Auxin biosynthesis and cellular efflux act together to regulate leaf vein patterning

Abstract

Our current understanding of vein development in leaves is based on canalization of the plant hormone auxin into self-reinforcing streams which determine the sites of vascular cell differentiation. By comparison, how auxin biosynthesis affects leaf vein patterning is less well understood. Here, after observing that inhibiting polar auxin transport rescues the sparse leaf vein phenotype in auxin biosynthesis mutants, we propose that the processes of auxin biosynthesis and cellular auxin efflux work in concert during vein development. By using computational modeling, we show that localized auxin maxima are able to interact with mechanical forces generated by the morphological constraints which are imposed during early primordium development. This interaction is able to explain four fundamental characteristics of midvein morphology in a growing leaf: (i) distal cell division; (ii) coordinated cell elongation; (iii) a midvein positioned in the center of the primordium; and (iv) a midvein which is distally branched. Domains of auxin biosynthetic enzyme expression are not positioned by auxin canalization, as they are observed before auxin efflux proteins polarize. This suggests that the site-specific accumulation of auxin, as regulated by the balanced action of cellular auxin efflux and local auxin biosynthesis, is crucial for leaf vein formation.

Auxin, auxin biosynthesis, auxin canalization, auxin transport, leaf, mathematical modeling, vein patterning

Introduction

Vascular systems are continuous networks of conductive cells which connect a wide range of tissues. In leaves, the vasculature comprises veins which form characteristic patterns and support photosynthesis in the surrounding mesophyll cells. As the vascular tissue matures in the young leaf primordium, its cells differentiate and acquire specialized functions. All leaf vascular cells have their origin in the pre-provascular cell which is identified by the expression of PIN1, an auxin efflux carrier (Scarpella et al., 2006). This expression is significant as auxin is a key regulator of vascular cell development (Bruck and Paolillo, 1984). The expression of an auxin transporter in pre-provascular cells is particularly important because flux-dependent increases in cell permeability to auxin lead to the formation of self-reinforcing auxin streams which drive cell differentiation (Mitchison, 1980).

Experimental studies on this channeling of auxin (generally referred to as auxin canalization) have been corroborated with theoretical models based on the hypothesis that PIN1-dependent auxin efflux is primarily responsible for cellular auxin efflux in the primordium, and that its distribution in the plasma membrane determines the direction of auxin streams. Modeling auxin fluxes in a fixed domain of leaf tissue can create channels of auxin that resemble the vein patterns observed in leaves (Rolland-Lagan and Prusinkiewicz, 2005; Cieslak et al., 2015; Abley et al., 2016). However, the robustness of canalization models is reduced when applied to a growing tissue or to certain observed architectural features such as vein loops (the enclosure of mesophyll cell islands by vascular strands) (Lee et al., 2014; Feller et al., 2015).

The canalization hypothesis predicts that the position of auxin sources in the leaf primordium is critical for vein patterning (Rolland-Lagan and Prusinkiewicz, 2005). Although it has been proposed that these sources are defined by the focal points of epidermal auxin flux (Scarpella et al., 2006; Abley et al., 2016), auxin is also synthesized in the lamina of the growing primordium (Müller-Moulé et al., 2016), an observation which has, to date, not been considered by any models. Elsewhere in the plant, heterogeneous landscapes of auxin biosynthesis drive the formation of diverse tissues (Brumos et al., 2018), a paradigm which is also likely to apply to leaf vein formation, as here, genetically reducing auxin biosynthesis rates has a strong impact (Cheng et al., 2006; Brumos et al., 2018).

In this study, we use a combination of experiment and theory to ask whether auxin biosynthesis and auxin transport interact during vein initiation. We conclude that, in addition to PIN1, sites of future vein growth are marked by expression domains of auxin biosynthetic enzymes. Our key observation is that when auxin biosynthesis is decreased, a simultaneous decrease in auxin efflux has a compensatory effect on vascular development, recovering a wild-type (WT) leaf vein phenotype. This is consistent with a model in which both processes act antagonistically on cellular auxin concentration in pre-provascular cells. We propose that the relative strength of both processes defines the capacity of incipient vascular cells to develop. Furthermore, our model identifies a third factor, the mechanical constraints under which expanding pre-provascular cells are placed, which shapes the developing vein network, spontaneously organizing vascular cells into a distally branched midvein.

Materials and methods

Plant material and growth conditions

Columbia Arabidopsis ecotype (Col-0) as WT, YUCp::GUS reporter lines, and single and multiple combinations of yuc mutants were as previously described (Cheng et al., 2006; Ditengou et al., 2008; Tao et al., 2008). Details of pTAA1::TAA1-GFP, wei8-1, wei8-1tar2-1, and wei8-1tar1-1tar2-1 and pin1 mutants are as previously described (Galweiler et al., 1998; Stepanova et al., 2008). wei8 (Stepanova et al., 2008) is also known as sav3 (Tao et al., 2008). To more accurately reflect the corresponding protein’s function, WEI8 is referred to here as TAA1 TRYPTOPHAN AMINOTRANSFERASE OF ARABIDOPSIS1. YUC4p::GFP seeds were obtained from Yunde Zhao. Seeds were surface-sterilized and sown on solid Arabidopsis medium [2.3 g l^–1 Murashige and Skoog (MS) salts, 1% sucrose, 1.6% agar–agar (pH 6.0) adjusted with KOH]. After vernalization for 2 d at 4 °C, seeds were germinated under a long-day period (16 h light, 8 h darkness). In polar auxin transport inhibitory experiments, seedlings were grown on media supplemented with either 5 µM or 10 µM naphthylphthalamic acid (NPA) for 0–10 d.

Immunocytodetection

Seedlings of different ages were fixed with 4% paraformaldehyde in phosphate-buffered saline (PBS; pH 7.3) and used for whole-mount in situ immunolocalization as previously described (Ditengou et al., 2008; Pasternak et al., 2015). PIN1 was detected using a mouse anti-PIN1 monoclonal antibody (1:100) and TAA1–green fluorescent protein (GFP) with a rabbit anti-GFP antibody (1:600) (Molecular Probes). YUC4p::GFP was detected with a rabbit anti-GFP polyclonal antibody (1:200). Samples were incubated with secondary antibodies (Alexa 488 goat anti-mouse and Alexa 555 goat anti-rabbit from Invitrogen, both at 1:1000 dilution).

Microscopy and analysis

Histological detection of β-glucuronidase (GUS) activity was performed according to (Scarpella et al., 2004). For analysis of vascular patterns, seedlings were cleared in 100% ethanol overnight. They were re-hydrated and dissected under 50% glycerol, then mounted in chloral hydrate:glycerol:water (8:3:1, w/v/v). Leaf surface area and vein length were calculated using ImageJ software (https://imagej.nih.gov/ij/). Leaf vein density was calculated by dividing the total vein length by leaf surface area. GFP plants were fixed with 4% formaldehyde at room temperature and mounted in Prolong Gold antifade reagent (Molecular Probes). For light microscopy, samples were observed with a Zeiss Axiovert 200M MOT (Carl Zeiss, Goettingen, Germany) for high magnification pictures. In contrast, low magnification views were taken with a Zeiss Stemi SV11 Apo stereomicroscope (Carl Zeiss) and viewed under differential interference contrast (DIC) optics or dark field illumination. Fluorescent proteins were analyzed with a Zeiss LSM 5 DUO scanning microscope or AZ-C1 Macro Laser Confocal Microscope (NIKON GmbH, Düsseldorf, Germany). GFP was excited using the 488 nm laser line in conjunction with a 505–530 nm band-pass filter. To simultaneously monitor Alexa 488 and Alexa 555 fluorescence, we used multitracking in frame mode and the emission was separated using the online unmixing feature of the Meta spectral analyzer. Images were extracted and analyzed with the Zen2009 software (Carl Zeiss MicroImaging) and are representative of at least 20 individual plants.

In this work, we use the following definitions: pre-provascular cells are isodiametric and express PIN1 in a non-polar fashion. Provascular cells express both TAA1 and PIN1; during this stage cells elongate and PIN1 becomes polarized. Vascular cells do not express TAA1, are fully elongated, and display a sharply polarized PIN1.

Real-time RT–PCR

The effect of auxin on gene expression was quantified by real-time quantitative reverse transcription–PCR (qRT–PCR). Arabidopsis seeds were deposited on 6–7 mm filter paper strips lying at the surface of solid Arabidopsis growth medium (see above). This procedure facilitates the transfer of seedlings on the filter paper from agar to liquid Arabidopsis growth medium (not containing agar) supplemented with or without indole acetic acid (IAA). Three-day-old Arabidopsis seedlings were transferred to medium containing either mock (control), 1 µM IAA, or 5 µM IAA. After treatment, the filter paper strips with seedlings were transferred to RNA-later solution (Ambion). Total RNA was extracted from shoot tissues using the RNeasy Micro Kit (Qiagen). Reverse transcription was performed using 1 μg of total RNA and RevertAid M-MulV reverse transcriptase (Fermentas, St Leon-Rot, Germany) according to the manufacturer’s instructions. qRT–PCR was performed using the Maxima SYBR Green kit (Fermentas) on a Light cycler480 Real-Time system (Roche, Mannheim, Germany). The gene-specific primers for qRT–PCR are listed in Supplementary Table S2. The efficiency of each primer pair was determined by examination of a standard curve using serial dilutions of genomic DNA. The PCR was performed using a three-step protocol including melting curve analysis. The relative gene expression was analyzed using the Δ–Δ cycle threshold method. ACTIN2 served as a reference gene; three biological replicates and three technical replicates were used to evaluate gene expression.

Image post-processing

All images depicting PIN1/TAA1 double labeling or PIN1/DAPI are 3D reconstructions of optical sections with Imaris 7.4.0 (Bitplane AG, Zurich, Switzerland). All images were assembled using Microsoft PowerPoint 2013 or Adobe^® illustrator CS6.

Computational model description

A cell-based modeling framework, Virtual Leaf, that couples vertex dynamics and chemical dynamics, was adapted to study the vein patterning in leaves in silico (Merks et al., 2011). We simplified the model system by representing the longitudinal section of leaf primordia by a 2D network of interconnected polygons (cells), specified by the surrounding vertices. Each cell in this framework is characterized by an energy function which describes the balance between turgor pressure and cell wall stiffness,

$${\displaystyle \begin{array}{l}{\displaystyle E={\lambda }_A{\displaystyle \sum _{\alpha }}{(A_{\alpha }-A_{\alpha }^0)}^2+{\lambda }_s{\displaystyle \sum _{<ij>}}{(l_{ij}-l_{ij}^0)}^2}\end{array}}$$

(1)

A_α is the area of the cell α and A⁰_α is the preferred (rest) area of that cell. l_ij is the length of the cell wall linking nodes i and j of the polygons, and the sum over <ij> is over all links. λ _A is a parameter setting the cell resistance to compression or expansion, and λ _s describes the cell wall stiffness. Cell behaviors, such as expansion, division, and active shape changes are described by minimization of the energy function E. The minimization depends on the dimensionless ratio λ _Al₀²/λ _s, where l₀ is a cell size-dependent length scale. The value of the dimensionless ratio used in our simulations was 0.01, in accordance with Merks et al. (2011), Hervieux et al. (2017), and Zhou et al. (2020).

To study vein patterning in developing leaf primordia, we created a leaf template to resemble the tissue of a leaf primordium, as in Fig. 2A. Cells at the base of this template (cells colored in gray in Fig. 2A) were assigned with very high cell stiffness λ _s, in order to implement proper boundary conditions, namely the effect of the firm attachment of the leaf primordia to the shoot apical meristem. The outermost layer of cells in the leaf primordia represents the stiff epidermal layer. To incorporate these properties of the epidermal cells in our model, the outermost cell walls of the cells at the perimeter of the leaf primordium were assumed to have higher stiffness than the internal cell walls. We used a value for the ratio of perimeter to internal cell wall stiffness within the experimentally reported range, λ _s (perimeter)=12λ _s (interior) (Gibson et al., 1988; Onoda et al., 2015). In our model, auxin is synthesized locally in only a few cells and subsequently auxin diffuses across the neighboring cells in the tissue. The leaf primordia in our model consists of three different cell types: (i) auxin-synthesizing cells (colored in dark green in Fig. 2A); (ii) cells that do not synthesize auxin (colored in light green in Fig. 2A); and (iii) petiole cells (colored in gray in Fig. 2A). The latter is a computational cell type aimed to simulate the firm attachment of the leaf to the meristem and the drainage of auxin.

Fig. 2.

In silico modeling of spontaneous patterning of vascular tissue in leaf primordium. (A) Left: leaf template resembling a leaf primordium. The model assumes local auxin synthesis (dark green colored cells), leaf petiole cells (gray colored cells), a relatively stiff outermost surface of the epidermal layer, non-directional transport of auxin, and auxin-dependent cell growth; middle: TAR2-driven GUS expression in a 2-day-old control leaf primordium; right: PIN1 subcellular localization in 2-day-old-leaf; scale bar=20 µm. (B) Left: simulation result of our model with the parameter values as given in Supplementary Table S1. The model was able to reproduce a realistic midvein made up of elongated cells and branched vasculature, as observed in a 4-day-old control leaf primordium (see C). Forces on the vertices of a cell in the leaf tissue are shown by an arrow representing the force vector, in the case of a control leaf; magenta colored arrows represent forces on the vertices of midvein cells, and blue arrows represent forces on the vertices of other cells in the leaf primordia. Inset shows cellular auxin concentration in the leaf primordia. (C) TAR2-driven GUS expression in a 4-day-old control leaf primordium. (D) Simulation result with a 25-fold lowered rate of auxin transport. All the other parameter values were kept the same as in (C). Inset shows cellular auxin concentration in the leaf primordium. (E) TAR2-driven GUS expression in 3-day-old NPA-treated leaf primordia. (F) Simulation result for reduced auxin production and reduced transport rate. A realistic midvein could be produced in simulation only when the overall cell area growth rate was decreased, consistent with NPA treatment rescue of an auxin biosynthesis mutant. (G) In silico modeling of PIN1-based auxin transport. The model incorporates PIN1-based transport of auxin along with auxin diffusion, and is able to reproduce a realistic midvein. The model reproduces a realistic midvein even in the case of no diffusion (d=0) and PIN1-based transport only. Inset shows cellular auxin concentration in the leaf primordia. The arrows indicate the net direction, but not the magnitude of the auxin flux. The lack of clear polarity of the auxin flow in the epidermal cells is likely to be due to the absence of any external source of epidermal auxin.

The general dynamic equation governing the amount of auxin n_α in a cell is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{n}_{\alpha }}{\mathrm{\partial }t}=s(\alpha )+d{\displaystyle \sum _{\beta }}{l}^{\alpha \beta }(\frac{n_{\beta }}{A_{\beta }}-\frac{n_{\alpha }}{A_{\alpha }})}\end{array}}$$

(2)

Here l^αβ is the length of the cell wall separating cells α and β, across which the auxin diffusion takes place, and A_α is the area of the cell α. The first term describes auxin production at a constant rate s(α)=s₀δ _α,p, non-zero only for the auxin producer cells p. Auxin production is limited to only a few cells (cell colored in dark green in Fig. 2A). The second term accounts for auxin spreading in the tissue of leaf primordia by a diffusion process, where d is a constant that measures the speed of intercellular auxin transport and is related to the auxin diffusion coefficient. Auxin is drained through cells located at the base of leaf primordia. To simulate this auxin drainage, from these cells (termed ‘petiole cells’ and colored in gray in Fig. 2A), we implement perfectly absorbing boundary condition at the petiole cell walls.

At time t=0 the amount of auxin is n=0 in all the cells in the leaf primordia. For t>0, auxin is produced only in auxin-synthesizing cells at a constant rate s₀. Motivated by experimental images of a leaf at 2 days after germination (DAG), in our simulation we start initially with only four auxin-synthesizing cells, but our qualitative results are not dependent on the initial geometry (Fig. 2A left and right panel; Supplementary Fig. S4). All the cells in the leaf primordia grow by increasing their target area at the same constant rate g₀=610^–14 m² s^–1 and divide over their shortest axis once their area is doubled. However, in any cells except the auxin-synthesizing cells, if the auxin concentration increases beyond a threshold value c*=2.4×10⁹ m^–2, then the growth rate of that cell increases to a value g>g₀. We tested a broad range of cell area growth rates, g, and found that main vein formation and main vein bifurcation is a robust feature appearing for the values of ratio g/g₀ ≥3 and up to at least several hundreds. Simulation results of our model with the parameter values are given in Supplementary Table S1.

The force F(X_i) on each node i at vector position X_i of a cell is given by:

$${\displaystyle \begin{array}{l}{\displaystyle F\left(X_i\right)=-\frac{\mathrm{\partial }E}{\mathrm{\partial }{X}_i}}\end{array}}$$

(3)

$${\displaystyle \begin{array}{l}{\displaystyle =-{\lambda }_A{\displaystyle \sum _{\alpha ,i\in \alpha }}(A_{\alpha }-A_{\alpha }^0){\left[R(X_{i+1}^{\alpha }-X_{i-1}^{\alpha })\right]}^{}+{\lambda }_s{\displaystyle \sum _j}{(l_{ij}-l_{ij}^0)}^{}{{\displaystyle \stackrel{\mathbf{~}}{\mathbf{u}}}}_{X_i{X}_j}}\end{array}}$$

(4)

where $R$ is the 90 degree rotation matrix:

$${\displaystyle \begin{array}{l}{\displaystyle R=\left[\begin{array}{ll}0& {\displaystyle -1}\\ 1& {\displaystyle 0}\end{array}\right],}\end{array}}$$

${\stackrel{\mathbf{~}}{\mathbf{u}}}_{X_i{X}_j}$ is the unit vector joining the vertices at position X_i and X_j, the summation ${\sum }_{\alpha ,i\in a}^x$is over all cells that contain node i, and $X_{i+1}^{\alpha }$ (⁠$X_{i-1}^{\alpha }$⁠) denote the subsequent (antedecent) to node i as the nodes in cell α are traversed clockwise.

The typical cell size in the simulation has a length of about l₀=10 µm. We used the experimentally reported range, 10^–5–10^–8 ms^–1, for the value of speed of auxin transport d (Mitchison, 1980).

The force vector field of an in silico growing leaf primordium is shown by arrows in Fig. 2B, D, F. Magenta colored arrows represent net force acting on the vertices of auxin-producing cells, whereas blue colored arrows represent net force acting on the vertices of non-auxin-producing cells in the leaf primordia. The values for the model parameters are shown in Supplementary Table S1.

Compensation of vein patterning

In our model, the area growth rate of non-auxin-producing cells depends on the cellular auxin concentration in which leaf veins are formed by a dividing clonal population of cells, and not a broadening domain of newly differentiated cells. When the auxin concentration in a non-auxin-producing cell crosses a specified threshold, its area growth rate increases; as a result, these cells grow faster than the auxin-producing cells, inducing transverse forces (perpendicular to the axis of the midvein), acting laterally on the walls of midvein cells, much larger in magnitude than the forces acting on other cells in the leaf primordia. This prevents the midvein cells from proliferating, thus resulting in a thin vascular strand of elongated cells.

In this section, using our model, we will see how reducing the auxin production rate requires a simultaneous reduction in auxin transport rate as well as overall cell area growth rate in order to recover WT midvein patterning. Our model therefore proposes a mechanism through which a reduction in auxin biosynthesis can rescue the NPA-induced defects caused by auxin accumulation in the midvein.

Continuous auxin production in auxin-synthesizing cells and intercellular auxin transport sets up an auxin concentration gradient in the growing leaf primordia. Therefore, the auxin concentration threshold, beyond which non-auxin-producing cells start to grow faster, depends upon several kinetic parameters that also determine the auxin concentration gradient.

The auxin concentration of a cell in a growing leaf lamina changes according to the following equation:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{c}_{\alpha }(x,t)}{\mathrm{\partial }t}=s(\alpha )+D\frac{{\mathrm{\partial }}^2{c}_{\alpha }(x,t)}{\mathrm{\partial }{x}^2}}\end{array}}$$

(5)

where $c_{\alpha }(x,t)$ is the auxin concentration of cell $\alpha$ at position $x$ and time $t$ in the leaf primordia, $s(\alpha )=s_0{\delta }_{\alpha ,p}$⁠, describes auxin production at a constant rate $s_0$⁠, non-zero only for auxin producing cells $p$⁠, and $D$ is the auxin diffusion. Since leaf primordia is growing in area due to an irreversible growth in the area of an individual cell, as a result, the cellular auxin concentration $c=n/A$⁠, where $n$ is the number of auxin molecules in a cell of area $A$⁠, is effected,

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{c}_{\alpha }(x,t)}{\mathrm{\partial }t}=\frac{\frac{\mathrm{\partial }{n}_{\alpha }}{\mathrm{\partial }t}}{A_{\alpha }}-\frac{n_{\alpha }}{{A_{\alpha }}^2}\frac{\mathrm{\partial }{A}_{\alpha }}{\mathrm{\partial }t}.}\end{array}}$$

(6)

The first term in Equation 6 describes the change in auxin concentration due to auxin transport, and is given by Equation 2. The second term describes the change in auxin concentration as result of dilution of auxin due to an increase in cell area due to cell area growth. A similar situation of an area growth-induced dilution of a chemical species, growth hormone, morphogens, etc. is encountered in other biological systems (Wartlick et al., 2011; Romanova-Michaelides et al., 2015).

The change in auxin concentration of cell α at position x and time t can thus be written down as:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{c}_{\alpha }(x,t)}{\mathrm{\partial }t}=s\left(\alpha \right)+D\frac{{\mathrm{\partial }}^2{c}_{\alpha }}{\mathrm{\partial }{x}^2}-g{c}_{\alpha }}\end{array}}$$

(7)

where, $g$ is the area growth of a cell. The steady state solution of Equation (S5) for a delta source of auxin is given by:

$${\displaystyle \begin{array}{l}{\displaystyle c=\frac{s_0}{\sqrt{gD}(1-e^{-2\lambda L})}(e^{-2\lambda L}{e}^{\lambda x}+e^{-\lambda x})\mathrm{f}\mathrm{o}\mathrm{r}x\ge 0.}\end{array}}$$

(8)

where λ=√g/D. Equation 8 shows the dependence of auxin concentration threshold c* on auxin production rate s₀, auxin diffusion rate D, and cell area growth rate g. The auxin diffusion rate is related to the auxin transport rate d, which is the actual parameter used in our model to describe auxin transport. It can be seen that lowering auxin production s₀, as in auxin biosynthesis mutants, requires a simultaneous lowering of auxin transport rate d, similar to NPA treatment, as well as overall cell area growth rate g (Fig. 2F; Supplementary Fig. S5).

The model’s predictions were tested in planta (Supplementary Fig. S2). Plants were grown for 20 d on either l-kynurenine (l-Kyn; 2.5 µM), NPA (10 µM), or both chemicals in combination. Leaves were dissected, imaged, and quantified as described above.

PIN1-based auxin transport

In addition to the passive diffusive transport of auxin, we also studied, in our model, the effect of active transport of auxin via PIN1 only. The general dynamic equation governing the amount of auxin n_α in cell α in this case is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{n}_{\alpha }}{\mathrm{\partial }t}=s_0\left(\alpha \right)+\tau {\displaystyle \sum _{\beta }}(P_{\beta |\alpha }\frac{n_{\beta }}{A_{\beta }}-P_{\alpha |\beta }\frac{n_{\alpha }}{A_{\alpha }}).}\end{array}}$$

(9)

Here P_β|α is the amount of PIN1 on the cell wall of cell β neighboring cell α, τ is the constant related to the auxin transport via PIN1, and A_β is the area of cell β. The above equation is an extension of Equation 2 with d=0, the only addition being the second term on the right hand side of Equation 9, which describes the intercellular transport of auxin via PIN1.

The rate of attachment of PIN1 to a given cell wall that separates cell α and cell β depends on the amount of auxin in that particular cell, α and its neighboring cell β, as well as the amount of cytoplasmic PIN1, P_α, in that cell. The flux of PIN1 molecules attaching to a given cell wall is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \begin{array}{lll}{\phi }_{\alpha |\beta }& =(n_{\alpha }-n_{\beta }),& {\displaystyle \mathrm{i}\mathrm{f}{n}_{\alpha }>n_{\beta }}\\ & =0,& {\displaystyle \mathrm{i}\mathrm{f}{n}_{\alpha }\le n_{\beta }}\end{array}}\end{array}}$$

Equation governing the amount of PIN1 on the wall of a given cell is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{d{P}_{\alpha |\beta }}{dt}=k_{on}{\phi }_{\alpha |\beta }{P}_{\alpha }.}\end{array}}$$

(10)

Here, P_α is the amount of cytoplasmic PIN1 in cell α and k_on is the rate of association of cytoplasmic PIN1 with the wall of cell α neighburing cell β.

The dynamic equation governing the amount of cytoplasmic PIN1, P_α, in cell α is given by:

$${\displaystyle \begin{array}{l}{\displaystyle \frac{d{P}_{\alpha }}{dt}=-{\displaystyle \sum _{\beta }}{k}_{on}{\phi }_{\alpha |\beta }{P}_{\alpha }+p_0.}\end{array}}$$

(11)

where p₀ is the rate of PIN1 production in each and every cell.

Figure 2G shows the simulation result of our model including the PIN1-based auxin transport, described by Equations 9–11, with the parameter values given in Supplementary Table S3.

Our simulation results show the formation of a normal midvein also in the case of no diffusion (d=0) and PIN1 only transport, see Fig. 2G.

Results

The position of leaf veins is defined by the coordinated cellular polarization of PIN1 within the developing primordium, and the ensuing emergence of instructive patterns of auxin streams (Scarpella et al., 2006). The canalization hypothesis, which is commonly used to explain this process, requires auxin streams to be self-reinforcing, with the rate and site of cellular PIN1 polarization being influenced by the strength and direction of auxin flow. Auxin biosynthesis also affects vein formation; a reduced capacity for auxin biosynthesis simultaneously decreases auxin content in the primordium and reduces the density, organization, and continuity of leaf vein networks (Supplementary Fig. S1) (Stepanova et al., 2008; Nishimura et al., 2014). As a simultaneous reduction in rates of auxin efflux and auxin biosynthesis could theoretically counteract each other’s influence by respectively increasing and decreasing the auxin concentration in a cell, we set out to test whether both processes act in opposite directions on the same leaf-vein-building developmental program. We judged this hypothesis to be plausible as reducing auxin transport leads to the formation of thick leaf veins with high auxin signaling maxima in both epidermis and leaf lamina, whereas reducing auxin biosynthesis results in leaves with sparse vein coverage (or with no veins at all) and a reduced capacity for auxin signaling (Cao et al., 2019). Inhibiting either process leads to a reduction in leaf surface area.

Auxin transport and auxin biosynthesis interact to regulate vein patterning

We induced the simultaneous suppression of auxin biosynthesis and polar efflux by treating auxin biosynthetic mutant plants with the polar auxin transport inhibitor NPA. To reduce auxin biosynthesis, we turned to two families of enzymes which act in series to catalyze the indole-3-pyruvic acid (IPA)-dependent production of IAA: tryptophan aminotransferases (TAA1/WEI8, TAR1, and TAR2) and YUCCA flavin monooxygenases (Mashiguchi et al., 2011; Stepanova et al., 2011; Zhao, 2012). Genotypes deficient in members of these enzyme families can be radically compromised in their ability to synthesize auxin. Efflux carrier-mediated polar auxin is reduced by the application of 10 µM NPA (Thompson et al., 1973; Okada et al., 1991). We measured the effect on leaf vein structure of applying 10 µM NPA by assigning leaves to one of three groups of increasing phenotypic severity, with group III containing leaves with severely thickened fused vascular bundles towards the periphery of the leaf (Fig. 1). In the WT, after NPA application, 75% of leaves fell into group III, whilst no leaves of class I (an NPA-untreated phenotype, as shown in Supplementary Fig. S1) were observed. After the same 10 µM NPA treatment, yucca or wei8xtar2 loss-of-function plants either behaved like WT plants or displayed modest shifts in distribution toward class I leaves. Tar2 showed the largest shift after NPA treatment when compared with treated WT leaves, with 35% of their leaves falling into class I. When NPA was applied to the double knockout combinations yuc1xyuc4 and wei8xtar2 leaves, WT vascular patterning was restored (class I) in 44% and 35% of leaves in each genotype. In both cases, only 5% of leaves (versus 75% for WT plants) fell into the most severe phenotypic class. To explore further the relationship between auxin transport and auxin biosynthesis, we used L-Kynurenine (L-Kin), a competitive inhibitor of TAA1/TAR activity, to artificially reduce auxin biosynthesis (He et al., 2011), while inhibiting auxin transport with NPA. Leaves of WT plants treated with 2.5 µM l-Kyn phenocopied taa/tar mutant leaf defects such as reduced leaf surface area and simplified vein patterning (Supplementary Fig. S2). When simultaneously applied with NPA, l-Kyn restored leaf vein patterning of WT untreated plants, 73% of leaves falling into class I, whereas class II and class III represented only 16% and 10%, respectively (Supplementary Fig. S2). However, NPA was only able to rescue vein patterning but not leaf size. We conclude that if rates of polar auxin transport are reduced, vein patterns returned to normal when auxin biosynthesis is genetically or chemically suppressed.

Fig. 1.

Compensation of vein patterning in auxin biosynthesis mutants by NPA. (A) Vein patterning in 10-day-old wild-type (WT) leaf. (B) WT leaf after NPA treatment. (C) wei8-1tar2-1 leaf treated with (right) or without (left) 10 µM NPA. Leaves were classified according to their vein architecture. Class I, indistinguishable from untreated WT; leaves contain a single midvein reaching distally to the tip of the leaf. Class II, fused midveins. Class III, fused midveins and increased frequency of fused higher order veins. (D) Genotype-dependent distribution of Class I, II, and III leaves in seedlings treated with NPA. Scale bar=500 µm.

Therefore, the regulation of vein formation by (i) efflux-dependent polar auxin transport and (ii) auxin biosynthesis may be viewed as interacting mechanisms whose influences overlap during auxin-dependent vascularization in the developing leaf primordium.

The observation that the NPA-treated leaf phenotype can be partially rescued by lowering rates of auxin biosynthesis is seemingly at odds with the canalization hypothesis, which requires the directional transport of auxin through files of cells for the increasingly selective and asymmetrical permeabilization of those cells to auxin. We therefore tested, via computer simulation, whether realistic vein patterning is theoretically possible when auxin maxima are caused by local auxin biosynthesis instead of by the directional movement of auxin between neighboring vascular cells.

In silico modeling of spontaneous patterning of vascular cells in the leaf primordium

In order to test whether leaf vein development may be affected by cell-autonomous combinatory effects of auxin production and efflux, we developed a theoretical model and performed a series of in silico experiments. We adapted Virtual Leaf, an open-source cell-based modeling framework that describes cells of the leaf lamina as a 2D layer of interconnected polygons and accounts for mechanical properties of the tissue (Merks et al., 2011). In the following model, cell growth (an irreversible increase in cell area) proceeds in a quasi-static way at a rate which is defined by the antagonistic influence of cell turgor pressure and cell wall loosening, a relationship that is influenced by auxin. A cell divides each time its area doubles, which invariably results in tissue growth over time. Cell wall stiffness and mechanics are included in the model by the definition of a resting length for each cell wall element. The shape of the tissue at each stage of growth is determined by minimizing the generalized energy function that contains a cell area term and a cell wall elasticity term.

To model leaf vein patterning in a 2D growing leaf, our initial leaf tissue template closely resembled a 2-day-old leaf primordium, with a proximal area of auxin-producing, auxin-exporting incipient vascular cells surrounded by a small number of lamina cells which do not produce auxin (Fig. 2A). These auxin producer cells corresponded to cells which expressed TAR2, as defined by the pattern of TAR2-driven GUS expression in 2-day-old control leaf primordium (Fig. 2A, middle panel). The mechanical constraint given by the attachment of the leaf to the plant is modeled by including a row of petiole cells at the proximal portion of the leaf. These cells also act as auxin sinks, and modeled auxin drainage from the leaf into established veins in the stem. Further assumptions of our model were as follows: (i) auxin transport from immature isodiametric vascular cells is non-directional, following the observed non-polar PIN1 localization of cells at this developmental stage (Fig. 2A, right panel; Tsugeki et al., 2009); (ii) every cell in the leaf primordium grows by increasing its area at the same rate; but (iii) when auxin concentration crosses a threshold, the cell starts expanding at a higher rate (as auxin is known to promote cell expansion in aerial tissues) (Perrot-Rechenmann, 2010; Fendrych et al., 2016); accordingly, (iv) the rate of expansion of auxin-producing cells is set at its own maximum rate; and (v) the epidermal cell layer is relatively stiff (Onoda et al., 2015).

All simulations run with this simple model used experimentally reported parameter values (Mitchison, 1980; Supplementary Table S1), and proceeded until the in silico leaf primordium had grown to a size of either ~150 cells (Fig. 2B) or 900 cells (Supplementary Fig. S3). For a wide range of parameter values, and without including polarized auxin transport, this model reproduced four key features of leaf development: (i) higher frequency of vascular cell divisions in a distal region of leaf primordia; (ii) a central midvein position; (iii) coordinated vascular cell elongation; and (iv) a distally branched midvein (Fig. 2B, C; Supplementary Fig. S3; Supplementary Video S1). The shape and placement of the midvein was robust when either the initial geometry or the number of auxin-synthesizing cells was altered (Supplementary Fig. S4). Furthermore, our simulations showed that discrete subregions of auxin synthesis, followed by apolar PIN1-mediated auxin transport, were able to account for the establishment of a time-dependent auxin gradient across the leaf primordium which caused cells to grow at different growth rates (inset of Fig. 2B). A graded variation in growth rate then induced transverse forces which acted laterally on the walls of midvein cells. These forces were much larger in magnitude than those which acted on other cells in the leaf lamina (Fig. 2B) and resulted in non-trivial strain and force distributions in the tissue that prevented the midvein cells from proliferating. This led to the development of the characteristic thin vascular strand of elongated cells.

Modulation of vascular pattern by surrounding tissues

Experimentally, an NPA-dependent reduction in auxin efflux led to a change in leaf vascular patterning and an expansion of the TAR2 expression domain, suggesting an extension of auxin biosynthesis sites (Fig. 2E). Therefore, we next studied how midvein development in the model responded to a reduction in (still non-directional) auxin efflux rates. Simulations showed that a 25-fold lowering of the rate of auxin transport altered the formation of the midvein, and resulted in an increased proliferation of auxin-producing cells, as was experimentally observed (Figs 2D, 3B–G). Reducing the rate of auxin efflux in this way led to higher auxin concentrations in auxin-producing cells and steeper auxin gradients across primordia (lower panel in Fig. 3H; inset in Fig. 2D). In this case, lamina cells received less auxin from auxin-producing cells, and hence expanded more slowly when compared with leaves for which auxin transport rates were unmodified. As a result, the growth-induced forces exerted by the neighboring cells on the midvein cells were not strong enough to prevent the proliferation of auxin-producing cells (Fig. 2D). Accordingly, the model predicted that a reduction in cellular auxin efflux would result in a midvein with a high auxin concentration which was several cells wide. This correlated exactly with the midvein of primordia of plants grown on inhibitory concentrations of NPA (Fig. 3C–H). Furthermore, the midvein cells of the ‘NPA-treated’ model leaf did not expand in the same way as ‘untreated’ simulations (Fig. 2B and inset; Supplementary Video S2). It is therefore possible that the physical environment of each primordium cell plays a major role in defining the pattern of vascular cell development. As was observed experimentally by treating wei8xtar2 plants with NPA, simultaneously reducing rates of auxin production and cellular auxin efflux restored a WT leaf pattern in the model (Fig. 2F; Supplementary Fig. S5). We also studied, in silico, the effect of PIN1-only dependent polar auxin transport. We used a flux-based polarization model to study PIN1-mediated auxin transport as described in Stoma et al. (2008). Our simulations show that polar auxin flux in provascular cells results in the development of a midvein in much the same way as in the case of non-polar auxin transport (Fig. 2G).

Fig. 3.

Effect of NPA on leaf vein patterning, auxin biosynthesis, and response. (A) Vein patterning in 12-day-old leaves grown in the presence of increasing concentrations of NPA. (B–F) Immunodetection of both TAA1–GFP and PIN1 in 4-day-old leaves grown in the presence of NPA. Boxed inset in (D) shows polar PIN1 at the plasma membrane of vascular cells marked with an arrow in (D) and (E). (F) Transversal sections of leaves presented in (B) and (C), respectively. White dashed lines indicate leaf boundaries. Yellow dashed lines indicate pre-provascular cells expressing both TAA1 (red) and PIN1 (green). Note the proliferation of pre-provascular cells in a concave shape in the 10 µM NPA-treated leaf. (G) Percentage of cells expressing both PIN1 and TAA1 in leaf lamina, by dividing by the total number of cells constituting leaf primordium. Asterisks indicate significant difference from control at P<0.01 (t-test). Data are means (n=40±SE). (H) DR5::Venus (auxin response) and PIN1p::PIN1–GFP in 3-, 4-, and 5-day-old leaves. Top panel: control leaves. Lower panel: leaves treated with 10 µM NPA. Scale, 500 µm (A), 200 µm (F), and 10 µm (H).

Expression of auxin biosynthesis genes

Both experiments and modeling suggest that the site of auxin production could be a critical factor in fixing the site of vein formation (Supplementary Fig. S1) (Stepanova et al., 2011). This relationship would also predict a relationship between the site of auxin production within the leaf primordium and the future site of vascular cell development. We therefore prepared time-resolved expression maps of the developing primordium to define the sites of auxin biosynthesis.

Four YUCCA enzymes are responsible for auxin biosynthesis in shoots: YUC1, YUC2, YUC4, and YUC6 (Cheng et al., 2007). We therefore visualized the expression pattern of these genes in the leaf primordium alongside TAA-type aminotransferases at cellular resolution using well-established standard techniques (i.e. the imaging of GFP fusion proteins and promoter GUS fusions), which faithfully reflect gene expression in the leaf. As predicted, at between 2 and 5 DAG, enzymes of YUCCA and TAA families showed highly localized expression patterns in which the future sites of vascular cells can be clearly seen (Fig. 4; Supplementary Figs S6, S7). Over this time period, expression domains of both classes of enzymes were restricted to the regions of leaf primordia in which veins are formed, and the border between the abaxial and adaxial sides (lower panel in Fig. 4A; Supplementary Fig. S7). TAA-type aminotransferases and YUCCA genes were both expressed in partially overlapping domains (Fig. 4; Supplementary Figs S6, S7). Because TAA and YUC family enzymes are sufficient for the biosynthesis of auxin from tryptophan, overlapping expression domains of the corresponding genes are likely to lead to localized auxin production (Fig. 4A). In 2-day-old leaf primordia, the expression domains of TAA1, TAR2, and YUC4 include cells in the early stage of midvein development (Fig. 4; Supplementary Fig. S7), suggesting that auxin production and vein development coincide within cells of the young leaf primordia.

Fig. 4.

Time-resolved auxin biosynthesis map in Arabidopsis leaf. Combinatorial expression of auxin biosynthetic genes (TAA1, TAR2, YUC1, YUC2, and YUC4) at 2 DAG (C–H), 3 DAG (I–M), and 4 DAG (N–Q). (A) Longitudinal (above) and transverse (below) leaf sections. The blue dashed line defines the separation between adaxial (ad) and abaxial (ab) leaf polarity. (B) Schematic representation of the major auxin biosynthesis pathway (Trp, tryptophan; IPA, indole 3-pyruvic acid; IAA, indole 3-acetic acid). (C–Q) Time-resolved expression pattern of pTAA1::TAA1-GFP (C, D, I, J, N), TAR2::GUS (E, L, P), YUC2::GUS (F, M, Q), YUC4::GFP (G, H, K, O). (D), (H), and (J) are optical transversal sections of the leaf (areas indicated by a yellow dashed line in C, G, L). Asterisks: yellow asterisks indicate TAA1 and YUC4 expression in the epidermis; white asterisks indicate TAA1 and YUC4 expression in the lamina; black asterisks show TAR2 and YUC2 expression in provascular tissues. White and black dashed lines indicate the limits of the leaf primordia. SAM indicates the position of the shoot apical meristem. Scale bar=20 µm unless otherwise indicated. DAG, days after germination. The data supporting auxin biosynthesis in 5 DAG leaf are presented in Supplementary Figs S6 and S7.

Influence of auxin transport on auxin biosynthesis

Inhibiting auxin transport in Arabidopsis, either genetically or by the application of chemicals, leads to the formation of small, round leaves with a wide midvein and fused vascular bundles close to the periphery of their distal end (Fig. 3A; Supplementary Fig. S1F) (Mattsson et al., 1999, 2003). Considering the importance of auxin efflux to the development of leaf vasculature, and the presence of PIN1 in pre-provascular cells (Scarpella et al., 2006), we next hypothesized that auxin biosynthesis is influenced by auxin transport. To test this hypothesis, we examined the expression of auxin biosynthetic enzymes (or their transcripts) in the leaf primordium after the reduction of auxin efflux (by treatment with 10 µM NPA). In these conditions, both auxin biosynthetic enzymes and PIN1 continued to be expressed (Fig. 3B–F; Supplementary Fig. S8). Notably, provascular tissues (the site of TAA1 and PIN1 expression) were expanded, presumably due to a lateral increase in the number of cells that they contained (Fig. 3B–G), whilst auxin signaling (as indicated by DR5::Venus) was confined to the distal end of the leaf (Fig. 3H). These observations suggest that polar auxin transport is not necessary for auxin biosynthesis or for PIN1 protein expression in provascular cells. They also support the hypothesis that blocking auxin efflux causes auxin-producing provascular cells to accumulate auxin and therefore to become more competent for proliferation. If this were indeed the case, a simultaneous drop in the number of auxin-receiving cells would also be expected due to the lower amount of auxin leaving vascular cells. A corresponding lowering of division rates in these auxin-receiving non-vascular cells would then lead to the smaller leaves (Supplementary Fig. S1G) and thick leaf vascular bundles (Fig. 3B–F) observed in NPA-treated plants. The altered localization of DR5-dependent gene transcription after NPA treatment suggests that auxin accumulation in vascular cells is probably due to synthesis in situ and not to the import of auxin into the leaf primordia by dedicated auxin efflux proteins (Avsian-Kretchmer et al., 2002; Abley et al., 2016). We next asked whether auxin biosynthesis could be induced by its transport in a canalization-type positive feedback loop. If this were the case, then we would expect: (i) the expression of auxin biosynthetic genes to be induced by auxin (as PIN1 expression is up-regulated by auxin) and (ii) that PIN1 localization would be affected in auxin biosynthetic mutants as the presumed feedback loop would be broken. Wei8-1xtar2-1 and yuc1xyuc4 double mutants exhibited less formation of provascular tissues expressing PIN1 than WT plants (Supplementary Fig. S9A–C). However, more importantly, polar PIN1 localization in these mutants was indistinguishable from the WT, suggesting that PIN1 polarization is not affected by a de0crease in auxin content. Furthermore, neither TAA/TARs nor YUCCA expression was induced by IAA (Supplementary Fig. S9D). These observations are supported by public databases; here, the expression of TAA1, TAR1, TAR2, YUC1, YUC2, YUC4, and YUC6 genes is not induced by auxin (Paponov et al., 2008). Similarly, in our experiments, exogenously applied auxin did not significantly induce the expression of YUC1, YUC2, YUC4, and YUC6 (Supplementary Fig. S9D) (for YUC2 we even measured a repression), suggesting that auxin does not induce its own biosynthesis. Other studies have even reported a negative influence of auxin on its own biosynthesis (Suzuki et al., 2015). We next analyzed co-expression of TAA1 and PIN1 during the early stages of higher order vein development (Supplementary Fig. S10). Here, PIN1 was expressed and first localized in a non-polar fashion (dashed inset in Supplementary Fig. S10A–C expanded in D–F). However, unlike in the midvein and second-order veins (the first loops), TAA1 was also strongly expressed (solid inset in Supplementary Fig. S10A–C expanded in G–I). In polarizing cells (arrowheads in Supplementary Fig. S10G, I), TAA1 continued to be expressed. However, this expression reduced as the cell matured and is totally lost in the fully differentiated cells of the midvein (yellow inset in Supplementary Fig. S10C). Our data therefore strongly suggest that there is no positive feedback loop which connects IAA biosynthesis and polar IAA efflux and that tissue-specific auxin biosynthesis is a prerequisite for leaf vein development.

Discussion

In plants, polarized auxin transport is crucial for the initiation and regulation of several developmental programs. Though PIN-mediated auxin flux is widely accepted to be responsible for vascular patterning in leaves, with vascular differentiation following routes of auxin streams (Sieburth, 1999; Scarpella et al., 2006; Mazur et al., 2020), vein development also requires tissue-specific local auxin biosynthesis. This conclusion is supported by the fact that in mutants defective in local auxin biosynthesis, vein density is severely reduced (Supplementary Fig. S1) (Cheng et al., 2006; Stepanova et al., 2008). Inhibiting auxin transport in Arabidopsis leads to plants developing small, round leaves which display a characteristically fused venation pattern (Mattsson et al., 1999; Sieburth, 1999). Treating PIN-expressing cells with NPA causes them to accumulate auxin (Petrásek et al., 2006; Petersson et al., 2009). Therefore, since auxin is synthesized in both vein cells and epidermis cells, and the plasma membrane of both cell types contains PIN1, applying NPA could potentially increase cellular auxin concentration in both areas (Fig. 2). In roots, auxin induces the division of pericycle but not of epidermal cells (Himanen et al., 2002; Mähönen et al., 2014; Pacheco-Villalobos et al., 2016). We may use this model to explain the observation in leaf primordia that NPA treatment does not lead to an increase in leaf size. In this case, auxin would not increase the rate of cell division in the epidermis, but would present a similar landscape of mechanical properties to those identified by our model (Fig. 2B). These data support the assertion that the patterning of organ growth and development is constrained and directed by the physical environment of the cells, with mechanical stresses in particular influencing a broad range of developmental pathways (Sampathkumar et al., 2014). Taking into account auxin biosynthesis in leaf lamina cells and auxin transport in epidermal cells in future models will enable us to understand how leaf shape and vein patterning are coordinated.

Our model is consistent with some hitherto problematic aspects of leaf development, some of which are not easily resolved with models which rely exclusively on the canalization hypothesis (Kierzkowski et al., 2013). Such observations include a high density of vascular tissue in leaves treated with high concentrations of NPA (Fig. 3A; Supplementary Fig.S1H) (Mattsson et al., 1999), the relative insensitivity of vascular cell initiation to NPA when compared with a reduction in auxin biosynthesis, and the strong auxin response maxima observed in differentiating vascular cells (Heisler et al., 2005; Bayer et al., 2009). Using the simple model presented, we suggest a unifying hypothesis consistent with these observations. Our analysis suggests the existence of an auxin equilibrium in developing leaf vascular cells which is maintained by the concerted action of two processes acting on cellular auxin concentration: its lowering by PIN1-dependent efflux and its increase by in situ biosynthesis. This scheme is supported by the requirement for context-specific auxin biosynthesis to complement multiple yucca mutants (as opposed to exogenous application) (Cheng et al., 2006) and the fact that severe auxin-deficient phenotypes are caused by surprisingly small changes in auxin amounts, when the plant is taken as a whole (Stepanova et al., 2011).

Our model replicated the observation that reducing auxin biosynthesis in the presence of NPA partially restored normal venation patterns in wei/tar and yucca mutants. Indeed, treating auxin biosynthesis mutants with NPA restored WT-like vein patterning (Fig. 1). However, since NPA also affects the shape of the leaf, it confirms that auxin transport in the epidermal cells plays an important role in maintaining leaf shape, as previously proposed (Izhaki and Bowman, 2007; Scarpella et al., 2010).

Our model leads us to the hypothesis that mechanical forces exerted on the cells synthesizing auxin are sufficient to direct the development of vascular strands. Indeed, the importance of geometrical and mechanical constraints during vascular tissue development in the Arabidopsis embryonic root has already been underlined (De Rybel et al., 2014). Our data show that, unlike PIN1 whose expression is clearly stimulated by auxin, the transcription of biosynthetic genes is either insensitive to, or repressed by, auxin in leaves (Supplementary Fig. S9B). Moreover, (i) TAA1 expression is reduced, not increased, as veins mature (Supplementary Fig. S10A); (ii) PIN1 expression is strongly down-regulated in leaves of auxin biosynthesis mutants (Supplementary Fig. S10B); and (iii) whilst TAA1 can be seen in domains in which PIN1 is absent (Supplementary Fig. S10C), in differentiating higher order veins, the opposite case is never observed. Taken together, these observations suggest that local auxin biosynthesis in leaf primordia is not part of the auxin canalization positive feedback loop, but still plays an integral role in leaf vasculature development. Together, transport and synthesis could therefore stabilize an auxin concentration equilibrium in developing vascular cells and would be mutually indispensable for vein patterning.

So far, our mathematical model only considers midvein development. Understanding the initial branching stages of secondary and tertiary vein development is likely to require more complex models, which are outside the scope of this work. These models will also need to accommodate at least TIR1/AFB-mediated auxin signaling processes; by the careful observation of higher order pin and auxin signaling knockouts it has recently been shown that vein patterning is not under the exclusive control of auxin efflux (Verna et al., 2019; Mazur et al., 2020). However, whether the auxin signaling processes recently described which are necessary for vein patterning exert their influence by altering the mechanical properties of a cell remains to be established. Nevertheless, the data presented here indicate that the search for factors which drive vascular patterning be widened to encompass factors which influence the patterns of auxin biosynthesis as well as cellular auxin efflux.

Supplementary data

The following supplementary data are available at JXB online.

Table S1. Choice of parameters for modeling of wild-type vascular patterning.

Table S2. Primer pairs used for real-time quantification of expression of auxin biosynthetic genes (qRT–PCR).

Table S3. Choice of parameters for modeling of PIN1-based auxin transport.

Fig. S1. Vein development in mutants defective in auxin biosynthesis and transport.

Fig. S2. Compensation of vein patterning in NPA-treated leaves by l-kynurenine.

Fig. S3. In silico leaf vasculature development in a long-term simulation.

Fig. S4. Dependence of the final shape of the midvein on initial midvein geometry.

Fig. S5. Auxin biosynthesis mutants treated with auxin.

Fig. S6. Dynamic changes in pattern of auxin biosynthesis during leaf development.

Fig. S7. Auxin biosynthesis in the leaf is significant for vascular patterning.

Fig. S8. Change in auxin biosynthesis pattern in 10 µM NPA-treated leaves.

Fig. S9. Auxin does not regulate PIN1 polarization or its own biosynthesis.

Fig. S10. Gradual subcellular localization of PIN1 in dynamic TAA1 expression domains.

Video S1. In silico modeling of spontaneous patterning of vascular tissue in the leaf primordium (see Fig. 4B)

Video S2. In silico modeling of spontaneous patterning of vascular tissue in the leaf primordium treated with NPA (see Fig. 4D)

Acknowledgements

This work could not have been accomplished without the help of colleagues, collaborators, and friends who provided support, suggestions, and materials. We would particularly like to thank Jose Alonso and Yunde Zhao for sharing materials. We also gratefully acknowledge the excellent technical support from Beata Ditengou and Katja Rapp. This work was supported by the Baden-Württemberg Stiftung, Deutsche Forschungsgemeinschaft (SFB 746), the Excellence Initiative of the German Federal and State Governments (EXC 294), Bundesministerium für Forschung und Technik (BMBF SYSTEC, PROBIOPA, MICROSYSTEMS), Deutsches Zentrum für Luft und Raumfahrt (DLR 50WB1022), the Freiburg Initiative for Systems Biology, the European Union Framework 6 Program (AUTOSCREEN, LSHG-CT-2007-037897), the National Science Foundation (USA), and Japan Society for the Promotion of Science KAKENHI (grant no. 24570047, 16K07396). EK acknowledges support from the Burroughs Wellcome Fund.

Author contributions

FAD, WDT, IK, and KP conceived and designed the experiments. FAD and IK performed the experiments. JD performed the mathematical modeling. FAD, IK, WDT, EK, RT, and KP analyzed the data. FAD, WDT, EK, and KP wrote the paper. All authors discussed the results and commented on the manuscript. Correspondence and requests for material should be addressed to FAD ([e-mail]).

Data availability

The data that support the findings of this study are freely available at Dryad Digital Repository: https://doi.org/10.5061/dryad.vq83bk3px (Kneuper et al., 2021).

Author notes

These authors contributed equally to this work.