2.1. Spatial control of ROP GTPase during pavement cell morphogenesis

Rho guanosine triphosphate (GTP)ases are a family of membrane-bound, small signalling G-proteins that are known regulators of the cytoskeleton in eukaryotes. In plants, there is only one family of Rho GTPases referred to as ‘Rho of plants’ (ROP). Members of the ROP family have been identified as important signalling molecules in many cellular responses and developmental processes. They are known to do so often by interaction with members of the ‘ROP‐interactive CRIB motif‐containing’ (RIC) protein family (Fu et al., Reference Fu, Gu, Zheng, Wasteneys and Yang2005, Reference Fu, Xu, Zhu, Wen and Yang2009; Lin et al., Reference Lin, Ren and Fu2015; Xu et al., Reference Xu, Wen, Nagawa, Fu, Chen, Wu, Perrot-Rechenmann, Friml, Jones and Yang2010). In pavement cells, two specific ROP-RIC pairs (ROP2/ROP4-RIC4 and ROP6-RIC1) have been shown to respectively govern the formation of alternating protrusions and indentations (Figure 1; Fu et al., Reference Fu, Gu, Zheng, Wasteneys and Yang2005; Reference Fu, Xu, Zhu, Wen and Yang2009; Lin et al., Reference Lin, Ren and Fu2015; Xu et al., Reference Xu, Wen, Nagawa, Fu, Chen, Wu, Perrot-Rechenmann, Friml, Jones and Yang2010; Reference Xu, Nagawa and Yang2011). These ROPs are thought to be activated in alternating domains in the plasma membrane along the anticlinal face (i.e., perpendicular to the leaf surface). Each ROP then sets in motion a separate downstream signalling pathway through interaction with their respective RIC. ROP2 and ROP4 act redundantly and interact with and activate RIC4, which then induces the local assembly of F-actin at the plasma membrane. This was then proposed to promote/facilitate exocytosis, thereby enabling local cell outgrowth resulting in the formation of a protrusion. ROP6, on the other hand, activates RIC1,which then goes on to deploy the microtubule-severing protein KATANIN (KTN). KTN activity promotes the local assembly of microtubules, which results in the deposition of stiff cellulose microfibrils (CMFs) in similar regions of the anticlinal wall, leading to the formation of an indentation by locally suppressing outgrowth. These two pathways, ROP2/ROP4-RIC4 and ROP6-RIC1, have also been shown to laterally inhibit each other, reinforcing the alternated nature of their domains (Fu et al., Reference Fu, Gu, Zheng, Wasteneys and Yang2005; Reference Fu, Xu, Zhu, Wen and Yang2009; Lin et al., Reference Lin, Ren and Fu2015). ROP2-RIC4 signalling inhibits RIC1 activity preventing microtubule assembly in ROP2-governed domains. At the same time, the presence of microtubules in ROP6-governed regions inhibits RIC4 activity in those domains preventing F-actin assembly. In this way, the alternation of protrusions and indentations is reinforced and maintained. However, it is yet unknown how the microtubules cause inhibition of RIC4 activity, although it seems likely to involve some microtubule-associated proteins. Interestingly, Sugiyama et al. (Reference Sugiyama, Wakazaki, Toyooka, Fukuda and Oda2017) report that an IQ-domain protein (IQD13) fulfils such a role during xylem formation. During xylem formation, cells produce thick secondary cell walls except in the secondary wall pits, which form lateral connections between mature xylem vessels (Oda & Fukuda, Reference Oda and Fukuda2012a). The formation of these pits is governed by ROP11 which counteracts microtubule assembly in those domains through a pathway with MICROTUBULE DEPLETION DOMAIN1 (MIDD1) and KINESIN-13A (Oda & Fukuda, Reference Oda and Fukuda2012b, Reference Oda and Fukuda2013; Sugiyama et al., Reference Sugiyama, Wakazaki, Toyooka, Fukuda and Oda2017). Sugiyama et al. (Reference Sugiyama, Wakazaki, Toyooka, Fukuda and Oda2017) found that IQD13 interacts with both the plasma membrane and microtubules. It promotes microtubule rescue and inhibits ROP11 activity. In this way, IQD13 restricts ROP11 activity to its confined domains (Sugiyama et al., Reference Sugiyama, Wakazaki, Toyooka, Fukuda and Oda2017). Whether IQD13 or a related protein would fulfil such a role in pavement cell morphogenesis formation remains to be investigated.

Figure 1. Process of symmetry breaking in pavement cells. The illustration on the left-hand side highlights cell wall based modific ation and ROP signaling pathways that initiate the formation indentations (neck) and protrusions (lobe) in pavement cells.

ROPs can switch between an active GTP-bound form, present in the plasma membrane, and an inactive guanosine diphosphate (GDP)-bound form, detached from the plasma membrane based on the binding of its regulators. GUANINE NUCLEOTIDE EXCHANGE FACTORs (GEFs) facilitate the exchange of the GDP to GTP promoting the binding of the active ROPs to its downstream effectors (Hodge & Ridley, Reference Hodge and Ridley2016), whereas GTPase ACTIVATING PROTEINs (GAPs) boost the hydrolysis of the bound GTP rendering the ROPs inactive. In Arabidopsis, two groups of ROPGAPs are present: one that contains the Cdc42/Rac-interactive binding (CRIB) domain and another group that contains an amino-terminal pleckstrin homology, instead of the CRIB, domain called PHGAPs (Hwang et al., Reference Hwang, Vernoud, Szumlanski, Nielsen and Yang2008). Recently, PHGAPs were found to accumulate on the anticlinal face of the cell along indentations and inactivate ROP2, resulting in differential activation of ROPs necessary for the formation of the puzzle-piece morphology of pavement cells (Figure 1; Lauster et al., Reference Lauster, Stöckle, Gabor, Haller, Krieger, Lotz, Mayakrishnan, Späth, Zimmermann, Livanos and Müller2022).

2.2. Hormonal control of pavement cell morphogenesis

Phytohormones are also associated with pavement cell development via their regulation of ROPs. Brassinosteroids, a group of steroid hormones, have been shown to influence pavement cell shape by stabilising microtubules by a negative regulation of a brassinosteroid response GSK3-like kinase BR-INSENSITIVE2 (BIN2) (Liu et al., Reference Liu, Yang, Wang, Wang, Fu and Wang2018). More recently, BIN2-based phosphorylation of PHGAPs was shown to stabilise PHGAP presence in the indenting domains influencing ROP2 function (Figure 1; Zhang et al., Reference Zhang, Lauster, Tang, Houbaert, Zhu, Eeckhout, De Smet, De Jaeger, Jacobs, Xu, Müller and Russinova2022). Cytokinins, another group of plant hormones, were also shown to act upstream of the ROPs, with both signalling and biosynthetic mutants showing changes in pavement cell shape (Li et al., Reference Li, Xu, Lin, Wen, Xie, Duclercq, Bielach, Kim, Reddy, Zuo, Benková, Friml, Guo and Yang2013).

Auxin signalling is well studied and proposed to be involved in the formation of puzzle-piece shapes and over the past decade there has been progress in understanding the signalling mechanisms (Belteton et al., Reference Belteton, Sawchuk, Donohoe, Scarpella and Szymanski2018; Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020; Nagawa et al., Reference Nagawa, Xu, Lin, Dhonukshe, Zhang, Friml, Scheres, Fu and Yang2012; Pan et al., Reference Pan, Fang, Liu, Senay-Aras, Lin, Zheng, Zhang, Guo, Manor, Van Norman, Chen and Yang2020; Platre et al., Reference Platre, Bayle, Armengot, Bareille, Marquès-Bueno, Creff, Maneta-Peyret, Fiche, Nollmann, Miège, Moreau, Martinière and Jaillais2019; Xu et al., Reference Xu, Wen, Nagawa, Fu, Chen, Wu, Perrot-Rechenmann, Friml, Jones and Yang2010). It has been shown before that auxin triggers ROP activation, as is true for ROP2 and ROP6, and that this is mediated by a complex of AUXIN BINDING PROTEIN 1 (ABP1) and TRANSMEMBRANE KINASE 1 (TMK1, Lin et al., Reference Lin, Ren and Fu2015; Xu et al., Reference Xu, Wen, Nagawa, Fu, Chen, Wu, Perrot-Rechenmann, Friml, Jones and Yang2010; Reference Xu, Nagawa and Yang2011; Reference Xu, Dai, Chen, Nagawa, Cao, Li, Zhou, Chen, De Rycke, Rakusová, Wang, Jones, Friml, Patterson, Bleecker and Yang2014). In this complex, ABP1 acts as the extracellular auxin receptor where TMK1 transfers the signal to the cell’s interior. In fact, an earlier review by Lin et al. (Reference Lin, Ren and Fu2015) already suggested a model where this signalling complex is integrated with the ROP-RIC pathways to explain the molecular regulation of pavement cell lobe and neck formation. This model may now be expanded upon by introducing the process of nanoclustering. Several components already in the model have been observed to form auxin-induced nanoclusters at the plasma membrane. In one study, auxin presence resulted in reduced diffusion rates of TMK1 particles promoting the formation of nanoclusters in future indentation domains of pavement cells (Pan et al., Reference Pan, Fang, Liu, Senay-Aras, Lin, Zheng, Zhang, Guo, Manor, Van Norman, Chen and Yang2020). These TMK1 nanoclusters signal clustering and activation of ROP6 at the plasma membrane (the precise mechanism of which remains yet unclear) resulting in cortical microtubule changes (Figure 1; Pan et al., Reference Pan, Fang, Liu, Senay-Aras, Lin, Zheng, Zhang, Guo, Manor, Van Norman, Chen and Yang2020). The microtubules would then be involved in stabilising these TMK1 nanoclusters closing a positive feedback loop for formation of indented domains (Pan et al., Reference Pan, Fang, Liu, Senay-Aras, Lin, Zheng, Zhang, Guo, Manor, Van Norman, Chen and Yang2020). This is supported by an earlier study where auxin signalling triggers the nanoclustering of ROP6 on the plasma membrane of root cells. Here the process is shown to be mediated by the nanoclustering of phosphatidylserine (Platre et al., Reference Platre, Bayle, Armengot, Bareille, Marquès-Bueno, Creff, Maneta-Peyret, Fiche, Nollmann, Miège, Moreau, Martinière and Jaillais2019), indicating a possible role for specific plasma membrane components.

Further evidence for auxin involvement can be found when studying auxin transport. Indeed, many reports show auxin transporter proteins as a necessary component in pavement cell morphogenesis (reviewed in Liu et al., Reference Liu, Jobert, Rahneshan, Doyle and Robert2021). Specifically, PIN-FORMED (PIN) auxin transporters have been indicated in relation to the ROP signalling pathways (Nagawa et al., Reference Nagawa, Xu, Lin, Dhonukshe, Zhang, Friml, Scheres, Fu and Yang2012). In detail, ROP2-RIC4 signalling is shown to promote PIN localisation at the (future) protrusion regions through local inhibition of endocytosis (Nagawa et al., Reference Nagawa, Xu, Lin, Dhonukshe, Zhang, Friml, Scheres, Fu and Yang2012), which should increase auxin efflux at the tip of the protrusion. The extracellular auxin may then bind locally to ABP1 on the protrusion surface as well as on the indenting domains of the neighbouring cell. This may then trigger ROP2 signalling in these protrusions, creating positive feedback, and at the same time trigger ROP6 signalling in the indentations of the neighbouring cell. This model provides a mode of communication between neighbouring cells to coordinate the placement of their protrusions and indentations. However, in this case, it still seems to be dependent on the alternating ROP domains.

Several studies show a functional role of auxin during pavement cell development. However, automated detection of protrusions in pavement cells found no significant changes in their number or overall cell shape in a range of PIN mutants (Belteton et al., Reference Belteton, Sawchuk, Donohoe, Scarpella and Szymanski2018). Grones et al. (Reference Grones, Majda, Doyle, Van Damme and Robert2020) took advantage of a subpopulation of leaf epidermal cells known as the stomatal lineage ground cells (SLGCs) generated by stomatal meristemoids to re-investigate the role of auxin in formation of puzzle-shaped cells in a time-resolved fashion. SLGCs are generated by asymmetric cell division in a spiralling fashion around the meristemoid cell, together forming an anisocytic spiral complex. The SLGCs are initially polygonal and an ascending auxin response gradient influences the generation of indentations (Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020). In an early stage of anisocytic spiral complexes, these gradients were predominantly ascending from the centre outward, showing the lowest auxin response in the meristemoid cell and the highest in the oldest SLGC of the complex (Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020). The formation of these gradients was shown to be dependent on auxin transport, rather than auxin biosynthesis, and involves several different auxin transporters like PINs, AUX1, and ABCB (Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020). Mutations of these transporters lead to cell size and cell shape defects in pavement cells of true leaves but not in cotyledons (Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020). This implies that regulatory mechanisms of morphogenesis may differ between pavement cells of cotyledons and of true leaves (Grones et al., Reference Grones, Majda, Doyle, Van Damme and Robert2020).

3.1. The plant cell wall

The local mechanical properties of the cell wall are a major determinant of local cell growth and thus directly affect the overall morphology of the plant. The differential distribution of microtubules mentioned above has consequences in how the stiffest component of the cell wall, the CMFs, is deposited. Cellulose is synthesised directly at the plasma membrane by membrane-spanning cellulose synthase complexes (CSCs), which are transported to the plasma membrane via the Golgi and trans-Golgi network, where they are delivered in proximity to microtubules (McFarlane et al., Reference McFarlane, Döring and Persson2014). Within the plasma membrane, CSCs move at a constant speed. However, their direction of movement is governed by the direction of cortical microtubules to which the CSCs are coupled by CELLULOSE SYNTHASE INTERACTING 1 (CSI1/POM2) proteins (Bringmann et al., Reference Bringmann, Li, Sampathkumar, Kocabek, Hauser and Persson2012; Paredez et al., Reference Paredez, Somerville and Ehrhardt2006). Ongoing synthesis of new CMFs continuously extrudes stiff cellulose fibres into areas with older CMFs, causing mechanical tension to build up within them that is relieved by the displacement of CSCs within the plasma membrane along cortical microtubules (Diotallevi & Mulder, Reference Diotallevi and Mulder2007; Morgan et al., Reference Morgan, Strumillo and Zimmer2013). Partial disruption of microtubules leads to changes in CSC trajectories at the plasma membrane, and complete removal of microtubules by the drug oryzalin leads to the uniform movement of CSCs at the plasma membrane. Although cortical microtubules are required for the directional movement of CSCs at the plasma membrane, they do not affect CSCs motility or cellulose synthesis (Sugimoto et al., Reference Sugimoto, Himmelspach, Williamson and Wasteneys2003). Unlike cellulose, other cell wall matrix components are pre-synthesised in the Golgi apparatus, transported to the plasma membrane and extruded into the cell wall (Lerouxel et al., Reference Lerouxel, Cavalier, Liepman and Keegstra2006).

CMF–CMF interactions are widely present in the cell wall and in such regions, the presence of hemicellulose xyloglucan is also observed. It is proposed that the activity of expansion on those sites might mediate the remodelling of the nanostructure of the cell wall, thereby influencing the mechanics locally (Cosgrove, Reference Cosgrove2014).

3.2. Microtubule and cellulose-based models on pavement cell morphogenesis

Long-term kinematic imaging of microtubules in developing pavement cells combined with microtubule de-polymerisation experiments suggest that microtubules play an important role in shape emergence (Armour et al., Reference Armour, Barton, Law and Overall2015), whereas other independent studies suggest that microtubules are not consistently present at prospective sites of invaginations along the anticlinal walls (Belteton et al., Reference Belteton, Sawchuk, Donohoe, Scarpella and Szymanski2018). A two-step mechanism in which pectin-mediated changes to wall mechanics occur first and initiate the symmetry-breaking event followed by the presence of microtubules in such domains enhancing the morphology by guiding cellulose synthesis along similar directions (Altartouri et al., Reference Altartouri, Bidhendi, Tani, Suzuki, Conrad, Chebli, Liu, Karunakaran, Scarcelli and Geitmann2019). Studies until now proposed a growth restriction-based model of pavement cell morphogenesis, where anticlinal microtubules mediate growth restriction in the indentations and actin mediates growth promotion in the protrusions (Sapala et al., Reference Sapala, Runions, Routier-Kierzkowska, Das Gupta, Hong, Hofhuis, Verger, Mosca, Li, Hay, Hamant, Roeder, Tsiantis, Prusinkiewicz and Smith2018). However, these studies lacked detailed monitoring of growth behaviour at the subcellular scale. Tracking of fluorescently labelled plasmodesmata along the cell-to-cell interface of developing pavement cells disputes growth restriction-based models. The results of the study showed no differences in rates of expansion in subdomains of both indentations and protrusions measured based on the displacement of existing plasmodesmata. The authors of the same work propose that microtubules that transition from the periclinal to the anticlinal face (transfacial microtubules) could promote the initiation of indentations by regulating CMF deposition in such domains (Belteton et al., Reference Belteton, Li, Yanagisawa, Hatam, Quinn, Szymanski, Marley, Turner and Szymanski2021). Such patterns of CMF deposition could restrict expansion of cell height promoting expansion along the plane parallel to the leaf face facilitating morphogenesis.

3.3. Mechanical stress-based models on pavement cell morphogenesis

Atomic force microscopy-based measurements of the cell wall stiffness causally link the patterns of molecular effectors (microtubules) and differences in wall mechanics (Sampathkumar et al., Reference Sampathkumar, Krupinski, Wightman, Milani, Berquand, Boudaoud, Hamant, Jönsson and Meyerowitz2014). Regions of stable microtubule presence correlate with stiffer domains of the cell wall. Finite element model of these cellular geometric features in combination with turgor pressure shows increased magnitude and highly anisotropic mechanical tensile stresses (herein referred to as mechanical stresses) being present in the indenting regions (Bidhendi et al., Reference Bidhendi, Altartouri, Gosselin and Geitmann2019; Sampathkumar et al., Reference Sampathkumar, Krupinski, Wightman, Milani, Berquand, Boudaoud, Hamant, Jönsson and Meyerowitz2014; Sapala et al., Reference Sapala, Runions, Routier-Kierzkowska, Das Gupta, Hong, Hofhuis, Verger, Mosca, Li, Hay, Hamant, Roeder, Tsiantis, Prusinkiewicz and Smith2018). In addition, micromechanical manipulation of existing stresses results in concomitant changes to microtubule organisation along the newly predicted direction of stress (Sampathkumar et al., Reference Sampathkumar, Krupinski, Wightman, Milani, Berquand, Boudaoud, Hamant, Jönsson and Meyerowitz2014). These results demonstrate the existence of a mechanical feedback loop where cell geometry-driven subcellular stresses regulate microtubule organisation that promotes stiffening of indented regions resulting in the maintenance of cell shape (Chebli et al., Reference Chebli, Bidhendi, Kapoor and Geitmann2021; Jonsson et al., Reference Jonsson, Hamant and Bhalerao2022; Sampathkumar, Reference Sampathkumar2020; Trinh et al., Reference Trinh, Alonso-Serra, Asaoka, Colin, Cortes, Malivert, Takatani, Zhao, Traas, Trehin and Hamant2021). More recently, the transfacial alignment of the CMFs from the periclinal to the anticlinal wall was proposed to generate mechanical stresses hotspots that promote microtubule – cellulose-mediated modification to the mechanics of the cell wall that promotes the initiation of the puzzle-shaped morphology (Figure 1; Belteton et al., Reference Belteton, Li, Yanagisawa, Hatam, Quinn, Szymanski, Marley, Turner and Szymanski2021; Jonsson et al., Reference Jonsson, Hamant and Bhalerao2022). Such complex morphological features are shown to be necessary to reduce mechanical stresses in cells that attain large volumes while growing in an isotropic fashion (Sapala et al., Reference Sapala, Runions, Routier-Kierzkowska, Das Gupta, Hong, Hofhuis, Verger, Mosca, Li, Hay, Hamant, Roeder, Tsiantis, Prusinkiewicz and Smith2018).

3.4. Pectin-based models on pavement cell morphogenesis

Apart from cellulose and hemicellulose, pectin is an important cross-linking component of the cell wall that influences its stiffness. The major component of pectin is homogalacturonan and the esterification status of which determines the cell wall stiffness. This is regulated by the activity of PECTIN METHYLESTERASE and its inhibitor PECTIN METHYLESTERASE INHIBITOR. Finite element models suggest that initiation of indentations occurs due to mechanical heterogeneities present along the anticlinal wall between two adjacent cells (Figure 1; Majda et al., Reference Majda, Grones, Sintorn, Vain, Milani, Krupinski, Zagórska-Marek, Viotti, Jönsson, Mellerowicz, Hamant and Robert2017). Such mechanical heterogeneities are proposed to be created by the polar distribution of demethylated pectin and other cell wall components such as galactans and arabinans based on immunolabelling experiments. Presence of both esterified and non-esterified pectin was detected at the interface of the periclinal and anticlinal walls of protrusions in some angiosperm and fern species (Sotiriou et al., Reference Sotiriou, Giannoutsou, Panteris, Galatis and Apostolakos2018). This potentially indicates a high amount of turnover and remodelling of pectin in the protruding domains that could contribute to shape changes. Super resolution-based imaging indicates that pectin exists as nanofilaments along the anticlinal wall in pavement cells (Haas et al., Reference Haas, Wightman, Meyerowitz and Peaucelle2020). Further, an increase in pectin nanofilament width and spacing occurs due to the demethylation of the pectin by PMEs. The same study used finite element models to propose that the indentations of straight anticlinal walls occur due to changes in the ultra-structure of the pectin nanofilaments independent of turgor pressure or CMF-based reinforcements. However, such nanofilaments are absent in the periclinal wall (i.e., parallel to the cell surface) and both models (Haas et al., Reference Haas, Wightman, Meyerowitz and Peaucelle2020; Majda et al., Reference Majda, Grones, Sintorn, Vain, Milani, Krupinski, Zagórska-Marek, Viotti, Jönsson, Mellerowicz, Hamant and Robert2017) fail to consider the mechanics of the periclinal wall. It was also shown that the addition of the periclinal cell wall eliminated minor cell wall bending occurring along the anticlinal cell wall (Bidhendi & Geitmann, Reference Bidhendi and Geitmann2019). The presence of demethylated pectin correlated with stiff domains of the periclinal cell wall at prospective sites of indentation further highlighting the importance of periclinal wall mechanics (Figure 1; Bidhendi et al., Reference Bidhendi, Altartouri, Gosselin and Geitmann2019). The study also proposes that pectin-mediated shape changes generate stress differentials that attract microtubules to such sites further enhancing the morphological features (Bidhendi et al., Reference Bidhendi, Altartouri, Gosselin and Geitmann2019). However, it is still unknown what facilitates the heterogeneous presence/activity of the pectin modifying enzymes.

5.1. PaCeQuant: Curvature-based method (Möller et al., Reference Möller, Poeschl, Plötner and Bürstenbinder2017)

PaCeQuant is an ImageJ-based plugin that measures many geometric and biologically interpretable parameters of projected Z-stacks (images). These include (a) cell features such as area, perimeter and circularity, (b) contour-based features such as local curvature on which indentation and protrusion detection is based and (c) skeleton-based features such as longest path and number of branches, and finally cell-specific features such as the number of lobes, lobe length and lobe width. The tool includes the option to perform segmentation, which allows the user to start directly with confocal images. Due to the computationally efficient quantification of many features, large datasets can be analysed relatively quickly with this tool. The drawback is that each cell is analysed on its own, and thus, no reference can be made between neighbouring cells and intracellular entities (e.g., microtubules or other proteins). Furthermore, the tool can distinguish between true lobes versus lobes associated with tri-cellular junctions, but their position is not output directly.

5.2. Contour analysis: A curvature-based method (Eng et al., Reference Eng, Schneider, Matz, Carter, Ehrhardt, Jönsson, Nikoloski and Sampathkumar2021; Schneider et al., Reference Schneider, Ehrhardt, Meyerowitz and Sampathkumar2022)

This is a MATLAB-based tool that uses 2.5D projections of Z-stacks (images). However, multiple colour channels can be used in parallel, which additionally contain information about fluorescently labelled proteins. The tool performs watershed segmentation to first extract digitised cell outlines. It is one of the few tools that then correlate the abundance of fluorescently labelled proteins near the inner cell boundary with the local curvature. Curvature is determined by fitting a circle to the local cell boundary. Local curvature and mechanical strain are closely related, as confirmed by Eng et al. (Reference Eng, Schneider, Matz, Carter, Ehrhardt, Jönsson, Nikoloski and Sampathkumar2021). Therefore, it is worthwhile to use a modified, local version of the LEC (localLEC), which determines for each point along the contour the largest possible circle that just fits inside the cell at that location (Figure 3e,f). Also incorporated into this tool is the FibrilTool algorithm (Boudaoud et al., Reference Boudaoud, Burian, Borowska-Wykręt, Uyttewaal, Wrzalik, Kwiatkowska and Hamant2014) that extracts anisotropy and orientation of fibrillar structures both locally in relation to curvature and globally over the entire cell surface. The contour analysis tool can be downloaded from GitHub (https://github.com/DrReneSchneider/Matlab-Contour-Analysis). An optional 2.5D projection method developed by the same team of researchers that applies a smooth manifold algorithm to 3D Z-stacks to extract fluorescent signals from the cell surface can be downloaded also from GitHub (https://github.com/DrReneSchneider/Smooth-Manifold-Projection-Tool) and used as a preprocessing step prior to contour analysis. As with the other techniques, this tool does not directly provide the positions of lobes, necks and tri-cellular junctions. As the cell contour is analysed in terms of curvature, a clear distinction between lobes and tri-cellular junctions is problematic. However, the ability to sort out cell walls adjacent to stomata is useful.

5.3. LobeFinder: Envelope-based method (Wu et al., Reference Wu, Belteton, Pack, Szymanski and Umulis2016)

LobeFinder is a special application that aims to measure the complexity of cells based on the number of indentations and protrusions. It uses projected Z-stacks (images) and requires manual segmentation of cell outlines. The method is based on the determination of a convex envelope (hull) for 2D cell outlines (Figure 3). Since the envelope runs along protrusions but cannot follow the indentations, the distance between the envelope to the actual cell boundary can be used as a position indicator for these features. Although this method has been used in time-lapse imaging, it is not particularly sensitive to dynamic changes in the cell contour. Thus, an emerging curvature is likely to be overlooked due to the initially small distance to the envelope and detected only when the change is already substantial. Nevertheless, the method is a robust tool to describe the emergence/increase of complexity during growth and between mutants. Again, the problem arises that these features (indentations and protrusions) are analysed individually for each cell and no correlation can be established either to the neighbouring cells or to the cellular content (e.g., microtubules). The possibility of misinterpretation is therefore considerable.

5.4. GraVis: A network-based method (Nowak et al., Reference Nowak, Eng, Matz, Waack, Persson, Sampathkumar and Nikoloski2021)

This method employs ideas stemming from network theory, namely visibility graphs and closeness centrality to extract, quantify and compare shape descriptors. A visibility graph is a map of a network of ‘visible’ locations, for example, points along the cell contour. Each node of the network represents a point on the contour, and each edge represents a ‘line of sight’ between them. By using closeness centrality, GraVis can reliably determine whether a node along the contour is a protrusion (i.e., seen by only a few other contour points) or an indentation (i.e., seen by many other contour points). GraVis can determine these features similarly well as PaCeQuant, but only if both methods are manually parameter optimised. With default settings, GraVis performs much closer to the ground truth generated for GraVis and PaCeQuant with a group of >20 and >4 individuals, respectively. By comparing the position of lobes with the position of tri-cellular junctions, GraVis is one of the few methods that use information from neighbouring cells to obtain a better prediction accuracy of cellular shape features. However, relating shape features to intracellular entities is also lacking in this method. GraVis comes as an open-source R package and can be downloaded from GitHub as an executable file for all major operating systems (https://github.com/jnowak90/GraVisGUI).

5.5. LOCO-EFA: Elliptical Fourier analysis-based method (Sánchez-Corrales et al., Reference Sánchez-Corrales, Hartley, van Rooij, Marée and Grieneisen2018)

Elliptic Fourier analysis (EFA) decomposes segmented cell outlines into a series of ellipses, called modes or harmonics. This is comparable to conventional Fourier analysis, which decomposes a linear signal into its frequency components. In EFA, however, the deviation of the cell outline from an ellipse is used as the ‘signal’ and the complexity of this original signal is iteratively replaced by ellipses of increasing order that capture progressively finer details of the cell outline. The linear combination of all modes then allows the recovery of the original cell outline. The problem with this method as pointed out by Sánchez-Corrales et al. (Reference Sánchez-Corrales, Hartley, van Rooij, Marée and Grieneisen2018) that the EFA coefficients are redundant (there are more parameters than necessary to reproduce the outline) which can render the comparison of cell shapes using principal component analyses problematic. An advancement of this method termed ‘lobe contribution’ EFA (LOCO-EFA) decomposes each harmonic into a clockwise and a counterclockwise rotating signal, thereby generating a unique set of parameters. This approach thus provides unambiguous measurements of cell shape, often in the form of a spectrum of directional components that make up the complex cell boundary. This method can be used to quantify the increase in complexity of cell shapes during growth. The disadvantage is that each cell is analysed on its own and thus no reference can be made between neighbouring cells and cellular contents (e.g., microtubules). Also, the number and location of indentations and protrusions are not an output of this method. LOCO-EFA is delivered as open-source C code.