approximate instantaneous light capture under mostly direct light and are not intended to express fitness in the broad sense.

Problem Statement

Determine whether the functional interpretation of phyllotaxis proposed by Strauss et al. (2020) — that multiple Fibonacci-like divergence angles are essentially equivalent for light capture, so that selection, if acting through light capture at all, must target the organ-initiation spacing mechanism rather than a specific angle — holds once physiological realism and fitness are included. This single objective comprises four linked components that jointly decide it. (1) Light regime and carbon gain: replace the projection-based LCA/LCE measures under mostly direct, orthographically projected light with realistic radiative transfer — direct plus diffuse light, sunflecks, canopy dynamics, leaf optical properties, temporal variation — coupled to photosynthesis with within-leaf saturation and acclimation, and establish whether the reported equivalence of the multiple optimal divergence angles persists, weakens, or breaks when the outcome measure is integrated carbon gain rather than intercepted area. (2) Architectural generality: extend the analysis beyond rosette and short-internode plants across growth forms, seasonal stem elongation trajectories, and three-dimensional canopy interactions, since internode length is already known to dampen the angle dependence. (3) Costs and selectability: incorporate mechanical and allocation costs (for example, longer petioles or narrower laminae trading off with support and photosynthetic surface) and ecological interactions, so as to establish whether the multiple near-optimal angles are equivalently achievable and selectable in nature rather than only equivalent in intercepted area. (4) Empirical test of the selection hypothesis: provide field or controlled-environment evidence linking variation in the organ-initiation mechanism and divergence-angle distributions across taxa or genotypes to lifetime fitness, deciding whether selection acts on the mechanism, on a specific angle, or through light capture at all. The objective is resolved only by combining a coupled modeling study for components (1)–(3) with field or controlled-environment evidence (existing or newly generated) for component (4), since component (4) is not resolved by a modeled fitness proxy alone. An answer that fixes only one component leaves the objective open.

References

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