**1. Introduction**

Agnes Arber [1] in "The Natural Philosophy of Plant Form" comprehensively described the development of plant morphology from the ancient philosophers—Plato, Aristotle and Theophrastus—to the more recent Cambridge botanical tradition that extends from William Turner, Nehemiah Grew and "Robin" Hill to the present. William Turner (1508–68), father of English botany. published the first herbal in English (1551) as a Fellow of Pembroke College; Nehemiah Grew (1641–1712), another Pembroke graduate, father of plant anatomy published "The Anatomy of Plants" (1682) depicted in the exquisite stained-glass windows of the college library. Finally, the Hill reaction demonstrated the photolysis of water as the source of atmospheric oxygen and established molecular botany as a new level of scientific enquiry. Arber's historical perspective may help resolve some long-standing problems of plant morphogenesis. Thus, Arber [1] presented "The mechanism of plant morphology" and an insightful approach to the pivotal role of the cell wall and the stress–strain of cell expansion that results in *"form conditioned by pressure"* where *"even a minor [cell wall] alteration may be associated with striking changes in the external form."* In Northcote's laboratory, those ideas catalyzed the first Ph.D. dissertation devoted to the primary cell wall and the discovery of cell wall proteins as a new field of study. These hydroxyproline-rich glycoproteins, especially the arabinogalactan proteins (AGPs), are involved in a hypothetical Hechtian growth oscillator. It involves transduction of the wall

stress–strain to the plasma membrane where an auxin-activated proton pump dissociates AGP-Ca2<sup>+</sup>. Elevated cytosolic Ca2+-activates exocytosis thus regulating plant growth. Discussion of the Hechtian Oscillator vis-a-vis the role of the primary cell wall in plant morphogenesis [2] suggests extrapolating the oscillator to phyllotaxis based on the premise that presence of the oscillator components implies the presence of a functional Hechtian Oscillator. Indeed, recent work suggests the mechanotransduction of stress relocates auxin efflux PIN proteins that generate new protoderm primordia. However, the precise biochemical mechanisms involved in stress transduction and the role of auxin and calcium homeostasis remain to be elucidated. Here, we invoke Hechtian adhesion and AGPs as essential components that lead us to propose a novel biochemical algorithm for floral phyllotaxis and an explanation of its strong tendency towards a periodic series first described by Fibonacci (1170–1240). This approach contrasts with many previous studies with an overwhelming mathematical bias. Indeed, many observations in Nature involve periodicity and the probable underlying oscillations

Oscillatory plant growth, known since Darwin [3], was subsequently confirmed by rapid tip growth of pollen tubes and root hairs [4]. Plant morphogenesis also involves periodicity strikingly displayed by the pattern of leaves and floral organs [5] that often appear as Fibonacci spirals typified by whorls of 3, 5, 8, 13, 21 and 34 petals [6]. Hypothetically, such periodicity depends on an underlying oscillator such as the recently formulated Hechtian growth oscillator [2,7] that involves auxin-driven Ca2<sup>+</sup> release from arabinogalactan proteins (AGPs) of the cell surface; this hypothesis accounts for the origin of oscillations in molecular detail absent from previous models of tip growth [8]. Here, we extrapolate the Hechtian Oscillator to the challenging problem of phyllotaxis and the generation of primordia in the protoderm, the outermost cell layer of the stem apical meristem SAM. Earlier work emphasized physical factors and a mathematical approach was comprehensively reviewed in [6,9,10]. However, more recent work emphasizes a cell wall stress vector generated by rapid cell expansion in the protoderm that re-orientates auxin efflux PIN proteins of neighboring cells and thus directs auxin transport (and the inferred generation of Ca2<sup>+</sup> waves) that regulate growth and differentiation (e.g., [11–14]). The present paper complements these and more recent models of [15] but with the notable exception of [16]; none consider a possible role for cell surface AGPs. However, *"Nature keeps some of her secrets longer than others"* [17]. That includes the elusive molecular function of classical AGPs [18–20]. Identified some fifty years ago [21–23], AGPs remained "A Great Puzzle" until the recent demonstration that AGP glycomodules bind Ca2<sup>+</sup> specifically [24]. They form a cell surface AGP-Ca2<sup>+</sup> capacitor that involves the interaction of three essential ions auxin, H<sup>+</sup> and Ca2+. These "morphogens" of the Ca2<sup>+</sup> signal transduction pathway (Figure 1) interact and thus regulate cell expansion and growth.

The pathway begins with the transduction of the cell wall stress vector to the plasma membrane, via AGP57C [25] as the likely molecular basis of Hechtian adhesion between the cell wall and the plasma membrane. Further transmission of a biochemical signal to the cytoplasm involves stretch-activated proton and Ca2<sup>+</sup> ion fluxes of the plasma membrane generated by the Hechtian growth oscillator [7]. The cytoplasmic response to Ca2<sup>+</sup> influx presumably involves exocytosis of wall plasticizers and precursors including redirection/reorientation of auxin efflux PIN proteins, eponymously named after their mutant pin-shaped phenotype. These auxin transport proteins channel auxin flow away from slow expansion towards rapid expansion thus generating auxin waves with maxima corresponding to the periodicity of nascent primordia. Turing's classic paper [26] postulated only two morphogens sufficed to generate spiral phyllotactic periodicity. The sections below expand on Turing's original suggestion with recent experimental evidence. Turing's insight was much closer to reality than the "two interacting morphogens" he envisaged.

(**a**)

**Figure 1.** (**a**) The Hechtian Oscillator ion fluxes regulate growth. Depicts a simplified version of the Hechtian Oscillator in [2]. This figure shows stills from the animation in Supplemental Data. Membrane and ion fluxes are analogous to a molecular "pin-ball machine." KEY: protons: red, Ca2<sup>+</sup> ions: yellow, auxin: green, stretch-activated Ca2<sup>+</sup> channels; Ca2<sup>+</sup> trickle initiates proton pump activity: (**b**) **Phase I. Quiescent: [7s]** Proton pump minimally active; Ca2<sup>+</sup> channels closed with minimal Ca2<sup>+</sup> influx. **Phase II. Activation: [6s]** Turgor increases cell expansion and thus wall stress that increases demand for auxin and opens stretch-activated Ca2<sup>+</sup> channels; Ca2<sup>+</sup> trickle initiates auxin binding by the proton pump, initiating low-level oscillator activity leading to Phase III. **Phase III. Fully Activated: [12s]** high auxin levels fully activate proton pump. Proton extrusion dissociates periplasmic glycomodule AGP-Ca2<sup>+</sup>. Entry via Ca2<sup>+</sup> channels generates cytosolic Ca2<sup>+</sup> waves that activate: exocytosis of: cell wall precursors, wall plasticizers and redirect auxin efflux "PIN" proteins. **Phase IV: [9s]** Returns to Quiescent state: Stress relaxation closes Ca2<sup>+</sup> channels. Auxin dissociates from proton pump; cytosolic Ca2<sup>+</sup> recycles to recharge glycomodules and determine phyllotaxis periodicity as follows.

The ingenuity of Mother Nature exceeds our human imagination by involving three interacting ions, auxin, protons and Ca2<sup>+</sup> (Figure 1) as the master regulator of plant growth. Although ion accumulation studied for more than 80 years [27] has generally assumed the relative immobility of Ca2<sup>+</sup> ionically bound to the cell wall, non-intuitively Ca2<sup>+</sup> bound by cell surface AGPs now appears to be the major source of dynamic cytosolic Ca2<sup>+</sup>. Counter-intuitively, the mechanism for the release of dynamic Ca2<sup>+</sup> from ionically bound AGP-Ca2<sup>+</sup> is not obvious. However, the paired glucuronic carboxyls of AGP glycomodules explain the remarkable stoichiometric Ca2+-binding properties of periplasmic AGP-Ca2+; its dissociation by an auxin-activated proton pump predicts an essential role of AGPs in Ca2<sup>+</sup> homeostasis [24].
