The biological basis of primordial age-dependent inhibitory power in the EDC2 phyllotaxis model: mechanisms, species-specific parameters, and empirical tests of orixate pattern formation

Inhibitory-field models explain phyllotaxis (the ordered arrangement of leaves around a stem) by assuming each existing leaf primordium inhibits the initiation of new primordia nearby. Yonekura, Iwamoto, Fujita and Sugiyama (2019) showed that the classic Douady–Couder models cannot generate orixate phyllotaxis, a rare four-cycle pattern known from Orixa japonica and a few other taxa, but that an expanded model (EDC2), in which each primordium's inhibitory power increases sigmoidally and slowly at a late stage of its maturation, reproduces orixate patterns within a narrow parameter window and places them adjacent to spiral and decussate regions of parameter space. The hypothesis that the inhibitory power of a primordium grows with its age — interpreted as an expanding range of auxin removal (auxin 'drain' activity) around the maturing primordium — remains purely theoretical. The open problem is to give this hypothesis a biological foundation: identify the molecular and cellular mechanisms that expand the spatial range and strength of auxin removal during primordium maturation with the time-course the model infers, and determine what developmental-genetic factors set the species-specific model parameters. Whether orixate solutions survive added biological realism and stochasticity, and whether the model-implied transitions between neighboring pattern classes such as spiral, orixate and decussate can be tested empirically, are related directions that go beyond the open question posed by the source.

Background

Phyllotaxis is the regular arrangement of lateral organs (leaves, flowers, or floral organs) around the stem, conventionally described by the divergence angle between successive organs and the plastochron, the time interval between successive organ initiations. Organs are produced at the shoot apical meristem (SAM), a dome of stem cells at the growing tip; new leaf primordia arise one at a time (or in whorls) at the SAM periphery. The dominant mechanistic paradigm derives from Douady and Couder's inhibitory-field models (DC1, DC2), in which each existing primordium emits a constant inhibitory power whose effect declines with distance, and a new primordium forms where the summed inhibitory field on the SAM periphery drops below a threshold (Douady and Couder 1992, 1996). At the molecular level, organ position is widely believed to be set by the plant hormone auxin: the membrane-localized efflux carrier PIN1, polarized in epidermal cells toward incipient organ sites, concentrates auxin into convergence points that act as organ-initiation foci and as auxin sinks ('drains') that deplete surrounding tissue (Reinhardt et al. 2003; Smith et al. 2006). Most phyllotaxis research has focused on major patterns — distichous (180° alternation), decussate (opposite pairs rotated 90°), and spiral patterns with divergence angles near the golden angle of about 137.5°, which follow Fibonacci counts of parastichies (the visible spiral rows of organs arranged along a helix). Minor patterns received far less theoretical attention. Orixate phyllotaxis, found in Orixa japonica and a handful of other species in distant taxa, is a special alternating pattern with orthogonal tetrastichy (four vertical rows): the divergence angle cycles approximately as 180°, 90°, −180°, −90°. Yonekura, Iwamoto, Fujita and Sugiyama (2019) first confirmed by exhaustive simulation that the original DC models generate no orixate pattern at any parameter setting. They then expanded DC1 and DC2 into EDC1 and EDC2 by allowing each primordium's inhibitory power to change sigmoidally with the primordium's age. In EDC2, which allows simultaneous initiation of multiple primordia, the inhibitory power of a primordium of standardized age $t = T V_0/R_0$ — where $T$ is the time since the primordium's emergence and $V_0$ and $R_0$ are the meristem's radial growth velocity and radius scale, so that $t$ is measured in units of the meristem growth scale rather than in calendar or plastochron time — is multiplied by the sigmoid factor $F(t) = 1/(1+\exp(-A(t-B)))$, in which $A$ sets the rate (steepness) of the age-dependent increase of the inhibitory power, $B$ sets the timing of that increase, $\Gamma$ scales the inhibition range relative to the meristem periphery, $\alpha$ controls how steeply the inhibitory effect declines with distance, and a fifth parameter $N$ encodes the flatness of the shoot apex. Orixate patterns arise only in a narrow window — roughly $A \approx 3$–$7$ and $B \approx 0.4$–$1$ in the source's simulations at $\Gamma = 1$, $2$ or $3$ with $\alpha = 1$ and $N = 1/3$ fixed, and the window shifts when $\Gamma$, $\alpha$ or $N$ change — representing a late and slow increase of inhibitory power, and in the EDC2 parameter space the orixate region lies adjacent to decussate, Fibonacci-spiral and Lucas-spiral (Lucas numbers form a Fibonacci-like series) regions, so that small parameter changes convert neighboring patterns into orixate — consistent with the variety of patterns found among close relatives such as Kniphofia species. The authors interpret the age-dependent increase of inhibitory power biologically as an expansion, during primordium maturation, of the range over which the auxin convergence at the primordium absorbs auxin, and they hypothesize that the timing and rate of this expansion differ greatly among species and act as critical determinants of which phyllotactic pattern a species produces. This interpretation, however, is untested: EDC2 is an abstract, phenomenological model, and the paper itself closes by calling for investigation of the regulatory properties of auxin polar transport during primordium development as the route to the molecular mechanisms underlying the presumed age-dependent increase of inhibitory power and to the variation and limitation of phyllotactic patterns.

Problem Statement

The EDC2 model of Yonekura et al. (2019) accounts for orixate phyllotaxis by assuming that the inhibitory power of each leaf primordium increases sigmoidally with primordial age, as $F(t) = 1/(1+\exp(-A(t-B)))$ in standardized age $t$ — late and slowly, within a narrow parameter window ($A \approx 3$–$7$, $B \approx 0.4$–$1$ at the source's simulation settings $\alpha = 1$, $N = 1/3$ and $\Gamma \in \{1, 2, 3\}$; the window shifts when $\Gamma$, $\alpha$ or $N$ change) — and interprets this biologically as an expanding range of auxin removal around the maturing primordium. The open problem is to establish, or refute, the biological basis of this hypothesis. Concretely: (1) identify the molecular and cellular mechanisms — candidates include PIN1 localization dynamics, auxin sink strength of the developing primordium, and primordium growth — that expand the spatial range and/or strength of auxin removal during primordium maturation with the sigmoidal time-course the model infers (rate $A$ and timing $B$ in the ranges above, expressed in standardized primordial age); relevant evidence may include, for example, direct measurement of primordium-scale auxin transport range or inhibitory influence as a function of primordium age, perturbation or genetic studies, or model-based inference from comparative developmental data; and (2) determine which developmental-genetic factors, if any, set the species-specific values of the model parameters $\Gamma$, $\alpha$, $A$, $B$ and $N$, and test whether orixate taxa occupy the corresponding orixate window of the model while their close relatives (e.g., within Kniphofia, where related species show spiral, orixate or semi-decussate patterns) fall in neighboring pattern-class regions of parameter space. These components jointly serve one objective: to convert the EDC2 hypothesis that an age-dependent, species-specifically timed increase of primordial inhibitory power generates orixate phyllotaxis (and, more generally, contributes to interspecific phyllotactic diversity) from a phenomenological model assumption into a validated or refuted biological mechanism. A resolution requires both components (1) and (2) to be addressed; a study that answers only one of them does not resolve the task, whether the hypothesis is ultimately supported or refuted. Two further directions follow from the model's structure but are not part of the open question posed by the source, and are noted only as related context: whether orixate solutions survive added biological realism (cell geometry, molecular stochasticity, mechanical interactions, realistic PIN1 dynamics), and whether the transitions between neighboring pattern classes (spiral ↔ orixate ↔ decussate) implied by the parameter-space adjacency can be tested empirically.

References

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