To characterise patterns of epidermal PIN polarity associated with the development of kan1kan2 outgrowths, a PIN1::PIN1:GFP reporter (Benková et al., 2003) was imaged on the abaxial surface of wild-type (WT) and kan1kan2 leaves. Here we focus on PIN1 since this is the predominant epidermally expressed PIN that shows polar intracellular distributions in wild-type leaves (Guenot et al., 2012). In young kan1kan2 leaf primordia, the epidermal PIN1 polarity pattern was similar to that observed in WT: PIN1 polarities were oriented towards the leaf tip in both cases (Figure 1A,B). At later stages of development, WT leaves retained a proximo-distal polarity pattern, and then no-longer showed detectable expression of the PIN1::PIN1:GFP reporter (Figure 2A). The loss of PIN1::PIN1:GFP expression did not occur uniformly throughout the WT leaf because some patches of cells retained expression for longer than other regions of the leaf (Figure 2A iii). By contrast to WT, kan1kan2 leaves maintained PIN1 expression until later stages of development and formed centres of PIN1 polarity convergence, located at the tips of emerging ectopic outgrowths (Figure 2B iv and C iv). Cells at the centres of convergence had highly polarised PIN1, oriented towards the interface between three or four neighbouring cells (Figure 2C iv). To determine whether the centres of convergence preceded outgrowth emergence, leaves were imaged over several days and cell lineages that gave rise to convergences traced back through the time-course of the experiment (Figure 2C, yellow dots, arrows and lines). This revealed that cells closest to the centre of convergence at the outgrowth tip descended from one or two cells, which were already at a centre of polarity convergence prior to outgrowth emergence (Figure 2B,C). Taken together, these findings show that, prior to kan1kan2 outgrowth emergence, centres of polarity convergence develop within a proximodistally oriented polarity field.

Figure 1

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Figure 2

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To investigate how the pattern of PIN1 polarity is related to the dynamics of intracellular auxin accumulation, a DR5::GFP reporter (Benková et al., 2003) was imaged in kan1kan2 leaves over the course of outgrowth development. At early stages, DR5::GFP signal was detected exclusively at the leaf tip (Figure 3A), similar to the pattern described for the WT leaf (Mattsson et al., 2003; Scarpella et al., 2006). Approximately one day prior to the first observation of an outgrowth, locally elevated DR5::GFP signal was detected in groups of epidermal cells in the proximal region of the lamina, where outgrowths typically form (Figure 3B i, yellow arrow). At later stages, these regions of DR5::GFP expression persisted, and could be found throughout emerging outgrowths (Figure 3C,D, yellow arrows). Thus, both intracellular auxin activity maxima and centres of PIN1 polarity convergence precede and predict sites of outgrowth emergence in kan1kan2 leaves.

Figure 3

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cuc2 mutants fail to form centres of PIN1 polarity convergence in the leaf margin and do not develop serrations (Bilsborough et al., 2011; Nikovics et al., 2006). To investigate whether, like leaf serrations, kan1kan2 outgrowths depend on the CUC2 transcription factor, we generated kan1kan2cuc2 mutants. Leaves of kan1kan2 mutants consistently produce ectopic, finger-like outgrowths from the abaxial leaf surface (97% of 160 leaves observed had at least one outgrowth, and the mean number of outgrowths per leaf was 12.4) (Figure 4A,B). By contrast, mature leaves of kan1kan2cuc2 mutants only occasionally developed such outgrowths (only 8% of leaves observed developed outgrowths, and the mean number of outgrowths was 1.3) (Figure 4C,D). Some kan1kan2cuc2 leaves also developed ridge-like thickenings of the abaxial surface or serrations in the leaf margin (Figure 4—figure supplement 1A,B).

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To test if CUC2 is required for the formation of epidermal sites of PIN1 convergence, we generated kan1kan2cuc2 plants with the PIN1::PIN1:GFP reporter. Time-lapse imaging of PIN1::PIN1:GFP in the abaxial epidermis of the first two leaves of this mutant revealed that centres of convergence did not form (Figure 5). At early stages of leaf development, a proximodistal PIN1 polarity field was observed (Figure 5 i), but similar to WT leaves, at later stages the expression of PIN1::PIN1:GFP was lost throughout most of the abaxial epidermis (Figure 5B i–iii). Expression of the reporter was maintained in groups of epidermal cells which did not show centres of PIN1 convergence (Figure 5B ii).

Figure 5

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In upper leaves of kan1kan2cuc2 mutants (leaves 3 to 6), ectopic primordium-like bumps were occasionally observed on the abaxial leaf surface (in approximately 10% of leaves observed, n = 50) (Figure 5C). These bumps were associated with centres of PIN1 polarity convergence linked with sub-epidermal PIN1 strands. The frequency of centres of PIN1 convergence in kan1kan2cuc2 mutants was much lower than in the kan1kan2 background where approximately 95% of leaves 3–6 were observed to have at least one centre of PIN1 polarity convergence (n = 50). Thus, formation of kan1kan2 outgrowths and polarity convergence sites is largely dependent on CUC2, although this requirement can occasionally be circumvented.

To gain insight into how CUC2 might promote the formation of epidermal centres of PIN1 convergence, we characterised the expression pattern of CUC2 at the time when centres of PIN1 polarity convergence form in kan1kan2 leaves. The CUC2::RFPer reporter (which causes expression of endoplasmic reticulum-localised RFP) was time-lapse imaged together with PIN1::PIN1:GFP in the abaxial epidermis of the first two kan1kan2 leaves. At early stages of leaf development, when PIN1 polarities are oriented distally, elevated expression of CUC2::RFPer was detected throughout the proximal half of the lamina (Figure 6A,B). Before outgrowth emergence, expression of the reporter was lost from cells towards the base of the leaf, and from early centres of PIN1 polarity convergence (Figure 6C ii,D ii). This resulted in groups of cells within the proximal half of the lamina with CUC2::RFPer expression, surrounded by non-CUC2::RFPer expressing cells. Later on these groups of CUC2::RFP expressing cells were seen to be at the distal side of each epidermal PIN1 polarity convergence (Figure 6D iii) and at the distal base of emerging outgrowths (Figure 6D iii, side view, blue arrow head). This region is the boundary between the outgrowth and the abaxial lamina and is analogous to the sinus region of serrations where CUC2 is also expressed (Nikovics et al., 2006; Bilsborough et al., 2011).

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We next investigated how the expression pattern of CUC2::RFPer is related to the polarity of PIN1 in emerging outgrowths. Inspection of PIN1 polarities using the PIN1::PIN1:GFP reporter revealed that in developing outgrowths, cells close to the regions of elevated CUC2::RFPer expression frequently had PIN1 polarities oriented away the regions of high CUC2 expression (Figure 6E,F). On the proximal side of the CUC2::RFPer expression domain (i.e. in cells of the emerging outgrowth), PIN1 polarities were oriented proximally, away from cells with high CUC2::RFP expression and towards a centre of convergence at the outgrowth tip; while on the distal side of the CUC2::RFPer expression domain, polarity appeared to point distally.

In contrast to kan1kan2 leaves, in WT, CUC2::RFPer expression was mainly restricted to the leaf base and to regions of the margin associated with centres of PIN1 convergence involved in serration development (Figure 6G; Bilsborough et al., 2011).

Thus, ectopic outgrowth development in kan1kan2 leaves is associated with the ectopic expression of CUC2 in the abaxial lamina. Similar to leaf serrations, the normal development of kan1kan2 outgrowths and the centres of PIN1 polarity convergence that precede them is dependent on CUC2. We next compare different models of PIN1 polarity for their ability to capture the observed PIN1 polarity patterns and for their consistency with the role of CUC2 in PIN1 polarity convergence formation.

To test possible mechanisms that could underlie the observed epidermal polarity patterns, we compare the abilities of up-the-gradient, flux-based and indirect coupling models to account for them. We first characterise basic model behaviours, and then investigate the assumptions needed for each model to capture the observed polarity patterns and how they may be tested experimentally.

The flux and up-the-gradient models are implemented as in previous publications, using the simplifying assumption that auxin moves directly from cell to cell and omitting an explicit representation of the cell wall (Bilsborough et al., 2011; Feugier et al., 2005; Jönsson et al., 2006; Rolland-Lagan and Prusinkiewicz, 2005; Smith et al., 2006; Stoma et al., 2008). By contrast, we explicitly represent the cell wall in our implementations of the indirect coupling model. In this model, we also represent cell membranes and walls with several compartments per cell edge (allowing lateral diffusion of components to be simulated). However, as in previous work, flux and up-the-gradient models are implemented with a single compartment per edge and no lateral diffusion of components along the cell edge (Bilsborough et al., 2011; Feugier et al., 2005; Jönsson et al., 2006; Rolland-Lagan, 2008; Smith et al., 2006; Stoma et al., 2008). With the flux-based model, we assume a linear feedback between flux and PIN1 allocation, which tends to generate broad regions of coordinated polarisation rather than narrow canalised strands that arise with super-linear feedback (Feugier et al., 2005; Stoma et al., 2008). Details of all the model assumptions are given in the Materials and methods.

Previously simulations of up-the-gradient and flux-based models have considered polarity patterns in fields of cells all of which have the potential to polarise (Jönsson et al., 2006; Rolland-Lagan, 2008; Smith et al., 2006; Stoma et al., 2008). However, to get a clearer view of the properties of each model, we begin by considering a group of cells having initially uniform auxin concentration and PIN1 distribution, and with only the central cell having the ability to relocate its PIN1 according to the rules specified by each model (Figure 7A–C). We find that the central cell becomes polarised with both the up-the-gradient and flux-based models, assuming that there are small random fluctuations (noise) in the initial concentrations of PIN1 in the cell membranes and that auxin flux from the central cell can modify the initially homogeneous auxin concentration in the neighbouring tissue (Figure 7A and B). In the up-the-gradient model, a cell edge with elevated PIN1 causes an increase in auxin concentration in the neighbouring cell. This elevated auxin concentration in the neighbour feeds back to cause an increase in PIN1 allocation to the cell edge (Figure 7A ii, purple arrow). Assuming a limited pool of PIN1 in the cell, there will be a corresponding reduction in PIN1 allocation to other cell edges. In the flux-based model polarity arises because if one cell edge has slightly elevated PIN1 (due to small random fluctuations), this edge will have a higher rate of auxin efflux, which feeds back to favour further recruitment of PIN1 to the given cell edge (Figure 7B ii, purple arrow). Recruitment of PIN1 to other cell edges is inhibited because the different edges of a cell compete to export auxin into surrounding cells.

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As previously described, the indirect coupling model also generates a polarised distribution of PIN1 in the central cell (Figure 7C). This is due to the auto-activating and mutually inhibitory activities of membrane bound A* and B* polarity components, combined with relatively fast diffusion of their cytoplasmic forms, which generates cell polarity. PIN1 is then recruited to the membrane with high A* (Marée et al., 2006; Jilkine et al., 2007).

Thus, all three models have the potential to generate polarity in a single polarisable cell surrounded by non-polarisable neighbours in an initially uniform field of auxin concentration. For the indirect coupling model, this polarity arises through partitioning of the polarity components, while for the flux-based and up-the-gradient models it arises because small fluctuations in PIN1 distribution create variations in auxin flux or concentrations which feed back to reinforce the polarity. There is, however, a key difference between the up-the-gradient and other models. If auxin concentration in the surrounding medium remains fixed throughout the simulation (equivalent to voltage-clamping in neurophysiology), then the up-the-gradient model does not generate polarity (Figure 7—figure supplement 1). In contrast, both the flux-based and indirect coupling model present polarity even under such settings (Figure 7—figure supplement 1).

In a previous paper we overlooked the potential effect of feedback in the flux-based and up-the-gradient models for single polarisable cells, and implicitly assumed that such cells would not polarise (Abley et al., 2013). Moreover, we did not distinguish between lack of pre-established asymmetric cues versus lack of being able to establish asymmetric cues (e.g. through auxin transport). The above simulations show that this distinction needs to be taken into account to appreciate that, whereas the flux-based model can present intracellular partitioning (i.e. establish polarity in the absence of asymmetric cues or polarisable neighbours), the up-the-gradient model relies on its ability to change its surrounding to establish and maintain polarization. Thus, both the flux-based and up-the-gradient model share commonalities with the indirect coupling model regarding intracellular partitioning, albeit through a more indirect mechanism involving auxin transport between cells.

We next consider an array of polarisable cells in the absence of pre-established asymmetries. Noise is present in initial auxin concentrations (for the up-the-gradient and flux-based models), or in the A*-B* polarity components (for the indirect coupling model), and all cells have the ability to relocate PIN1 (Figure 7D–F). In this situation the up-the-gradient model generates convergent polarities, in which spaced groups of cells orient their PIN1 polarity towards a central region (Figure 7D) (Jönsson et al., 2006; Smith et al., 2006). This is because fluctuations lead to competing centres of high auxin concentration which orient polarities towards them.

By contrast, the flux-based model tends to generate swirled patterns of polarity, in which polarities are coordinated in tandem between neighbouring cells (Figure 7E). Convergent polarities are disfavoured because if two cells have PIN1 oriented towards each other, both cells would experience a low net auxin efflux across their PIN1-rich ends due to transport towards them from the opposing neighbour. The low net auxin efflux would cause removal of PIN1 from the membranes and relocalisation to edges juxtaposed with a PIN1-free edge of a neighbouring cell, promoting tandem alignments.

Similar to the flux-based model, indirect coupling also generates swirled patterns of polarity (Figure 7F, Abley et al., 2013). Convergent alignments are disfavoured because if two cells have PIN1 oriented towards each other, both cells transport auxin to the intervening extracellular space. Accumulation of auxin in the extracellular space then inhibits A*, and therefore PIN1, in adjacent membranes. This destabilises convergent polarities and favours tandem alignments.

Thus, in the absence of pre-established asymmetries, the models spontaneously generate two types of polarity pattern. The flux-based and indirect coupling models both generate tandem alignments, and are subsequently referred to as tandem alignment models; while the up-the-gradient model generates convergent alignments, and is subsequently referred to as a convergent alignment model.

We next explore the assumptions that need to be added to these models for them to account for the epidermal PIN1 polarity patterns observed in WT and kan1kan2 leaves. With each model, we attempt to capture the initial proximodistal polarity field observed in both WT and kan1kan2, where polarities are aligned in tandem along the proximodistal leaf axis with high intracellular auxin at the tip (Figure 1). We then explore the assumptions required for the formation of centres of PIN1 convergence with elevated intracellular auxin, such as those observed on the abaxial side of the main lamina at later stages of kan1kan2 leaf development (Figure 2B,C, Figure 3).

The convergent alignment model has been proposed to account for the initial centre of convergence at the distal end of the leaf primordium (Jönsson et al., 2006; Smith et al., 2006). For a proximodistal polarity field to be maintained within the growing primordium, the convergent alignment model requires maintenance of an increasing auxin concentration gradient from the leaf base to the leaf tip. One way this can be achieved is through net auxin removal at the leaf base. Net auxin removal from a region of tissue could be achieved through a decreased rate of auxin biosynthesis, an increased rate of transport away from the region, or an increased rate of auxin degradation or conjugation. Here net removal is achieved using elevated auxin degradation rates (Figure 8A). Combining this assumption with an initially elevated intracellular auxin concentration at the leaf tip, a proximodistal polarity field can be maintained within an array. The intracellular auxin concentration is kept high at the distal end of the array through transport towards this region (Figure 8A), similar to experimental observations using the DR5::GFP reporter (Figure 3, Mattsson et al., 2003; Scarpella et al., 2006). As the primordium increases in size (Figure 8B), the auxin concentration tends to become shallow at a distance from the leaf tip. This can result in the spontaneous formation of a centre of convergence with elevated intracellular auxin within the proximal region of the lamina (Figure 8B), similar to that observed in kan1kan2 mutants (Figure 2B,C). The length of the tissue at which the polarity reversal occurs depends on the parameters values used in the model. The failure to form such a convergence centre in WT leaves could be explained by the PIN1 polarity pattern becoming fixed at early stages of development, preventing polarity reorientation. However, how this could be achieved is unclear, although it has been proposed that absence of CUC2 activity may play a role in fixing polarity (Bilsborough et al., 2011).

Figure 8

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In the flux-based model, polarities become oriented away from auxin sources and towards auxin sinks. A proximodistal polarity pattern may therefore be generated with net auxin production (a source) proximally and elevated net auxin removal (a sink) distally (Figure 9A). Here we simulate net auxin production in proximal regions with an increased auxin biosynthesis rate, but it could also be achieved with a reduced auxin degradation rate or increased auxin influx from (or decreased auxin efflux to) tissues beyond those represented in the simulation. Likewise, net removal from the leaf tip is simulated using elevated degradation, but could also occur through reduction in biosynthesis, increase in conjugation, or via transport into underlying tissue layers (Bayer et al., 2009; Scarpella et al., 2006).

Figure 9

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With these assumptions the flux-based model generates a proximodistal pattern of polarities. However, intracellular auxin concentrations tend to be lowest at the distal end of the tissue (Figure 9A), inconsistent with experimental observations. This inconsistency can be avoided if the leaf tip acts as a weak instead of a strong auxin sink. The initial drop in intracellular auxin at the tip is then followed by a rise to a high level because of the transport towards this region (Stoma et al., 2008; Walker et al., 2013). Another possibility is that the leaf tip has an elevated auxin import rate as well as an elevated rate of auxin removal (Figure 9B). Elevated auxin import in the distal-most cells causes increased auxin flux towards them, encouraging polarities to point distally. The elevated import also causes the distal-most cells to accumulate intracellular auxin. Although auxin is synthesised at the highest rate in the proximal-most files of cells, in the final state of the simulation, maximal auxin concentrations are found at the distal end (Figure 9B). This is due to polar transport of auxin away from the base, towards the leaf tip.

Similar to the flux-based model, with indirect coupling an initial proximodistal alignment of polarities can be generated if the proximal region (leaf base) has net auxin production (here due to elevated auxin biosynthesis generating high extracellular auxin), while the distal region (leaf tip) has elevated net auxin removal (here due to elevated auxin degradation generating low extracellular auxin). As in the flux-based model, elevated auxin degradation at the distal end of the tissue can lead to low intracellular auxin in this region (Figure 9C). Combining an elevated rate of auxin import at the leaf tip with the elevated degradation ensures low extracellular auxin concentrations in this region (encouraging polarities to point distally) and allows distal cells to accumulate intracellular auxin (Figure 9D). Thus, the two tandem alignment models both have similar requirements for the generation of a proximodistal polarity field with an intracellular auxin maximum in the distal region.

The predictions of the tandem alignment models for the generation of a proximodistal polarity field differ from those of the convergent alignment model. Tandem alignment models predict that the leaf base has net production of auxin, whilst the convergent alignment model predicts that this region has net auxin removal. Also, tandem alignment models require that the leaf tip has an elevated rate of auxin import and removal to account for high auxin in this region, whereas the convergent alignment model has no such requirement.

A further difference is that unlike the convergent alignment model, the two tandem alignment models do not spontaneously generate centres of polarity convergence upon an increase in tissue size (the results of simulations shown in Figure 9A–D are similar for a range of tissue sizes). Thus, with tandem alignment models, further assumptions are needed to generate centres of PIN1 polarity convergence. With tandem alignment models, a site of convergence could arise by a cell having higher levels of auxin removal, caused by export to underlying cells. This cell could be positioned in regions of elevated intracellular auxin, as proposed by Stoma et al, 2008. However, induction of auxin removal alone can result in a drop in intracellular auxin concentration before convergence formation (Stoma et al., 2008), and it is unclear how such regions of elevated intracellular auxin could emerge in the proximal half of the kan1kan2 abaxial lamina.

One possible explanation for the generation of convergence sites in tandem alignment models is that the proximal region of the abaxial lamina has an extended region of auxin biosynthesis. This region corresponds to the region of ectopic CUC2::RFP expression in the proximal region of young kan1kan2 primordia (Figure 6A,C i). In the tandem alignment models, auxin biosynthesis in this region leads to a shallow peak in intracellular auxin proximal to the distal limit of auxin biosynthesis (Figure 10A,C). This peak arises as a consequence of elevated auxin biosynthesis in this domain, together with polarised transport which shifts the position of the peak distally. If intracellular auxin above a threshold concentration activates auxin import, a region with elevated import arises, leading to a centre of polarity convergence (Figure 10B,D). Elevated auxin removal needs to arise in addition to elevated auxin import since import alone causes very high levels of intracellular auxin to accumulate at the centre of convergence. This can disrupt convergence formation by preventing the maintenance of low extracellular auxin (in the indirect coupling model), and by increasing total auxin efflux from the central cell (in the flux-based model). Unlike the model of Stoma et al., 2008, where only auxin removal is induced by high intracellular auxin, the combination of elevated auxin import and removal helps to ensure that a dip in intracellular auxin concentration does not occur prior to convergence formation.

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D6 protein kinases are polarly localized in cells independently of PIN proteins and their phosphorylation of PIN proteins is required for PIN-mediated auxin efflux (Barbosa et al., 2014, Zourelidou et al., 2009). We added a representation of D6 kinase activity to the indirect coupling model, as cell polarity in this system does not depend on PIN activity, and found that this does not qualitatively affect model behavior (Figure 10—figure supplement 1).

This comparison of models gives rise to specific predictions that may be used to distinguish between them experimentally. Tandem alignment models require that, at the leaf tip and at the tips of outgrowths (where centres of convergence are located), the epidermis has elevated rates of auxin import and auxin removal (which could be removal into underlying tissue layers). Locally elevated auxin import is not required for the convergent alignment model, as sites with slightly elevated intracellular auxin reinforce themselves. The tandem alignment models predict that the epidermis towards the leaf base has elevated net auxin production, whilst the convergent alignment model predicts that it has net auxin removal. Moreover, a broader band of auxin biosynthesis at the base of the abaxial side of the kan1kan2 leaf provides one possible mechanism for inducing convergence site formation with tandem alignment models.

To test whether auxin import is elevated at centres of polarity convergence, we determined the expression pattern of auxin importers in kan1kan2 mutant leaves. The AUX/LAX family of auxin importers includes four genes in A.thaliana (AUXIN RESISTANT 1 (AUX1), LIKE AUXIN RESISTANT 1 (LAX1), LIKE AUXIN RESISTANT 2 (LAX2), and LIKE AUXIN RESISTANT 3 (LAX3)) which encode proteins that actively transport auxin from the extracellular space into the cytoplasm (Parry et al., 2001; Péret et al., 2012; Yang et al., 2006). We focus on AUX1 and LAX1 expression since both genes are expressed in the leaf epidermis and therefore may contribute towards the auxin import activity predicted by the models (Kasprzewska et al., 2015). Consistent with previous reports, expression of LAX2::GUS was detected in leaf vascular tissue (Figure 11—figure supplement 1A) and expression of LAX3::GUS was absent from leaves but present in vascular tissue of hypocotyls and roots (Bainbridge et al., 2008; Kasprzewska et al., 2015) (Figure 11—figure supplement 1B,C).

As expected from the tandem alignment models and consistent with a previous report, LAX1::GUS was expressed at the tips of young wild-type leaf primordia, at the tips of serrations, and in the midvein, but was absent from the rest of the leaf lamina (Kasprzewska et al., 2015) (Figure 11A). In kan1kan2 leaves, LAX1::GUS was expressed in the same regions as in WT, but was also ectopically expressed in groups of a few cells in proximal regions of the abaxial epidermis (Figure 11B). These groups of cells were at the tips of developing outgrowths (Figure 11B ii and iii). To investigate whether LAX1::GUS expression preceded outgrowth emergence, sections of kan1kan2LAX1::GUS seedlings stained to reveal GUS activity were imaged. This revealed that local epidermal sites of elevated LAX1::GUS expression were present in the abaxial lamina prior to outgrowth emergence (Figure 11C i), consistent with a role of LAX1 in the generation of centres of polarity convergence. Transverse sections also confirmed that LAX1::GUS expression at the tips of outgrowths is located in the epidermis (Figure 11C ii).

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Figure 11—source data 1

Counts of outgrowths used to generate Figure 11G.

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In wild-type primordia, AUX1::AUX1:YFP was expressed in all cells of the abaxial epidermis, with strongest expression at the leaf margin (Figure 11Di). On the adaxial leaf surface, expression was excluded from most of the epidermis, but detected in cells close to and within the leaf margin (Figure 11Dii). kan1kan2 leaves showed a similar pattern of AUX1::AUX1:YFP expression on their adaxial surface, but the expression pattern in the abaxial epidermis differed from that in wild type. Expression was found throughout distal regions of the leaf, but was concentrated in cells of emerging outgrowths in proximal regions (Figure 11E). Time-lapse confocal imaging revealed that, prior to outgrowth formation, AUX1::AUX1:YFP expression was absent from proximal regions (Figure 11Fiii,iv, yellow arrow), but became detectable as outgrowths emerged (Figure 11Fv–iv, yellow arrows). Since centres of PIN1 polarity convergence form at least one day before outgrowth emergence, these observations suggest that strong AUX1 expression is not involved in their initial development. However, local elevation of AUX1 expression at outgrowth tips could play a role in the stabilisation and maintenance of convergent polarities.

To investigate whether AUX/LAX auxin importers are required for the generation of ectopic outgrowths, we generated kan1kan2aux1lax1lax2lax3 hextuple mutants. Loss of all four AUX/LAX genes caused a reduction in the mean number of outgrowths per rosette leaf from 12.4 (in kan1kan2 mutants) to 1.6 (in kan1kan2aux1lax1lax2lax3 mutants) (Figure 11G). To determine the contribution of lax2 and lax3 to the loss of outgrowths, kan1kan2aux1lax1 quadruple and kan1kan2aux1lax1lax2 quintuple mutants were also generated. Loss of function of aux1 and lax1 alone caused a reduction in the number of outgrowths compared with kan1kan2 plants (Figure 11G). This reduction was further increased by a loss of LAX2 activity (kan1kan2aux1lax1lax2 mutants have a further loss of outgrowths [Figure 11G]). Since LAX2::GUS expression was only detected in sub-epidermal tissue, this finding supports a role of sub-epidermal auxin importer expression in the generation of outgrowths. LAX2-mediated sub-epidermal auxin import below centres of convergence could contribute to removal of auxin from the centre of convergence. Despite the absence of LAX3::GUS reporter expression in leaves, loss of LAX3 caused a further reduction in the number of outgrowths (kan1kan2aux1lax1lax2 mutants generated more outgrowths than kan1kan2aux1lax1lax2lax3 mutants, [Figure 11G]). This suggests that LAX3 may function redundantly with other members of the AUX/LAX family in the generation of ectopic outgrowths, or that differences in genetic background between the multiple mutants has an effect on outgrowth frequency (see Materials and methods).

The findings that auxin importers are expressed in the epidermis at the tips of leaves and emerging kan1kan2 outgrowths, and play a role in outgrowth development, meet the requirement of tandem alignment models (Figures 9 and 10). In its most parsimonious form, there is no need for elevated auxin import at the leaf tip or at subsequent centres of convergence with the convergent alignment model (Figure 8B). However, this does not make the model incompatible with elevated auxin import at these sites. If it is assumed that intracellular auxin above a threshold concentration induces elevated auxin import and removal, elevated import and removal may be induced after convergence formation (Figure 11H). This does not cause disruption of the centres of convergence. Distally oriented polarities near the leaf tip may also be achieved if these cells have an elevated rate of auxin import and removal (rather than an elevated initial auxin concentration) (Figure 11H). Thus, induction of auxin import by elevated intracellular auxin concentrations is compatible with the convergent alignment model, though it renders the model less parsimonious.

To test whether CUC2 could promote the formation of kan1kan2 outgrowths by influencing expression of auxin importers, we examined the expression of LAX1::GUS and AUX::AUX1:YFP reporters in the kan1kan2cuc2 background. Sites of detectable LAX1::GUS expression were lost from the kan1kan2cuc2 lamina (Figure 12A). Additionally, in kan1kan2cuc2 leaves, AUX1::AUX1:YFP was expressed uniformly throughout the abaxial lamina (Figure 12B), suggesting that CUC2 represses the expression of AUX1 in the region surrounding outgrowths. In support of this conclusion, AUX1::AUX1:YFP and CUC2::RFP were found to be expressed in mutually exclusive domains surrounding outgrowths, with AUX1::AUX1:YFP expression being excluded from the region of high CUC2::RFP expression in the outgrowth axil (Figure 12C).

Figure 12

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These findings suggest that CUC2 is required for both the regions of elevated LAX1 expression at outgrowth tips and the regions of low AUX1::AUX1:YFP expression surrounding emerging outgrowths. Thus, CUC2 contributes to locally elevated auxin import at the outgrowth tip. These findings are compatible with a role for CUC2 in promoting convergence formation in the context of a tandem alignment polarity mechanism.

We next test whether centres of convergence have elevated auxin removal rates as well as elevated rates of import. One way that auxin could be removed from epidermal centres of convergence is through transport into underlying tissue layers. In the shoot apical meristem, and in serrations, epidermal centres of PIN1 convergence are linked with sub-epidermal strands of PIN1 expression with PIN1 oriented auxin away from the epidermis (Hay et al., 2006; Reinhardt et al., 2003; Scarpella et al., 2006). To test whether this is also the case for kan1kan2 outgrowths, we immuno-localised PIN1 in transverse cross-sections of kan1kan2 leaves before (Figure 13A) and after (Figure 13B) outgrowth emergence. At both time points, epidermal centres of convergence were coupled with sub-epidermal strands of cells with elevated PIN1 expression (Figure 13). PIN1 in these strands was polarised away from the epidermis, towards underlying tissue (Figure 13A), suggesting that the strands provide a route for the removal of auxin from the epidermis. This observation provides a mechanism for auxin removal required for the tandem alignment models (Figure 10B,D). It is compatible with the convergent alignment model (Figure 11H) but not required (Figure 8B).

Figure 13

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The epidermis at the leaf base is predicted to have net auxin production by the tandem alignment models, and one way this could be achieved is through elevated auxin biosynthesis. To identify likely locations of elevated auxin production in the leaf, we analysed the expression patterns of genes encoding YUCCA auxin-biosynthesis enzymes in wild-type and kan1kan2 leaves. A.thaliana has six YUCCA genes, three of which (YUCCA1 (YUC1), YUCCA2 (YUC2) and YUCCA4 (YUC4)) are expressed in the leaf and are redundantly required for kan1kan2 outgrowth development (Cheng et al., 2007, 2006; Wang et al., 2011). YUC1::GUS was previously reported to be expressed at the base of wild-type leaf primordia, in all cell layers (Wang et al., 2011). Using altered staining conditions to restrict the diffusion of the coloured GUS product, we found that the expression of this reporter was restricted mainly to outer cell layers at the base of the wild-type leaf, with strongest expression on the adaxial side (Figure 14A i,ii). This adaxial expression was detected in very young wild-type leaf primordia emerging from the shoot apical meristem (Figure 14B), and expression at the leaf base persisted during leaf development (Figure 14C).

Figure 14

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The expression of YUC1::GUS in the epidermis at the base of primordia matches the predictions of the tandem alignment models. When this region of auxin production is added to the convergent alignment model along with auxin import and removal at the leaf tip, the formation of a proximo-distal polarity field may be disrupted. A divergent polarity field may be established, with proximally oriented polarities towards the base of the tissue, and distally oriented polarities towards the tip (Figure 14D). This pattern emerges because cells with sufficiently elevated rates of auxin synthesis at the leaf base acquire increased intracellular auxin concentrations, causing polarities to orient towards them. However, it is possible that auxin biosynthesis in regions of YUC1 expression does not contribute sufficiently to influence polarity.

Compared to wild-type leaves, kan1kan2 mutants tended to have a broader distribution of YUC1::GUS in the proximal half of young leaf primordia (Figure 15A i). Transverse sections through these primordia revealed that the expression domain extended further along the proximodistal axis on the abaxial side of the leaf (the region where centres of convergence form) than on the adaxial side (Figure 15A ii and iii). This is consistent with the tandem alignment models for convergence formation, where a broad domain of auxin biosynthesis in the proximal region of the leaf induces cells with elevated auxin import and removal, causing convergence formation (Figure 10). As in WT, YUC1::GUS expression in the basal-most cells of the leaf persisted and could still be detected in leaves of around 500 µm in width (Figure 15B iv). Consistent with a role of auxin biosynthesis in inducing cells with elevated auxin import (Figure 10), the majority (98%) of kan1kan2yuc1yuc4 mutant rosette leaves failed to form outgrowths and lacked sites of elevated LAX1::GUS expression within the lamina (Figure 15C, compare with Figure 11B).

Figure 15

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Before kan1kan2 outgrowths were detected, YUC1::GUS was expressed in an additional band, a few cells wide, running across the abaxial side of the leaf, approximately one-third of the way from the leaf base (Figure 15B ii, black arrow). As outgrowths emerged, the band of YUC1::GUS expression was present at their base on their distal side (the outgrowth axil) (Figure 15B iii and iv, black arrow) and absent from the outgrowth tips (Figure 15B iii, black arrow head).

The expression pattern of YUC1::GUS is similar to that of CUC2::RFP (Figure 6). To investigate whether CUC2 functions upstream of YUC1 expression in kan1kan2 leaves, we generated kan1kan2cuc2 mutants with the YUC1::GUS reporter. In young kan1kan2cuc2 leaves, the proximal domain of YUC1::GUS expression was reduced in size (Figure 15D i) and at later stages of leaf development the ectopic band of YUC1::GUS expression was lost (Figure 15D ii). Thus, CUC2 is required for the ectopic expression of YUC1::GUS in kan1kan2 leaves.

To test the potential effects of the pattern of YUC1::GUS expression close to outgrowths on PIN1 polarity patterns, we introduced a pre-pattern of auxin biosynthesis similar to that observed for YUC1 into simulations of the different models. In tandem alignment models, a band of cells with elevated auxin production introduced after the establishment of a proximodistal polarity field generates a region of divergent polarities centred on the band with elevated production (Figure 16A,C). If the band of elevated auxin biosynthesis is introduced together with a cell with elevated auxin import and removal located on the proximal side of the band, a polarity convergence tends to form centred on the cell with elevated import and removal (Figure 16B,D). This behaviour is consistent with the finding that centres of PIN1 polarity convergence are located at the tips of outgrowths, where auxin importer expression is elevated.

Figure 16

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In contrast to the tandem alignment models, when a band of elevated auxin synthesis is introduced in the convergent alignment model, polarities tend to orient towards the band (Figure 16E). This may occur even in the presence of a cell with locally elevated auxin import and removal on the proximal side of the band if the rate of auxin import in this cell is not sufficiently high (Figure 16F). This behaviour is inconsistent with the observation that PIN1 polarities orient towards regions with elevated LAX1 and AUX1 expression, even in the presence of a band with elevated YUC1 expression in the outgrowth axil.

According to these simulations, the patterns of YUC1-mediated auxin biosynthesis close to kan1kan2 outgrowths are more compatible with the tandem rather than convergent alignment models. However, the presence of YUC expression in regions of polarity divergence at the leaf base and outgrowth axil may be reconciled with the convergent alignment model if it is assumed that YUC expression appears as a consequence of low auxin in these regions following polarity reorientation. Consistent with this possibility, the expression of YUC1, YUC2, YUC4 and YUC6 has been shown to be down-regulated by auxin, and upregulated by expression of an auxin biosynthesis inhibitor (Suzuki et al., 2015). Given such negative feedback acting upon YUC expression, it is possible that local auxin-biosynthesis by YUCs does not raise auxin concentrations sufficiently to affect PIN1 polarities.

YUC2::GUS and YUC4::GUS were previously reported to be expressed at the tips of wild-type leaves, in contrast to expression of YUC1::GUS which is expressed at the base (Wang et al., 2011). This expression pattern of YUC2::GUS and YUC4::GUS is more consistent with the convergent alignment model. However, at early stages of leaf development, when YUC1::GUS is expressed at the leaf base, YUC4::GUS and YUC2::GUS were found to be absent from the leaf tip (Wang et al., 2011). The proximo-distal PIN1 polarity pattern was only observed in young WT primordia (Figure 1A, Figure 2A). This suggests that, at the time that the proximodistal PIN1 polarity field is observed, the leaf base is the main site of YUC-mediated auxin biosynthesis.

YUC2::GUS and YUC4::GUS were also previously reported to be expressed at the tips of kan1kan2 outgrowths (whether this expression domain is epidermal or sub-epidermal was not shown). However, whether expression of these genes in regions of polarity convergence precedes outgrowths or appears later was not determined. We therefore performed time-lapse imaging of a YUC4::GFP reporter to capture the dynamics of YUC4 expression (Figure 17). This revealed that, approximately one day prior to outgrowth emergence, when centres of PIN1 convergence were previously determined to form, this reporter was expressed in the epidermis, distal to where outgrowths subsequently emerged (Figure 17 i,ii). As outgrowths emerged, they maintained epidermal expression of YUC4 in their axils, the region where YUC1 is also expressed (Figure 17 iii,iv, compare with Figure 15B ii–iv). Emerging outgrowths also showed a second region of expression of YUC4::GFP in epidermal and sub-epidermal tissue at the outgrowth tip (Figure 17 iii,iv). These findings suggest that, at the time that centres of PIN1 polarity convergence form, YUC4 is expressed in the epidermis distal to the centre of convergence. YUC4 expression at the outgrowth tip then appears after convergence formation. The epidermal expression domain distal to polarity convergences in the outgrowth axil is similar to that inferred for YUC1 and is more consistent with tandem than convergent alignment models. Similar to YUC1, expression of YUC4::GFP within the abaxial lamina was lost in the kan1kan2cuc2 background (Figure 17B).

Figure 17

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All kan1kan2 mutants carry kan1-2 and kan2-1 alleles in the Landsberg erecta (Ler-0) background (Eshed et al., 2001) except in the kan1kan2yuc1yuc4 background, where plants carry kan1-11 and kan2-5 alleles (Wang et al., 2011). The cuc2 mutant allele used is cuc2-3 and was initially in the Columbia (Col-0) background (Hibara et al., 2006). All marker lines were originally in the Col-0 background. PIN1::PIN1:GFP, DR5::GFP (Benková et al., 2003); LAX1::GUS, LAX2::GUS, LAX3::GUS (Bainbridge et al., 2008); YUC1::GUS, YUC2::GUS, YUC4::GUS (Cheng et al., 2006) were described previously. The AUX1::AUX1:YFP line corresponds to the AUX1-YFP116 construct described by (Swarup et al., 2004). The CUC2::RFP line was kindly provided by Patrick Laufs prior to publication.

The YUC4::GFP reporter was provided by Yunde Zhao and Youfa Cheng. The YUC4 promoter was amplified with the following primers: Y4pro3p_XmaI: cccggggtcgactaataaaagcgaaagagag and Y4pro5p_HindIII: aagcttatgtccaacatgcatgcg. The 2916 bp PCR product was digested with HindIII and XmaI, and cloned into pBJ36-ERGFP vector. The promoter was subcloned from pBJ36-ERGFP into the pART27 vector and transformed into Col-0 background by floral dipping and subsequently selecting for kanamycin resistant transgenic plants.

All images of marker construct expression in WT were obtained using the original marker lines in the Col-0 background. The kan1kan2cuc2 mutant was generated by crossing cuc2 mutants with kan1kan2+/- plants (kan1kan2 mutants are sterile). In the F2 of this cross, kan1kan2+/- cuc2 individuals were identified by screening for the kan1kan2+/- abberent fruit phenotype, and by PCR-based screening for cuc2-3 homozygous individuals. The presence of a T-DNA insertion in the CUC2 locus was detected by using a forward primer, which binds to the T-DNA: 5’-TCCATAACCAATCTCGATACAC-3’ and a reverse primer which binds within the CUC2 locus downstream of the T-DNA insertion (referred to as CUC2-R): 5’- GGAGGCTAAAGAAGTACCATTC-3’. The presence of a WT copy of CUC2 was detected using a forward primer CUC2-F: 5’-AATATCCATCCACATTATTACCAC-3’, which binds to CUC2 upstream of the site of the T-DNA insertion in cuc2-3, along with CUC2-R. One quarter of the offspring of kan1+/- kan2 cuc2 individuals were kan1kan2cuc2 mutants, and could be identified at the seedling stage (and therefore selected for experiments) based on the upwardly curled cotyledon phenotype of kan1kan2 mutants.

Marker constructs were introduced into the kan1kan2 and kan1kan2cuc2 backgrounds by carrying out the corresponding crosses using kan1kan2+/- or kan1kan2+/-cuc2 mutants. The kan1kan2aux1lax1lax2lax3, kan1kan2aux1lax1lax2 and kan1kan2aux1lax1 mutants were generated by crossing kan1kan2+/- plants with the aux1lax1lax2lax3 quadruple mutant (Bainbridge et al., 2008). In segregating F2s from this cross, kan1kan2+/- plants with desired mutant alleles of aux/lax genes were selected as described in Bainbridge et al., 2008.

The kan1kan2+/- line used to generate the kan1kan2aux1lax1lax2lax3 mutants was in the Ler-0 background, and the aux/lax quadruple mutant was in the Col-0 background. To check that the loss-of outgrowth phenotype in hextuple mutants was not due to the introduction of an unknown genetic element from the Col-0 background, we determined whether the reduction in the number of outgrowths segregated with the mutant LAX1 allele. An F3 family of plants was used that segregated for mutations in KAN2 and LAX1, but was homozygous for kan1, aux1, lax2 and lax3 mutant alleles. Of 156 kan1kan2aux1lax2lax3 plants segregating for lax1, 120 (77%) developed many outgrowths and 36 (23%) had a reduced number and size of outgrowths. All of the plants with few outgrowths were homozygous for the lax1 mutation, whilst all siblings with many outgrowths that were genotyped (40 in total) all had a wild-type copy of LAX1. Such segregation analyses were not carried out for kan1kan2aux1lax1 and kan1kan2aux1lax1lax2 mutants thus it is possible that the reduction in outgrowths in these lines is due to introduction of an unknown genetic element during the cross between Col-0 and Ler backgrounds. However, two independent families of kan1kan2aux1lax1lax2 plants (derived from different F2 plants) had similar losses of outgrowths (data provided in source data file for Figure 11G). The loss of outgrowths in kan1kan2aux1lax1lax2lax3 plants was also observed in two independent families quantified in two different experiments (see source data file for Figure 11G).

Plants were sown on plates for confocal imaging experiments and to select those to be used for crosses. Seeds were surface-sterilized before plating on Petri dishes with solid Murashige and Skoog medium as described (Sauret-Güeto et al., 2013) and grown in controlled environment room at 20°C in long day conditions (8 hr dark and 16 hr light under fluorescent white light at a photon fluence rate of 100 µmol·m⁻²·s⁻¹). When appropriate, medium was supplemented with Kanamycin 50 µg/ml (to select for PIN1::PIN1:GFP, AUX1::AUX1:YFP, LAX1::GUS, LAX2::GUS, LAX3::GUS, YUC1::GUS and YUC4::GFP constructs).

kan1kan2 plants carrying aux/lax or cuc2 mutant alleles were sown on soil prior to characterisation of their phenotype. Seeds were stratified directly (without sterilisation) and then sown in John Innes Centre A.thaliana Soil Mix (Levington F2 compost with Intercept and grit at a 6:1 ratio) and were grown under long-day conditions (16 hr light and 8 hr dark) in a glasshouse, supplemented with artificial light, at approximately 22°C.

The kan1kan2yuc1yuc4 mutants used are as described in (Wang et al., 2011). kan1kan2+/-yuc1+/-yuc4 LAX1::GUS plants were selected in an F2 of a cross between kan1kan2+/- yuc1+/- yuc4 plants and LAX1::GUS plants. F2 seedlings with the LAX1::GUS construct were by selected for sowing seeds on plates containing 50 µg/ ml Kanamycin. Those that were kan1kan2+/- mutants were identified based on their fruit phenotype and the presence of the yuc1+/-yuc4 genotype was selected for by PCR-based genotyping. The yuc1 and yuc4 alleles have T-DNA insertions, and mutant alleles were screened for using the following primers for yuc1: LBb1: 5’-GCGTGGACCGCTTGCTGCAAC-3’ and YUC1-R: 5’-CCTGAAGCCAAGTAGGCACGTT-3’ and the following primers for yuc4: LBb1: 5’-GCGTGGACCGCTTGCTGCAAC-3’ and YUC4-R: 5’-GCCCAACGTAGAATTAGCAAG-3’. WT copies of YUC1 were screened for using YUC1-F: 5’-GGTTCATGTGTTGCCAAGGGA-3’ and YUC1-R and WT copies of YUC4 were screened for using YUC4-F 5'-CCCTTCTTAGACCTACTCTAC-3’ and YUC4-R. To analyse the expression of LAX1::GUS in kan1kan2yuc1yuc4 mutants, the offspring of kan1kan2+/-yuc1+/-yuc4 LAX1::GUS or kan1kan2+/yuc1+/-yuc4 LAX1::GUS+/- plants were sown on plates containing 50 µg/ ml Kanamycin and seedlings with a kan1kan2 leaf phenotype (narrow, upwardly pointing leaves) were selected, genotyped for YUC1 and YUC4, and stained to reveal GUS activity. Of these seedlings, one quarter did not have outgrowths, and genotyping showed they were kan1kan2yuc1yuc4 mutants.

For tracking experiments, seedlings were first grown on plates (until 5 days after stratification for kan1kan2 and kan1kan2cuc2 mutants, and until 4 days after stratification for WT seedlings). Seven seedlings were then transferred into a tracking chamber (Sauret-Güeto et al., 2013; Calder et al., 2015) where they were kept for the duration of the imaging experiment. During this period, there was a constant flow of liquid media (1/4 strength Murashige and Skoog, 0.75% sucrose, 1.1 μg / ml MES, pH 5.8) at 1 μl /s through the growth chamber. Seedlings in the chamber were imaged using a Zeiss EXCITER Laser Confocal Microscope every 6 to 24 hr (times are provided for specific experiments in figure legends). To image GFP and YFP markers, a 488-nm line of an argon ion laser was used. Emitted light was filtered through a 500-550-nm band-pass filter. To image RFP markers, a 543 nm helium-neon laser was used and emitted light was filtered using a 560-615-nm band-pass filter. A 40x oil objective was used for all experiments. Between imaging, the chamber containing seedlings was kept at 20°C, with 16 hr light, 8 hr dark cycles. In confocal imaging experiments without tracking, seedlings were mounted in water on microscope slides.

Confocal z-stacks were converted into individual PNG images using Bioformats converter (http://cmpdartsvr3.cmp.uea.ac.uk/wiki/BanghamLab/index.php/BioformatsConverter). Z-stacks of PNG images were then rendered in 3D using Volviewer (http://cmpdartsvr3.cmp.uea.ac.uk/wiki/BanghamLab/index.php/VolViewer#Download). All snapshots shown in figures were taken from Volviewer. Leaf widths were measured from maximum intensity projections in image J and thus do not take into account the 3D curvature of the leaf.

OPT imaging was performed as described by (Lee et al., 2006, Sharpe et al., 2002). Following PI staining (described below), plants were embedded in 1% low melting point agarose and embedded samples were kept in methanol overnight to dehydrate. To clear the agarose before scanning, the methanol was replaced with 2 parts (v/v) Benzyl Benzoate, 1 part Benzyl Alcohol and samples were left for 12 hr until almost transparent. A prototype OPT device was used to image embedded leaves (Lee et al., 2006; Sharpe et al., 2002). A 20-W halogen lamp connected to the OPT device was used to collect visible light transmission images. Images were reconstructed into png slices and visualised in 3D using VolViewer.

For propidium iodide staining, dissected leaves were placed into 10% (v/v) acetic acid, 50% (v/v) methanol solution and kept at 4°C overnight. Samples were then washed with water and dehydrated with two washes each in 40% (v/v) ethanol, 60% (v/v) ethanol and then 80% (v/v) ethanol. The samples were boiled in 80% (v/v) ethanol at 80°C for 2–10 min in a water bath and then rehydrated with two washes each in 60% (v/v) ethanol, 40% (v/v) ethanol, 20% (v/v) ethanol, and water. Samples were then treated with alfa-amylase [3 mg of alpha-amylase (SIGMA, UK) in 10 ml of 20 mM phosphate buffer (pH 7), 2 mM NaCl, 0.25 mM Ca₂Cl] overnight at 37°C, followed by three water washes, and then treated with 1% periodic acid [1% in solution from SIGMA, UK] (an oxidising agent) for 40 min at room temperature. Following two washes with water, PI staining solution [333 mM sodium metabisulphite, 0.5 M HCl, 148 μM PI] was added and incubated for 1–2 hr at room temperature until the material appeared pink in colour. PI staining solution was then removed and the samples were washed with water.

GUS staining with and without wax embedding was performed as described by Sessions et al. (1999), but here using 6 mM Potassium Ferrocyanide and 6 mM Potassium Ferricyanide and with an 18 hr incubation with the GUS staining solution.

Immuno-localisation of PIN1 was performed as described (Conti and Bradley, 2007), using a goat anti-PIN1 (aP-20) polyclonal primary antibody (Santa Cruz Biotechnology, TX, USA) (RRID:AB_670528) diluted 1:300 and a Cy3 conjugated rabbit anti-goat secondary antibody (Jackson Immunoresearch, PA, USA), diluted 1 in 300.

All models were created using the VVe modelling environment, an extension of the VV system (Smith et al., 2003), which in turn is an extension of L-systems. All models are provided as supplementary source code files. Each model can be run using L-studio with VVe, which can be downloaded from http://algorithmicbotany.org/virtual_laboratory/. Instructions on how to run models are available as supplementary information (Supplementary file 1).

The indirect coupling model was implemented based on the description in Abley et al., 2013 except in some simulations we use rectangular rather than hexagonal cell geometries and we introduce an explicit representation of PIN1 to all models. The details of these differences are described below. All parameter values are as described in Abley et al., 2013, unless otherwise detailed in Table 1.

Each rectangular cell is represented by a single central cytoplasmic compartment, 22 membrane compartments (6 membrane compartments for each long edge and 5 membrane compartments for each short edge), and surrounding cell wall compartments. Each cell wall compartment links two membrane compartments located in adjacent cells.

The system is initialised with a default concentration of PIN1 in the cytoplasmic compartment of all cells. The rate of change of PIN1 concentration in a given membrane compartment depends on a background binding rate of PIN1 from the cytoplasm to the membrane, plus default unbinding from the membrane into the cytoplasm. It is also assumed that A* in a given membrane compartment promotes the binding of PIN1 to that membrane compartment and that PIN1 can diffuse between adjacent membrane compartments of the same cell. The equation describing the rate of change of PIN1 concentration in a given membrane compartment is:

where PIN is the concentration of PIN1 in the focal membrane compartment, with units of Arbitrary units (A_u)/ µm and PIN_C is the concentration of PIN1 in the cytoplasmic compartment of the same cell as the focal membrane compartment, with units of A_u/µm². ρ_PIN is the default binding rate of PIN1 to the membrane, with units of µm/s, and τ is a constant describing the rate at which membrane-bound A* promotes the binding of PIN1 to the membrane, with units of µm²/A_u.s. A* is the concentration of the A* polarity component in the focal membrane compartment, with units of A_u/ µm. µ_PIN is the default unbinding rate of PIN1 from the membrane into the adjacent cytoplasmic compartment, with units of /s. D_PIN is the diffusion constant of PIN1 in the membrane, with units of µm²/s.

The corresponding equation describing the rate of change of PIN1 concentration for a given cytoplasmic compartment is:

Where PIN_C is the concentration of PIN1 in the focal cytoplasmic compartment and PIN_n is the concentration of PIN1 in the membrane compartment n, in the neighbourhood of the cell c (N(c)). R_c is the area of the cytoplasmic compartment and l_n is the length of the nth membrane compartment. A^*_n is the concentration of A* in the membrane compartment n. ρ_PIN, τ and µ_PIN are as described above for Equation 1.

The export of auxin from a cytoplasmic compartment to a wall compartment depends on the concentration of PIN1 in the intervening membrane compartment. The equation presented by Abley et al., 2013, describing the rate of change of auxin (referred to as mediator (M) in Abley et al., 2013), in cytoplasmic compartment i becomes:

Where Aux_i is the concentration of auxin in the cytoplasmic compartment i, Aux_w is the concentration of auxin in the wall compartment neighbouring the membrane compartment n in the neighbourhood of the cell c (N(c)), ρAux is the production rate of auxin with units of A_u/µm².s, R_c is the area of the cytoplasmic compartment, and µ_Aux is the degradation rate of auxin with units of /s. PIN_n is the concentration of PIN1 in the nth membrane compartment of the cell. ν_in is the background permeation rate of auxin into the cytoplasm from the wall with units of µm/s and ν_out is the background permeation rate of auxin into the wall from the cytoplasm with units of µm/s. l_w is the length of the cell wall compartment into/out of which auxin flux is occurring and R_c is the area of the cytoplasmic compartment. ψ _PIN is the rate of PIN1-dependent active efflux of auxin from the cytoplasm into the wall with units of µm²/ A_u.s.

The equation describing the rate of change of auxin in wall compartments becomes:

where R_w is the area of the wall compartment, l_w is the length of the wall compartment, Aux_w is the concentration of auxin in the wall compartment, A_n* is the concentration of A* in the membrane compartment n in the neighbourhood of the given wall compartment w (N(w)) and Aux_c is the concentration of auxin in the cytoplasm of the same cell as the membrane compartment n. ψ _PIN is as described above and PIN_n is the concentration of PIN1 in the membrane neighbor, n, in the neighbourhood of the wall compartment w ((N(w)). D_Aux is the diffusion constant for auxin within the cell wall with units of µm²/s (this constant relates to lateral diffusion between wall compartments as it is assumed that the concentration of auxin is uniform across the thickness of the wall).

In the simulation used to generate Figure 10—figure supplement 1, we introduce D6 kinase-like activity into the model by assuming that only a fraction of PINs are active and able to export auxin, and that this fraction depends on the local concentration of A* in the membrane. A* therefore represents D6 kinase-like activity. In this scenario, the term describing PIN1-mediated auxin efflux from the cell to the wall in Equations 3 and 4 is modified to become:

Where A*_n is the concentration of A* in the same membrane compartment as PIN, h is the Hill coefficient (here a value of 2 was used, giving a sigmoidal function), and c is the concentration of A* at which half of the PINs are activated. ψ _PIN, PIN_n and Aux_c are as described above.

In all simulations of the indirect coupling model, there is no flux of auxin across the boundaries of the tissue.

In the simulation used to generate Figure 10C and D (positioning of cells with elevated auxin import and removal), in the first 6500 s of the simulation, the production rate of auxin, ρ_Aux, in all cells in the proximal half of the array (except those in the proximal-most row) is 1.5 × 10⁻⁴ A_u/ µm².s (in the proximal most row ρ_Aux = 2 × 10⁻⁴A_u/µm².s). Then, between 6500 and 10 000 s of the simulation, the auxin production rate of all cells in the proximal half of the array (including those in the proximal-most file) is increased at every time step of the simulation:

Where ρ_Aux is the auxin production rate of a given cell and β is a constant describing the rate of increase of the auxin production rate, with units of A_u/ µm².s² (β = 5 × 10⁻⁶ A_u/ µm².s²). As a consequence of this increase in the auxin production rate, after 10 000 s of the simulation, in the proximal-most file of cells, ρ_Aux = 3.75 × 10⁻⁴ A_u/ µm².s, and in the other cells in the proximal-most half of the leaf, ρ_Aux = 3.25 × 10⁻⁴ A_u/ µm².s.

After 6 500 s of the simulation, if the auxin concentration of a cell exceeds a threshold, T_import, the inwards permeability of the cell to auxin is increased 50-fold (v_in = 37.5 µm /s) and the auxin degradation of the cell is increased 10-fold so that µ_Aux = 0.1 /s. T_import = 0.0285 A_u/ µm².

In the simulation used to generate Figure 10C and D, a noise term is added to the concentrations of auxin every 0.05 s of the simulation, according to the following equation:

where Aux_n is the concentration of auxin in a given cytoplasmic compartment after the addition of noise with units of A_u/µm², Aux is the concentration of auxin in that compartment before the addition of noise (with the same units), θ_Aux is a random number drawn from a normal distribution, with mean 0, and standard deviation 5 × 10⁻⁴. Noise is added to the concentration of auxin in proportion to the square root of the auxin concentration.

The flux-based model was implemented using a similar discretisation of cells as in previous implementations (Rolland-Lagan and Prusinkiewicz, 2005; Stoma et al., 2008). Each cell is represented by a single central cytoplasmic compartment surrounded by six peripheral compartments representing cell edges (one for each cell-cell interface). There is no representation of the cell wall. Each edge compartment of a cell is connected to the cytoplasmic compartment of the same cell and to the juxtaposed edge compartment of the neighbouring cell, unless the vertex is at a border of the tissue.

The model is implemented based on previously described implementations (Rolland-Lagan and Prusinkiewicz, 2005; Stoma et al., 2008). At the beginning of all simulations, the system is initialised with auxin in cytoplasmic compartments and PIN1 at edge compartments.

The system may be initialised with noisy auxin concentrations in each cytoplasmic compartment, i:

where Aux_i(t = 0) is the initial concentration of auxin in a given cytoplasmic compartment, i, with units of A_u/ µm² and c_A is the default initial concentration of auxin at an edge compartment. θ_Aux is a random number uniformly distributed between an upper and lower limit, ɛ_Aux.

The system may also be initialised with noisy PIN1 concentrations at each cell edge:

where PIN_edge(t = 0) is the initial concentration of PIN1 at a given edge compartment, with units of A_u/ µm and c_PINedge is the default initial concentration of PIN1 at an edge compartment. θ_PIN is a random number uniformly distributed between an upper and lower limit, ɛ_PIN.

PIN1 is recruited to a cell edge depending on the rate of auxin efflux across that edge, and PIN1 is removed from the edge at a background rate. The equation describing the rate of change of PIN1 concentration for a given cell edge, between cells i and j, is

Where PIN_ij is the concentration of PIN1 in cell i, at the cell edge between cells i and j, α is a dimensionless constant determining the extent to which flux promotes PIN1 allocation to the membrane and ${\Phi }_{i\to j}$ is the auxin flux across the interface between cells i and j, with units of A_u / µm.s. Outgoing fluxes, from cell i, to cell j are considered to be positive, and incoming fluxes are considered to be negative. γ is the unbinding rate of PIN1 from the cell edge, with units of /s.

As previously described (Rolland-Lagan and Prusinkiewicz, 2005), we assume that once the concentration of PIN1 at the cell edge reaches a threshold value, P_max, PIN1 can no-longer be allocated to the edge. This is equivalent to assuming that there is a maximum density of PIN1 proteins which may be present in the plasma membrane.

In all versions of the model, the flux across a given cell edge, from cell i to cell j, at a given time point depends on passive permeability of auxin between cells and PIN-mediated auxin export. In the simulations used to generate Figures 9B, 10A,B and 16A,B, we assume that auxin may also be actively imported into cells, and that the rate of import depends on the auxin import rate of a given cell, and the auxin concentration in its neighbour. The flux across a given cell edge, from cell i to cell j, at a given time point is therefore given by

Where ${\displaystyle {\mathit{\Phi }}_{i\to j}}$ is the flux across a given cell-cell interface, from cell i to cell j, D is a constant describing the passive permeation rate of auxin, with units of µm / s, A_i is the concentration of auxin in cell i, A_j is the concentration of auxin in cell j, PIN_ij is the concentration of PIN1 in cell i, at the cell edge between cells i and j, and PIN_ji is the concentration of PIN1 in cell j, at the cell edge between cell j and i. T is a constant describing the rate of PIN-mediated auxin transport, with units of µm²/ A_u.s. I_j and I_i are parameters describing the auxin import rates of cells j and i respectively, with units of µm / s.

The rate of change of auxin concentration in the cytoplasm of a given cell, i, depends on the production and degradation rates of auxin, and its flux into adjacent cells and is given by:

Where ρ is the production rate of auxin, with units of A_u / µm².s, µ is the degradation rate of auxin, with units of /s, $A_i$ is the concentration of auxin in the cytoplasm of cell i and R_i is the area of the cell i. ${\displaystyle {\mathit{\Phi }}_{i\to j}}$ is the flux of auxin across the interface between cells i and j, in the neighbourhood of the cell i ((N(i)) and l_ij is the length of the cell edge at the interface between cells i and j.

In the simulations used to generate Figure 7B,E and Figure 7—figure supplement 1B, there is no flux of auxin across the boundaries of the tissue. To minimise boundary effects, in the simulations used to generate Figures 9A,B, 10A,B and 16A,B, the left and right boundaries of the tissue are connected so auxin flux can occur between them. All changes in concentrations were solved numerically using an explicit Euler integration method.

In all simulations where the row of cells at the base of the tissue has an elevated rate of auxin production (orange cells at the base in Figures 9A,B, 10A,B and 16 A,B), the auxin production rate in these cells = 0.0008 A_u /µm².s. In Figure 16A,B, in the band of three cells with elevated auxin production, the auxin production rate = 0.032 A_u /µm².s. In the simulation used to generate Figure 9A, in the distal row of cells, the auxin degradation rate = 0.5/s. In the simulations used to generate Figures 9B, 10A,B and 16 A, B, in the distal most row of cells with elevated auxin import and removal, I = 300 µm / s and μ = 0.04/s. In the simulation used to generate Figure 10B, in the induced cells with elevated auxin import and removal, I = 30 µm / s and μ = 0.5 /s. In the simulation used to generate Figure 16B, I = 100 µm / s and μ = 0.8 /s in the single cell with elevated auxin import and removal.

In the simulation used to generate Figure 10A and B (positioning of cells with elevated auxin import and removal), in the first 165 s of the simulation, the production rate of auxin, ρ_Aux, in the proximal-most file of cells, = 0.0008 A_u/ µm².s. In this first phase of the simulation, the production rate of auxin in all other cells in the proximal half of the array is 0.0005A_u/ µm².s. Then, between 165 and 200 s of the simulation, the auxin production rate of all cells in the proximal half of the array (including those in the proximal-most file) is increased at every time step of the simulation according to Equation 6. Here,$$β = 0.0001 A_u/ µm².s². As a consequence of this increase in the auxin production rate, after 200 s of the simulation, in the proximal-most file of cells, ρ = 0.0043 A_u/ µm².s, and in the other cells in the proximal-most half of the leaf, ρ = 0.004 A_u/ µm².s.

After 165 s of the simulation, if the auxin concentration of a cell exceeds a threshold, T_import, the auxin import of the cell is increased so that I = 30 µm / s and the auxin degradation rate is increased so that μ = 0.5 /s. T_import = 0.09 Au/ µm² .

After 20 s of the simulation, noise is added to the concentrations of auxin every 0.01 s of the simulation, according to Equation 7. In this case, θ_Aux is a random number drawn from a normal distribution, with mean 0, and standard deviation 5 × 10⁻³.

The up-the-gradient model is implemented based on the description by Bilsborough et al., 2011. The tissue is represented as described for the flux-based model. The system is initialised with auxin in cytoplasmic compartments and PIN1 at edge compartments. In some simulations, the system is initialised with noisy PIN1 concentrations at each cell edge, as described by Equations 9a,b, or noisy auxin concentrations in each cell, as described by Equations 8a,b.

PIN1 only exists in cell edge compartments and each cell has a fixed total concentration of PIN1 available to its cell edges. This total concentration of PIN1 is distributed between edge compartments at each time step according to an exponential function of the auxin concentrations in neighbouring cells. The equation describing the concentration of PIN1 in a given edge compartment in cell i, at the interface between cell i and j, at a given time step is:

Where PIN_ij is the concentration of PIN1 at the cell edge between cells i and j, in cell i. PIN_i is the total PIN1 concentration available to be divided between all the edges of cell i (with units of A_u/ µm), A_j is the concentration of auxin in the neighbour j and A_k is the auxin concentration in the neighbour k of cell i. The exponentiation base, b, controls the extent to which the auxin distribution in neighbouring cells influences PIN1 protein distribution at cell edges.

Previous simulations have differed in the implementation of auxin production. Some assume that intracellular auxin feeds back to inhibit its own production (Smith et al., 2006; Bayer et al., 2009; Bilsborough et al., 2011; Feugier et al., 2005), whilst other assume that auxin production occurs independently of intracellular auxin concentration (Rolland-Lagan and Prusinkiewicz, 2005; Jonsson et al., 2006; Stoma et al., 2008). We use the latter assumption in all models.

The rate of change of auxin concentration for a given cytoplasmic compartment depends on the rates of auxin production and degradation, and the rates of auxin flux between the given cell and its neighbours. This is described by the following equation:

Where A_i is the auxin concentration of cell i, ρ is the auxin production rate, with units of A_u /µm².s, µ is the auxin degradation rate, with units of / s and R_i is the area of the cell i, with units of µm². ${\displaystyle {\mathit{\Phi }}_{i\to j}}$ is the auxin flux (with units of A_u / µm.s) out of cell i, across the interface between cells i and j, in the neighbourhood of cell i ((N(i)). l_ij is the length of the cell edge at the interface between cells i and j, with units of µm.

The flux of auxin between neighbouring cells is calculated as described by Equation 11. Changes in auxin concentrations were solved numerically using an explicit Euler integration method.

As for the flux-based model, in the simulations used to generate Figure 7A and D and Figure 7—figure supplement 1A, there is no flux of auxin across the boundaries of the tissue. To minimise boundary effects, in the simulations used to generate Figures 8A,B, 11H, 14D and 16 E,F, the left and right boundaries of the tissue are connected so auxin flux can occur between them.

In simulations used to generate Figure 8A,B and Figure 16E,F, where the distal-most row of cells has an elevated initial auxin concentration (row of orange cells in Figures 8A,B), the initial auxin concentration in these cells is 0.5 A_u /µm². In all simulations where the proximal-most row of cells has a low initial auxin concentration and an elevated rate of auxin degradation (blue cells in bottom row in Figures 8A,B, 11H, 16E,F), the initial auxin concentration in these cells is 0 A_u /µm² and their auxin degradation rate = 0.05/s.

In the simulation used to generate Figure 14D, the initial auxin concentration in the distal-most row of cells = 0.5 A_u /µm², whilst in all other cells, the initial auxin concentration is as stated in Table 3. In the distal most row of cells, the rates of auxin import and degradation are elevated (I = 20 µm / s and µ = 0.01 /s). In the proximal most file of cells, the auxin production rate is elevated so that ρ = 0.001 A_u /µm².s

In the simulation used to generate Figure 11H, in the distal most row of cells, auxin import and degradation are elevated, as described for Figure 14D. When the auxin concentration exceeds a threshold, T_import, in cells other than those at the proximal and distal boundaries, the rates of auxin import and degradation are increased to be equal to those in the distal most row of cells (I = 20 µm / s and µ = 0.01 /s). T_import = 0.08 A_u /µm².

In Figure 16E,F, in the band of three cells with elevated auxin production the auxin production rate = 0.04 A_u /µm².s. In Figure 16F, in the cell with elevated auxin import and removal, I = 5 µm / s and µ = 0.03 /s.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Youfa Cheng and Yunde Zhao for sharing the unpublished YUC4::GFP line, Patrick Laufs for providing the unpublished CUC2::RFP line and John Bowman for providing kan1kan2+/- seeds and for helpful discussions. Thanks to three anonymous reviewers for their help with improving the manuscript. Thanks to all to members of the Coen lab for discussions and to Xana Rebocho Christopher Whitewoods and Annis Richardson for comments on the manuscript. Additionally, special thanks to Xana Rebocho for advice on experimental work and to Samantha Fox for confirming genotypes of kan1kan2aux/lax mutants. Thanks to Przemysław Prusinkiewicz, Mik Cieslak, Adam Runions and Pau Formosa Jordan for comments on the manuscript and discussions on the relationships between models. Thanks to Pierre Barbier de Reuille for development of the VVe modelling environment and to Pierre Barbier de Reuille and Pascal Ferraro for preparing L-studio/VVe for distribution. We also thank Grant Calder for confocal microscope support. Thanks to the John Innes Horticultural Services team for help with plant care. Many thanks to Andrew Bangham for his support and encouragement and for many useful discussions about this work.

© 2016, Abley et al.

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