The bilaterally symmetric Snapdragon (Antirrhinum majus) corolla has five petals united for part of their length to form a tube with five separate lobes, which diverge at their sinuses (Figure 2A–B). The upper part of the corolla comprises two dorsal petals, while the lower part consists of a pair of lateral petals flanking a ventral petal (Figure 2A). The corolla functions like a mouth, with the lower corolla articulated at a hinge to open or shut (hinge, Figure 2B). Several genes controlling dorsoventral asymmetry and flower shape in Snapdragon have been characterised. CYCLOIDEA (CYC), DICHOTOMA (DICH) and RADIALIS (RAD) encode dorsal-specific transcription factors that repress the ventral identity gene DIVARICATA (DIV) (Almeida et al., 1997; Corley et al., 2005; Luo et al., 1999, 1996).

Figure 2

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The most striking feature of the wild-type lower corolla is a wedge-shaped fold (Figure 2C–D), represented in a simplified form in Figure 2E. The wedge comprises a platform (palate) that slopes up towards a ridge (rim) and a descending lip (Figure 2C–E). The rim rises to form two small peaks at the junctions between lateral and ventral petals (foci, Figure 2A and C–E). DIV is required for the development of the wedge-shaped fold. In div mutants, the wedge-shape of the lower corolla is lost and two small domes form at the foci (Figure 2F–J).

To determine the regional origins of the wedge of wild type, we created a fate map (Figure 3) using morphological landmarks such as sinuses between petals (blue arrows), main veins (yellow and orange lines) and trichomes (purple ellipse, from 14 DAI = Days After petal Initiation) (Figure 3A–G). The first visible sign of wedge formation was a shallow furrow proximal to the petal sinuses at 12 DAI (yellow bracket, Figure 3B). Prior to this, the lower corolla showed no out-of-plane deformation (Figure 3A). During the next 5 days, the furrow became more pronounced and the regions that will form the palate and lip could be identified as two crescents by 14 DAI (light and dark pink, Figure 3C). These crescents spanned the ventral petal together with half the lateral petal on each side, referred to as the flanks (boundary marked by the lateral midvein, Figure 3A–G). The other halves of the lateral petals grew less in width and formed the hinge (green shading, Figure 3C–G). Over the next seven days the crescent bulged further to form the wedge with a slope and steep lip (Figures 3E–G and 2C–D). Thus, the wedge initiates from a narrow strip of tissue at the tube-lobe boundary that undergoes an out-of-plane deformation through a defined series of morphogenetic events.

Figure 3

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In contrast to wild type, the mature div flower has an open mouth and lacks an extended palate (Figure 2F–G). The ventral corolla has two small domes where the tube-lobe boundary (yellow dashed line) intersects the ventral-lateral petal junctions (blue dashed line) (Figure 2H–J). The centres of intersection correspond to the foci (asterisks in Figure 2H–J). Initially, the div lower petals showed no out-of-plane deformation (Figure 3H).The domes first became evident as small out-of-plane adaxial bulges at 14 DAI, and were clearly visible by 17 DAI (Figure 3I–J). At these stages, we mapped the lip region to a crescent-shaped region distal to the domes (Figure 3I–J).

To determine how these out-of-plane deformations might relate to cellular behaviours, we analysed the pattern of cell files in div and wild type, at different stages of development by staining the petal tissue with calcofluor white. Calcofluor stains older walls less brightly than younger walls, and reveals a range of different cell patterns (Figure 4). For example, old walls (highlighted in blue) may surround a region of cells subdivided by randomly oriented recent walls (highlighted in green or white, Figure 4A). Or old walls may surround a region subdivided by a ladder of perpendicular recent walls (Figure 4B), or a region subdivided by recent perpendicular walls (green), which are in turn further subdivided by more recent walls (white) perpendicular to these (Figure 4C).

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To understand the origin of these patterns, we modelled cell divisions using the Growing Polarised Tissue (GPT framework), in which the tissue is treated as a connected continuous material, termed the canvas (Kennaway et al., 2011). This is the same framework used to model the deformations illustrated in Figure 1. Within the GPT framework, specified anisotropic growth is oriented by a polarity field that propagates through the canvas. Regional gene activities modify local specified growth rates parallel (K_par) or perpendicular (K_per) to the polarity field. Plant cells can be readily imposed on this framework because cell topology is preserved: once a vertex is created by division, its relation to other vertices is maintained through subsequent growth (plant cells do not move relative to each other). We assume that when the area of a cell exceeds a threshold, it is divided along the shortest path through its centroid (Errera, 1886; Besson and Dumais, 2011). The new wall is then shortened slightly to give more realistic angles (Nakielski, 2000). Cell wall colour and thickness reflect wall age (Figure 4D–F).

If specified growth was isotropic and uniform, the resulting pattern of walls resembled that of Figure 4A, with the blue outline highlighting the clone derived from the original cell (Figure 4D). Within the clone, cell walls were oriented randomly (Figure 4D). If specified growth was uniformly anisotropic, with growth oriented proximodistally (vertically in Figure 4), a pattern similar to that seen in Figure 4B was generated, where division walls were largely perpendicular to the main orientation of growth (Figure 4E). If specified growth was initially oriented proximodistally and then switches to mediolateral, a pattern similar to that in Figure 4C was generated (Figure 4F). Similar patterns of division occurred irrespective of initial cell geometries, although the shape of the final clone varied (Figure 4—figure supplement 1). Thus, the calcofluor pattern gives an indication of the history of cell divisions, and orientations of growth. Lines perpendicular to the cell walls reflect growth oriented parallel (red lines) or perpendicular (yellow lines) to the proximodistal axis of the tissue (Figure 4G–H).

Calcofluor staining of developing div and wild-type petals showed that at 10 DAI, growth was largely oriented parallel to the proximodistal axis (red lines in Figure 5A–B). In the next three days, cells in regions flanking the presumptive foci showed a switch to mediolateral growth (yellow lines in Figure 5C–D, Figure 5—figure supplement 1A–B and F–G). This growth pattern continued so that by 15 DAI a clear orthogonal pattern of cell files was observed in the div mutant, with cell files being elongated medially at the tube-lobe boundary flanking the foci, proximodistally at the petal junctions and mostly isotropic at the intersection (Figure 5E). In wild type, the orthogonal pattern of cell files at 15 DAI was less clear than in div as the rim flanking the ventral-lateral petal showed a mixture of cell wall orientations, indicating greater proximodistal growth (compare Figure 5F to Figure 5E). There was also a region of enhanced mediolateral growth in the lateral (lilac box, Figure 5F) and ventral lip (green box in Figure 5—figure supplement 1H). In addition, diagonally oriented cell files were observed near the sinus (orange lines in blue box in Figure 5F). The observed pattern of growth at 15 DAI was maintained through later stages of div and wild-type development (Figure 5—figure supplement 1C–D and I) and correlated with the regions of the petal that formed out-of-plane deformations (Figure 5—figure supplement 1E and I). Thus, there was a switch to mediolateral growth in regions flanking the foci prior to and during the out-of-plane deformation, which was modulated by the activity of DIV. The timing of this switch was in line with the repatterning of growth predicted previously (Green et al., 2010).

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To determine whether this growth repatterning was specific to the ventral-lateral junctions, we analysed cell files at the lateral-dorsal junction (Figure 5—figure supplement 2). An orthogonal pattern of cell files was again observed at a similar stage, but in contrast to the ventral-lateral junction, the distal arm of the orthogonal pattern did not grow. Thus, repatterning occurred at all petal junctions of the lower corolla but with varying growth dynamics.

The orthogonal patterns of anisotropic growth observed in wild type and div suggests that an orthogonal directional conflict may be involved in generating these out-of-plane deformations. To explore this idea further, we simulated orthogonal conflicts using a square initial canvas, similar to that used in Figure 1. We introduced several transcription factors with different expression domains into the initial canvas: a factor expressed in a vertical strip (JUN, Figure 6A), a horizontal strip (RIM, Figure 6B), the right half (HALFSIDE, Figure 6C) and upper half (DISTALSIDE, Figure 6D). We first assumed a polarity field converging at the centre, similar to that employed in Figure 1I. To generate an orthogonal pattern of specified anisotropy, while keeping areal growth rate uniform across the canvas (Figure 6E), we assumed both RIM and JUN promoted specified growth parallel to the polarity (K_par) and inhibited specified growth perpendicular to the polarity (K_per) (the ratio of K_par to K_per is indicated by Kaniso in Figure 6). Growth was specified to be isotropic where RIM and JUN overlap. This set of assumptions created an orthogonal pattern of specified anisotropic growth (red and yellow regions Figure 6F) next to quadrants of specified isotropic growth (green regions, Figure 6F). Running a model based on these assumptions led to the formation of a dome (Figure 6G, Video 6), with clones more elongated within the orthogonal domains of high specified anisotropy than in adjacent quadrants (red ellipse, Figure 6G). The anisotropy exhibited by the quadrants was the result of passive deformation rather than being specified directly.

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Video 6

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The contribution of the orthogonal directional conflict to shape could be further dissected by removing components. Removing one of the regions of high specified anisotropy by introducing dependence of RIM activity on HALFSIDE, gave an asymmetric dome (Figure 6H–I, Video 7). Removing two regions of anisotropy, by making JUN activity dependent on DISTALHALF, and RIM activity dependent on HALFSIDE (Figure 6J), produced a less pronounced asymmetric dome (Figure 6K, Video 8). Removing both horizontal regions of high anisotropy by removing RIM activity, gave a ridge (Figure 6L–M, Video 9). Removing all regions but one, by making specified anisotropy dependent on the combination of JUN with DISTALHALF, resulted in a slightly arched and symmetric ridge (Figure 6N–O, Video 10). Thus, various types of out-of-plane deformation may be generated by varying the pattern and extent of directional conflict.

Video 7

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Video 10

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To determine other ways of generating orthogonal directional conflicts, we considered a parallel rather than convergent polarity field. An orthogonal conflict could be generated by JUN promoting K_par (and inhibits K_per), while RIM promoted K_per (and inhibits K_par) (Figure 6P). A dome was again generated, with the polarity field being deformed by growth so that it passed over and around the dome (Figure 6Q, Video 11). Virtual clones were oriented parallel to the polarity in the JUN domain (red ellipse, Figure 6Q) and perpendicular to the polarity in the RIM domain (black ellipse in Figure 6Q).

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In all the above cases, the axial information provided by the polarity field was critical for the resulting shape. Boosting isotropic growth in an orthogonal pattern (areal conflict) generated a series of bulges rather than a dome (Figure 6R–S, Video 12).

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Orthogonal directional conflict resolution thus provides a mechanism for generating out-of-plane deformations, and can be specified in several ways through a combination of orthogonal regional identities and polarity fields (convergent or parallel). Further dome shapes can be generated by combining the orthogonal directional conflict with areal and surface conflicts (Figure 6—figure supplement 1A–C), or by combining surface and areal conflicts (Figure 6—figure supplement 1D).

In the above examples, we modelled deformations as arising through differential growth. In animals, contractions as well as growth may be involved in generating out-of-plane deformations. In early animal embryogenesis it is common for deformations to occur without overall change in size. To determine how principles of tissue conflict resolution could apply in these contexts, we modelled deformations that preserve overall size. For these purposes, contraction could be considered as equivalent to negative growth.

For surface conflict, high specified isotropic growth of the upper surface was counterbalanced by an equal negative specified growth rate (contraction) for the lower surface, and gave a folded shape similar to that based on differential growth (compare Figure 7A–B to Figure 1E–F). For areal conflict, negative specified isotropic growth at the periphery was counterbalanced by high specified growth in the centre, and gave a shape again similar to that based on increased growth alone (compare Figure 7C–D to Figure 1G–H). For directional conflict, enhanced growth parallel to polarity (positive K_par) was counterbalanced by negative specified growth perpendicular to the polarity (negative K_per), and gave a similar result to overall positive growth (compare Figure 7E–H to Figure 1I–J and Figure 6P–Q). Such a balance between growth in one orientation and contraction in the perpendicular orientation is equivalent to the process of convergent-extension (Keller et al., 2008). Surface, areal and directional conflict resolutions, thus provide possible mechanisms for generating out-of-plane deformations with or without overall growth.

Figure 7

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Given the cell file data, we hypothesise that orthogonal directional conflict is involved in generating the out-of-plane deformation in div and wild type. This hypothesis requires a polarity field for orienting the growth in each region. To test whether a polarity field is present and determine its pattern of orientations, we analysed PIN1 auxin transport protein distribution as this is a possible readout of cell polarity (Steinmann et al., 1999). The Snapdragon genome has 11 PIN genes, three of which are PIN1 orthologues (PIN1a, PIN1b and PIN1c). RNA in situ hybridisation revealed that the three PIN1 genes showed detectable expression in developing wild-type petals, with all PIN1 orthologues strongly expressed in the vascular tissue and epidermis in the palate region of the lower petals (Figure 8—figure supplement 1). To determine protein localisation, we raised peptide antibodies against each of the three proteins, as well as a generic antibody against a conserved domain in all three. The antibody raised to the PIN1a-based peptide gave the clearest signal. As this peptide shares some amino acids with the other PIN1s, we cannot be sure that the signal derives entirely from PIN1a and therefore refer to the signal obtained as PIN1 signal. Immunolocalisations on sectioned tissue showed that PIN1 signal is localised at the distal end of epidermal cells in early wild-type petal primordia, with abaxial and adaxial polarity converging at the tip of the petal primordia (day 9, Figure 8—figure supplement 2A–C). This is in agreement with PIN1 patterns observed in emerging primordia in other species (O'Connor et al., 2014; Reinhardt et al., 2003).

To analyse PIN1 polarity in regions undergoing out-of-plane deformations, we developed a whole-mount immunolocalisation protocol for plant tissue sheets, and software to quantify the distribution of PIN1 signal for each cell (PinPoint software). Prior to the repatterning of corolla growth in div and wild type, PIN1 is strongly expressed near the petal sinuses and margins, and is oriented proximodistally (Figure 8A–D and Figure 8—figure supplement 2D–E). This pattern extends over more cells proximally when the orthogonal pattern of cell files begins to form (13 DAI, Figure 8E–H and Figure 8—figure supplement 2F–G), overlapping with the distal region showing proximodistal cell files (Figure 8I–J). PIN1 is more extended in wild type than in div. Within the extended region, polarity is oriented proximodistally along the petal junction (3–4 central files within green dashed line in Figure 8G, Figure 8—figure supplement 2G). Either side of this, polarity points diagonally, towards the central files and nearby margins (Figure 8G, Figure 8—figure supplement 2G). In both div and wild type, the region with diagonal PIN1 polarity corresponds to that showing diagonal cell files (yellow boxes, Figure 8I–J). Strong PIN1 signal was not observed in the regions showing mediolateral cell files (orange boxes in Figure 8I–J). By 15 DAI, PIN1 expression domain became restricted to the petal margins (Figure 8—figure supplement 2H–I). The upregulation of PIN1 in wild type was not observed at the lateral-dorsal sinuses (Figure 8—figure supplement 2J–K), where DIV was not strongly expressed at this stage (Galego and Almeida, 2002). Thus, PIN1 reveals an early proximodistal polarity pattern near the sinus which is modulated by DIV to become extended and deflected during repatterning.

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In light of the data above on fate mapping, cell files and PIN1 localisation, we modelled corolla morphogenesis, beginning with the simpler deformation observed in div. Rather than modelling the entire div corolla, we first modelled the region forming the out-of-plane deformation.

Based on the fate map experiments, the div and wild-type out-of-plane deformations develop from a curved strip of tissue at the tube-lobe boundary, encompassing the presumptive palate and lip (Figure 3). We therefore used a strip of canvas as the initial configuration for our minimal model (Figure 9A). The strip was convex on the abaxial and concave on the adaxial surface, as observed for this region of the corolla before out-of-plane deformation. Growth rates were modulated by regional identity factors distributed along three axes – proximodistal, mediolateral and dorsoventral – similar to those employed previously (Green et al., 2010). The dorsoventral factor was a diffusible signal derived from the dorsal region, sRAD, and specified the hinge region of the lateral petal (red shading in Figure 9A). The proximodistal factors included PALATE (PLT), RIM and LIP (Figure 9A), which corresponded to distinct regions and cell types in the flower (Keck et al., 2003). RIM was a diffusible signal that activated the expression of BRIM to subdivide the regions of PLT and LIP expression (Figure 9A). The mediolateral factors LAT and MED corresponded to the petal junctions and midvein regions, respectively (Figure 9A). All of these factors interacted combinatorially to control the specified rates of growth parallel (K_par) and perpendicular (K_per) to the polarity field (details of interactions are given in Supplementary Material and methods). As PIN1 cellular localisation was only observed around the sinus and was mainly proximodistal, we made the simplest assumption that polarity is maintained as proximodistal throughout the canvas (black arrows in Figure 9B). Alternatively, polarity outside the PIN1 upregulation domain may reorient to point towards the petal junction to create a region of mediolateral polarity flanking a channel of proximodistal polarity (Figure 6T–U, Video 13).

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Video 13

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For our minimal model of div, growth of the initial strip was specified to be higher parallel than perpendicular to the polarity (ratio of K_par to K_per is indicated by Kaniso in Figure 9B). K_par was also inhibited on the lower (abaxial) surface in the rim to promote out-of-plane bending (surface conflict, Figure 9C). Specified growth was further inhibited at the edges of the strip (hinge) through the activity of a dorsoventral gene (sRAD) (areal conflict, Figure 9D). K_par was also promoted in the distal lip (areal and directional conflict, Figure 9D). These assumptions led to the generation of a bilaterally symmetric canvas with short lateral edges by 11.5 DAI (Figure 9E). Orthogonal directional conflict was introduced at 12 DAI through combinatorial interactions that enhanced K_par relative to K_per at the petal junctions, and enhanced K_per relative to K_par along the rim (Kaniso panel in Figure 9E showing orthogonal pattern marked by dashed white lines). At the hinge, growth in the distal part of the orthogonal domain was inhibited by sRAD (red dashed lines in Figure 9E). In addition, K_per was inhibited by MED (Figure 9E). Running this model generated two domes centred on the foci, and an extended lip, matching the shape observed in div (Figures 9F and 2J, Video 14).

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This model involved three types of conflict: areal, directional and surface. To evaluate the contribution of each type of conflict to the final shape, we removed each individually. Removing surface conflict (by equalising growth rates on the two surfaces) led to the out-of-plane deformations protruding abaxially instead of adaxially (Figure 9G and Video 15). This was because the direction of bulging reflected the curvature of the initial strip of tissue, which was abaxially convex (Figure 9A). Removing areal conflicts, by normalising the growth rates across the canvas, resulted in a crumpled strip with long hinge regions (Figure 9H, Video 16). Removing all directional conflict by eliminating polarity, while maintaining the patterns of areal and surface conflicts, generated a curved strip (Figure 9I, Video 17). Removing orthogonal directional conflict by removing the enhanced mediolateral growth at the rim generated a relatively flat shape with a bend at the rim (Figure 9J, Video 18). Thus, the out-of-plane deformation of div depends on all three types of conflict, with orthogonal directional conflict playing a key role in creating the domes.

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The interactions from the above minimal model were incorporated into a model of the entire Snapdragon corolla to see if they could account for the overall final shape. We first modified the published full corolla model (Green et al., 2010) to create a ground state for the lower petals (see Supplementary Material and methods). This produced a corolla with a wild-type looking upper corolla, but a lower corolla lacking out-of-plane deformations (Figure 9—figure supplement 1A). Incorporating the interactions of the minimal div model above into this ground state produced a corolla with an open mouth, a reduced palate, two domes tipped by the foci and a flat extended lip, similar to the div phenotype (Figure 9K 2F, Video 19). As with the minimal model, removing orthogonal directional conflict generated a corolla with a bend at the rim for the lower corolla rather than two domes (Figure 9—figure supplement 1B).

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To explore the role of DIV, we incorporated its activity within the minimal model. In accordance with previous experimental data, we assumed that DIV expression was graded, from low levels at the lateral-dorsal junctions to high levels in the ventral petals (Figure 10A) (Galego and Almeida, 2002). Unlike the div mutant, the wild-type corolla has an extended palate and lip (forming the slope of the wedge, Figure 2E). We therefore assumed that DIV promotes K_par in the palate and lip regions from an early stage (Kaniso, Figure 10B; K_area, Figure 10D). DIV also enhanced surface conflict by increasing the extent of differential growth parallel to the polarity between the two surfaces near the rim, consistent with differential expression of cell division genes in this region (Figure 10D; (Gaudin et al., 2000)). These assumptions generated a canvas similar to the div model at 11.5 DAI but with an extended palate (compare 11.5 DAI, Figure 10E with Figure 9E). To simulate the observed deflection of polarity towards the wild-type sinus at 12 DAI, we assumed DIV activates expression of a new polariser sink at the sinus (blue arrows, Figure 10E). To account for the observed patterns of orthogonal cell files in the wild type, we assumed that at 12 DAI DIV extends the region of high K_per flanking the foci into the lip region (white dashed lines, Figure 10E), enhancing directional conflict. K_per was also inhibited at the distal and proximal boundaries as a proxy for the restraining effect of adjacent tissue.

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With these assumptions, a wedge shape was generated that resembled that observed in wild type (Figure 10F, Video 20). The pattern of growth could be compared with that of a wild-type flower by generating virtual clones, induced at 12 DAI. The pattern obtained was similar to that seen by mapping experimentally induced clones on the final wedge shape (Figure 2E). In both cases, clones were predominantly proximodistal throughout the palate and distal lip regions, but mediolateral in the flanking lip regions (compare Figure 10F to Figure 2E).

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As with the minimal model of div, we evaluated the contribution of each type of conflict to the final shape. Removing surface conflict led to the out-of-plane deformation protruding abaxially instead of adaxially (Figure 10G, Video 21). Removing areal conflicts resulted in a crumpled strip with long hinge regions and protruding domes with a narrow palate (Figure 10H, Video 22). Removing all directional conflict by eliminating polarity generated a curled canvas with two large domes in the lateral petal (Figure 10I, Video 23). Reducing orthogonal directional conflict by removing the enhanced mediolateral growth at the rim generated a sharp bend at the rim and a narrow lip region (Figure 10J, Video 24). Thus, as for div, morphogenesis of wild type depends on all three types of conflict, with directional conflict playing an essential role.

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Integrating the activity of DIV into the full corolla model of div described above generated a corolla with a typical Snapdragon shape (Figures 10K and 2A, Video 25). Removing dorsoventral gene activities (i.e. DIV, CYC, DICH) from the model generated phenotypes resembling those of the respective mutants (Figure 11A–D, Videos 26 and 27). These phenotypes were distinct from virtual mutants obtained by removing each tissue conflict from the wild-type model (Figure 9—figure supplement 1C and E–F). Removing the deflection of polarity at the sinus resulted in an overarching lower corolla (Figure 9—figure supplement 1D), suggesting that the deflection of polarity could play a role in refining corolla shape.

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Figure 12

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Antirrhinum majus (snapdragon) wild type (JI7 and JI2) and div-13 (Almeida et al., 1997) were grown in the greenhouse at the John Innes Centre.

Using the published staging reference (Vincent and Coen, 2004) for JI2 flower development, we adapted it to JI7. Although these stocks are related, bud emergence and plastochron counts are slightly different in JI7. Therefore, we developed a JI7 staging reference by dissecting and photographing several flower bud series and relating bud features such as carpel, stamen and petal morphology between stocks as well as measurements such as petal width and lobe length. Imaging of buds and flattened petals was done using a stereomicroscope (Leica M205C). Bud age is referred to as Days After petal Initiation (DAI) of the flower meristem, and corresponds to the published reference system (Vincent and Coen, 2004). We used the JI7 reference system to stage the mutants. As the div-13 phenotypes have modified lower petals, we used the length and width of the dorsal petals to stage the mutant flowers at different stages of development.

Snapdragon buds were dissected from inflorescences, sepals removed and the corolla opened and flattened on a drop of prostatic adhesive (technovent) placed onto a glass slide. At early stages the whole petal could be flattened onto the glue. At later stages, when wedge formation was deforming the lower petals, we tried to keep the 3D shape of the petals by only flattening the distal lobe and palate-tube regions.

To follow the development of the wedge region from emergence until maturity in div-13 and wild-type, we used dissected and flattened material from staging experiments, as well as the OPT bud series, to mark different morphological landmarks that delimit the wedge, and made measurements on how this region grew. The measurements were done in imageJ. We used the same landmarks for div-13 and wild-type (exceptions indicated below). We used the lateral petal midvein as the mediolateral limit of the wedge, which was visible at early stages just after petal emergence at 8 DAI. To mark the upper and lower limits of the wedge along the proximodistal axis, we used the lip-distal lobe boundary and the palate-distal tube boundary, respectively. The petal-distal lobe boundary ran along the sinus of the petals, so to mark this boundary we drew lines from the D-L petal sinus to the V-L sinus as well as the two V-L sinuses. This morphological landmark was easily marked from the time of petal emergence. The proximal wedge limit could only be marked after 14 DAI, when palate trichomes emerged. This limit could not be marked in div-13 as the lower petals did not develop palate trichomes (due to absence of ventral identity). The rim position was visible from 14 DAI, as a furrow in flattened petals.

Snapdragon flower buds were dissected from inflorescences, sepals removed and the buds prepared for OPT as previously described (Lee et al., 2006). Scanning and reconstructions were performed using a prototype OPT Scanner at the John Innes Centre. The 3-D images were visualised and processed using VolViewer software (http://cmpdartsvr3.cmp.uea.ac.uk/wiki/BanghamLab/index.php/Main_Page).

Snapdragon inflorescences were harvested and hand cut to 1 cm thick, before fixing in phosphate-buffered saline (PBS) pH 7.5 containing 4% paraformaldehyde (RT-15710, EMS), 0.1% Triton-X100 and 0.1% Tween-20, overnight at 4°C. After dehydration (50% and 70% ethanol), samples were embedded in paraffin (wax paramat-361148C, VWR) using a Tissue-TEK VIP processor (Sakura). 8 µm sections were cut, mounted on polysine slides (VWR) and dried overnight at 42°C. Slides were dewaxed in histoclear (2x for 10 min) (HS-200, National Diagnostics) and hydrated through a decreasing ethanol series (made in 1x saline solution: 0.85% (w/v) NaCl). Slides were washed in PBS solution before digesting the tissue with Pronase (0.125 mg/ml, P6911 Sigma-Aldrich) for 12 min. A 0.2% (v/v) Glycine-PBS solution was used to block the Pronase activity (3 min, RT), followed by a PBS wash and another fixation step with 4% paraformaldehyde in PBS for 10 min. After a 2 × 10 min PBS washes, the tissue charges were neutralised using a 0.5% (v/v) acetic anhydride solution made in 0.1M triethanolamine pH 8.0, to minimise unspecific binding and background. Slides were washed in PBS and dehydrated though an increasing ethanol series (made in 1x saline solution) and left to dry for at least 15 min. To generate specific digoxigenin-labelled riboprobes probes, the ORF of AmPIN1a (F, 5’ATGATAACTTTATCTGATTTTTACCTG-3’ and R, 5’-TCATAGTCCCAGCAAAATATAGTAG-3’), AmPIN1b (F, 5’-AAAAATGATATCTTTATCTGATTTTTACC-3’ and R, 5’TCAAGTCCCAACAAAATATAGTAGATG-3’) and AmPIN1c (F, 5’ATGATCACTTTAACAGACTTATATCACG-3’ and R, 5’-CTAGAGTCCAAGCAAGATGTAGTAG-3’) were amplified using TAQ DNA polymerase PCR (201205, Qiagen) and cloned into pCR4-TOPO TA vector (K4575-01, Life Technologies) following manufacturer’s instructions. The cloned fragments were amplified using forward specific primers (AmPIN1a, 5’-GAGAAAGTGAAAGTGATGCCTCC-3’, AmPIN1b, 5’-CAATGAAGATGGTTACTTGGAG-3’ and AmPIN1c, 5’CCAAACCAACTGCAATGCCACC −3’) and the M13F primer (pCR4-TOPO kit), column purified (28104, Qiaquick PCR purification) followed by a phenol/chloroform extraction. Antisense probes were obtained by RNA transcription using the T7 promoter (10881767001, Roche) and DIG-UTP (11209256910, Roche), according to manufacture instructions. The AmPIN1a, AmPIN1b and AmPIN1c probes were hydrolysed at 60°C using 200 mM carbonate buffer pH 10.2 solution, for 80, 8 and 60 min, respectively. The hydrolysed probes were boiled in 50% formamide (4311320, Life Technologies) solution for 2 min and left on ice to cool, before adding the hybridisation buffer (per 5 ml: 625 µl 10x Salts (3M NaCl, 0.1M Tris-HCl pH6.8, 0.1M NaPO4, 50 mM EDTA), 2500 µl Deionized formamide (4311320, Life Technology Ltd), 62.5 µl tRNA (R4251, Sigma-Aldrich), 125 µl Denhardt’s(30915, Sigma-Aldrich), 437.5 µl RNase-free water, 1250 µl 50% Dextran Sulphate (D8906, Sigma-Aldrich)). Two microliters of hydrolysed probe plus 100 µl of hybridisation buffer were used per slide, and covered with Hybrislip hybridisation cover (HS6024-CS, Grace biolabs). Slides were hybridised in a humid box overnight at 50°C. Next day, slides were washed 3 times in 0.2x SSC solution at 55°C for 25 min. Single stranded RNA was digested with RNaseA (0.02 mg/ml, R4875 Sigma-Aldrich) in NTE buffer (10 mM Tris-HCl pH 7.5, 1 mM EDTA and 500 mM NaCl) at 37°C for 30 min. Slides were washed in 100 mM Tris.NaCl pH7.5 solution and then blocked using a freshly made 0.5% (w/v) blocking reagent (11096176001, Roche) in the same buffer, for 1 hr at room-temperature (RT). After a 30 min wash in 1% (w/v) BSA (A7906, Sigma-Aldrich) (made in 100 mM Tris.NaCl pH7.5 with 0.3% Tritonx100), slides were hybridised with a 1:3000 dilution of Anti-digoxigenin-AP antibody (11093274910, Roche) for 90 min. Slides were washed in 100 mM Tris.NaCl pH7.5 with 0.3% (v/v) tritonx100, 4 times for 25 min. Triton-x100 was removed from slides by washing them in 100 mM Tris.NaCl pH7.5 for 5 min. Before detection, the slides were washed in a 100 mM Tris.NaCl pH9.5 to equilibrate the pH. Detection was performed overnight using a NBT/BCIP (Promega) in 100 mM Tris.NaCl pH9.5 plus 50 mM MgCL2. The in situ experiments were performed three times.

Protocol adapted from Conti and Bradley (Conti and Bradley, 2007). For section immunolocalisation, snapdragon inflorescences were harvested and hand cut to 2.5 mm thickness. Tissue was fixed in a 3.7% (v/v) formaldehyde (F8775, Sigma-Aldrich)-ethanol-acetic acid solution (FAA) with 1% (v/v) DMSO and 0.5% (v/v) Triton-x100, overnight at 4°C. After dehydration (50% and 70% ethanol), samples were embedded in paraffin (wax paramat-361148C, VWR) using a Tissue-TEK VIP processor (Sakura). 8 µm sections were cut, mounted on polysine slides (VWR) and dried overnight at 42°C. Slides were dewaxed in histoclear (2x for 10 min) (HS-200, National Diagnostics) and hydrated through a decreasing ethanol series. To improve antigen retrieval, slides were boiled in 10 mM citrate (pH 6) for 15 min and then left to cool for 30 min at room temperature. Slides were rinsed in deionized water and blocked in BB (5% (w/v) non-fat dry milk in PBS pH 7.5) at RT for 3 hr. Immunohybridisation of AmPIN1 was carried out overnight at 4°C using 1:500 primary anti-AmPIN1 antibody in BB. After washing three times (10 min) with PBS containing 0.3% (v/v) Triton-x100, the slides were hybridised with Alexa594-conjugated secondary antibody (Jackson ImmunoResearch, 1:500) for 3 hr. After incubation, slides were washed as after primary antibody incubation. Finally, slides were counterstained with 0.1% (w/v) calcofluor (F3543, Sigma-Aldrich) and mounted and imaged in 1% (w/v) DABCO (D27802, Sigma-Aldrich)/PBS/50% glycerol solution. For whole-mount immunolocalisation, buds were dissected and fixed flat onto a glass slide by glue as described in the dissection and flattening of lower petals section. The tissue was then fixed in a 3.7% (v/v) formaldehyde (F8775, Sigma-Aldrich)-ethanol-acetic acid solution (FAA) with 1% (v/v) DMSO and 0.5% Triton-x100, overnight at 4°C, after a short vacuum treatment. After two washes with 50% ethanol, the slides were washed with deionized water and boiled in 10 mM citrate (pH 6) for 15 min and then left to cool for 30 min at room temperature. Slides were rinsed in deionized water and blocked in BB (3% (w/v) BSA in PBS pH 7.5) at RT for 3 hr. Primary and secondary antibody hybridisation, washes and calcofluor staining were performed as described above. Tissue was mounted and imaged in 1% (w/v) DABCO/PBS solution. Number of replicas for calcofluor staining and whole mount PIN1 immuno of div mutant lower petals at various stages of development = 6. Number of replicas for calcofluor staining and whole mount PIN1 immuno of wild-type mutant lower petals at various stages of development = 11. Replicas gave similar results for similar developmental stages.

The AmPIN1a, AmPIN1b, AmPIN1c and AmPIN1 (conserved peptide between all three PIN1 proteins) antibodies were produced in rabbit by Cambridge Research Discovery (CRB) using the following predicted antigenic peptide sequences present in the cytosolic loop domain. Peptide sequence for AmPIN1a -SRGPTPRPSNFEEE, AmPIN1b – HRGNNEDGYLERDEL, AmPIN1c – FSPASTKKKGENGKD and AmPIN1 -IYSMQSSRNPTPRGS. The clearest signal was obtained with the AmPIN1a antibody. The AmPIN1 antibody gave similar results but with more background.

Scanning electron microscopy was carried out as described in Vincent and Coen (Vincent and Coen, 2004). In situ hybridised material was imaged using a Leica DM6000 and Nomarski settings, x10 dry lens, and a DFC420 digital camera was used to photograph in situ sections. All measurements made from in situ images were calculated using ImageJ software (http://imagej.nih.gov/ij/). Fluorescent immunolocalisation results were imaged using a Leica SP5 II confocal microscope and x25 water dipping lens. Images were acquired in a format of 1024 × 1024, with a line average of 4, scan speed of 600 Hz and pinhole of 1 airy unit. Excitation for Alexa 594 on laser 20% was with a 561 laser line, with a detection band of 570–630 nm. Calcofluor was imaged with a 405 diode laser and detection at 450–500 nm. Z stack images were at 1 µm/section. Image reconstruction was performed using the ImageJ software.

PIN1 polarity was quantified using the PinPoint software, developed in MATLAB to analyse gene expression in cell boundaries. A composite stack of Anti-PIN1-Alexa594 and calcofluor staining was loaded into the program. For each cell segmentation, we determined which cell the PIN signal belonged to by manually scrolling through the z-stack. In ambiguous cases this determination was aided by PIN signal following the curvature of cell walls (Figure 8—figure supplement 2). A representative segmentation plane was chosen near to the cell’s midplane and the cell boundary was manually segmented. The segmentation was used to create an image mask of the cell. A second image mask was created for each cell by morphologically eroding the cell boundary with a disk-shaped structuring element of radius five pixels (≈1.5 µm). A difference image was created by subtracting the eroded mask from the original cell mask. For each pixel in the difference image, a vector was calculated using the PIN1 pixel intensity as magnitude and angle measured from the cell centroid to the pixel, relative to the x-axis as orientation. The sum of these vectors was used to represent the polarity of each cell. Average vector fields were calculated by summing vectors for each cell within a 30 × 30 pixel window.

Mathematically, growth of a region is described by a growth tensor which can have both a strain (symmetric) and rotational (skew-symmetric) component (Hejnowicz and Romberger, 1984). A specified growth tensor has a strain component but no rotational component, as we assume that each tissue region exerts no intrinsic rotational force. However, a resultant growth tensor may have both strain and rotational components. If all regions are specified to grow in a similar manner and no external forces act on the tissue, specified and resultant growth are the same. This is a conflict-free situation and no local rotations are generated (the resultant tensor has no rotational component). By contrast, if specified growth varies across a tissue it may create situations of potential conflict which are resolved or reduced through local rotation (i.e. the resultant growth tensor has a rotational component). Local rotations provide degrees of freedom not part of the specified growth which can allow potential stresses to be reduced. We refer to this mechanism for generating local tissue rotations as tissue conflict resolution.

In some cases, the potential conflicts in growth may be fully resolved through rotations, in which case the only difference between specified and resultant growth tensors is in the rotational component and there is no residual strain. For example, a conformal map is a planar deformation in which local rotations are generated through a gradient in isotropic specified growth without producing residual strain (Alim et al., 2016; Mitchison, 2016). If the tissue did not rotate locally to accommodate the gradient in specified growth, residual stresses would be generated. Thus, local rotations fully resolve the potential tissue conflict. In most cases, however, local rotations only partially resolve the conflicts and the specified and resultant growth tensors differ in the strain component (residual strain) as well as the rotational component. Some residual strain is inevitable in cases of tissue buckling (curvature through areal or directional conflicts), as the two surfaces of the tissue grow to a different extent, even though their specified strains are identical.

See Supplementary Material and methods for a full description of tissue-level modelling. All models used can be downloaded from:

http://cmpdartsvr1.cmp.uea.ac.uk/downloads/software/OpenSourceDownload_Elife_Rebocho_2016/GPT_TissueConflicts.zip

Models are based on the Growing Polarised Tissue framework (GPT-framework) (Green et al., 2010; Kennaway et al., 2011). Tissue, is considered as a continuous sheet, termed a canvas, with two surfaces (A and B). Identity and signalling factors can be specified throughout the canvas and interact through a gene regulatory network (GRN). A growth regulatory network (KRN) controls the specified growth parallel (K_par) and perpendicular (K_per) to the local polarity, established by taking the gradient of a diffusible factor POLARISER (POL). The production and degradation of POL depends on a polarity regulatory network (PRN). To visualise specified anisotropy, we calculate Kaniso = ln (K_par /K_per) and display its local value using a colour scale. If growth is isotropic (K_par = K_per), Kaniso = 0. Specified areal growth rate is given by K_area = K_par + K_per.

To illustrate how various types of tissue conflict resolution lead to curvature, we use an initial square with very slight curvature and marked with circular clones.