Phyllotaxis, the arrangement of leaves on a plant stem, is well known because of its beautiful geometric configuration, which is derived from the constant spacing between leaf primordia. This phyllotaxis is established by mutual interaction between a diffusible plant hormone auxin and its efflux carrier PIN1, which cooperatively generate a regular pattern of auxin maxima, small regions with high auxin concentrations, leading to leaf primordia. However, the molecular mechanism of the regular pattern of auxin maxima is still largely unknown. To better understand how the phyllotaxis pattern is controlled, we investigated mathematical models based on the auxin–PIN1 interaction through linear stability analysis and numerical simulations, focusing on the spatial regularity control of auxin maxima. As in previous reports, we first confirmed that this spatial regularity can be reproduced by a highly simplified and abstract model. However, this model lacks the extracellular region and is not appropriate for considering the molecular mechanism. Thus, we investigated how auxin maxima patterns are affected under more realistic conditions. We found that the spatial regularity is eliminated by introducing the extracellular region, even in the presence of direct diffusion between cells or between extracellular spaces, and this strongly suggests the existence of an unknown molecular mechanism. To unravel this mechanism, we assumed a diffusible molecule to verify various feedback interactions with auxin–PIN1 dynamics. We revealed that regular patterns can be restored by a diffusible molecule that mediates the signaling from auxin to PIN1 polarization. Furthermore, as in the one-dimensional case, similar results are observed in the two-dimensional space. These results provide a great insight into the theoretical and molecular basis for understanding the phyllotaxis pattern. Our theoretical analysis strongly predicts a diffusible molecule that is pivotal for the phyllotaxis pattern but is yet to be determined experimentally.
Living organisms often form periodic patterns with spatial regularity in a self-organizing manner [1,2]. One such well-known example is phyllotaxis, the arrangement of leaves on a plant stem. The phyllotaxis exhibits various types of patterns depending on the plant species and this has attracted many scientists because of its beautiful geometric configuration [3]. Phyllotaxis is originated at the shoot meristem, in which leaf primordia are periodically formed by maintaining a constant distance from each other [4–7]. This spatial regularity is established by the mutual interaction between a mobile plant hormone auxin and its efflux carrier membrane protein PIN1, which cooperatively generate small regions with high auxin concentrations called auxin maxima that are involved in leaf primordia. In the process of the auxin maxima formation, auxin accumulates at the presumptive position of a future primordium while PIN1 is polarized toward the center position [8–11]. According to this experimental observation, the auxin maxima pattern is often explained by the concept of “up-the-gradient” in which auxin is transported by PIN1 against its own gradient while PIN1 is polarized toward higher auxin [11–13]. Based on this concept, a class of mathematical models (corresponding to Model O in this paper) has been proposed, in which PIN1 is localized to a cell membrane depending on the auxin concentration of neighboring cells. Because these models can successfully reproduce the spatial regularity of auxin maxima and various phyllotaxis types, they are excellent models for understanding the nature of phyllotaxis pattern formation [11,12,14–16]. On the other hand, they are highly simplified and abstract models, and could have various problems when considering the molecular mechanism. With respect to spatial structure, these can be distilled into two major points. First, extracellular space (i. e. , apoplast space) is absent in these models, in which auxin moves directly between cells by PIN1 and diffusion. However, in plant tissues, auxin is transported between cytoplasm and apoplast because cells do not contact each other but are separated by apoplast space. Second, it remains unclear how cells sense auxin concentrations of neighboring but separated cells for PIN1 polarization. Sahlin et al. [16] showed that, despite the extracellular space, self-organized patterns can be generated by the “up-the-gradient” concept that PIN1 polarization depends on the auxin concentration of neighboring cells. However, it has still not been clarified how the information on auxin concentration is transmitted between neighboring cells. Conversely, although the apoplast space is also considered by Webnik et al. [17] and Cieslak et al. [18], these reports describe the canalization pattern generated by the “with-the-flux” concept, which is different from auxin maxima by the “up-the-gradient” concept. In this paper, therefore, we investigated how the spatial regularity of phyllotaxis pattern is controlled under realistic conditions, that is, in the presence of extracellular space. We first confirmed that the introduction of extracellular space has a disruptive effect on the spatial regularity in the conventional model (Model O), even in the presence of direct auxin diffusion between cytoplasm or apoplast spaces. This result strongly suggests that an unknown molecular mechanism is required for phyllotaxis pattern formation. We also found that the spatial regularity can be restored by assuming a diffusible molecule that mediates the feedback signaling from auxin to PIN1 polarization. This theoretical analysis strongly predicts a diffusible molecule that is critical for phyllotaxis pattern but remains to be found.
Change of auxin concentration of cell i (ai) is described by daidt=Ga (A−ai) −∑jfi, j+∑jDa (aj−ai) (1) fi, j=Ep (pi, jai−pj, iaj) (2) where cell j is a neighbor of cell i, A is related to the synthesis rate, Ga is the degradation rate, Da is the diffusion coefficient, Ep is the efficiency of PIN1 efflux carrier, and fi, j (= −fj, i) is the net flow of auxin by PIN1 from cell i to cell j, consisting of auxin efflux and influx. Auxin is constantly synthesized and degraded at a constant rate (the first term of the right-hand side of Eq 1), is transported by PIN1 (the second term), and diffuses between neighboring cells (the third term). On the other hand, change of PIN1 density (pi, j) is described by dpi, jdt=Gp (Kpφ0 (aj) ∑jφ0 (aj) −pi, j) (3) where Gp is the degradation rate, K is the number of neighboring cells, p is a constant related to PIN1 density, and φ0 (aj) is the regulatory function for PIN1 polarization (Fig 1A). PIN1 is localized to cell membrane depending on the auxin concentration of neighboring cells and is degraded at a constant rate. The total PIN1 amount of cell i, Pi ≡ ∑jpi, j, satisfies dPi/dt = Gp (Kp − Pi), indicating that Kp is the stable equilibrium of Pi. Thus, equilibria of ai and pi, j are given respectively by aeq=Aandpeq=p (4) When Gp is sufficiently large, pi, j quickly approaches equilibrium: pi, j=Kpφ0 (aj) ∑jφ0 (aj) (5) Therefore, Eq 5 can be used instead of Eq 3 in a simplified version of Model O. Model O in this study is equivalent to models reported previously in references [11] and [12]. Model equations of the simplest form in [11] can be described by daidt=−∑jT (pi, jai−pj, iaj) +∑jD (aj−ai) (6) pi, j=Paj∑jaj (7) where T, D, and P are constants. Eqs 6 and 7 are identical to the simplified version of Model O (Eqs 1,2 and 5) with Ga = 0, Da = D, Ep = T, p = P/K, and φ0 (aj) = aj. Conversely, equations used in [12] are somewhat complicated compared to those in [11]. However, phyllotactic patterns can be generated under the condition of fixed total PIN1 concentration (i. e. , [PIN]i is constant in Eq 2 of [12]), no saturation of auxin synthesis (i. e. , κIAA = 0 in Eq 5), and the linear dependence of the flux on auxin concentration (i. e. , replacement of [PIN]i2 and [PIN]j2 by [PIN]i and [PIN]j, respectively, in Eq 3). In addition to these conditions, by considering a regular cell lattice (i. e. , cell side length li→j is constant in Eq 2), model equations can be simplified by daidt=ρIAA−μIAAai−∑jfi, j+∑jD (aj−ai) (8) fi, j=T (pi, jai1+κTaj−pj, iaj1+κTai) (9) pi, j=Pbaj∑jbaj (10) where ρIAA, μIAA, D, T, κT, P, and b are constants. Eqs 8–10 are identical to the simplified Model O (Eqs 1,2 and 5) with Ga = μIAA, A = ρIAA/μIAA, Da = D, Ep = T, p = P/K, and φ0 (aj) =baj, except for the saturation effect of auxin from neighboring cells on the flux in Eq 9. This effect would negatively affect the pattern formation in a manner that the saturation effect becomes strong and accordingly the pattern tends to disappear as κT increases. Therefore, this effect is not essential for generating a phyllotactic pattern, indicating that Eqs 8–10 are equivalent to Model O. We used one-dimensional arrays of N = 200,50, or 40 cells and two-dimensional sheets of 20 × 20 or 14 × 14 hexagonal cells in the numerical simulations. Initial values of auxin, PIN1, and molecule X are given by their equilibrium with 1. 0% fluctuation. The numerical simulations were performed using the Euler method with time step Δt = 0. 001 under the periodic boundary condition. Equations and regulatory functions used are summarized in S1 Table. Parameter values used are described in figure legends. To evaluate the spatial scale of auxin patterns generated by the numerical simulations, we used wavelength of auxin maxima (L1) and average size of auxin maximum (L2) as indices of the spatial scale (Fig 1G). In a one-dimensional array of N cells in the periodic boundary condition, auxin concentration of cell n or apoplast (n, n+1) is denoted here by cn, where n = 0,1, ⋯, N and c0 ≡ cN.
The phyllotaxis pattern of auxin maxima has been explained by a class of simplified mathematical models based on the feedback dynamics between auxin and PIN1. Because these models do not consider extracellular region (i. e. , apoplast), auxin moves directly between cells by PIN1-dependent directional transport and passive diffusion to change its distribution (Fig 1A). On the other hand, PIN1 is asymmetrically localized to the cell membrane, preferentially toward neighboring cells with high auxin concentrations. We used Model O (Eqs 1–3), which is one of the simplest representations of such dynamics, to examine the spatial regularity control of auxin maxima pattern. As in previous reports [11,12], we confirmed that Model O can form auxin maxima patterns with spatial regularity, an essential characteristic of phyllotaxis, focusing on its spatial scale.
In phyllotaxis pattern formation, auxin maxima involved in leaf primordia are formed in a self-organizing manner while maintaining a constant distance from each other. However, the molecular mechanism generating the spatial regularity remains unclear. This spatial regularity has been explained by simple mathematical models (Model O; Figs 1A, 2 and 3A), in which PIN1 is polarized preferentially toward neighboring cells with higher auxin concentrations [11–13]. But these models have two major problems concerning spatial structure: one is the absence of extracellular space and the other is how cells perceive auxin concentrations of neighbors for PIN1 polarization. In this study, therefore, we intensively investigated how the spatial regularity of the phyllotaxis pattern is controlled under appropriate conditions for plant cells. We showed theoretically and numerically that auxin maxima patterns with large spatial scale are completely eliminated by introducing extracellular space (Model A; Figs 1B, 3B, 4 and 5). This strongly suggests that phyllotaxis pattern requires an unknown molecular mechanism as well as auxin–PIN1 mutual interaction. Furthermore, we found that regular patterns can be restored by the simple and plausible assumption that a diffusible molecule mediates the feedback from auxin to PIN1 polarization (Model B6; Figs 3C and 8). Although we mostly investigated in one-dimensional space, the same can be applied to the case of two-dimensional space. Model O can generate regular patterns of auxin maxima (Fig 3A). This spatial regularity is completely disrupted by considering extracellular space in Model A (Fig 3B), but is restored by introducing a diffusible molecule in Model B6 (Fig 3C). This diffusible molecule plays the role of transmitting auxin concentration to neighboring cells. Auxin reportedly enhances the PIN1 localization at the cell membrane [27–29]. AUXIN-BINDING PROTEIN 1 (ABP1) might act as an apoplastic auxin receptor in the signaling pathway of PIN1 polarization although the function of ABP1 has recently contended [28,30–32]. However, ABP1 probably makes no contribution to the phyllotaxis pattern formation regardless of whether it is an actual auxin sensor or not, because our study strongly suggests that the auxin maxima pattern cannot be established by the direct regulation of auxin on PIN1 polarization (Figs 4–6 and S1–S3). Although our study showed that a diffusible factor can restore regular patterns that are disrupted by the presence of extracellular space, this finding does not rule out other possibilities for the spatial communication between cells. One such possible mechanism is mechanical force, including stress and strain, which affects the morphogenesis of plants and animals [33–36]. Mechanical force could stabilize the outgrowth of leaf primordia by feedback mechanism in which mechanical stress induces alignment of microtubules, enhancing cell elongation and primordial outgrowth, which reinforces the stress field [37]. In contrast, experimental evidence that mechanical force is involved during auxin maxima formation has not yet been obtained [4,7]. However, auxin could alter the mechanical properties of the extracellular matrix by inducing cell-wall loosening [5,6, 38,39], suggesting that mechanical force may contribute to the pattern formation. Our model predicts that the spatial scale of generated patterns (L*) becomes large by increasing ν*, which follows ν* ∝ (1 + Ra) /aeq|θ′ (aeq) |, where Ra ≡ Da/Epp and θ (ai) is the regulatory function of auxin on X synthesis (Model B6; S1 Text, Eq 44). Therefore, if molecule X predicted in this paper exists, L* is affected by the regulatory activity of auxin on the expression of X as well as by the amount of PIN1. That is, under the conditions of a constant amount of PIN1 and constant activity of PIN1 recycling between cytosol and cell membrane, it is expected that, as the gene expression control by auxin becomes weak, the spacing between auxin maxima gradually increases and patterns suddenly disappear under a threshold of the control strength. This prediction could be used to experimentally validate whether or not our model is correct. Auxin affects the expression of many genes by cooperating with the TRANSPORT INHIBITOR RESISTANT 1/AUXIN SIGNALING F-BOX (TIR1/AFB) F-box proteins, the AUXIN/INDOLE-3-ACETIC ACID (Aux/IAA) transcriptional coregulators, and sequence-specific binding proteins called AUXIN RESPONSE FACTORs (ARFs) [40–42]. Because these factors are possible candidates that control the expression of molecule X, our model could be verified using plants showing various expression activities by genetically manipulating these factors. Our study could predict a diffusible factor that is essential for phyllotaxis pattern but remains to be found. This factor (s) X must satisfy the following requirements: Although such factors like X are not yet known, several diffusible molecules affecting PIN1 polarization have been reported. Strigolactone is a mobile plant hormone and cooperates with auxin to control shoot branching of plants. Auxin positively regulates the transcript of strigolactone biosynthesis genes and, in turn, strigolactone signaling triggers PIN1 depletion from the plasma membrane [43–45]. On the other hand, GOLVEN (GLV) genes encode small secretory peptides that are involved in root gravitropic responses and meristem organization in Arabidopsis. Transcription of GLV genes is rapidly induced by auxin, and the GLV peptide treatment stimulates the localization of auxin efflux carrier PIN2 at the cell membrane [46,47]. Another mobile plant hormone cytokinin plays important roles in various developmental events through crosstalk with other plant hormones including auxin [48–50]. For example, in vascular differentiation, cytokine affects the orientation of PIN proteins in cell membrane while auxin regulates cytokinin signaling [51,52]. Besides, also during lateral root organogenesis, cytokinin enhances the PIN1 depletion from cell membrane to affect PIN1 polarization [53–55]. Furthermore, it is reported that the localization of PIN proteins is affected by diffusible molecules such as jasmonate and narciclasine [56,57]. It is not yet clear whether these molecules are involved in the phyllotaxis pattern or not. However, in near future, we hope that a molecule predicted theoretically in this study will be revealed experimentally.
Self-organization of spatially regular patterns is critical for development and differentiation in multicellular organisms. Phyllotaxis, the arrangement of leaves on a plant stem, shows diverse patterns depending on plant species, which attracts many people because of its beautiful geometric configuration. In particular, it is well known that the spiral phyllotaxis is closely related to mathematical concepts such as the golden ratio and Fibonacci sequence. The phyllotaxis pattern is established by the mutual interaction between a diffusible plant hormone auxin and its efflux carrier PIN1, but its molecular mechanism is still largely unknown. To understand how phyllotaxis the pattern is controlled, we have theoretically and numerically investigated mathematical models based on the auxin–PIN1 interaction. Our theoretical analysis predicts a diffusible molecule that is critical for the phyllotaxis pattern but is yet to be determined experimentally. Furthermore, we predict the molecular mechanism of this molecule, which must mediate the feedback signaling from auxin to PIN1 polarization. Our study provides a detailed insight into the theoretical and molecular basis for understanding the phyllotaxis pattern, leading to the creation of plant varieties with various phyllotaxis patterns in the future.
plant anatomy
apoplastic space
hormones
simulation and modeling
developmental biology
plant science
plant hormones
cellular structures and organelles
morphogenesis
pattern formation
research and analysis methods
extracellular space
leaves
cell membranes
cytoplasm
biochemistry
plant biochemistry
cell biology
biology and life sciences
auxins
Spatial regularity control of phyllotaxis pattern generated by the mutual interaction between auxin and PIN1