Martin Janda

Séverine Planchais

Nabila Djafi

Jan Martinec

Lenka Burketová

Olga Valentová

Alain Zachowski

Eric Ruelland
email: eric.ruelland@upmc.fr

Severine Planchais

Lenka Burketova

Olga Valentova

Á J Martinec

Á L Burketova

Á O Valentova

S Planchais

Á N Djafi

Á A Zachowski

Á E Ruelland

Phosphoglycerolipids are master players in plant hormone signal transduction

Keywords: Phospholipase, Phosphoglycerolipids, Lipid kinase, Abscisic acid, Salicylic acid, Auxin, Signalling, Phosphatidic

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Importance of hormones for plant physiology and development

Hormones are molecules produced by plant cells in response to environmental stresses and during plant development. The chemical natures of plant hormones are diverse. Some are related to adenine (cytokinins), others are terpenoids (gibberellins; abscisic acid, ABA) or polyhydroxysteroids (brassinosteroids) or are related to phenol derivatives (salicylic acid, SA) or derived from oxidised fatty acid (jasmonate). Some peptide hormones have also been described. Some plant hormones are volatile such as ethylene and some jasmonates. All hormones have in common that they act at extremely low concentrations. The first step of their action initiates when they are recognised by a receptor, thus triggering intracellular transduction and hormone signal transducers such as protein phosphatases or NADPH oxidases. Finally, we will discuss how the pathways can be activated or inhibited downstream of hormone perception. This review will thus propose mediating lipids to be master players of cell signalling, modulating, if not controlling, major transducing steps of hormone signals.

Overview of phosphoglycerolipid metabolic pathways

Phosphoglycerolipids are built from a glycerol backbone that is acylated by fatty acids on hydroxyls at positions sn-1 and sn-2. The hydroxyl group in the sn-3 position is esterified by phosphoric acid which is further esterified by another alcohol such as ethanolamine, serine, choline or myo-inositol (Fig. 1a). Signalling phosphoglycerolipids are generated through the action of phospholipases or lipid kinases (Fig. 1b). Structural phosphoglycerolipids such as phosphatidylcholine (PC), phosphatidylglycerol (PG) or phosphatidylethanolamine (PE) can be hydrolysed by phospholipase D (PLD, EC 3.1.4.4) into phosphatidic acid (PA). PA can also be formed in the reaction catalysed by diacylglycerol kinase (DGK,EC 2.7.1.107). The diacylglycerol (DAG) substrates of DGK can be generated by the action of phospholipase C enzymes (PLCs) of which there are two kinds. Phosphoinositide-specific PLC (PI-PLC, EC 3.1.4.11) hydrolyses phosphatidylinositol-4,5-bisphosphate (PI-4,5-P 2 ) into DAG and inositol-1,4,5-triphosphate (IP3). IP3 can be phosphorylated into higher phosphorylated forms of soluble inositol by inositol kinases (IPK). Phosphatidylinositol-4-phosphate (PI4P) may be an in vivo substrate of PI-PLC in plants as it is known to be an in vitro substrate and PI4P is 10-to 100-fold more abundant than PI-4,5-P 2 in plants [START_REF] Delage | Signal transduction pathways involving phosphatidylinositol 4-phosphate and phosphatidylinositol 4,5-bisphosphate: convergences and divergences among eukaryotic kingdoms[END_REF]. The so-called nonspecific PLC (NPC,EC 3.1.4.3) hydrolyses structural phosphoglycerolipids into the corresponding phosphoalcohol and DAG [START_REF] Pokotylo | The plant non-specific phospholipase C gene family. Novel competitors in lipid signalling[END_REF]. Phospholipase A (PLA) catalyses the hydrolysis of ester bonds at the sn-1 and/or sn-2 positions of phosphoglycerolipids, thus liberating a free fatty acid (FFA) and forming a lysophosphoglycerolipid. PLAs may act specifically at the sn-1 or the sn-2 positions, and are named PLA1 (EC 3.1.1.32) or PLA2 (EC 3.1.1.4) accordingly [START_REF] Ryu | Phospholipid-derived signaling mediated by phospholipase A in plants[END_REF][START_REF] Scherer | Patatinrelated phospholipase A: nomenclature, subfamilies and functions in plants[END_REF]). PI4P and PI-4,5-P 2 result from the sequential phosphorylations of phosphatidylinositol (PI) catalysed by PI-4kinases (PI4K, EC 2.7.1.67) and EC 2.7.1.68). PI can also be phosphorylated into phosphatidylinositol 3-phosphate (PI3P) by a PI-3-kinase (PI3K, EC 2.7.1.137). PI3P can then be phosphorylated into PI(3,5)P 2 [START_REF] Delage | Signal transduction pathways involving phosphatidylinositol 4-phosphate and phosphatidylinositol 4,5-bisphosphate: convergences and divergences among eukaryotic kingdoms[END_REF]. Finally, PA can be phosphorylated into diacylglycerol pyrophosphate (DGPP) by PA kinase (PAK) or dephosphorylated by phosphatidate phosphatase (PAP,EC. 3.1.3.4) or lipid phosphate Plant Cell Rep amplification of the signal. This will ultimately lead to cellular and organ responses.

The importance of hormone in plant physiology can be briefly illustrated by a few examples. Pathogen attacks induce an increase in SA that then triggers a response leading to systemic acquired resistance. ABA is synthesised during drought stress and promotes stomatal closure and inhibits stomatal opening, thereby reducing transpiration and water loss. ABA therefore has a major role in protection against dehydration. In development, auxins, for instance, are well known to stimulate cell elongation, stimulate differentiation of phloem and xylem, mediate the tropic response of bending in response to gravity and light, to stimulate lateral root development and delay leaf senescence [START_REF] Taiz | Plant physiology[END_REF].

These few examples clearly illustrate that hormones intervene at many levels during the plant's lifetime. As a consequence, studying hormone signal transduction has always been considered primordial in plant biology. Hormone production, signalling and responses can affect vegetative growth, plant health and development of flowers, fruits and seeds, all aspects that are of high importance for agriculture.

For all hormone signalling pathways, major advances were achieved using genetics, mainly but not exclusively with Arabidopsis, where the focus is on the regulation of gene expression, the protein/protein interactions, the protein post-translational modifications and, more recently, the role of noncoding RNAs. For instance, now classical studies of promoter analysis allowed the identification of ABA-responsive element motifs (ABRE; [START_REF] Marcotte | Abscisic acidresponsive sequences from the Em gene of wheat[END_REF]; the first step is to identify ABRE binding factors and their regulators. The isolation of ABA-insensitive mutants made it possible to show the importance of protein phosphorylation status in ABA responses [START_REF] Leung | Arabidopsis Aba response gene Abi1features of a calcium-modulated protein phosphatase[END_REF]. Those major findings and the ease of molecular biology tools using protein encoding genes eclipsed the fact that lipids were found early on to be signal mediators in hormone transduction [START_REF] Scherer | A rapid response to a plant hormone: auxin stimulates phospholipase A2 in vivo and in vitro[END_REF]. Recent studies have nevertheless put the production and involvement of lipids in hormone signalling back in the spotlight. Nevertheless, knowledge on lipids needs to be better integrated into some of the genetically defined hormone pathways.

We attempt this here by reviewing the current knowledge of the role of phosphoglycerolipids in hormone signalling. For each hormone signalling cascade known to require a phosphoglycerolipid pathway, we will describe which specific lipid signalling pathways are activated or inhibited. Ideas on how these pathways, and the molecules they generate, act and interact will be discussed. In doing so, lipid mediators, and in particular phosphatidic acid, will be shown to be involved in controlling well-established phosphatases (LPP,EC 3.1.3.76). DGPP can also be a substrate of LPPs [START_REF] Pleskot | Turnover of phosphatidic acid through distinct signaling pathways affects multiple aspects of pollen tube growth in tobacco[END_REF]. Phosphates in the inositol group of either soluble polyphosphate-inositol or in lipid phosphoinositides can be hydrolysed by myo-inositol polyphosphate phosphatases (EC 3.1.3.57) and PI phosphatases (EC 3.1.3.64/66) that are more or less specific for the soluble or lipid inositol forms [START_REF] Williams | Mutations in the Arabidopsis phosphoinositide phosphatase gene SAC9 lead to overaccumulation of PtdIns(4,5)P2 and constitutive expression of the stress-response pathway[END_REF].

Gene expression in response to hormones

A great deal of data implicating phosphoglycerolipids in hormone signal transduction is coming from transcriptome analysis [START_REF] Lin | DNA chip-based expression profile analysis indicates involvement of the phosphatidylinositol signaling pathway in multiple plant responses to hormone and abiotic treatments[END_REF]). If a gene encoding an enzyme of the phosphoglycerolipid signalling pathways is upregulated in response to a hormone, it could mean that the encoded protein has a role in transducing the hormone signal. We analysed the relative levels of expression of genes encoding enzymes involved in phosphoglycerolipid pathways in Arabidopsis thaliana using Genevestigator [START_REF] Hruz | Genevestigator v3: a reference expression database for the meta-analysis of transcriptomes[END_REF]. We chose experiments in which RNA was sampled no later than 4 h after hormone treatment (Fig. 2). The transcript level of most genes encoding enzymes involved in the phosphoglycerolipid pathway changed in response to hormone. Within individual families of enzymes, some isoforms are induced and some repressed by the same hormone treatment, which suggests that different isoforms have different roles. For instance, PLDbs, c1, c3, f1 and a1 are repressed by ABA. They are also repressed by the cytokinin zeatin, but induced by SA. The fact that ABA and zeatin have the same effect, which is opposite to that of SA, cannot be generalised though. PLDd and f2 are induced by ABA but are repressed by zeatin. It is striking that transcript levels of some PLA1 and plastid LPP genes are very much controlled by hormones, while their role in hormone transduction in plants is as yet poorly documented (Fig. 2).

Abscisic acid

The most documented description of phosphoglycerolipids in hormone signal transduction is for ABA. In Arabidopsis thaliana suspension cells, ABA plasmalemma perception results in a transient stimulation of phospholipase D (PLD) activity that is necessary for the activation of anion channels in the pathway leading to RAB18 expression [START_REF] Hallouin | Plasmalemma abscisic acid perception leads to RAB18 expression via phospholipase D activation in Arabidopsis suspension cells[END_REF]. This PLD activity leads to a transient increase in PA that is followed by an increase in DGPP, showing that a PAK is active. While exogenous PA activates -2.5 -2 -1.5 -1 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

Fig

Involvement of phosphoglycerolipids in hormone signalling

While transcriptome studies can indicate possible involvement, it is important to show that enzymes of these pathways are actually active and/or activated or inhibited in response to hormones. Mostly in vivo lipid labelling has been used for which suspension cells are the most amenable and relevant models. The availability of mutant plants is helpful to further narrow down the exact role of the in vivo activities measured.

plasmalemma anion currents, it does not trigger the expression of RAB18 or RD29A. On the contrary, exogenous DGPP mimics ABA in stimulating similar changes in both anion currents and gene expression. This suggests that DGPP could be the lipid mediator resulting from ABA activation of the PLD pathway in Arabidopsis suspension cells [START_REF] Zalejski | Diacylglycerol pyrophosphate is a second messenger of abscisic acid signaling in Arabidopsis thaliana suspension cells[END_REF][START_REF] Zalejski | Induction of abscisic acid-regulated gene expression by diacylglycerol pyrophosphate involves Ca 2? and anion currents in Arabidopsis suspension cells[END_REF]. Guard cells generate reactive oxygen species (ROS) in response to ABA, which triggers stomatal closure. Inhibiting the production of PI3P or PI4P, or sequestrating them by the expression of PI3P-binding or PI4P-binding domains, inhibits ABA-induced stomatal closure [START_REF] Park | A role for phosphatidylinositol 3-phosphate in abscisic acid-induced reactive oxygen species generation in guard cells[END_REF]. The role of PI4P could be linked to its role as a precursor of substrates for PI-PLC. Indeed, endogenous IP3 levels were found to increase within 2 min of ABA application to stomata [START_REF] Lee | Abscisic acid-induced phosphoinositide turnover in guard cell protoplasts of Vicia faba[END_REF]. The induction of some ABAresponsive genes was reduced in transgenic lines expressing an antisense AtPLC1 [START_REF] Sanchez | Arabidopsis PLC1 is required for secondary responses to abscisic acid signals[END_REF]. The signalling molecules generated by the PI-PLC pathway activated in stomata in response to ABA are probably soluble phosphorylated inositols (polyphosphoinositols) as microinjection of IP3 into stomata initiates their closure [START_REF] Gilroy | Elevation of cytoplasmic calcium by caged calcium or caged inositol triphosphate initiates stomatal closure[END_REF]). Reducing the level of polyphosphoinositols by overexpressing inositol polyphosphate phosphatases decreases the induction of some ABA-responsive genes [START_REF] Sanchez | Arabidopsis PLC1 is required for secondary responses to abscisic acid signals[END_REF]. By contrast, Arabidopsis fiery1 (fry1), in which an inositol polyphosphate 1-phosphatase is mutated, accumulates more IP3 than the wild type upon ABA treatment with stronger induction of ABA-responsive genes [START_REF] Xiong | FIERY1 encoding an inositol polyphosphate 1-phosphatase is a negative regulator of abscisic acid and stress signaling in Arabidopsis[END_REF].

PLDs are also involved in ABA signal transduction in guard cells. PLDs can use primary alcohols as substrates to form a phosphatidylalcohol (in the so-called transphosphatidylation reaction) to the detriment of PA production. Secondary or tertiary alcohols are not substrates of the PLD-catalysed reaction. n-Butanol (n-ButOH) inhibits ABA-induced stomatal closure, pointing to a role for PLDproduced PA in this process. Similarly, in an Arabidopsis plda1 pldd double mutant, ABA-induced stomatal closure was suppressed. Accumulation of ROS and nitric oxide (NO), alkalisation of the cytosol, and inhibition of inwardrectifying K ? channel currents that occur in wild-type ABA-treated guard cells are all inhibited in this double mutant. However, the calcium response to ABA is not altered in double-mutant guard cells [START_REF] Uraji | Cooperative function of PLD delta and PLD alpha 1 in abscisic acid-induced stomatal closure in Arabidopsis[END_REF]). While the above results suggest PLDs are upstream of the NO response, PLDd could also be downstream of NO as pldd mutants fail to close their stomata in response to NO [START_REF] Diste ´fano | Phospholipase Dd is involved in nitric oxide-induced stomatal closure[END_REF]. The mode of action of the PA produced by PLDa1 in response to ABA is detailed in ''Phosphatidic acid targets''.

PA and DGPP levels increase when seeds are treated with ABA. In Arabidopsis, LPP-coding genes AtLPP2 and AtLPP3 are expressed during seed germination. AtLPP2 insertion mutants are hypersensitive to ABA in terms of germination inhibition. This is correlated with greater accumulation of PA during mutant germination compared to WT, showing that PA relays the ABA effect on germination. Similarly, in rice, the expression of a-amylase, a key step in germination, is inhibited when seeds are treated with PA. Double-mutant analysis in Arabidopsis showed that ABA-insensitive 4 (ABI4) is epistatic to AtLPP2. PA thus appears to be upstream of the activity of ABI4, an AP2-type transcription factor. The origin of PA involved in ABA inhibition of germination might be due to PLD activities. In Oryza sativa, mutation of PLDb1 inhibits the repression of germination by ABA and leads to a reduction in the repression of a-amylase by ABA (Li et al. 2007). PI-PLC could also participate in ABA inhibition of germination. Germination of seeds overexpressing the AtPLC1 gene is not inhibited by ABA [START_REF] Sanchez | Arabidopsis PLC1 is required for secondary responses to abscisic acid signals[END_REF].

Here again the active molecule generated by PI-PLC in seeds might be polyphosphoinositol. Seeds with a mutated inositol polyphosphate phosphatase gene, which have increased levels of IP3, have an enhanced sensitivity to ABA [START_REF] Gunesekera | Inositol polyphosphate 5-phosphatases 1 and 2 are required for regulating seedling growth[END_REF]).

Salicylic acid

In Arabidopsis suspension cells, we showed that SA stimulation led to the rapid activation of PLD. In the presence of primary alcohols that quench PA formation by PLDs, transcriptomic changes due to the hormone were strongly affected [START_REF] Krinke | Phospholipase D activation is an early component of the salicylic acid signaling pathway in Arabidopsis cell suspensions[END_REF]. Interestingly, in the same model, SA also activated a type-III PI4K, leading to the formation of PI4P and PI-4,5-P 2 . There was considerable overlap between PLD-controlled genes and PI4Kcontrolled genes. Amongst the genes controlled similarly by both pathways were WRKY38 and PR1. This led us to hypothesise that SA activates PI4K, increasing PI4P and PI-4,5-P 2 , with the latter lipid activating some PLD isoforms. However, NPR1 induction by SA was abolished by including n-ButOH but not by inhibiting type-III PI4K, suggesting that in response to SA, both PI-4,5-P 2 -dependent PLDs and PI-4,5-P 2 -independent PLDs are brought into play (Krinke et al. 2007a[START_REF] Krinke | Phospholipase D activation is an early component of the salicylic acid signaling pathway in Arabidopsis cell suspensions[END_REF]. Identifying the PLDs activated in response to SA will help in validating this hypothesis. In Arabidopsis, the PI-4,5-P 2 -dependent PLDs, PLDb2 and PLDcs, are induced by SA (see ''Gene expression in response to hormones''), which might point to them having a role in SA transduction.

Do the pathways involved in the suspension cell SA response have the same role in plants? Possibly. The addition of primary alcohols blocks PR1 expression in response to SA in Arabidopsis plantlets (Janda et al., unpublished). PI-PLC could also have a role in SA signal transduction in plants. PR1 is overexpressed in Arabidopsis plants expressing a mammalian inositol polyphosphate 5-phosphatase in control conditions [START_REF] Perera | Transgenic Arabidopsis plants expressing the type 1 inositol 5-phosphatase exhibit increased drought tolerance and altered abscisic acid signaling[END_REF]. [START_REF] Lee | Phospholipase A(2) is required for PIN-FORMED protein trafficking to the plasma membrane in the Arabidopsis root[END_REF]). It has also been suggested that an At-PLDf2 mutant was affected in PIN cycling (Li and Xue 2007). Finally, auxin synthesis might be controlled through PI-PLC. Myo-inositol and indole-3-acetonitrile, a precursor of the auxin indole-3-acetic acid, were up-regulated in the rotunda1-1 (ron1-1) mutant. RON1 gene is identical to FRY1, mentioned in ''Abscisic acid'', which encodes an enzyme with inositol polyphosphate 1-phosphatase activity [START_REF] Robles | The RON1/FRY1/SAL1 gene is required for leaf morphogenesis and venation patterning in Arabidopsis[END_REF].

Which molecules act as mediators

Phosphoglycerolipid pathways generate molecules that can act as mediators by binding and thus altering the activity of effectors, or targets. Altering may imply activating or inhibiting the target and also sequestering it in a membrane, thus preventing the effector from reaching its site of action. Identifying these targets is a field of research in full swing. Although some targets have been identified, their roles in hormone signal transduction may not necessarily have been described. We will summarise current knowledge of the targets of molecules generated by phosphoglycerolipid pathways, always bearing in mind how these pathways may act in hormone signal transduction and how new actors may be identified.

Phosphatidic acid targets

PA is the common product of PLC/DGK and PLD pathways. Besides its involvement in hormone signalling as reviewed above, it has been shown to be involved in response to various abiotic and biotic stresses. Some PA binding proteins have been found in plants [START_REF] Dubots | Role of phosphatidic acid in plant galactolipid synthesis[END_REF][START_REF] Wang | Signaling functions of phosphatidic acid[END_REF]. Table 1 lists a selection of plant proteins that bind PA and also have a role in signalling. The list is not exhaustive as protein kinases, protein phosphatases, lipid kinases and lipid phosphatases also bind PA in a way that modifies their activity.

The role of some of these proteins has been well described in studies of ABA responses. PLDa1 produces PA that binds to the ABI1 protein phosphatase 2C, a negative regulator of ABA action. This interaction inhibits the function of ABI1 by inhibiting its phosphatase activity and tethering it to the plasma membrane. PA formation by PLDa1 thus alleviates the inhibiting effect of ABI1 and allows gene expression through ATHB6. PA produced by PLDa1 has also been shown to interact with and stimulate NADPH oxidases. Interestingly, the interaction between PA and ABI1 is not required for the ABA-induced production of ROS and NO but is important for mediating the ROS effect on stomatal closure [START_REF] Zhang | Phospholipase dalpha1 and phosphatidic acid regulate NADPH oxidase activity and production of reactive oxygen species in ABA-mediated stomatal closure in Arabidopsis[END_REF]. Finally another PA binding protein related to ABA signal Plant Cell Rep It is also important to understand how SA activates PI4K. In in vitro enzymatic assays of microsomes extracted from Capsicum chinense Jacq. suspension cells 30 min after SA treatment, it was found that SA provoked an increase in lipid kinase activities leading to PI4P and PI-4,5-P 2 and a decrease in PI-PLC activity [START_REF] Altuzar-Molina | Phospholipidic signaling and vanillin production in response to salicylic acid and methyl jasmonate in Capsicum chinense J. cells[END_REF]. This fits well with what we observed in vivo in Arabidopsis suspension cells. The very short period in which these in vitro changes occur after SA treatment most likely implies that PI4K and/or PI-PLC are subject to posttranslational modifications, which will be confirmed by further investigation.

Auxins

As early as 1989, Scherer and Andre ´ showed that when auxins were added to soybean suspension cells prelabelled with radioactive ethanolamine or choline, an increase in radioactive lysoPE or lysoPC occurred within 5 min. This was a strong indication that a PLA was activated. Interestingly, some PLA2 inhibitors were found to inhibit the growth of etiolated Arabidopsis hypocotyls, an auxin-driven mechanism [START_REF] Holk | A cell wall protein downregulated by auxin suppresses cell expansion in Daucus carota (L.)[END_REF], as well as the auxininduced activation of the promoter of the DR5 gene [START_REF] Scherer | A role for phospholipase A in auxin-regulated gene expression[END_REF]. Plant PLA2 enzymes are encoded either by patatin-related PLA (pPLA) genes or by secreted PLA2 (sPLA2) genes. All single Arabidopsis pPLA mutants show delays in the up-regulation of early auxininduced genes compared to WT. Thus, ppla mutants are considered to be auxin-signalling mutants [START_REF] Labusch | Changes in pattern and timing of expression for early auxin-induced genes in phospholipase A mutants imply their involvement in the regulation of auxin responses[END_REF]. Overexpression of AtsPLA2b promotes cell elongation, whereas silencing of AtsPLA2b expression retards it. AtsPLA2b is also rapidly induced by auxin treatments. To explain the elongation phenotypes, it is proposed that AtsPLA2b has a role in transducing auxin [START_REF] Lee | Secretory low molecular weight phospholipase A2 plays important roles in cell elongation and shoot gravitropism in Arabidopsis[END_REF]. PLAs are not the only phospholipases that have been shown to be involved in passing on the auxin signal. In cucumber, a PLD was activated, and PI4P and PI-4,5-P 2 also accumulated within 1 min of auxin treatment. Interestingly, the auxin-induced accumulation of these signalling lipids is dependent on auxin-induced NO production [START_REF] Lanteri | Nitric oxide triggers phosphatidic acid accumulation via phospholipase D during auxin-induced adventitious root formation in cucumber[END_REF].

When considering auxin signalling, it is important to bear in mind that auxin has an asymmetrical distribution. This asymmetry is achieved through polar transport of auxin by influx or efflux auxin carriers. PINs are the best-characterised proteins controlling auxin efflux. Auxin distribution is altered in an inositol polyphosphate 5-phosphatase 13 mutant [START_REF] Wang | The role of Arabidopsis 5PTase13 in root gravitropism through modulation of vesicle trafficking[END_REF], whereas PIN distribution is affected in AtPLA 2 a and AtPIP5K2 mutants transduction is sphingosine kinase (SPHK). The interplay between PLD and SPHK pathways could be complex. PA is involved in the activation of SPHK, but activation of PLDa1 requires SPHK activity, suggesting that SPHK and PLDa1 are co-dependent in amplification of the ABA response-mediating stomatal closure in Arabidopsis [START_REF] Guo | Connections between sphingosine kinase and phospholipase D in the abscisic acid signaling pathway in Arabidopsis[END_REF].

It would be interesting to know whether the same protein targets play the same roles in other hormone signalling pathways involving the production of PA. [START_REF] Kalachova | Involvement of phospholipase D and NADPH-oxidase in salicylic acid signaling cascade[END_REF] showed that Arabidopsis NADPH oxidase AtRbohD is also downstream of PA production in guard cells challenged with SA.

The other proteins listed in Table 1 have not been directly linked to hormone signal transduction, but they are very good candidates as mediators of hormone signals that lead to PA accumulation. Many of these proteins belong to multigene families, so other members of the families may also be involved in hormone signalling.

Diacylglycerol

Very few proteins that bind DAG have been identified in plants [START_REF] Dong | Does diacylglycerol serve as a signaling molecule in plants?[END_REF]. In mammals, DAG binds protein kinase C (PKC) isoforms through their cysteine-rich C1 domain. Higher plants lack genes encoding PKCs. However, some kinases having the enzymatic properties of PKC have been detected, and some of them are activated by DAG, such as a Brassica campestris kinase [START_REF] Nanmori | Purification and characterization of protein kinase C from a higher plant, Brassica campestris L[END_REF]. Plant genomes also have genes encoding proteins with C1 domains. For instance, the Arabidopsis thaliana genome encodes 30 confirmed or putative C1 domain proteins (Supplemental figure S1), including 2 DGKs. It is possible that, for some of these proteins, the C1 domain is an actual DAG binding domain. More data are necessary to decipher the exact role of DAG in hormone signal transduction.

Phosphorylated inositols

IP3 is produced through the action of PI-PLC on PIP2. In mammals, IP3 is well documented to bind to some calcium channels, thus allowing the release of calcium from internal stores into the cytosol. No IP3-binding channels have been found in land plants (Krinke et al. 2007b), although they do occur in Chlamydomonas [START_REF] Wheeler | Ca 2? signalling in plants and green algae-changing channels[END_REF]. It has been suggested that the more highly phosphorylated forms of inositol could be mediators in higher plants. For instance, inositol hexakisphosphate (IP6, also named phytic acid) has been shown to act as a more potent endomembrane calcium-release agent than IP3 in guard cells of Vicia faba [START_REF] Lemtiri-Chlieh | Inositol hexakisphosphate mobilizes an endomembrane store of calcium in guard cells[END_REF].

The auxin receptor transport inhibitor response 1 (TIR1) was unexpectedly found to have crystallised with IP6 [START_REF] Tan | Mechanism of auxin perception by the TIR1 ubiquitin ligase[END_REF], as was the jasmonate receptor COI1 [START_REF] Sheard | Jasmonate perception by inositol-phosphate-potentiated COI1-JAZ co-receptor[END_REF]. Does this mean that they are polyphosphoinositide receptors and that PI-PLC signalling controls auxin and jasmonate perception? If so, this would be very interesting. However, we need to keep in mind that the PI-PLC pathway is not the only source of phosphorylated inositols. Glucose-6-P can be converted into IP3 by myoinositol monophosphate synthase and IP3 can then be sequentially phosphorylated [START_REF] Raboy | Approaches and challenges to engineering seed phytate and total phosphorus[END_REF].

Phosphoinositides

Several phosphoinositide-binding domains are known, such as the pleckstrin homology (PH), FYVE, HEAT, C2 and Phox homology (PX) domains [START_REF] Catimel | The PI(3,5)P2 and PI(4,5)P2 interactomes[END_REF]. These domains are more or less specific for phosphoinositides as ND not determined a Dephosphorylates 3 0 phosphate group of PI3P, PI-3,4-P 2 and PI-3,5-P 2 b 14-3-3 Proteins are regulatory proteins that bind client proteins in a phosphorylation-dependent manner, but PA binding significantly hampers the interaction with the client protein tested

Conclusion

Open questions

Clearly lipid signalling is of major consequence in plant hormone signal transduction. Yet many questions remain open that will define the immediate future of this fascinating research field.

Lipid signalling in the signal transduction of other hormones Phosphoglycerolipid signalling is now well documented for ABA, SA and auxins. Does it also play a role in the transduction of other hormone signals? Very promising data suggest so. Methyljasmonate has been shown to activate PLD in Brassica napus, Taxus cuspidata and Silybum marianum cells [START_REF] Profotova | Involvement of phospholipases C and D in early response to SAR and ISR inducers in Brassica napus plants[END_REF][START_REF] Yang | Lipidomic analysis reveals differential defense responses of Taxus cuspidata cells to two elicitors, methyl jasmonate and cerium (Ce 4? )[END_REF][START_REF] Madrid | Silymarin secretion and its elicitation by methyl jasmonate in cell cultures of Silybum marianum is mediated by phospholipase D-phosphatidic acid[END_REF][START_REF] Sa ´nchez-Sampedro | Elicitation of silymarin in cell cultures of Silybum marianum: effect of subculture and repeated addition of methyl jasmonate[END_REF]. In Arabidopsis plantlets, mechanical wounding known to induce jasmonate led to a 4-to 5-fold increase in IP3 with concomitant depletion in phosphoinositides. No such IP3 increase was observed in dde2-2 mutant plants that are deficient in JA biosynthesis [START_REF] Mosblech | Phosphoinositide and inositolpolyphosphate signalling in defense responses of Arabidopsis thaliana challenged by mechanical wounding[END_REF].

Cytokinins activate PLD as a step in the control of gene expression in Arabidopsis [START_REF] Romanov | A rapid cytokinin response assay in Arabidopsis indicates a role for phospholipase D in cytokinin signalling[END_REF] and Catharanthus roseus suspension cells [START_REF] Amini | Effects of phosphatidic acid on cytokinin signal transduction in periwinkle cells[END_REF]).

In Amaranthus seedlings, a PI-4,5-P 2 -dependent PLD controls the accumulation of amaranthin (a pigment) that follows cytokinin application [START_REF] Kravets | Rapid activation of specific phospholipase(s) D by cytokinin in Amaranthus assay system[END_REF]. Ethylene activates PLD in carrot suspension cells [START_REF] Lee | Ethylene-mediated phospholipid catabolic pathway in glucose-starved carrot suspension cells[END_REF]. PLDa-deficient transgenic plants have slower rates of senescence induced by ethylene than WT [START_REF] Fan | Antisense suppression of phospholipase D alpha retards abscisic acid-and ethylene-promoted senescence of postharvest Arabidopsis leaves[END_REF]. Gibberellic acid (GA) leads to a rapid and transient IP3 accumulation in barley aleurone [START_REF] Villasuso | Gibberellin signal in barley aleurone: early activation of PLC by G protein mediates amylase secretion[END_REF]. Interestingly, in rice seeds, ABA prevents the gibberellininduced formation of IP3, although ABA alone does not alter the IP3 level [START_REF] Kashem | Possible involvement of phosphoinositide-Ca 2? signaling in the regulation of a-amylase expression and germination of rice seed (Oryza sativa L.)[END_REF]. Brassinosteroid treatment of Arabidopsis seedlings leads to increased expression of NPC3 and NPC4 monitored by GUS activity in leaves [START_REF] Wimalasekera | Plant phosphatidylcholine-hydrolyzing phospholipases C NPC3 and NPC4 with roles in root development and brassinolide signaling in Arabidopsis thaliana[END_REF]. LysoPE, a product of PLA2 activity, collaborates with brassinosteroids during stimulation of Arabidopsis root growth and development [START_REF] Jeong | Effect of lysophosphatidylethanolamine and brassinosteroids on development of Arabidopsis roots[END_REF].

How are the lipid signalling pathways activated or inhibited in response to hormones?

The upstream events leading to the regulation of phospholipid signalling pathways are far less well understood than the downstream processes.

Calcium has an important role in controlling the activity of enzymes involved in lipid signalling. PI-PLCs are Plant Cell Rep some can also bind to other anionic phospholipids or proteins. The Arabidopsis thaliana genome encodes ca. 168 proteins with at least one of these domains (Supplemental figure S2). Many of these proteins are involved in signal transduction, in cell fate determination or intracellular trafficking. For most of these proteins, however, it has yet to be established whether they act in hormonal signal transduction.

Some phosphoinositides control cellular processes. In animal cells, a wide range of channels are controlled by PI-4,5-P 2 .In Nicotiana tabacum protoplasts isolated from cultured cells with genetically altered plasma membrane levels of PIP2, the net steady-state outward K ? current was inversely related to the plasma membrane PIP2 level. Increasing PIP2 levels in controls or in ABA-treated cells with high PI levels, using the PI-PLC inhibitor U73122, decreased the outward-rectifying potassium channel (NtORK) activity. These results would be consistent with the negatively charged PI-4,5-P 2 in the inner plasma membrane leaflet inhibiting NtORK [START_REF] Ma | Phosphatidylinositol (4,5)bisphosphate inhibits K ? -efflux channel activity in NT1 tobacco cultured cells[END_REF].

Tubby and Tubby-like proteins (TLPs) were first discovered in mammals where they are involved in the development and function of neuronal cells. The binding of the carboxyl-terminal Tubby domain to PI-4,5-P 2 sequestrates these proteins to the plasma membrane. When PI-PLC is activated, PIP2 is hydrolysed and thus TLPs are released from the plasma membrane. TLPs are conserved across eukaryotic kingdoms. Hydrogen peroxide stimulates the release of AtTLP3 from the plasma membrane in a mechanism involving PI-PLC [START_REF] Reitz | The subcellular localization of Tubby-like proteins and participation in stress signaling and root colonization by the mutualist Piriformospora indica[END_REF]). Therefore, Tubby proteins could be a link between PI-PLC and oxidative stress, and hence to hormone signalling. However, in Arabidopsis, no phytohormone treatment (with methyljasmonate, ABA, SA, auxin, brassinosteroid, or GA) leads to relocalisation of AtTLP3. Therefore, the possible role of other Arabidopsis TLPs in hormone transduction in conjunction with PI-PLC is uncertain.

Lysophosphoglycerolipids and free fatty acids

Lysophosphoglycerolipids and FFA are released by the action of PLA. Both types of molecules have mediating roles. For instance, oxidised fatty acids in potato suspension cells in vivo and saturated fatty acids in vitro stimulate protein kinase activity. Unsaturated fatty acids inhibit MP2C, a protein phosphatase, while polyunsaturated fatty acids can modulate K ? channels. Oleic acid activates PLDd. The lysophosphoglycerolipid lysoPE inhibits a PLD in fruit senescence. It has been postulated that lysoPC could activate an H ? /Na ? exchange vacuolar transporter, thus mobilising the vacuolar proton pool for pH signalling (references in [START_REF] Scherer | Patatinrelated phospholipase A: nomenclature, subfamilies and functions in plants[END_REF]. Clearly, more work is necessary to identify other targets of FFA and lysophosphoglycerolipids. strictly dependent on calcium for their activity, except AtPLC4 that retains 20 % of its activity with no calcium. Most PLDs need calcium for their activity and only PLDf isoforms have shown to be calcium independent. PLDa enzymes need higher concentrations of calcium than other calcium-dependent PLDs, which may indicate that this is an important regulatory mechanism. Some PLAs, both pPLA2 and sPLA2 isoforms, have shown to be calcium dependent [START_REF] Chen | Biology and biochemistry of plant phospholipases[END_REF].

Some enzymes of the phosphoglycerolipid pathway may be post-translationally modified in a way that modulates their activities. In Brassica oleracea, a PIP2-dependent PLD is phosphorylated and this phosphorylation might positively regulate its activity [START_REF] Novotna | Plant PIP2-dependent phospholipase D activity is regulated by phosphorylation[END_REF]. Arabidopsis PLDa1 also seems to be regulated by this mechanism (Novotna ´, unpublished results). Arabidopsis patatin-related PLA2 pPLA-IId and pPLA-IIe are phosphorylated by calcium-dependent protein kinases in vitro [START_REF] Rietz | Roles of Arabidopsis patatin-related phospholipases a in root development are related to auxin responses and phosphate deficiency[END_REF]. Regulation by phosphorylation is probably a major component of the activation/inhibition of lipid pathways. The PhosPhAt database curates the sequences of experimentally detected phosphorylated peptides obtained in Arabidopsis phosphoproteomics experiments (http://phosphat.mpimp-golm.mpg.de/download.html; [START_REF] Durek | PhosPhAt: the Arabidopsis thaliana phosphorylation site database. An update[END_REF], and includes many that originate from enzymes of the phosphoglycerolipid pathway (Supplemental Table 2). However, the significance of these phosphorylations in relation to hormone signalling has not been investigated.

An important mechanism in phosphoglycerolipid signalling seems to be the coupling with heterotrimeric G protein [START_REF] Tuteja | Plant signaling in stress[END_REF]. Potential G-protein activators and inhibitors, such as toxins, nonhydrolyzable guanine nucleotide analogues, and alcohols, affect plant PLD [START_REF] Munnik | G protein activation stimulates phospholipase D signaling in plants[END_REF][START_REF] Ritchie | Abscisic acid stimulation of phospholipase D in the barley aleurone is G-protein-mediated and localized to the plasma membrane[END_REF], and PLD can also control G proteins. The direct interaction between AtPLDa1 and the a-subunit of heterotrimeric G proteins (GPA1) has been characterised using protein coprecipitation. PLDa1 binds to GPA1 and activates the intrinsic GTPase activity that converts active Ga-GTP to inactive Ga-GDP. In turn, Ga-GDP binds to PLDa1 and attenuates its activity. Thus, the PLDa1-GPA1 interaction mutually regulates both partners [START_REF] Mishra | A bifurcating pathway directs abscisic acid effects on stomatal closure and opening in Arabidopsis[END_REF]). An intriguing possibility that follows on from this observation is that PLD might be regulated by G protein-coupled receptors. G proteins also regulate PI-PLC. In tobacco cells overexpressing either Arabidopsis G protein-coupled receptor (GCR1) or the Ga-subunit of the heterotrimeric G protein, PI-PLC is active and IP3 levels are elevated [START_REF] Apone | The G-protein-coupled receptor GCR1 regulates DNA synthesis through activation of phosphatidylinositolspecific phospholipase C[END_REF]. The G-protein activator Mas7 is able to trigger a-amylase secretion in barley aleurone just as strongly as GA does. U73122 inhibits this Mas7 effect, linking the Mas7 effect with PI-PLC activity [START_REF] Villasuso | Gibberellin signal in barley aleurone: early activation of PLC by G protein mediates amylase secretion[END_REF]. Pisum sativum Ga interacts with the C-terminal C2 domain of pea PLCd [START_REF] Misra | Heterotrimeric G-protein complex and G-protein-coupled receptor from a legume (Pisum sativum): role in salinity and heat stress and cross-talk with phospholipase C[END_REF]. It is still not known whether all PI-PLC and PLD isoforms are dependent on G protein activity.

An intriguing aspect of lipid signalling regulation may be the role played by the interaction of phospholipases and signalling phospholipids with components of the cytoskeleton. In tobacco, monomeric G-actin inhibits and polymeric F-actin enhances the PIP2-dependent activity of NtPLDb1, while PIP2-independent PLDa activity and oleate-activated PLDd activity are unaffected by either form of actin [START_REF] Pleskot | Mutual regulation of plant phospholipase D and the actin cytoskeleton[END_REF]. PA has also been shown to modulate actin dynamics both in vitro and in vivo by binding to capping protein [START_REF] Li | Capping protein modulates the dynamic behavior of actin filaments in response to phosphatidic acid in Arabidopsis[END_REF][START_REF] Pleskot | Turnover of phosphatidic acid through distinct signaling pathways affects multiple aspects of pollen tube growth in tobacco[END_REF].

Finally, it should be noted that the different phosphoglycerolipid pathways are likely to be intertwined. For example, plant PLDs need PI-4,5-P 2 as cofactors, the exceptions being the a-type PLDs. Therefore, PI-PLC activity might indirectly modulate PLD activity, by modulating the level of phosphoinositide cofactors.

What is the link with known hormone receptors?

The first step in signalling is when a receptor perceives the hormone. However, when phosphoglycerolipid signalling is known to be involved in a specific hormone signalling pathway, the link to the hormone receptor(s) has not been made. It has recently been proved that the PYR/PYL/ RCAR family of soluble proteins in Arabidopsis are ABAbinding receptor-like proteins. Upon ABA binding, these PYR/PYL/RCAR receptors bind to and inhibit PP2C allowing the phosphorylation and activation of SnRK2 proteins. It is not certain how PA fits into this scenario given that PA interacts with ABI1. Is PLDa1 activated upon ABA binding to PYR/PYL/RCAR soluble receptors? Or does PLDa1 activation involve a yet-to-be-discovered receptor? We note that a membrane impermeant ABAbovine serum albumin conjugate, which cannot enter the cell, is able to activate PLD in Arabidopsis suspension cells [START_REF] Hallouin | Plasmalemma abscisic acid perception leads to RAB18 expression via phospholipase D activation in Arabidopsis suspension cells[END_REF]. Different proteins are known to bind SA, such as catalase, ascorbate peroxidase, chloroplast carbonic anhydrase, methyl salicylate esterase, some a-ketoglutarate dehydrogenases and some glutathione S-transferases [START_REF] Tian | The combined use of photoaffinity labeling and surface plasmon resonance-based technology identifies multiple salicylic acid-binding proteins[END_REF]. NPR1 [START_REF] Wu | The Arabidopsis NPR1 protein is a receptor for the plant defense hormone salicylic acid[END_REF] and its paralogues NPR3 and NPR4 [START_REF] Fu | NPR3 and NPR4 are receptors for the immune signal salicylic acid in plants[END_REF]) are proteins that have also been shown to bind SA. Thus, it needs to be elucidated whether or not lipid signalling is connected to these ''receptors'', and if so how (whether directly or via other proteins such as G proteins). Auxin is perceived either by ABP1 or TIR1/auxin-signalling F box proteins. It has been suggested that the receptor responsible for pPLA activation by auxin is ABP1, but more experimental data are necessary to establish this [START_REF] Scherer | Phospholipases and the network of auxin signal transduction with ABP1 and TIR1 as two receptors: a comprehensive and provocative model[END_REF].

How lipid mediators act

More work is needed to discern the modes of action of the lipid mediators. There is now quite a long list of PA binding proteins, but it is unlikely to be exhaustive. No DGPP binding proteins have been identified so far either in plants or in other biological kingdoms. Nevertheless, the specific role of DGPP as a hormone signal transduction player seems beyond doubt since DGPP has effect that PA has not [START_REF] Zalejski | Induction of abscisic acid-regulated gene expression by diacylglycerol pyrophosphate involves Ca 2? and anion currents in Arabidopsis suspension cells[END_REF]. Finding DGPP binding proteins is therefore a challenge ahead.

Most of the molecules generated by these pathways are lipids and stay in membranes. This gives rise to two questions. First, how do these lipids affect the physical state of the membrane-its fluidity, flexibility, or capacity to curve. PA, for instance, is considered to be a cone-shaped lipid. Lysophospholipids have inverted cone shapes. FFAs make membranes more fluid. Do such changes in the physical state of membranes have a role in hormone signal transduction as part of lipid signalling? It is believed that vesicle budding is favoured by PLA2 activity, because its action on a single leaflet of the bilayer asymmetrically increases the number of inverted cone-shaped lysophospholipids, thus enhancing membrane curvature towards the said leaflet side [START_REF] Brown | Phospholipase A2 (PLA2) enzymes in membrane trafficking: mediators of membrane shape and function[END_REF]. PLD products of PA could have similar effects. PA is a fusogenic lipid [START_REF] Jenkins | Phospholipase D: a lipid centric review[END_REF]. Vesicle formation, mobility and fusion are necessary for hormone signalling as exemplified by the trafficking of PIN proteins in auxin responses. Discovering the importance of lipid mediators not through binding proteins but through membrane effects will be a fascinating field of research.

In conclusion, phosphoglycerolipids appear to be major players in hormone transduction. The data obtained so far suggest they are involved in the signalling of most, if not all, plant hormones. Much more data will be necessary to understand how these pathways are regulated and how they act to transduce hormone signals. The same phosphoglycerolipid players are involved in the responses to different hormones and this could explain in part the crosstalk between hormones. However, the exact timing, duration and amplitude of the phosphoglycerolipid responses, and their interacting partners, probably differ going some way to explain the specificity of hormone responses.

Fig. 1

 1 Fig.1Structure of glycerophospholipids (a) and metabolic pathways generating phosphoglycerolipid signal mediators (b). Each phosphoglycerolipid class differs according to the nature of the polar head. The structure of inositol is indicated; its hydroxyl at position 1 is the one engaged in forming the phosphodiester bond; the other hydroxyls can be phosphorylated leading to different phosphoinositides. PI phosphatidylinositol, PI4P phosphatidylinositol-4-phosphate, PI3P phosphatidylinositol-3-phosphate, PI4,5P2 phosphatidylinositol-4,5bisphosphate, DAG diacylglycerol, IP3 inositol triphosphate, IP6

Table 1

 1 Proteins that bind PA and have a role in signalling

	Protein	Function	PA effect	References
	Protein kinase
	ZmCPK11 CDPK	Stimulation Klimecka et al.
				(2011)
	SnRK2.4	Class 1 SnRK2	ND	McLoughlin et al.
	SnRK2.10	kinases		(2012)
	PDK1	Protein kinase	Stimulation Anthony et al.
				(2004)
	MPK6	MAP-kinase	Stimulation Yu et al. (2010)
	CTR1	MAPKKK	Inhibition	Testerink et al.
				(2007)
	Protein phosphatase
	ABI1	PP2C	Inhibition	Zhang et al. (2004)
	PP1	Type 1 protein	Inhibition	Takemiya and
		phosphatase		Shimazaki (2010)
	Lipid kinase
	SPHK1	Sphingosine	Stimulation Guo et al. (2011)
	SPHK2	kinases
	PI4Kbeta	PI4 K	ND	Lou et al. (2006)
	Lipid phosphatase
	PTEN2a	P3P	ND	Pribat et al. (2012)
		phosphatase a
	Other mechanisms
	RbohD	NADPH oxidase Stimulation Zhang et al. (2009)
	RbohF
	14-3-3	14-3-3 proteins	Inhibition b Camoni et al. (2012)
	AtCP	Capping protein Inhibition	Huang et al. (2006)

Plant Cell Rep

Acknowledgments The authors acknowledge funding by Czech Science Foundation (grant no. 501/11/1654; OV, MJ, JM), by CNRS and AVCR bilateral convention (MSMT, grant no. 7AMB12FR018; JM, ER, OV, MJ) and the French Agence Nationale de la Recherche (Programme Blanc PANACEANT09_517917; ER). ND is a recipient of a Hubert Curien-Tassili PhD fellowship and MJ is a recipient of a Specific University Research PhD fellowship (MSMT No. 21/2011).


Lydie Richaume

Eric Andres

Gaëlle Largeteau-Skapin

Rita Zrour

Unfolding Level 1 Menger Polycubes of Arbitrary Size With Help of Outer Faces

Keywords: Polycube, orthogonal polyhedra, unfolding, high genus object

In this article, we suggest a grid-unfolding of level 1 Menger polycubes of arbitrary size with L holes along the x-axis, M the y-axis and N the z-axis. These polycubes can have a high genus, and most vertices are of degree 6. The unfolding is based mainly on the inner faces (that do not lie on the outer most envelope) except for some outer faces that are needed to connect lines or planes in the object. It is worth noticing that this grid-unfolding algorithm is deterministic and without refinement.

Introduction

Unfolding a geometric three-dimensional shape consists in providing a twodimensional flat non-overlapping and connected representation of it. This problem is as old as cartography and is still widely open with many potential applications including puzzle design [START_REF] Aloupis | Common Unfoldings of Polyominoes and Polycubes[END_REF], 3D paper craft [START_REF] Mitani | Making papercraft toys from meshes using stripbased approximate unfolding[END_REF], manufacturing [START_REF] Joseph | Folding and unfolding in computational geometry[END_REF][START_REF] Wang | Manufacturability-Driven Decomposition of Sheet Metal Products[END_REF]... Our interest lies in the grid edge-unfolding without refinement of a very specific class of polyhedra: the level 1 Menger polycube of arbitrary size (a generalisation of the Menger sponge [START_REF] Menger | Dimensionstheorie[END_REF]). A polycube is a set of face-connected voxels without inner cavity. An edge unfolding is an unfolding along the original edges of the polyhedron. A grid unfolding of a polycube is an unfolding that consists in creating a 2D net of all the faces of the polycube (faces that are face-connected to the complement) in a 2D orthogonal grid. Many grid unfolding algorithms found in the literature require refinements. A n × m refinement means that a polycube face is cut into n×m rectangular subfaces. Oftentimes, algorithms deal with orthogonal polyhedra rather than polycubes. An orthogonal polyhedron is a polyhedron where all the faces are parallel to Cartesian coordinate planes and can be seen as a generalisation of polycubes.

Edge-unfolding of a general polyhedron is not always possible [START_REF] Demaine | A survey of folding and unfolding in computational geometry[END_REF]. There are several examples of non-convex polyhedra that cannot be edge-unfolded. As for convex polyhedra, it is still an open question whether they always admit an edge unfolding. These last couple of years, the community tried to make some progress by focusing on various subclasses of orthogonal polyhedra that admit a grid edge-unfolding without refinement such as orthotubes [START_REF] Biedl | Unfolding Some Classes of Orthogonal Polyhedra[END_REF], well-separated orthotrees [START_REF] Damian | Unfolding well-separated orthotrees[END_REF], orthoterrains [START_REF] Joseph | Unfolding orthogonal terrains[END_REF], H-convex Manhattan towers... [START_REF] Richaume | Unfolding H-convex Manhattan Towers[END_REF] For some other subclasses, the known methods require the grid to be refined further by adding cutting planes in a constant or linear number of times: for example, 4 × 5 grid-unfolding for Manhattan towers [START_REF] Damian | Unfolding Manhattan Towers[END_REF], 1 × 2 grid-unfolding for orthostacks... [START_REF] Biedl | Unfolding Some Classes of Orthogonal Polyhedra[END_REF] More recently, the community has begun to focus on orthogonal polyhedra of genus greater than zero. In [START_REF] Liou | On edge-unfolding onelayer lattice polyhedra with cubic holes[END_REF], edge-unfolding of several classes of one-layered orthogonal polyhedra (one layer of voxels) with cubic holes are considered and it was shown that several special classes of them admit grid unfoldings. In [START_REF] Chang | Improved algorithms for grid-unfolding orthogonal polyhedra[END_REF], authors generalised the work of [START_REF] Liou | On edge-unfolding onelayer lattice polyhedra with cubic holes[END_REF] and offered a grid unfolding method suitable for a wider class of one layer polyhedra of genus greater than zero using a (2 × 1)-refinement. In [START_REF] Damian | Unfolding genus-2 orthogonal polyhedra with linear refinement[END_REF], an algorithm for unfolding all orthogonal polyhedra of genus 1 or 2 is proposed; it extends the idea of [START_REF] Chang | Unfolding orthogonal polyhedra with linear refinement[END_REF] to deal with holes, and both require a linear refinement. In [START_REF] Ho | Unfolding Some Classes of Orthogonal Polyhedra of Arbitrary Genus[END_REF], Ho et al. treat two subclasses of orthogonal polyhedra with arbitrary genus with 2 × 1 refinement. Our Menger polycube is a subclass of their well-separated orthographs, but our algorithm does not need any refinement.

The general question whether there always exists a grid unfolding without refinement for any polycube is an open question. A grid unfolding without refinement means that each polycube voxel face can be unfolded as a square 2D grid face (a pixel if you prefer). The final 2D net should be connected and it should be noted that two faces in the net are considered as neighbours iff the original corresponding faces in the 3D polyhedra share a common edge. Two 2D grid faces may be neighbours in the classical 4-connected sense and not be neighbours in the net.

One way of looking at unfolding problems is to consider 3D vertices. In three dimensions, a polycube vertex may have a degree from 3 to 7, meaning that there is a cycle of 3 to 7 faces around a vertex. However, once unfolded, a face can have a maximum degree of 4. All the faces around a vertex of degree over 4 cannot be placed together on the unfolding. Does the proportion of vertices of degree greater than 4 limit the unfoldability of a polycube? This question led us to the level 1 Menger polycube of arbitrary size with L × M × N holes [START_REF] Menger | Dimensionstheorie[END_REF]. The dimensions correspond to the number of holes in a direction: L holes along the x-axis, M along the y-axis and N along the z-axis. This represents a polycube with a potentially high genus and almost all the vertices in such a polycube are of degree 6 except for the 8 corner vertices that are of degree 3, some outer faces that are of degree 4 and the hole vertices on the outer envelope faces that are of degree 5.

In this paper we propose a deterministic algorithm for grid unfolding without refinement (edge unfolding) of level 1 Menger polycubes of arbitrary size. The idea is to focus on the "inner" faces (those that are not on the outer envelope). The outer envelope is defined as all the faces that have minimum or maximum x, y or z centroid coordinates. The inner faces is where the vertices of degree 6 can be found. Our unfolding method requires the help of some "outer" faces only at specific places. Once all the inner faces are unfolded with help of some outer faces, all the other outer faces can then be unfolded very simply.

The paper is organized as follows: Section 2 presents some preliminaries as well as the particular polycubes we are interested in. The unfolding method is then detailed. We finish the paper with a conclusion and prospects.

2 Unfolding Level 1 Menger Polycubes

Introducing Level 1 Menger Polycubes of Arbitrary Size

A Menger sponge is a fractal object [START_REF] Menger | Dimensionstheorie[END_REF]. In this paper, we only consider the level 1 of this construction: a simple level 1 Menger polycube is made of 20 unit cubes forming a greater 3 × 3 × 3 cube where the cubes in the centre of each face are missing as well as the cube in the very centre of the greater cube (figure 1a). This is what we call here a level 1 (1, 1, 1)-Menger polycube. The genus of this polycube is 5.

A level 1 (L, M, N )-Menger polycube is the Minkowski addition between a Cartesian (L,M,N) subset of points of (2Z) 3 and a structuring element that is simply the (1, 1, 1)-Menger sponge. This polycube has L holes along the x-axis, M holes along the y-axis and N holes along the z-axis. Each hole is the size of one unit cube and is separated from the holes around it by one unit cube. Such a polycube has a size of (2L + 1) × (2M + 1) × (2N + 1). An example is presented on figure 2a for the level 1 (1, 3, 1)-Menger polycube. This polycube is of genus 13. The general formulas for a level 1 (L, M, N )-Menger are:

number of voxels:

1 + 2(L + M + N ) + 3(LM + LN + M N ) + 4LM N -number of faces: 6 + 8(L + M + N ) + 10(LM + LN + M N ) + 12LM N -number of vertices: 8(1 + L)(1 + M )(1 + N ).
From which :

• 8 vertices of degree 3;

• 8(L + M + N ) vertices of degree 4

• 8(LM + M N + LN ) vertices of degree 5

• and 8LM N vertices of degree 6.

-genus: LM + LN + M N + 2LM N .
For instance, a (3, 3, 3)-Menger polycube (see figure 6) has a genus of 81, is composed of 208 voxels, 672 faces with 8 vertices of degree 3, 72 vertices of degree 4, 216 vertices of degree 5 and 216 vertices of degree 6. The ratio of vertices of degree 6 to other vertices climbs rapidly. For a (100, 100, 100)-Menger we have 8 million vertices of degree 6 and 242408 vertices of other degrees (and a genus of 2030000).

In the following, we start with handling the inner faces of such polycubes. These faces form a volume that, to unfold, we will divide in simpler structures: we can unfold a volume if we know how to unfold planes. In the same way, each plane is divided into lines and each line into crosses.

The outer faces (forming the outer envelope of the polycube) can be unfolded in a similar way as a simple cube as we will see in section 2.5.

Unfolding an Inner Line

We call inner cross (or simply cross) the inner faces of a (1, 1, 1)-Menger polycube, i.e. the faces around the central "hole". A complete cross has 6 branches (see figure 1a and1b). We can merge several crosses to form an inner line: the back branch of one cross is also the front branch of the next one (see figure 2b). An inner line is defined as the inner faces of a (1, M, 1)-Menger.

We first explain how to unfold a simple cross and then how to go on with another connected one and finally how to unfold the whole inner line.

Unfolding a cross: First we start with the simple cross which is the object obtained with the (1, 1, 1)-Menger polycube (i.e. the Menger sponge) inner faces (see figure 1b).

As we can see on figure 1d, unfolding one branch leads to four aligned faces. In figure 1d,"u,b,d,f, l, r" stands for "up, back, down, front, left, right" faces as they appear visually in figure 1b. We can navigate from one branch to the next by reversing the faces cycle: this way, the unfolding keeps progressing in the same direction, ensuring the non-overlapping property we seek. On figures 1c and 1d we propose an order to browse the branches and show the corresponding unfolding. Linking two crosses: As we can see on figure 2b, between two consecutive crosses, there is a common branch. On the pattern we used to unfold a single cross, the last and the first branch are opposite to each other. If we apply this pattern to each cross of the inner line, the first and last branches are the shared ones. This allows us to simply concatenate the unfoldings by merging the shared branches (see figure 2c).

Unfolding inner line results:

We now have all we need to unfold several crosses in a row: we simply have to concatenate the cross pattern M times to get a M sized line unfolding.

We can notice (see figure 2c) that the unfolding of each cross is on the right and slightly above the previous one, preventing any two faces from different crosses to overlap each other. The unfolding of the back branch of a cross is also the unfolding of the front branch of the next cross (presented by mixed colour on the figure). It allows linking every consecutive cross together.

Unfolding an Inner Plane

In the same way we formed a line by merging several crosses, we can now merge several lines side by side to form the inside of a (L, M, 1)-Menger plane, called an inner plane for what follows. As previously, some branches will be shared between two lines (see figure 3a). We will first see how to unfold an inner line without the side branches and then how to connect the nets of two consecutive inner lines to finally unfold an inner plane.

Unfolding a partial inner line: Between two consecutive inner lines, each cross shares a lateral branch with a cross from the other inner line. This time, the shared branches will not be used to link the unfoldings of the different inner lines. Since we cannot unfold the same faces twice, we choose to ignore the shared branches while unfolding the first inner line and unfold them with the following inner line and so forth until the last.

This means that, when we have M inner lines, each cross of the first M -1 inner lines will miss the left branch. The crosses of these lines have only 5 branches (see figure 3a and3b) instead of (these are partial inner lines)6 . Therefore a new pattern is needed. Indeed, the pattern we used in section 2.2 works for an inner line with crosses that have 6 branches.

A suitable pattern can be found in figure 3c. Once again, each two consecutive crosses from a same partial inner line share the front and back branches (in orange and red on the figures), and we will be using them to concatenate the unfoldings. We can see that, this time, the shared branch is not the last branch of the first cross, but we can nonetheless link the unfoldings of two consecutive partial crosses without overlap. The top branch is unfolded under the shared branch and under the next partial cross. We can thus concatenate any number of 5 branch crosses to create a partial inner line of any length.

Linking two lines in an inner plane: To link two partial inner lines together, we use an outer face. Indeed, we chose to unfold in such a way that the last visited branch of a partial inner line is a branch connected to the outer envelope of the polycube. We use the outer face as a bridge to reach the cross to the left of the previous one (see the orange faces on figure 3).

In a partial inner line, each cross unfolding is placed on the top of the preceding one. Following the same idea, each partial inner line unfolding is also placed strictly on top of the preceding one. This way, two different line-unfoldings cannot overlap. Each partial inner line is visited in a reverse order from the preceding one; this is why each inner line unfolding mirrors the previous one.

Unfolding inner plane results: A full L×M inner plane is composed of L-1 partial inner lines and one complete inner line. We just provided a way to unfold a partial inner line and the unfolding of an inner line (with 6 branches crosses) is treated in section 2.2. We simply have to link these different line-unfoldings as previously described to get a full inner plane unfolding. At each step, the unfolding of the different crosses and lines are placed on the top of the previous ones. This way we can ensure that there is no unwanted overlapping.

Unfolding an Inner Volume

We can now merge inner planes to build the inner faces of a (L, M, N )-Menger polycube volume. As for the previous case, some branches will be shared by two consecutive inner planes. To unfold a volume, we therefore consider that the lower planes miss their top branches (see figure 4a). This defines partial inner planes. The top plane is a full inner plane as discussed in section 2.3. Now, we explain how to unfold the partial inner planes. Then, we explain how to link two consecutive inner plane-unfoldings together.

Unfolding a partial inner plane: To unfold a partial inner plane (see figure 4), we have to unfold new types of partial inner lines where this time, the top and left branches are missing. This leads to unfolding crosses having 4 (for the first lines) or 5 (for the last line) branches.

When considering figure 3c, the pattern for 4 branch crosses is the same pattern as the one with 5 branches except that we drop the blue branch.

The last line of such planes has 5 branches, but this time it is the top branch that is missing. Globally the idea of the pattern is similar as for the 5 branch pattern in section 2.3 but the last branch is linked to the "left" of the red branch instead of the "up" (see figure 3c).

Figure 4 presents the faces of a partial inner plane of a Menger polycube. This plane misses every top branch. In blue and green, we can see two partial inner lines missing their top and one side branch. The last partial inner line, that misses only its top branches, is in red. To unfold such a partial inner plane, we use a face of the outer envelope (in orange) to go from one line to the next (similarly to what was described previously in section 2.3).

Linking two planes: As for the link from an inner line to the next inner line (see section 2.3), we use an outer face to link a partial inner plane to the above located inner plane. We exit a partial inner plane using a cross branch connected to a side of the polycube and re-enter it by the corresponding branch of the cross located above. The outer face used is coloured in yellow on figure 4.

Unfolding an inner volume results:

The whole inner volume unfolding is obtained by adding a full last inner plane to the N -1 partial inner planes unfoldings. We can see the unfolding of the inner volume of a (3, 3, 3)-Menger polycube on figure 6. It is the zigzag pattern unfolding on the top of the figure (in green and blue we have the two partial inner planes and in red, the full inner plane).

Unfolding the Remaining Outer Faces

The remaining faces all belong to the outer envelope of the polycube. We will now see that all of these faces remain connected, can be unfolded in the same fashion as a simple cube (see the bottom part of figure 6) and can be added to the unfolding of the inner faces without overlap. Connectedness of outer faces: While unfolding the inner parts of the polycube we use outer faces to link two inner lines either horizontally (for two consecutive inner lines of an inner plane) or vertically (for two inner lines of two consecutive inner planes). The way we unfold all the inner lines is similar to a sewing pattern starting on the bottom-front of the polycube, passing through to exit on the opposite side and coming back through the next inner line either on the left or on the top of the first one.

The outer faces used to link the inner unfoldings belong only to the front and the back sides of the polycube, the other four sides are untouched.

We can see on figure 5 that the unfolding of the outer faces remains connected even with the missing outer faces (in light blue and green) that were used to link the inner lines and planes together. Despite having used some outer faces to unfold the inner part, the outer faces remain connected and can be unfolded in the same way as a simple cube (see figure 6). A last face, coloured in orange, has been removed to be used as a bridge between the unfolding of the outer and inner part.

Linking the outer unfolding to the inner unfolding: When we unfold the inner faces, the first unfolded branch is placed to have the lowest ordinate value of the inner faces unfolding. This branch is linked to the outer envelope of the polycube. During the unfolding process, faces are always added either at the same level as the previous ones or above. Therefore, the outer faces can be unfolded under the inner unfolding.

If we keep strictly the unfolding of a cube, we cannot connect the outer faces to the inner faces. There is no free edge in the unfolded outer faces that is common with an inner face. To create this common free edge, we move one of the two faces of the front side of the polycube that are connected to both the front branch of the starting cross and a face from another outer side of the polycube. This face makes the bridge between the inner unfolding part and the outer unfolding part.

Final Result

Figure 6 shows the unfolding of a (3, 3, 3)-Menger polycube and gathers all the previously presented unfolding elements. We can see the unfolding of most of the outer faces on the bottom. We can see the unfolding results of the inner part: the zigzag unfolding on the upper part. Each inner line unfolding corresponds to a linear part. Each zigzag composed of three linear parts belongs to the unfolding of an inner plane. Each linear part is connected to the next using an outer face either to connect two inner lines in the same inner plane (orange faces) or to connect two inner planes (yellow faces). On the front of the (3, 3, 3)-Menger polycube in figure 6, one can see some of the outer faces (in orange and yellow respectively) that are used to link the inner lines and the inner planes. A similar set of outer faces is located on the back of the Menger polycube.

Conclusion

In this paper, we have proposed a deterministic method to unfold without refinement high genus polycubes made out of level 1 Menger sponges. This proves that despite the high rate of 6-degree vertices, there exists a non-overlapping flat unfolding for such a polycube. It is the first unfolding example of such a complex polycube object with a high genus without refinement.

As a short time prospect, we will try to unfold the inner part of the Menger polycubes without the help of outer faces. The goal is to prove that it is possible to unfold an object composed only of degree 6 vertices. In the long term, we want to study if our method can be adapted to other types of high genus polycubes and maybe come up with a deterministic unfolding for any type of polycube.

  (a) (1, 1, 1)-Menger polycube. (b) A cross: the inside of a (1, 1, 1)-Menger. (c) The order and rotating direction to unfold the branches. (d) Unfolded inner faces of a (1, 1, 1)-Menger polycube.

Fig. 1 :

 1 Fig. 1: The (1, 1, 1)-Menger polycube. The inner faces of this polycube and their deterministic edge-unfolding.

  (a) A Line (1, 3, 1)-Menger Polycube (b) The faces of an inner line Menger. The green cross and the red cross each share a branch with the blue cross. (c) Unfolded Inner Faces of an Inner Line (1, 3, 1)-Menger

Fig. 2 :

 2 Fig. 2: The line (1, 3, 1)-Menger polycube. The inner faces of this polycube and its deterministic edge-unfolding.

  (a) A Menger inner plane and, in orange, the exterior faces used to link two lines. (b) A cross missing its side branch (c) The unfolding of a 5-branch cross inner line. The red branch is used to link this 5-branch cross to the next one (in grey). (d) The unfolding of an inner (3, 3, 1)-Menger plane. Faces of a same colour are from the same cross. Orange faces are outer faces.

Fig. 3 :

 3 Fig. 3: The inner faces of a (3, 3, 1)-Menger polycube and their unfolding.

  (a) A Partial Inner Plane Missing Every Top Branches. (b) The unfolding of a partial plane of a (3, 3, N )-Menger inner volume. The yellow face will be used to link this partial inner plane to one above it.

Fig. 4 :

 4 Fig. 4: The inner faces of a partial inner plane from a (3, 3, N )-Menger polycube and their unfolding.

Fig. 5 :

 5 Fig. 5: The outer face unfolding in a cube unfolding fashion. The light-blue faces are used to link 2 lines, and the green faces are used to link 2 planes.

Fig. 6 :

 6 Fig. 6: The unfolding of the level 1 (3, 3, 3)-Menger polycube.