Olivier Ali

Ibrahim Cheddadi

Benoit Landrein

Yuchen Long
email: yuchen.long@nus.edu.sg

Revisiting the relationship between turgor pressure and plant cell growth

Keywords: turgor pressure, cell wall mechanics, mechanical signals, plant growth, tissue hydraulics, osmosis. I

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Revisiting the relationship between turgor pressure and plant cell growth

Introduction

Plant cells often have a high intracellular osmolyte content which, thanks to their rigid walls, means they can become turgid without risk of bursting. Conversely, under water stress, they lose turgidity and shrink. These observations have led some biophysicists to compare plant cells to balloons or bubbles: their surfaces are under tension and their interiors are under pressure (Fig. 1). This analogy, though simplistic, partially captures the mechanical interactions occurring during cell growth (reviewed by [START_REF] Beauzamy | Flowers under pressure: ins and outs of turgor regulation in development[END_REF]. Wall tension induced by turgor pressure is believed to underlie the irreversible cell expansion (reviewed by [START_REF] Landrein | Connected through the force: mechanical signals in plant development[END_REF]). Previous biomechanical studies on plant growth mainly focused on the pressure-driven, mechanical stress-induced expansion of the primary cell wall. Wall expansion is often considered to be a passive or a protein-mediated stress relaxation, occurring through de novo wall material synthesis and/or wall remodeling (discussed by [START_REF] Cosgrove | Building an extensible cell wall[END_REF]. Initial mechanical models described isotropic growth responses [START_REF] Lockhart | An analysis of irreversible plant cell elongation[END_REF][START_REF] Ortega | Augmented growth equation for cell wall expansion[END_REF][START_REF] Wei | Loss of stability-a new model for stress relaxation in plant cell walls[END_REF]. More recently the orientation of wall materials such as cellulose microfibrils were integrated into models, conceptualizing wall stress guidance [START_REF] Sampathkumar | Subcellular and supracellular mechanical stress prescribes cytoskeleton behavior in Arabidopsis cotyledon pavement cells[END_REF][START_REF] Zhang | Molecular insights into the complex mechanics of plant epidermal cell walls[END_REF]Oliveri et al., 2019). Other formalisms also include pectin dynamics. It has indeed been recently proposed that some walled cells, like Arabidopsis pavement cells, may expand over short timescales when plasmolyzed due to pectin nanofilament swelling [START_REF] Haas | Pectin homogalacturonan nanofilament expansion drives morphogenesis in plant epidermal cells[END_REF]. A similar "stored growth" phenomenon has been described in microbes [START_REF] Rojas | Regulation of microbial growth by turgor pressure[END_REF], suggesting that some walls may "grow" without turgor pressure under certain conditions.

Nonetheless, despite the key role of the wall in growth regulation, it is generally accepted that turgor pressure is the driving force of plant growth. Pressure is often treated as a passive, constant biophysical parameter, only required to initiate and maintain wall expansion. However, pressure and growth dynamics do not necessarily correlate, notably in pollen tubes (reviewed by [START_REF] Chebli | Mechanical principles governing pollen tube growth[END_REF]), and the relation between turgor and growth is thus complex. Recently, several studies indicate that turgor may be biologically regulated, hinting at a more active role of turgor pressure in cell growth regulation [START_REF] Cheddadi | Coupling water fluxes with cell wall mechanics in a multicellular model of plant development[END_REF][START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF][START_REF] Dumais | Mechanics and hydraulics of pollen tube growth[END_REF][START_REF] Mielke | Jasmonate biosynthesis arising from altered cell walls is prompted by turgor-driven mechanical compression[END_REF][START_REF] Creff | Endosperm turgor pressure both promotes and restricts seed growth and size[END_REF].

In this review, we first briefly describe the basic concepts of turgor pressure and related phenomena. We then summarize recent findings regarding its relationship with plant growth on different levels. Finally, we provide our perspectives on the study of turgor pressure and plant growth regulation.

II.

Turgor pressure and osmosis "Turgor pressure" is often confused with "osmotic pressure". To clarify the difference between these two parameters, consider a cytosol with high solute concentration and the extracellular environment with low solute concentration, separated by a semipermeable plasma membrane (Fig. 1A). Water flows towards the compartment with lowest "water potential" Ψ, defined as the chemical potential of water relative to a reference state. Cellular water potential is mainly determined by osmotic and pressure potentials. Assuming the initial cross-membrane pressure difference is 0, a solute concentration differential will create an unequilibrated chemical potential called the "osmotic potential" Ψ Π , and cross-membrane hydraulic fluxes will spontaneously occur to equilibrate the potential gradient. Since aqueous solutions have negative osmotic potentials, the positive term "osmotic pressure" Π=-Ψ Π is often used. Water influx increases cell volume, leading to a simultaneous stretching of the walls and a compression of the protoplast. This causes the emergence of a pair of "actionreaction forces" in the form of cell wall tensile stress σ and hydrostatic pressure P, or "turgor pressure". At equilibrium, the value of turgor pressure equals the osmotic pressure differential gradient, or "osmotic gradient" (P=ΔΠ). In unequilibrated cells, such as growing cells, the net fluid flux J is proportional to their water potential gradient ΔΨ=-(ΔΠ-P).

Textbox 1. Glossary

Anisotropy (dimensionless): The level of directional dependence a property, opposite to "isotropy" where this property is equal in all directions. For example, growth anisotropy describes how much an object (cell, tissue, organ) preferentially grows in one specific direction compared to other directions. A fully anisotropic growth is elongation.

Conductance (Pa -1 ⋅s -1 ): Quantifies the rate at which a material allows fluid to diffuse through it. In a cylindrical cell, cell conductance ϕ a = (A / V) L p is a parameter that describes the efficiency of fluxes as a function of the permeability per unit membrane area L p (unit m⋅Pa -1 ⋅s -1 ) and the surface-volume ratio of the cell, which is inversely proportional to the radius of the cylinder R. Note that conductance and wall extensibility (see below) are two parameters of completely different biophysical origins.

Curvature (m -1 ): Reciprocal of the radius of the osculating circle of a curve (a tangent circle closely adhering to a curve).

Extensibility: Quantifies the rate of deformation of a material when loaded with mechanical constraints. In plant cells, it quantifies the effect of chemical relaxation of cell wall stresses, wall material yielding, and/or the cell's ability to synthesize new wall materials upon wall stretching. In Eq.(1), the "effective cell wall extensibility" (unit Pa -1 ⋅s -1 ) is given as a function of the apparent wall material extensibility Φ (unit also Pa -1 ⋅s -1 ) and the ratio between cell radius and wall thickness R/h, depending on the exact cell geometry [START_REF] Cheddadi | Coupling water fluxes with cell wall mechanics in a multicellular model of plant development[END_REF].

Hydrostatic pressure (Pa): The pressure that a fluid at rest experiences. In plant cells, it is also called "turgor pressure" and corresponds to the isotropic stress in the cell interior that balances external forces and constraints.

Osmotic potential (Pa): An extensive (proportional to the concentration) quantity with the same unit as pressure, given by the Van't Hoff/Morse equation: , where i is the van't Hoff index of the compound (for example, one molecule of KCl dissolves into one K + and one Cl -ions, so i=2 for KCl), C is the concentration of the compound in mol/m 3 , is the absolute temperature in Kelvin and R the universal gas constant ~8.314 m 3 ⋅Pa⋅K -1 ⋅mol -1 .

Osmotic pressure (Pa): The opposite of the osmotic potential. A reasonable assumption could be that it is constant during growth, as cells may try to maintain osmotic pressure to ensure normal physiological functions. However, constant osmotic pressure is not guaranteed, as first evaluated by [START_REF] Lockhart | An analysis of irreversible plant cell elongation[END_REF], and equations describing osmolyte fluxes have been developed in e.g. pollen tube elongation [START_REF] Dumais | Mechanics and hydraulics of pollen tube growth[END_REF] and tumorous tissue growth [START_REF] Mcevoy | Gap junctions amplify spatial variations in cell volume in proliferating tumor spheroids[END_REF].

Quasi-staticity: Mechanical state very close to equilibrium where the forces experienced by a system almost totally compensate for each other. A quasi-static process happens slowly enough for the system to remain in internal physical equilibrium at each time-point considered; under this hypothesis, the pressure in an idealized cell containing only water is spatially homogeneous (but can vary in time).

Semipermeable membrane: A membrane that allows solvent molecules to pass through but retains solute molecules.

Tensile stress (Pa): Quantifies the internal force exerted on the cross section of a stretched object.

Tension (N): The pulling force in a stretched object.

Topology: The connectivity of geometric objects.

Water potential (Pa): The total energy potential of a fluid relative to pure water at rest state.

It depends on many components. At the cellular scale, the major contributions are osmotic potential and pressure potential, all other potentials being negligible.

III. Turgor pressure and growth are variable

a) The Lockhart model

In plant cells, growth relies on osmotic water intake and cell wall expansion, often depicted as an irreversible viscoplastic yielding mechanism. This coupling implies that cell growth rate and turgor pressure both depend on the balance between osmotic pressure and cell wall yielding threshold, Eq.(1b). Equation (1a) corresponds to the seminal Lockhart equation [START_REF] Lockhart | An analysis of irreversible plant cell elongation[END_REF], formulated in the case of an elongating cylindrical cell. We recall some of its properties and refer readers to a recent review [START_REF] Dumais | Mechanics and hydraulics of pollen tube growth[END_REF] for a more thorough exploration.

The entanglement between hydraulics and wall mechanics is omnipresent in Eqs.( 1 Lockhart's model is that turgor pressure, in the general case (when ), is not a control variable of growth, i.e. the variable P is not present in the right-hand-side of Eq.(1a). Eqs.(1) highlight that both the relative expansion rate and the turgor pressure are controlled by the osmotic potential, , the cell wall yielding threshold, , and the balance between extensibility and conductance. These equations illustrate that turgor and growth result from complex interactions between fluxes and wall expansion, and that they are not unequivocally correlated. In particular, dominant mechanisms underlying growth rate and turgor pressure might not always act equivalently depending on the spatial and temporal resolution considered. Thus, turgor does not determine the rate of growth, but it is an emergent property of this system, tightly linked to growth.

After analyzing an elongating cell with extreme growth anisotropy, we will now look at an isotropically-growing spherical cell.

b) Links between stress, turgor, and geometry

In general, plant cells do not only elongate as described in the Lockhart model, but tend to acquire rounder shapes. Combined with their pressurized status, cells are thus often compared to soap bubbles (Textbox 2). However, though intuitive, the bubble analogy is treacherous, because cell walls do not behave as fluid membranes. Plant cell mechanics are therefore more accurately grasped by pressurized shell models [START_REF] Ali | Physical models of plant development[END_REF][START_REF] Bozorg | Stress and Strain Provide Positional and Directional Cues in Development[END_REF]. Despite their differences, pressurized shells and bubbles share a common feature: a relationship between their shape, inner pressure ( ) and the tensile stress (σ) at mechanical equilibrium. In the simple case of an isotropic, homogeneous spherical shell, this relationship is formalized by the Young-Laplace equation:

, ( 2
)
where R denotes cell radius, h, wall thickness (Fig. 1B), and when the external pressure is zero. Assuming constant thickness , this equation implies that the ratio is proportional to , which is constant in the elongating cylinder in Lockhart's model. To examine the consequences of this difference, consider two extreme cases: first, if pressure is constant because of fast fluxes, stress increases with cell size. This could explain why plants make puzzle cells in leaves, as non-spherical shapes help decrease wall stress (Sapala et al., 2018). From a mechano-sensitive perspective, wall stress could allow cells to "measure" their shape and size, useful information in the context of morphogenesis. Second, if stress is constant, for instance because it is close to the growth threshold, then turgor decreases with size, as was observed in the growing shoot apical meristem (SAM) of Arabidopsis thaliana [START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF]. In general, growing tissues could be situated near either of the states described above, or somewhere in-between. As a result, the size and shape of growing cells should affect turgor, wall stress, and growth.

Having established the turgor-growth relationship in individual cells, we will next explore how multicellularity brings extra complexity to turgor-dependent tissue growth.

IV. Turgor pressure and growth are heterogenous

Turgor pressure can be different in specialized cells, such as stomatal guard cells (Nieves- et al., 2022), or during invasive growth of pollen tubes [START_REF] Chebli | Mechanical principles governing pollen tube growth[END_REF] or fungal hyphae [START_REF] Lew | How does a hypha grow? The biophysics of pressurized growth in fungi[END_REF]. Recent studies suggest that cells within the same tissue may also have different pressures. We discuss below how this could happen, and the implications for tissue growth.

Cordones

a) Turgor pressure and growth derived from cell arrangements

Besides physical properties, cells within a tissue can vary in size, geometry, and topology (number of neighbors). Experiments using excised Arabidopsis SAMs, cultured either under normal conditions, or treated with oryzalin to inhibit cell division and anisotropic growth by depolymerizing microtubules, show that cell geometry strongly correlates with turgor levels determined by atomic force microscopy (AFM). Thus smaller cells with fewer than six neighbors exhibit higher turgor than larger cells with more neighbors [START_REF] Corson | Turning a plant tissue into a living cell froth through isotropic growth[END_REF][START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF]. A recent model [START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF] proposed that geometry and topology are sufficient to generate these heterogeneities, even when applying the simplest assumption; homogeneous cellular properties (Fig. 2). This interpretation treats plant tissues rather like foams (see Textbox 2 for further discussion), and implies that models with more realistic cellular properties should include the emergent turgor heterogeneity as a biomechanical baseline.

b) Pressure from tissue shape and growth heterogeneities

In a tissue, plant cells are not mechanically isolated. A specific portion of the cell wall, the middle lamella, glues primary walls together. It prevents cell movement and enables propagation of mechanical stresses. Therefore, a consequence for the global quasi-staticity of the tissue is that the overall mechanical stress ( ) in any small portion (Fig. 2F) of the primary cell wall around a given cell encompasses cell-autonomous ( ) and non-cellautonomous ( ) contributions:

(6)

The cell-autonomous contribution corresponds to turgor-induced stresses imposed by the considered cell on its wall (Fig. 3C). The non-cell-autonomous contribution encompasses all the mechanical constraints generated by the surrounding tissues (Fig. 3C). These constraints include factors such as pressure-induced contributions from deeper cells and stresses generated by growth heterogeneities. Depending on the tissue and developmental stage considered, the effects of one contribution can override those of another. In the following section we will solely focus on pressure-induced contributions, consistent with the scope of this review.

In the absence of any significant cell-autonomous contribution, i.e. , non-cell-autonomous mechanical stresses can trigger a mechano-sensitive response. Such a mechanism has been suggested to play regulatory roles in various organogenic processes [START_REF] Landrein | Connected through the force: mechanical signals in plant development[END_REF]. It has been established that many plant outer tissues, such as those of the SAM, seed, sepal, stem, and various animal embryonic and tumorous tissues, are pressurized from below by a fluid or another tissue [START_REF] Dumais | New Evidence for the Role of Mechanical Forces in the Shoot Apical Meristem[END_REF][START_REF] Navis | Developing pressures: fluid forces driving morphogenesis[END_REF][START_REF] Beauzamy | Mechanically, the Shoot Apical Meristem of Arabidopsis Behaves like a Shell Inflated by a Pressure of About 1 MPa[END_REF][START_REF] Fourquin | Mechanical stress mediated by both endosperm softening and embryo growth underlies endosperm elimination in Arabidopsis seeds[END_REF][START_REF] Ruiz-Herrero | Organ size control via hydraulically gated oscillations[END_REF][START_REF] Robinson | Global Compression Reorients Cortical Microtubules in Arabidopsis Hypocotyl Epidermis and Promotes Growth[END_REF][START_REF] Tlili | Shaping the zebrafish myotome by intertissue friction and active stress[END_REF][START_REF] Zhao | Microtubule-Mediated Wall Anisotropy Contributes to Leaf Blade Flattening[END_REF][START_REF] Mcevoy | Gap junctions amplify spatial variations in cell volume in proliferating tumor spheroids[END_REF][START_REF] Asaoka | Stem integrity in Arabidopsis thaliana requires a load-bearing epidermis[END_REF][START_REF] Verge-Serandour | A hydro-osmotic coarsening theory of biological cavity formation[END_REF]. The Young-Laplace law, applicable at the cellular level, also applies to this wider, tissular scale. As Eq.( 7) states, cell-autonomous and non-cellautonomous components of the mechanical stress experienced by the primary wall within a curved tissue can be expressed similarly, but parametrized with different variables:

(7) P c and P in quantify respectively the turgor pressure within the considered cell and the pressure exerted by the compressed or expanding medium underneath the considered tissue. Similarly, the geometrical quantities R c and h w depict the radius of the cell and the thickness of its primary wall, while R e and h e account for the curvature radius of the pressurized tissue and its thickness (Fig. 2).

Mechano-sensitive cells embedded within a pressurized tissue can therefore have access to a mechanical stimulus (σ) directly proportional to a dimensionless geometric variable describing the shape of the tissue and quantifying its size: the ratio between its radius of curvature and its thickness: R e /h e . Moreover, in tissues displaying heterogeneous growth patterns, mechanical stresses arise between regions growing at different rates and/or in different directions [START_REF] Shraiman | Mechanical feedback as a possible regulator of tissue growth[END_REF], providing cells with dynamic information about their environment.

Textbox 2. Do (plant) cells act like soap bubbles?

Plant tissues have long been compared to soap foams, from cell arrangement [START_REF] Lewis | The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of cucumis[END_REF][START_REF] Besson | Universal rule for the symmetric division of plant cells[END_REF] to tissue mechanics and dynamics [START_REF] Corson | Turning a plant tissue into a living cell froth through isotropic growth[END_REF][START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF]. However, how comparable are plant cells to bubbles? And is it safe to rely on the bubble analogy?

First, although both bubbles and plant cells have tension at their surfaces and pressure within, the mechanical origins of these factors are not the same.

Bubbles and foams are composed of two elements: a gas phase surrounded by a liquid film. Due to inner cohesive forces, the film tends to minimize its surface. Meanwhile, entropic forces in the gas tend to expand it as much as possible. Equilibrium is reached with specific shapes called "minimal surfaces"; spheres in case of isolated bubbles and polyhedral tiling with regular angles of 2π/3 (120°) in case of 2D foams (Fig. 2B). Pressure in the gas volume arises to equilibrate the surface tension in the liquid film. These two constraints, and the geometry of the minimal surface of the system are related through the Young-Laplace equation, eq.( 2).

Mechanically, plant cells are more comparable to liquid-filled pressure tanks: a protoplast is embedded within a cell wall. While water influxes tend to increase the protoplast volume, the elastic properties of the surrounding walls, described as a visco-elastic shell, tend to oppose this expansion. Despite being very different in nature, an antagonistic equilibrium, similar to the one at play in soap bubbles, also emerges in plant cells. The conjunction of an expanding volume with a resisting surface generates similar constraints (i.e. pressure in volume and tensile stress on the surface).

The difference between both systems resides in the nature of the surface phases: Contrary to a liquid film, the solid shell surrounding protoplasts can display complex structures (inhomogeneous and anisotropic) and therefore sustain complex stress and deformation patterns. As a consequence the resulting equilibrium shapes do not correspond to minimal surfaces in the general case.

However, the theory of pressurized shells states that at equilibrium, a relationship between curvature, pressure and tensile stress also emerges. Moreover, in the simple cases where the shell material is homogeneous and isotropic, this relationship is exactly the Young-Laplace equation. So, from a quasi-static perspective, where cell wall composition is excluded, similar relationships between inner pressure, curvature and surface stress can be used to describe soap bubbles/foams and plant cells/tissues.

Second, the volumetric dynamics of bubbles and plant cells are not necessarily the same.

In bubbles, the liquid flows much faster in the membrane than the gas diffusion across the membrane, corresponding to α close to 0 in Eq.( 1b). The Laplace pressure causes gas to gradually diffuse out of the bubbles, and in a foam, higher-pressured small bubbles tend to lose volume to their bigger neighbors and eventually disappear (von Neumann-Mullins coarsening). In plant tissues, it is likely that water fluxes across plasma membranes occur faster than the expansion of the cell walls, and the classical measurements by [START_REF] Boyer | Control of the rate of cell enlargement: Excision, wall relaxation, and growth-induced water potentials[END_REF] and [START_REF] Cosgrove | Cell Wall Yield Properties of Growing Tissue: Evaluation by in Vivo Stress Relaxation[END_REF] correspond to an α = 0.9. Direct cell-to-cell fluxes through plasmodesmata are likely relatively fast too [START_REF] Rutschow | Regulation of Solute Flux through Plasmodesmata in the Root Meristem[END_REF]. Interestingly, thanks to the rheology of the cell wall, stress higher than the growth threshold can be constantly relaxed by wall expansion to keep wall stress relatively similar (simulated by [START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF], which is not a prerequisite for foam-like dynamics (see below). Unlike bubbles that lose gas and shrink, growing cells constantly absorb water from the apoplast by osmosis and shrinkage is rarely observed. And because plant cells may likely modulate α by, for example, mechanical feedback, cell growth rate may have different trends, as discussed in Section IV. So, foam-like dynamics represent a subset of plant tissue dynamics, despite their different mechanical properties.

Lastly, the foam-like analogy is also commonly used outside the plant field. In animal tissues, cell geometry, particularly membrane curvature and cell junction angles, have been frequently used to infer membrane tension and cellular pressure in animal tissues [START_REF] Roffay | Inferring cell junction tension and pressure from cell geometry[END_REF]. Geometrical inference without membrane curvature can also predict membrane tension and cellular pressure by energy minimization, as in foams [START_REF] Ishihara | Comparative study of non-invasive force and stress inference methods in tissue[END_REF]. In the plant field, energy minimization has also been classically used to predict and explain the positioning and geometry of the new cell wall formed during cell division (particularly cell wall curvature), which can be demonstrated by soap films [START_REF] Besson | Universal rule for the symmetric division of plant cells[END_REF]. The hidden assumption in the soap film system is that the Laplace pressure in the bubbles (equivalent to turgor pressure in plant cells) is related to the membrane curvature. Soap film membranes and the new cell walls were almost always curved as demonstrated by [START_REF] Besson | Universal rule for the symmetric division of plant cells[END_REF]. Experimental data on turgor measurement or turgor removal by cell ablation also showed strong correlation between anticlinal wall convexity and turgor differences between cells on either side of the wall [START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF][START_REF] Hoermayer | Wounding-induced changes in cellular pressure and localized auxin signalling spatially coordinate restorative divisions in roots[END_REF]. Importantly, none of the above examples require tension or to be homogeneous.

To sum up, even though bubbles and plant cells are completely different physical entities and have very different properties, the bubble analogy, although it should be used with caution, is still useful when thinking about cells, and has been used to derive useful predictions in the shoot apical meristem [START_REF] Long | Cellular Heterogeneity in Pressure and Growth Emerges from Tissue Topology and Geometry[END_REF], growing seeds [START_REF] Creff | Endosperm turgor pressure both promotes and restricts seed growth and size[END_REF], and even less "foam-like" tissues such as sepal primordia [START_REF] Zhao | Microtubule-Mediated Wall Anisotropy Contributes to Leaf Blade Flattening[END_REF].

However, not all plant tissues can be easily compared to foams. For example, the analogy is less obvious in roots, where cuboidal cells are tightly packed into files, or in the leaf epidermis where pavement cells acquire complex, jigsaw-like shapes. How, or if, the bubble analogy may apply to these tissues, remains to be analyzed.

V.

Turgor pressure and growth are actively regulated

In the previous section, we considered how nontrivial interactions between turgor and growth can emerge from uniform, constant physical parameters. Thanks to continuous cell wall modification and osmoregulation, growing cells may experience mechanical fluctuations.

Here we will discuss how dynamic changes in pressure and hydraulics can contribute to diverse growth patterns.

Pollen tubes are cylindrical cells that show oscillatory growth. Whereas growth is turgordependent, neither the growth rate nor the oscillations were found to correlate with turgor [START_REF] Benkert | The turgor pressure of growing lily pollen tubes[END_REF][START_REF] Zerzour | Polar growth in pollen tubes is associated with spatially confined dynamic changes in cell mechanical properties[END_REF][START_REF] Kroeger | Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth[END_REF]. Recently, [START_REF] Dumais | Mechanics and hydraulics of pollen tube growth[END_REF] proposed a "Lockhart differential equation" to explain the observed growth oscillation. First, the author proposed that growing lily pollen tubes are not at osmotic equilibrium, a proposition supported by experimental measurements. Then, by testing several potential feedback mechanisms, the author predicted that a hydraulic feedback is sufficient to generate and maintain oscillatory growth. This work clearly emphasized the importance of considering both mechanical and hydraulic regulation to explain growth dynamics [START_REF] Dumais | Mechanics and hydraulics of pollen tube growth[END_REF]Fig. 2I).

At the organ level, how the pressure of inner tissues affects organ growth has been studied in developing seeds, where the hydrostatic pressure of the endosperm, a zygotic compartment that fills up most of the internal space within early-stage seeds, is critical for seed growth [START_REF] Beauzamy | Endosperm turgor pressure decreases during early Arabidopsis seed development[END_REF]. Endosperm pressure decreases towards the end of the growth phase and it was hypothesized that pressure drop could trigger growth arrest.

However, endosperm pressure also generates tension in the testa, the developing seed coat, and it was proposed that this tension can be perceived by a specific layer of the testa, inducing wall stiffening, thus restricting growth [START_REF] Creff | A mechanically sensitive cell layer regulates the physical properties of the Arabidopsis seed coat[END_REF]. Using modelling and experimental approaches, it was shown that endosperm pressure promotes seed growth initially but also restricts growth by inducing testa stiffening. Therefore, somewhat counterintuitively, the reduction of endosperm pressure during the late growth phase does not trigger growth arrest, but allows prolonged seed growth by preventing precocious testa stiffening [START_REF] Creff | Endosperm turgor pressure both promotes and restricts seed growth and size[END_REF]Fig. 2G). This study illustrates the importance of regulating turgor during organ growth, although the associated mechanisms remain undeciphered.

The importance of regulating water fluxes during organ growth was highlighted during lateral root emergence. Lateral roots initiate deep within the root, and need to penetrate the endodermis, cortex and epidermis in order to emerge [START_REF] Teixeira | The Systems Biology of Lateral Root Formation: Connecting the Dots[END_REF]. Lateral root initiation and emergence are controlled by auxin, which notably represses the expression of aquaporin channel proteins that mediate cross-membrane water fluxes at specific interfaces between the lateral root primordium, the overlying tissue, and the vasculature. Assuming that increasing osmotic pressure in the primordium is necessary for growth, precise patterning of aquaporins likely permits rapid build-up of high turgor in the emerging lateral root, while the surrounding tissues lose volume to accommodate lateral root penetration [START_REF] Péret | Auxin regulates aquaporin function to facilitate lateral root emergence[END_REF] (Fig. 2H). Auxin-induced symplasmic isolation by callose deposition may also contribute to hydraulic repatterning during lateral root formation [START_REF] Benitez-Alfonso | Symplastic intercellular connectivity regulates lateral root patterning[END_REF][START_REF] Sager | Auxin-dependent control of a plasmodesmal regulator creates a negative feedback loop modulating lateral root emergence[END_REF].

VI. Perspectives

We have considered how complex relationships between turgor pressure and growth arise from tissue mechanics and hydraulics, and how physical modelling is crucial to permit understanding of the results of such complexity. We propose that turgor does not set the growth rate, but is an emergent property that drives growth. An important emerging question is to what extent this new vision of turgor pressure and tissue hydraulics adds to the existing understanding of how plant growth is regulated? As cell wall stress and turgor pressure are interdependent, it is likely that their sensing and regulation are also biologically connected.

In support of this, recent studies revealed that some wall defective mutants show turgor defects [START_REF] Mielke | Jasmonate biosynthesis arising from altered cell walls is prompted by turgor-driven mechanical compression[END_REF][START_REF] Bacete | The Mechanosensitive Ion Channel MSL10 Potentiates Responses to Cell Swelling in Arabidopsis Seedlings[END_REF], and that wall-induced growth defects can be rescued by turgor manipulations [START_REF] Verger | A tension-adhesion feedback loop in plant epidermis[END_REF][START_REF] Malivert | FERONIA and microtubules independently contribute to mechanical integrity in the Arabidopsis shoot[END_REF]. These findings suggest that wall mechanics and turgor are biologically coordinated and may share common regulatory mechanisms. Possible candidates in the perception of these mechanical cues include receptor-like kinases [START_REF] Bacete | The Mechanosensitive Ion Channel MSL10 Potentiates Responses to Cell Swelling in Arabidopsis Seedlings[END_REF][START_REF] Malivert | FERONIA and microtubules independently contribute to mechanical integrity in the Arabidopsis shoot[END_REF] or mechanosensitive channels [START_REF] Bacete | The Mechanosensitive Ion Channel MSL10 Potentiates Responses to Cell Swelling in Arabidopsis Seedlings[END_REF], and transduction to the cytoskeleton [START_REF] Colin | Cortical tension overrides geometrical cues to orient microtubules in confined protoplasts[END_REF]. Additionally, pressure is uniform in all directions, and its translation into biologically relevant signals during directional growth must rely on (bi-)directional cues such as anisotropy [START_REF] Sassi | An auxin-mediated shift toward growth isotropy promotes organ formation at the shoot meristem in Arabidopsis[END_REF], polarity [START_REF] Ali | Simulating Turgor-Induced Stress Patterns in Multilayered Plant Tissues[END_REF], or gradients [START_REF] Robbins | Growth is required for perception of water availability to pattern root branches in plants[END_REF]. Finally, turgor has the potential to integrate information from biomechanics, hydraulics, physiology and metabolism, as many osmotically active compounds are energy sources, biological building blocks, and physiological regulators [START_REF] Beauzamy | Flowers under pressure: ins and outs of turgor regulation in development[END_REF].

Examples in cotton fiber elongation revealed involvement of all these processes [START_REF] Ruan | The control of single-celled cotton fiber elongation by developmentally reversible gating of plasmodesmata and coordinated expression of sucrose and K+ transporters and expansin[END_REF], and it will be both informative and exciting to investigate how such information is coordinated in other growing systems.

Figure 1 .

 1 Figure 1. The hydraulics and mechanics of plant cells. (A) Top panel: The osmolarity differential between the cell (left hand-side) and the extracellular medium (right hand-side), separated by a semi-permeable plasma membrane (thin beige line) and a cell wall (thick pink line), induces a water flux ( ) toward the compartment with the highest osmolarity. Middle panel: This water influx increases cell hydrostatic pressure ( ) which, in turn, generates a water flux toward the right hand-side one ( ). Bottom panel: Cell wall deformation leads to the accumulation of tensile stress ( ) within the wall, which balances the hydrostatic pressure exerted on the wall. (B) The hydrostatic pressure ( ) within an

  ): the proportionality coefficient in the right-hand-side of Lockhart equation, Eq.(1a), is a symmetric function of the effective cell wall extensibility, , and the water conductance, . Their respective contributions can be directly compared by the dimensionless parameter , with corresponding to an equal contribution of the two parameters, i.e. . One major outcome of

Figure

  Figure. 2. A, B, C: Qualitative relation between topology (number of neighbors n), cell inner pressure ( ) compared to the pressure of neighboring cells ( ), and water fluxes (blue arrows). By analogy with foam, cells with less than six neighbors feature higher pressure than their surroundings and tend to exhibit convex walls. Adapted from (Cantat et al., 2013). D, E, F: Effect of turgor pressure across scales. (D) At the organ

Acknowledgements

The authors thank Gwyneth Ingram for her critical reading of the manuscript. We apologize to all colleagues whose works we could not cite in this review due to length restrictions. The authors declare no conflict of interests.


Marie Schuler

Deepti Duvvuru

Pascal Retailleau

Jean-Francois Betzer

Angela Marinetti
email: angela.marinetti@icsn.cnrs-gif.fr

New Access to Trisubstituted 3-Pyrrolines under Phosphine Catalysis

Conjugated dienes, properly activated by electron-withdrawing groups on both ends, are shown to be suitable substrates for phosphine-promoted organocatalytic processes. Their reactions with imines, under phosphine catalysis, afford a new and efficient synthetic approach to functionalized 3-pyrrolines.

Nucleophilic phosphines are known to be useful mild catalysts for the conversion of imines into nitrogen heterocycles via [3 + 2] 1 and [4 + 2] 2 cyclization reactions, in which either electron-poor allenes or alkynes behave as the three-or four-carbon synthons. 3 The synthetic potential of these reactions has been largely demonstrated by the (1) (a)

preparation of biologically relevant natural products 4 as well as by using them to access a range of pyrrolidines and piperidines for pharmaceutical applications. 5 In this context, with the aim of extending the scope of phosphine organocatalysis, we disclose here a new approach to 2,3,5trisubstituted 3-pyrrolines, based on the use of conjugated dienes as the three carbon units in annulation reactions on activated imines. This new method complements the previous ones, as far as it allows access to a different range of 3-pyrrolines from easily available new substrates.

Our strategy of using conjugated dienes as starting materials has been developed, based on the well-known activation of bis(enones) by phosphorus nucleophiles leading to the so-called intramolecular 6 when the enone functions are tethered by either two or three atoms chains. The postulated mechanism of these reactions is depicted in Scheme 1 (left). We reasoned that if the two Scheme 1. Intramolecular Rauhut-Currier Reaction (Left) and the Suggested Variant Involving Trapping of I by Imines (Right) electron-poor olefinic functions of the substrates were directly connected, as in conjugated dienes (n ) 0 in Scheme 1), the Rauhut-Currier type reactions would be disfavored due to the short distance between the carbanionic center of I and the potential olefinic acceptor. In this case, it should be possible to trap the zwitterionic intermediate I with electrophilic substrates, such as activated imines (step b in Scheme 1, right). This would create a three atom spacer between the negatively charged atom (the nitrogen center in II) and the enone function, thus enabling an intramolecular nucleophilic addition to take place (step c), following a classical Rauhut-Currier type process. Pyrrolines IV are expected to be quickly accessed in this way, from a reaction sequence which formally combines intramolecular Rauhut-Currier and aza-Morita-Baylis-Hillman 7 type reactions.

To test the feasibility of the catalytic manifold above, we examined the reactions between N-tosylbenzaldimine 3a and both symmetrical and unsymmetrical dienic substrates, i.e., the diester 1a, 8 the diketone 1b, 9 and the enone-ester 2a 10 in the presence of MePPh 2 as the Lewis base catalyst (Scheme 2).

Dimethyl trans-trans muconate 1a failed to react with the imine, while the anticipated cyclization products 4b and 5a were obtained from 1,4-dibenzoyl-1,3-butadiene 1b and the 6-oxo-6-phenylhexadienoic acid methyl ester 2a, respec- tively. Pyrrolines 4b and 5a were produced as mixtures of two isomers in about 9:1 ratios, tentatively assigned as the 2,5-syn and anti diastereomers. The syn isomer is the major product. 11 When starting from the unsymmetric substrate 2a, the cyclization might occur, in principle, on each one of the electron-poor C-C double bonds. The reaction proceeded, actually, with high regioselectivity, leading to pyrroline 5a.

The inertness of the diester toward the above cyclization reactions is in line with the known reluctance of bisenoates to undergo intramolecular Rauhut-Currier type cyclizations. 6b-d The behavior of the mixed enone-enoate substrate 2a is however rather surprising. Indeed, the isolated product 5a results from the initial addition of the phosphorus nucleophile to the enoate double bond (R 2 ) OMe, step a in Scheme 1), while the enone function serves as the Michael acceptor in the subsequent ring-closing step c. This contrasts with literature data showing that intramolecular Rauhut-Currier reactions on enone-enoates take place with the opposite regiochemistry: enones being the more electrophilic of the two Michael acceptors of the bifunctional substrates, they preferentially undergo the initial addition of the phosphorus nucleophile. 6b-d,12 The observed, unusual cyclization manifold leading to 5a might be determined by either electronic effects and orbitals distribution on the diene or the relative reaction rates of the two possible phosphinediene adducts with the imine in step b.

It must be pointed out here that cyclizations in Scheme 1 seem to be uniquely susceptible to phosphine catalysis since they did not proceed in the presence of nitrogen Lewis bases such as DABCO, DMAP, or Et 3 N.

After optimization of the reaction conditions, 13 the method appeared to be applicable to a wide range of substrates and notably to dienic enone-esters with various R 1 substituents. The 6-oxo-dienoates 2b-2o were conveniently accessed as E/Z mixtures from the commercially available (E)-4-oxobut-2-enoic acid ethyl ester, via Wittig-type reactions (Scheme 3).

Their reactions with N-tosylbenzaldimine have been performed with either PBu 3 or PMePh 2 , at 50 and 120 C, respectively. Neither PPh 3 nor PCy 3 were efficient catalysts for Scheme 3. Synthesis of the 6-Oxo-dienoates 2b-o these cyclizations. Depending on the substrate, yields and diastereomeric ratios may vary slightly, as a function of the phosphorus catalyst and the reaction temperature. For each substrate, the optimized conditions and results are reported in Table 1.

As shown in Table 1, these reactions delivered the

In most of the isolated products the syn:anti ratio is higher than 9:1, and it increases to 99:1 for pyrroline 5g as well as for the alkyl-substituted ketones of entries 12 and 13. The syn:anti ratios of 5 are irrespective of the E/Z isomer ratios of the starting dienes. The syn stereochemistry has been assigned to the major isomers of 5, based on X-ray diffraction studies on a representative compound, 5g (entry 6). Crystals of 5g have been obtained from a dichloromethane solution by slow addition of pentane. The ORTEP drawing is shown in Figure 1. Finally, to further establish the scope of this new cyclization reaction, a variety of N-Ts imines have been prepared and reacted then with the 6-oxo-hexadienoates 2b (R 1 = Ph) and 2m (R 1 = t-Bu). Representative results are given in Table 2, showing that the reaction may tolerate electron-rich and electron-poor aryl substituents on the imine carbon.

Most notably, not only aryl-substituted but also a representative alkyl-substituted imine 3h could be reacted with dienoates 2b,m to afford the corresponding 2-alkyl-substituted pyrrolines 6h and 7h in good yields.

As a general trend, the t-Bu substituted 6-oxo-dienoate 2m (entries 8-13) gives very clean reactions, almost total diastereoselectivity and higher yields than the benzoylsubstituted diene 2b (entries 1-7).

(11) The molecular structure and the syn stereochemistry of the major product have been established by X-ray diffraction studies (see Figure 1). The minor isomer has not been characterized; nevertheless, selected NMR signals have been assigned by COSY Assignments support the 3-pyrroline structure. For compound 5a (minor product), NMR data are the following: δ 3.32 (dd, 2 JHH = 18.3 Hz, 3 JHH = 9.3 Hz, 1H, CH2CO), 4.54 (dd, 2 JHH = 18.3 Hz, 3 JHH = 3.9 Hz, 1H, CH2CO), 5.39 (m, 1H, 3 J HH = 9.3 Hz, 4 J HH = 5.7 Hz, 3 J HH = 3.9 Hz, 3 J HH = 1.8 Hz, 4 4 expected pyrrolines in moderate to good yields, starting from aryl ketones with both electron-donating and -withdrawing substituents (entries 1-9), as well as from naphthyl (entry 10) and thienyl ketones (entry 11). Alkyl ketones may also be suitable substrates, and especially high yields were obtained from dienone 2m which displays a t-Bu substituted carbonyl function (entry 12). Decreasing yields are given by dienones with secondary and primary alkyl groups (entries 13 and 14).

NCHCH 2 ), 5.85 (dd, J HH = 5.7 Hz, J HH = 1.5 Hz, 1H, CHPh) ppm. The corresponding NMR data for the major product 5a are the following: δ 3.36 (dd, 2 JHH = 17.7 Hz, 3 JHH = 10.6 Hz, 1H, CH2CO), 4.04 (dd, 2 JHH = 17.7 Hz, 3 JHH = 3.8 Hz, 1H, CH2CO), 5.10 (ddt, 3 JHH = 10.6 Hz, 3 JHH = 3.8 Hz, 3 J HH = 4 J HH = 1.9 Hz, 1H, NCHCH 2 ), 5.58 (t, 4 J HH = 1.9 Hz, 1H, CHPh) ppm. (12) Brown, P. M.; Ka ¨ppel, N.; Murphy, P. J.; Coles, S. J.; Hursthouse, M. B. Tetrahedron 2007, 63, 1100-1106. (13) For full details on solvents and catalyst screenings, see Supporting Information.

(14) Crystallographic data for this compound have been deposited at the CCDC, Cambridge, UK, with the deposit number 741184.

(15) Guan, X.-Y.; Shi, M. J. Org. Chem. 2009, 74, 1977-1981.

  Scheme 2. Attempted Reactions of Conjugated Dienes withImines in the Presence of PMePh2

Table 1 .

 1 Synthesis of Pyrrolines 5b-o from Various 6-Oxodienoates a a Reactions performed at a 0.3 mmol scale, with a 1:1 diene:imine ratio, under Ar in toluene (1 mL) for 18 h. b Reaction temperature: 120 C. c Reaction temperature: 50 C. d In the crude reaction mixture.

  14

Figure 1 .

 1 Figure 1. ORTEP view of pyrroline 5g showing the relative syn arrangement of the ring substituents. Displacement ellipsoids are at the 30% level.

Acknowledgment. Authors warmly thank the Agence Nationale de la Recherche (ANR) for financial support to M.S. and the COST PhoSciNet Action for supporting research topics on phosphine organocatalysis.

dienes 2 as substrates gives access to a new range of 2,3,5trisubstituted 3-pyrrolines. The known approaches, which make use of 3-substituted allenoates or butynoates, 1c-f,h,5c,15 alkynyl ketones, 1g enones 3a or allylic compounds, 3d and aryl imines as reactants, have been applied so far to the synthesis of 2-aryl-substituted pyrrolines, with either simple alkyls (Me, Et, Pr, n-Bu, t-Bu, n-Pent, n-Hex, c-C 5 H 9 CH 2 ) or aryl groups in the 5 position. As shown in Tables 1 and2 above, the new method opens the way to both 2-alkyl-and 2-arylsubstituted pyrrolines with carbonylmethyl functions on the C-5 carbon.

In conclusion, we have developed a simple, general, and practical method for the synthesis of new 2,3,5-trisubstituted 3-pyrrolines from easily available dienic derivatives, under phosphine catalysis. Future studies will focus on the development of related organocatalytic transformations of dienic substrates, as well as on synthetic applications of the The above results show that, compared to other phosphinebased organocatalytic methods, the use of the conjugated functionalized pyrroline scaffolds above.